From landman%hanami@Sun.COM Mon Jun 27 16:16:35 1988 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Newsgroups: comp.music Keywords: arbitrary even-tempered tonal scales Reply-To: landman@sun.UUCP (Howard A. Landman) Organization: Sun Microsystems, Mountain View From: landman%hanami@Sun.COM (Howard A. Landman) Subject: good even-tempered scales Date: 23 Jun 88 00:31:34 GMT In article <171@ttl.UUCP> naisbitt@ttl.UUCP (Pete Naisbitt) writes: >Does anyone know of musical instruments which have been constructed for >the performance of music based on arbitrary tonal scales? On the first, self-titled album by the southern California vocal group "The Sound of Feeling", there is a piece called "Hex" which uses microtonal vocals and vibes, in chords called "hexanies". I have also heard of at least one instrument based on the 53-tone even-tempered scale. (The choice of 53 will be obvious from what follows.) >Apparently, the tonal scales which are part of the Fibonacci series (12, >19, 31, 50 .... note scales) offer a very large number of consonant >intervals (ie. that most closely approximate intervals such as the perfect >fifth, fourth, major third etc. etc.) for an evenly tempered scale. This is not totally wrong, but it's not totally right either. 12, 19, and 31 are pretty good, but both 41 and 53 are far better than 50: Interval 12 tone scale 50 tone scale 53 tone scale 612 tone scale 9/8=1.125000 2 1.122462 8 1.117287 9 1.124911 104 1.125008 8/7=1.142857 2 1.122462 10 1.148698 10 1.139720 118 1.142988 7/6=1.166667 3 1.189207 11 1.164734 12 1.169924 136 1.166529 6/5=1.200000 3 1.189207 13 1.197479 14 1.200929 161 1.200031 5/4=1.250000 4 1.259921 16 1.248331 17 1.248984 197 1.249972 9/7=1.285714 4 1.259921 18 1.283426 19 1.282084 222 1.285870 4/3=1.333333 5 1.334840 21 1.337928 22 1.333386 254 1.333329 7/5=1.400000 6 1.414214 24 1.394744 26 1.404996 297 1.399871 3/2=1.500000 7 1.498307 29 1.494849 31 1.499941 358 1.500005 8/5=1.600000 8 1.587401 34 1.602140 36 1.601302 415 1.600036 5/3=1.666667 9 1.681793 37 1.670176 39 1.665377 451 1.666623 9/5=1.800000 10 1.781797 42 1.790050 45 1.801323 519 1.800053 Note particularly (in 53) the near-perfect fifth (3/2) and fourth (4/3). The major third (5/4) isn't as good, but still better than the 50-tone one. So I don't believe that there's anything to this Fibonacci theory. >Obviously, the more notes there are in the scale, the more this is the case. Yes, but this has to be traded off against the complexity of more notes, especially for a performing instrument. The 612-tone even-tempered scale is really excellent (see above), but who'd want a keyboard with thousands of keys? Also, it's not a monotone function of the number of notes; the 13-note scale is MUCH worse than the 12-note scale, for example. >For example, a seventh on a 31 note scale seems to be able to conclude an >almost perfect cadence, which is not the case on a 12 note scale. Could you define what you mean by a "seventh", independent of the scale? For those who are interested, here are the "best" numbers of tones in even-tempered scales, where the criterion of goodness is a weighted (inverse-quadratic weight) average of the errors for small-integer ratios (natural harmonics and subharmonics) with the integers relatively prime and summing less than 100. That is, 1 < p/q < 2, gcd(p,q) = 1, p + q < 100. A different weighting scheme might lead to a slightly different list, as might extending the number of harmonics examined. Since fewer notes are always better, a number is only included on the list if it is better (smaller error) than anything that came before: tones error error * tones 1 0.14056452 0.14056452 2 0.05099084 0.10198167 3 0.03538255 0.10614765 4 0.03290493 0.13161973 5 0.01443912 0.07219560 7 0.01041083 0.07287579 10 0.00871246 0.08712462 12 0.00392017 0.04704209 19 0.00339107 0.06443038 22 0.00316588 0.06964938 24 0.00260458 0.06250982 29 0.00235854 0.06839780 31 0.00203190 0.06298881 34 0.00197648 0.06720037 36 0.00188306 0.06779015 41 0.00101100 0.04145101 53 0.00055207 0.02925980 94 0.00045192 0.04248033 106 0.00041718 0.04422083 118 0.00028779 0.03395911 159 0.00026530 0.04218348 171 0.00017211 0.02943017 224 0.00016597 0.03717802 270 0.00012576 0.03395515 342 0.00010882 0.03721524 388 0.00010479 0.04065987 400 0.00009092 0.03636905 441 0.00006367 0.02807997 494 0.00005682 0.02806880 612 0.00003473 0.02125778 I tested tones < 1000, so there's nothing between 613 and 999 that's better than 612. The "error * tones" column is the "error" column multiplied by the number of tones, thus weighting for fewer tones ("Amadeus" to the contrary notwithstanding :-). It is easy to see, in this column, why the 12-tone scale is so good. No scale with fewer than 41 tones is better by this measure. Note also that 612 = 12*51, so the 612-tone scale includes the 12-tone scale; but the table earlier shows that its goodness isn't a consequence of this, because none of the 12-tone notes are best fits in the 612-tone scale.. Pete's Fibonacci series gives 12,19,31,50,81,131,212,343,555,898. The first three numbers appear on my list also, but I don't really see any evidence for a correlation. If time permits, I'll dredge up the software I used to perform these analyses (several years ago), and post it here. Howard A. Landman landman@hanami.sun.com UUCP: sun!hanami!landman From sandell@batcomputer.tn.cornell.edu Mon Jun 27 16:17:02 1988 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Newsgroups: comp.music Keywords: arbitrary even-tempered tonal scales Reply-To: sandell@tcgould.tn.cornell.edu (Gregory Sandell) Organization: Cornell Theory Center, Cornell University, Ithaca NY From: sandell@batcomputer.tn.cornell.edu (Gregory Sandell) Subject: Re: good even-tempered scales Date: 23 Jun 88 22:45:10 GMT An article which beautifully characterizes the very special properties of 12-, 19- and 31-tone scales is: Balzano, Gerald J.(1980) The group-theoretic description of twelvefold and microtonal pitch systems. COMPUTER MUSIC JOURNAL 5, 66-84. Greg Sandell From landman%hanami@Sun.COM Mon Jul 4 11:52:25 1988 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Newsgroups: comp.music Keywords: arbitrary even-tempered tonal scales Reply-To: landman@sun.UUCP (Howard A. Landman) Organization: Sun Microsystems, Mountain View From: landman%hanami@Sun.COM (Howard A. Landman) Subject: Re: good even-tempered scales Date: 1 Jul 88 00:38:48 GMT ------------- 8< ------- CUT HERE ------- 8< ------------ /* scale - prints the tempered scale with n tones. */ /* Copyright (c) 1983, 1988 by Howard A. Landman. */ /* Permission is granted to copy, transmit, compile, */ /* execute, and/or modify this program for any */ /* non-commercial purpose, as long as this notice */ /* and the compiled-in copyright notice are retained. */ /* This program lets you look at how well a given well-tempered scale */ /* fits the various natural harmonics and subharmonics. All harmonics */ /* are expressed as fractions between 1.0 and 2.0, i.e. they are */ /* coerced to lie in a single octave. It uses the math library, so */ /* you need to give cc the -lm flag: */ /* cc -o scale scale.c -lm */ /* Then you just say "scale 12" to get the 12-tone scale. The most */ /* interesting scales are 12, 19, 31, 53, and 612. The more tones */ /* in the scale, the more harmonics are printed. BUG: For large n, */ /* this may result in some long lines. The output from "scale 12": */ /* SCALE Copyright (c) 1983, 1988 by Howard A. Landman */ /* Tempered scale with 12 tones. */ /* */ /* 0 1.000000 */ /* 1 1.059463 */ /* 2 1.122462 8/7=1.142857 9/8=1.125000 10/9=1.111111 */ /* 3 1.189207 6/5=1.200000 7/6=1.166667 11/9=1.222222 */ /* 4 1.259921 5/4=1.250000 9/7=1.285714 */ /* 5 1.334840 4/3=1.333333 */ /* 6 1.414214 7/5=1.400000 10/7=1.428571 */ /* 7 1.498307 3/2=1.500000 */ /* 8 1.587401 8/5=1.600000 11/7=1.571429 */ /* 9 1.681793 5/3=1.666667 12/7=1.714286 */ /* 10 1.781797 9/5=1.800000 */ /* 11 1.887749 13/7=1.857143 */ /* 12 2.000000 2/1=2.000000 */ /* The first column is the tone index, with the tonic as 0 and its */ /* octave as n. (BUG: If n is greater than 9999, your columns won't */ /* line up perfectly.) The second column is the frequency, based on */ /* 1.0 for the tonic. After that there may be one or more (or no) */ /* harmonics, given both as a fraction reduced to lowest terms */ /* (3/2 appears, but 6/4 doesn't), and as a real number (the frequency, */ /* which you would like to match with one of the tempered tones). */ /* Each harmonic follows the tempered tone which is closest to it. */ #include #include #define RATIO (((double) num) / ((double) denom)) extern char *malloc(); int gcd(m,n) int m,n; { /* Greatest common divisor using Euclid's algorithm. */ int tmp; m = abs(m); n = abs(n); if (m > n) { tmp = m; m = n; n = tmp; } while (m > 0) { tmp = n % m; n = m; m = tmp; } return n; } main(argc,argv) int argc; char **argv; { int num, denom, n, *i, *interval, tone; double two = 2.00000000000000000000, half = 0.50000000000000000000, logroot, lognote; /* Check usage. */ if (argc != 2) exit(argc); /* Read number of tones per octave from command line. */ (void) sscanf(argv[1],"%d",&n); /* Print header. */ printf("SCALE\tCopyright (c) 1983, 1988 by Howard A. Landman\n"); printf("Tempered scale with %d tones.\n\n",n); /* Initialize. */ logroot = log(two)/n; /* log of nth root of 2 */ /* Allocate a scratchpad table to hold results. */ interval = (int *) malloc((unsigned) ((sizeof(int) * 3 * n * n) / 4)); i = interval; /* Look at all "reasonable" integer ratios between 1.0 and 2.0. */ for (denom = 1; (denom < 10) || (denom < n/2) ; denom++) for (num = denom + 1; num <= denom * 2; num++) { /* Only track ratios that are relatively prime. */ if (gcd(denom,num) > 1) break; /* Find the closest interval on this scale, */ /* and save it, num, and denom in the table. */ lognote = log(RATIO); *(i++) = (int) (lognote/logroot + half); *(i++) = num; *(i++) = denom; } /* Mark end of table. */ *(i++) = -1; *(i++) = 0; *(i++) = 0; /* Now go through the tempered tones one by one. */ for (tone = 0; tone <= n; tone++) { printf("%4d %8.6f",tone,exp(logroot * tone)); /* Search the table for close harmonics. */ /* This takes quadratic time (total), while we */ /* could get n log n if we sorted, but I haven't */ /* found the program to be too slow yet! */ i = interval; while (0 <= *i) { if (tone == *i) { /* This one is close, print it. */ i++; num = *(i++); denom = *(i++); printf(" %d/%d=%8.6f",num,denom,RATIO); } else { /* Skip it. */ i += 3; } } printf("\n"); } } ------------- 8< ------- CUT HERE ------- 8< ------------ Howard A. Landman landman@hanami.sun.com UUCP: sun!hanami!landman From curt@dtix (Welch) Mon Feb 5 14:57:54 1990 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Newsgroups: rec.music.synth Message-ID: <921@nems.dt.navy.mil> References: <13492@phoenix.Princeton.EDU> Reply-To: curt@dtix.dt.navy.mil (Curt Welch) Organization: David Taylor Research Center, Bethesda, MD Lines: 52 From: curt@dtix.dt.navy.mil (Welch) Subject: Re: FREQUENCIES of chromatic scale? Date: 2 Feb 90 14:10:24 GMT In article <13492@phoenix.Princeton.EDU> greg@phoenix (greg Nowak) writes: >could someone give me a list of the frequencies of all the notes in >a chromatic scale? > Greg Nowak/Phoenix Gang/Princeton NJ 08540 This output: A 440.00 A# 466.16 B 493.88 C 523.25 C# 554.37 D 587.33 D# 622.25 E 659.26 F 698.46 F# 739.99 G 783.99 G# 830.61 A 880.00 Was produced with this C program: /* * scale.c - Print the frequencies of the notes in a chromatic scale. * * 2-2-90 Curt Welch * * Compile with: cc -o scale scale.c -lm */ #include char *note[] = { "A", "A#", "B", "C", "C#", "D", "D#", "E", "F", "F#", "G", "G#", "A" }; main() { double f; int i; for (i=0; i <= 12; i++) { f = 440.0 * pow(2.0, i/12.0); printf("%-2s %6.2f\n", note[i], f); } } Curt Welch curt@dtix.dt.navy.mil David Taylor Research Center (A Navy Lab) Bethesda, MD From ogata@leviathan (Jefferson Ogata) Fri Mar 15 10:24:16 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["16643" "" "14" "March" "91" "23:55:46" "GMT" "Jefferson Ogata" "ogata@leviathan.cs.umd.edu " nil "365" "Info on some scales (long)" "^From:" nil nil "3"]) Newsgroups: rec.music.synth,comp.music Keywords: scale tuning africa gamelan bali java chin hindu Message-ID: <31484@mimsy.umd.edu> Reply-To: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Followup-To: rec.music.synth Organization: U of Maryland, Dept. of Computer Science, Coll. Pk., MD 20742 Lines: 365 From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Info on some scales (long) Date: 14 Mar 91 23:55:46 GMT Well, I've been getting so many requests for this, I decided to just post it. It's long, and not comprehensive, but I hope it's useful to some people. I was surprised at the number of requests; it seems a lot of people are interested in different tunings. Feel free to email or post with comments or corrections. I'll be happy to include further information in this compilation, and I intend to add some more stuff myself later. Someone else has volunteered to send me notes on the Hindu scale. I didn't provide a bibliography, but the references are given in the text. - jeff As there has been some interest in non-(twelve-tone equal-tempered) scales lately, I decided to dig out some numbers. I have used a variety of scales myself, but some of the pitches I got out of books I don't own, and others were already programmed into synths. Not having a frequency counter, it would be difficult for me to get the actual pitches or deviations for these scales, so I dug up some books in the music library here at U of MD. The topics are in the following order: equal-tempered and Just twelve- tone scales, Pythagorean twelve-tone and Hindu 22-tone scales, Chinese and Japanese scales, Balinese and Javanese scales, scales of the Shona people of Zimbabwe. I will try to follow up this post with a bit more info in a few days. I have used several of the scales in discussion; I recorded some music in Just scales last year. I wish I could provide a table for this interesting 19-tone tuning on my Proteus/1, but I don't know what the values are (I'll see if I can find out). Some basic scale theory: The octave is the standard for measurement of intervals by Western scholars. An octave is the interval attained by multiplying the frequency of the lower pitch by exactly 2. Octaves are divided into cents. A cent is 1/1200 of an octave, or 1/100 of an equal- tempered semitone. To raise a pitch by one cent, multiply its frequency by 2 ** (1/1200), where ** denotes "to the power of". To raise a pitch by one semitone (== 100 cents) multiply its frequency by 2 ** (100/1200) (== 2 ** (1/12)). In general, to raise a pitch by n cents, multiply its frequency by 2 ** (n/1200). To determine how many cents frequency Y is above frequency X, take 1200 * log2 (Y/X). If you don't have a calculator with log2, take 1200 * (logx (Y/X)) / (logx 2) where logx is the log you do have on your calculator. The tables I give are all measured in cents. Where I give a table for a scale, I give a version where every pitch is in cents above the tonic. When tuning a synthesizer to one of these scales, pick a tonic for reference. For each note that you decide to map to one of the pitches in question, note how many semitones above the tonic it is. Now subtract 100 cents from the target interval for each semitone. For example: suppose the tonic is A, I need to map a pitch 340 cents above the tonic to the C. I subtract 300 cents and tune C 40 cents sharp. To map a pitch 685 cents above the tonic to the E, I subtract 700 cents and tune E 15 cents flat. The "standard" of the octave, while common, is not universal; check out the stuff on the Shona mbira tunings at the end. When two sustained notes are played together, a third implied tone arises, called a beat or interference beat. The frequency of the beat tone is equal to the difference in frequency between the two sounding tones. Frequently this beat lands in subsonic frequencies (i.e. < ~20 Hz), and people have traditionally avoided such beats because they are often disturbing to human physiology (perhaps it is a sign of an earthquake that triggers an emotional response). For an example of the problem of beats, play two notes together a semitone apart on a piano in a low octave. You may be able to perceive a low-frequency tone, and the overall sound will probably be unpleasant. Now play the same two notes in a high octave. The beat will no longer sound offensive, since it is no longer a subsonic; higher notes on an equal-tempered instrument are farther apart frequency-wise than lower notes. Intervals that are separated by a factor of low integer ratios (e.g. the Just perfect fifth 3/2) have beats that are easy to keep out of subsonic frequencies. For example, the 3/2 ratio always has a beat that is exactly one octave lower than the low tone. The 2/1 or octave ratio has a beat that is exactly equal to the low tone. This is why octaves sound so good; the beat reinforces the chord. Note that subsonics are not always unpleasant; some frequencies are very soothing; other frequencies sound good by themselves. Subsonics between ~1 and ~20 Hz often sound annoying when added to music. Here is a table from _The Gamelan Music of Java and Bali_ by Donald A. Lentz (1965, University of Nebraska Press LCCCN 65-10545) pp. 24-25. I have chopped this table into bits, as it is in a wide format in the book. The table is based on a tonic of C and gives a lot of cent values. The cent values are given with no fractional parts. Comments are mine. The equal-tempered twelve-tone (Tempered) scale is the standard for modern Western music. The Just (perfect) scale is the ancient beat-canceling integral-ratio scale commonly used until the eighteenth century in Western music. It is still used by some performers of ancient music for the sake of authenticity and in certain contexts by other performers. The actual pitches of the Just scale depend on which note is taken as the tonic of the scale. The ratio and cent values, however, remain constant regardless of the tonic. Lentz apparently made this table in C so he could provide note names. -I- -II- -III- Cents Equal Just above Tem- (Using Funda- pered C as a mental Tonic) Name Interval Ratio Interval Name 0 C Unison 1/1 Unison C 100 Db Half step 200 D Whole step 204 9/8 Whole step D 300 Db Minor 3rd 386 5/4 Major 3rd E 400 E Major 3rd 498 4/3 Perfect 4th F 500 F Perfect 4th 600 F# Aug. 4th 700 G Perf. 5th 702 3/2 Perf. 5th G 800 G# Aug. 5th 884 5/3 Maj. 6th A 900 A Maj. 6th 1000 A# Aug. 6th 1088 15/8 Maj. 7th B 1100 B Maj. 7th 1200 C Perf. 8ve 2/1 Perf. 8ve C The Pythagorean scale is derived from perfect fifths alone. The intonation of the Renaissance period used eight ascending fifths and three descending fifths. When 12 perfect 3/2 fifths have been ascended and the octave has been corrected back to the original register, the resulting ratio is 531441/524288 (== (3/2) ** 12 / 2 ** 7). This interval is known as a comma and is equal to 24 cents. The Hindu scale is based both on ascending fifths and ascending fourths. The eleven pitches from ascending fifths and the eleven pitches from ascending fourths are combined to produce a 22-tone scale. There are quite a few details about this scale that I won't try to summarize here. The Pramana is the distance between Srutis 4 and 5, which are the two whole-tone intervals of the Just scale: 9/8 (between I and II) and 10/9 (between II and III). -I- -IV- -V- Cents Pythagorean Hindu above Funda- mental Cycle Sruti No. Interval Name Ratio No. Name 0 0 Unison C 1/1 1 Sa 22 81/80 Pramana 24 12 Comma 90 Minor 2nd (Limma) 256/243 2 Ri 112 16/15 3 Ri 114 7 Aug. Prime C# 182 10/9 4 Ri 204 2 Maj. 2nd D 9/8 5 Ri 294 -3 Min. 3rd Eb 32/27 6 Ga 316 6/5 7 Ga 318 9 Aug. 2nd D# 386 5/4 8 Ga 408 4 Maj. 3rd E 81/64 9 Ga 498 -1 Perf. 4th F 4/3 10 Ma 520 27/20 11 Ma 590 45/32 12 Ma 600 6 Aug. 4th F# 64/45 13 Pa 702 1 Perf. 5th G 3/2 14 Fa 792 128/81 15 Dha 814 8/5 16 Dha 816 8 Aug. 5th G# 884 5/3 17 Dha 906 3 Maj. 6th A 27/16 18 Dha 996 -2 Min. 7th Bb 16/9 19 Ni 1018 9/5 20 Ni 1020 10 Aug. 6th A# 1088 15/8 21 Ni 1110 5 Maj. 7th B 243/128 22 Ni I am skipping Columns VI and VII of the table, which give blown fifths and string-length division values. The blown fifths are Chinese intervals arrived at by cutting bamboo tubes of particular lengths. These tubes produce small fifths that range between 670 and 680 cents. A study of these tubes by Dr. E. M. von Hornbostel claimed that the average interval is 678 cents. This produces a cycle of 23 fifths, and the comma after 23 fifths is 6 cents. Here are some quotes from pp. 27-28 of Lentz. "Two theoretical systems evolved in China, one derived from the Cyclic Pentatonic and the other from the division of string lengths. They are found combined in the highest form of Ch'in music. The Cyclic Pentatonic, arrived at mathematically, is important in Chinese musical thought. The conception of the twelve liis (tones) dates back to the Han dynasty. They are formed by building empirical fifths in a manner similar to that used in Pythagorean tuning. Methods of arriving at these fifths included the use of twelve tubes. Levis indicates that a stopped bamboo tube 230 millimeters long, 8.12 millimeters in diameter, and vibrating at 366 vibrations per second was the Yellow Bell (Huang Chong), the standard established by the Bureau of Weights and Measures in 239 B.C. Tube number two was made two-thirds the length of the reference tube; number three was made equal to two-thirds of number two and then doubled to bring the tone within the ambit of an octave. This process was repeated for each of the twelve tubes. The fifths produced by these tubes were small compared to Western fifths. Various musicologists place them between 670 and 680 cents as compared to the Just fifth of 702 cents." "In Java musicians and gamelan makers, when queried about the small fifth, explained it as coming from nature but gave no specific examples. At dawn one morning in a small village in Java, a bird was singing with a call of an octave and a small fifth. This I recorded. The same song was frequently heard later and again recorded for confirmation of interval. It is a fascinating bit of fancy that the fifth in the bird call when measured on the stroboscope varied only two or three cents from the one of 678 cents. There are also indications that the small fifth might have been a standard in Sumeria and Egypt." "In music for the ch'in, a zither-type instrument, the seven strings are tuned to the Cyclic Pentatonic. Each string has frets or nodes dividing it into the following lengths: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 4/5, 1/6, 5/6, 1/8, and 7/8. The resultant cent values are listed in column VII of Chart IV. Each string employs the principle of Just intonation, but many microtonic intervals result when this theory of string length division is combined with the above Cyclic Pentatonic procedure, which is used to tune the seven open strings. There is no counterpart in Western Tempered music. Many Oriental stringed instruments use frets which are movable. The accuracy of placing the fret, which is done by ear, can greatly affect the pitch and thus produce noticeable deviances in a system using natural intervals of microtonic size. The tones of the present-day scales of Japanese koto and gekkin music evolved with variance from the Ch'in principle. "Western musicians think of the fifth as being an interval of 700 to 702 cents. The deviation of only two cents betwen Just, Pythagorean, and Tempered fifths is so small that these sizes are accepted as being the true fifth. This is not the case in Oriental music, even though some Western musicologists try to explain their fifths as anomalies from the Western norm. The conception of the fifths is in many cases very different. For the most part they are smaller than the Western fifth. In Chinese music, another common theoretical fifth of 693 cents, as contrasted with the Cyclic fifth of 678 cents, results from combining three of the characteristic large seconds derived from string-length division (see column VII in Chart IV). Each of the large seconds has a value of 231 cents and a ratio of 8/7." "Fifths of varying sizes are produced on different pipes when the end-correction factor is not considered, thus not fitting a theoretical system. These convert to a standard when duplicated. This procedure of duplication is found in China along with the theoretical fifth, and although one cannot find positive documentation of it for the gamelans of Java and Bali, it is highly possible that it became a factor in the varying sizes of the fifth there too." On p. 33 the gamelan scale is described: "Three basic tones and two or four secondary tones are the background of the gamelan tonal system. The main tone, called dong in Bali, is supported by two tones, one a fifth above (called dang) and a the other a fifth below (called dung). The secondary tones are a fifth above (d`eng) and a fifth below (ding) the supporting tones. By bringing the five tones within an octave, the following scale results: dong, d`eng, dung, dang, ding. "...For convenience the tone names of Western notation will be used, with C arbitrarily chosen as a starting tone. But it should be recalled that the Oriental fifths are variable in size and in all probability will not correspond to a Western fifth. This results in a scale named C D F G Bb. "In the Balinese-Javanese five-tone scale, a large interval, approximately a minor third, which will vary in size from one gamelan to the next, occurs between the second and third and the fourth and fifth degrees." Using a fifth of 678 cents, we can generate one example of this scale: Degree Cents above tonic I 0 == unison II 156 == two fifths minus one octave III 522 == octave minus one fifth IV 678 == one fifth V 1044 == two octaves minus two fifths In _Musics of Vietnam_ by Pham Duy, Edited by Dale R. Whiteside (1975, Southern Illinois University Press) Duy explains that theoretically the Khmer scale of South Vietnam is divided into seven equally-spaced tones. This would make cent values as follows: Degree Cents above tonic I 0 II 171 III 342 IV 514 V 685 VI 857 VII 1028 However, Duy notes that in practice the intervals in the Khmer scale are not exactly equal. In _The Soul of Mbira_ by Paul F. Berliner (1978, 1981, University of California Press), Berliner describes a number of tunings used for the mbira, which is a kalimba-like African instrument common in Zimbabwe. Apparently each region has its own tuning, and different instrument makers tune their instruments differently. The prevailing theory of Shona mbira tuning is that "mbira makers and players use a distinctive, well defined scale, with only slight variation in different parts of the country....It can be described as a seven-note scale, with all the intervals equal." (p. 66) However, Berliner found in a sample of tunings that the variation was very large, varying between 37 to 286 cents between adjacent scale degrees, and not equal at all. The various mbira players select their instruments based on a variety of factors, including tuning, which they refer to collectively as the "chuning" of the instrument. "...I asked several musicians who owned these mbira...to select from a set of fifty-four forks tuned 4 c.p.s. apart (from 212 c.p.s. to 424 c.p.s.) the individual forks which each thought matched the tuning of the keys on his respective instrument. The fact that they sometimes said that the pitch of an mbira key fell between two tuning forks demonstrated that the musicians could discern fine variations in tuning." Here is one of the tunings Berliner gives: Between Mbira interval in cents C and D 185 D and E 204 E and F 204 F and G 163 G and A 158 A and B 137 B and C 251 This gives the following table: Degree Cents above tonic I 0 II 185 III 389 IV 593 V 756 VI 914 VII 1051 oct. 1302 Note that the octave is not a factor of two in this tuning. Apparently the "octaves" are highly variable in mbira tunings. Berliner gives some tables of the variations from a "true" octave he found in various mbiras. The octave also is not exact in Scottish and Irish bagpipe music, since the high overtone used for the octave on the canter pipe is somewhat off. The high notes of a bagpipe melody tend to come out weaker and a little bit flat. This concludes the summary of information I collected; I hope to follow this up with some other examples in a few days. Happy tuning. -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From quayster@cynic (Tony Chung) Sat Apr 13 22:43:09 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["872" "" "12" "April" "91" "15:16:28" "GMT" "Tony Chung" "quayster@cynic.wimsey.bc.ca " nil "22" "Re: Minor mode names (Re: Chord Families and Associate Scales)" "^From:" nil nil "4"]) Newsgroups: rec.music.makers,rec.music.synth,rec.music.bluenote Message-ID: <1991Apr12.151628.21590@cynic.wimsey.bc.ca> References: <1991Apr5.112528.13559@cynic.wimsey.bc.ca> <16788@prometheus.megatest.UUCP> Organization: Mad Artists' Technological Hangout Lines: 22 From: quayster@cynic.wimsey.bc.ca (Tony Chung) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 12 Apr 91 15:16:28 GMT To David Jones, thanks for naming those scales, and yes, maybe we should hold a "name the scales" contest. I think I know the answers to the I, II and V in harmonic minor: I = "harmonic minor" (not very original) :-) II = "locrian sharp 6" (because that's what it looks like) :-) V = "Spanish phrygian" (at least that's what the Improv teach says) Now, in my post, I made a most grevious error. I said that over a Cma7(#5) chord you use the A melodic minor, and Cma7(#5,b5) chord you use the A harmonic minor. They should be reversed. Sorry if someone else posted this correction earlier; I haven't been here of late! :-) -Tony Chung -- -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- "If you drive, don't drink." -- Tony Chung quayster@cynic.wimsey.bc.ca quayster@arkham.wimsey.bc.ca From sf@sco (Steve Finney) Mon Apr 15 11:34:36 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["773" "" "15" "April" "91" "00:07:26" "GMT" "Steve Finney" "sf@sco.COM " nil "18" "Re: Minor mode names (Re: Chord Families and Associate Scales)" "^From:" nil nil "4"]) Newsgroups: rec.music.makers,rec.music.synth,rec.music.bluenote Message-ID: <11383@scolex.sco.COM> Organization: The Santa Cruz Operation, Inc. Lines: 18 From: sf@sco.COM (Steve Finney) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 15 Apr 91 00:07:26 GMT In article <1991Apr12.151628.21590@cynic.wimsey.bc.ca> quayster@cynic.wimsey.bc.ca (Tony Chung) writes: >To David Jones, thanks for naming those scales, and yes, maybe >we should hold a "name the scales" contest. I think I know the >answers to the I, II and V in harmonic minor: > > I = "harmonic minor" (not very original) :-) > II = "locrian sharp 6" (because that's what it looks like) :-) > V = "Spanish phrygian" (at least that's what the Improv teach says) > Or go with middle eastern mode names. The V is _really_ common in middle eastern and balkan musics (greek, bulgarian, ...), and is referred to as "hejaz" (alternatively "hijaz"). Actually, it's a little trickier, since there are various forms of hejaz. The minor starting on the 4 is "nikris". sf -- From cbolton@csd475a (Chris Bolton) Fri Apr 19 09:27:59 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["871" "" "17" "April" "91" "23:43:05" "GMT" "Chris Bolton" "cbolton@csd475a.erim.org " nil "21" "Re: Minor mode names (Re: Chord Families and Associate Scales)" "^From:" nil nil "4"]) Newsgroups: rec.music.makers,rec.music.synth,rec.music.bluenote Message-ID: References: <17335@prometheus.megatest.UUCP> Organization: Environmental Research Institute of Michigan, Ann Arbor, Michigan Lines: 21 From: cbolton@csd475a.erim.org (Chris Bolton) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 17 Apr 91 23:43:05 GMT In article <17335@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: > )To David Jones, thanks for naming those scales, and yes, maybe > )we should hold a "name the scales" contest. I think I know the > )answers to the I, II and V in harmonic minor: > ) > ) I = "harmonic minor" (not very original) :-) > ) II = "locrian sharp 6" (because that's what it looks like) :-) > ) V = "Spanish phrygian" (at least that's what the Improv teach says) > ) > Whoa! This one blew by me the first time around. Ask your improv teach how > come a "phrygian" got a major third, eh? (Demand a refund.) If my memory serves me correctly, my improv. teacher said a Spanish phrygian was the same as a phrygian but with a major third as well. That is, it contains both a major AND minor third. -Chris Bolton cbolton@csd460a.erim.org From RICH@SUHEP (Richard S. Holmes) Wed Apr 24 09:57:17 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["3117" "" "22" "April" "91" "20:42:00" "GMT" "Richard S. Holmes" "RICH@SUHEP.BITNET " nil "67" "Pierce scale" "^From:" nil nil "4"]) Newsgroups: bit.listserv.emusic-l Message-ID: Lines: 67 Comments: Gated by NETNEWS@AUVM.AUVM.EDU Original_To: JNET%"emusic-l@auvm" Original_cc: RICH From: RICH@SUHEP.BITNET (Richard S. Holmes) Subject: Pierce scale Date: 22 Apr 91 20:42:00 GMT JSMOR@CONNCOLL.BITNET writes >Another note about the Pierce scale, I don't really understand its derivation, >but it sounds great. If anyone out there could explain it, it would be much >appreciated. I'll try to recall what very little I've read. Normally, two notes an octave apart (frequency ratio 2:1) sound "alike" because the upper note contains no harmonics not found in the lower one. However, for waveforms that have only odd harmonics this is not true; the first note for which it is true is the twelfth (3:1). That is, if all harmonics are present, your base note has fundamental and harmonics like this: 1 : 2 : 3 : 4 : 5 : ... and the note an octave higher is 2 : 4 : 6 : 8 : 10 : ... (no harmonics not in the base note). But for a tone with only odd harmonics the base note is 1 : 3 : 5 : 7 : 9 : ... and the note an octave higher is 2 : 6 : 10 : 14 : 18 : ... (no harmonics in common) and the note a twelfth higher is 3 : 9 : 15 : 21 : 27 : ... (no harmonics not in common). So for such tones, the scale starts to "repeat" after a twelfth, not an octave. A 13-degree E.T. scale would have steps of 3^(1/13) (rather than 2^(1/12) in our usual 12-degree octave-based E.T. scale); the frequencies are Degree Freq. ratio Cents Nearby rational interval 1 1.0000 0.0 1:1 ( 0.0 cents) 2 1.0882 146.3 13:12 ( 138.6) 3 1.1841 292.6 6:5 ( 315.6) 4 1.2886 438.9 13:10 ( 454.2) 5 1.4022 585.2 7:5 ( 582.5) 6 1.5258 731.5 3:2 ( 702.0) 7 1.6604 877.8 5:3 ( 884.4) 8 1.8068 1024.1 9:5 (1017.6) 9 1.9661 1170.4 2:1 (1200.0) 10 2.1395 1316.7 15:7 (1319.4) 11 2.3282 1463.0 7:3 (1466.9) 12 2.5335 1609.3 5:2 (1586.3) 13 2.7569 1755.7 11:4 (1751.3) Some of the above are not really very close to small rational intervals, but notice there are good approximations to the 7:5, 3:2 (perfect fifth), 5:3 (major sixth), 9:5 (minor seventh), 15:7 (minor ninth), 7:3, and 11:4. I'm sure there's more to it than this, but this is a start. Rich Holmes =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- | Richard S. Holmes | | | | Physics Department | (315)443-5973 | rich@suhep.bitnet | | Syracuse University | or -2701 | rich@suhep.phy.syr.edu | | Syracuse, NY 13244 | | | =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- DISCLAIMER: I have no opinions. From ST601909@BROWNVM (Tim Boulette) Wed Apr 24 10:00:45 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil t nil] ["886" "" "22" "April" "91" "16:13:10" "GMT" "Tim Boulette" "ST601909@BROWNVM.BITNET " nil "20" "Re: An Electronic Music Concert Review (semi-" "^From:" nil nil "4"]) Newsgroups: bit.listserv.emusic-l Message-ID: Lines: 20 Comments: Gated by NETNEWS@AUVM.AUVM.EDU From: ST601909@BROWNVM.BITNET (Tim Boulette) Subject: Re: An Electronic Music Concert Review (semi- Date: 22 Apr 91 16:13:10 GMT >From: JSMOR@CONNCOLL.BITNET >Another note about the Pierce scale, I don't really understand its derivation, >but it sounds great. If anyone out there could explain it, it would be much >appreciated. > >Jon Morris The Pierce scale is an analog of the equal-tempered chromatic scale, simply using a different "equivalence" (rather than the octave), and dividing the result into 13, rather than 12. Here, the basic equivalence is the twelfth (an octave and a fifth) in pythagorean tuning (that is, a "real" fifth, which is a frequency ratio of 3:1, rather than a equal-tempered fifth, which is a frequency ratio of 2 to the 7/12 power). In other words, a "semitone" in the Pierce scale has a frequency ratio of 3 to the 1/13 power, while a equal-tempered semitone has a frequency ratio of 2 to the 1/12 power. (Somebody correct my math if it's in error....) -Tim From djones@megatest (Dave Jones) Wed Apr 24 14:25:21 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["793" "" "19" "April" "91" "23:58:30" "GMT" "Dave Jones" "djones@megatest.UUCP " nil "14" "Re: Minor mode names (Re: Chord Families and Associate Scales)" "^From:" nil nil "4"]) Newsgroups: rec.music.makers,rec.music.synth,rec.music.bluenote Message-ID: <17911@prometheus.megatest.UUCP> References: Organization: Megatest Corporation, San Jose, Ca Lines: 14 From: djones@megatest.UUCP (Dave Jones) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 19 Apr 91 23:58:30 GMT >From article , by cbolton@csd475a.erim.org (Chris Bolton): > If my memory serves me correctly, my improv. teacher said a Spanish > phrygian was the same as a phrygian but with a major third as well. That > is, it contains both a major AND minor third. That makes more sense. That makes it the third mode of the so-called "major bebop scale". When played in eighth-notes, all the odd-numbered modes of that scale sound similar, because there are eight notes in the scale. Everything stays "lined up". In the Spanish phrygian as you define it, the major third is only a passing tone between the minor third, which occurs on a down-beat, and the four, which does same. (In the major bebop scale, the same note occurs between the five and the six.) From RICH@SUHEP (Richard S. Holmes) Wed Apr 24 14:27:45 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["2945" "" "23" "April" "91" "15:47:00" "GMT" "Richard S. Holmes" "RICH@SUHEP.BITNET " nil "55" "Re: Pierce scale" "^From:" nil nil "4"]) Newsgroups: bit.listserv.emusic-l Message-ID: Lines: 55 Comments: Gated by NETNEWS@AUVM.AUVM.EDU Original_To: JNET%"emusic-l@auvm" Original_cc: RICH From: RICH@SUHEP.BITNET (Richard S. Holmes) Subject: Re: Pierce scale Date: 23 Apr 91 15:47:00 GMT "William R(ay) Brohinsky" writes: >That's a good description of the pierce scale. However, as I have noted >in many postings on bix about the Lucy scale, approximations of 30 >cents is not a `good approximation'. awright awright... so it was a bit lame. Yeah, after I sent it off I saw I'd made some rather excessive claims for the fifth and octave... also there are some better rational intervals. Here's a slightly less lame table. PIERCE SCALE NEARBY RATIONAL INTERVALS DIATONIC DEGREE FREQ. RATIO CENTS RATIO CENTS ERROR INTERVAL 1 1.0000 0.0 1:1 0.0 -- Unison 2 1.0882 146.3 12:11 150.6 -4.4 -- 3 1.1841 292.6 6:5 315.6 -23.0 m3 4 1.2886 438.9 9:7 435.1 3.8 m7 - tritone 5 1.4022 585.2 7:5 582.5 2.7 tritone 6 1.5258 731.5 3:2 702.0 29.5 P5 7 1.6604 877.8 5:3 884.4 -6.6 M6 8 1.8068 1024.1 9:5 1017.6 6.6 m7 9 1.9661 1170.4 2:1 1200.0 -29.5 Octave 10 2.1395 1316.7 15:7 1319.4 -2.7 P12 - tritone 11 2.3282 1463.0 7:3 1466.9 -3.8 M6 + tritone 12 2.5335 1609.3 5:2 1586.3 23.0 M10 13 2.7569 1755.7 11:4 1751.3 4.4 -- Note that the basic interval is just 3.7 cents shy of 150 cents, i.e., in terms of a 12-degree octave based ET scale, a half step plus a quarter tone. Two of these put you 7.4 cents shy of 300.0 cents, a tempered minor third, which is itself 15.6 cents shy of the 6:5 just minor third, so this minor third is quite flat. The fifth degree is close to a diatonic tritone and the seventh is close to a diatonic major sixth. The fourth degree can be thought of as a tritone below a minor seventh. The sixth degree is a very sharp fifth. The other notes are just complements of these, but complementary with respect to the twelfth, not the octave. So actually the most consonant intervals here are major sixths and tritones! This would seem to imply some rather exotic harmonies. What's the Lucy scale? I must have missed your "many postings". Rich Holmes =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- | Richard S. Holmes | | | | Physics Department | (315)443-5973 | rich@suhep.bitnet | | Syracuse University | or -2701 | rich@suhep.phy.syr.edu | | Syracuse, NY 13244 | | | =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- DISCLAIMER: I have no opinions. From @ruuinf.cs.ruu.nl:LISTSERV@AUVM.BITNET Fri Jun 14 13:54 MET 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil t nil] ["20784" "Tue" "19" "March" "91" "10:41:00" "EST" "TUTTEROW@UNCG.BITNET" "TUTTEROW@UNCG.BITNET" nil "384" "Why divide by 12 and not 15?" "^From:" nil nil "3"]) Received: from ruuinf.cs.ruu.nl by alchemy.cs.ruu.nl with SMTP (15.11/15.6) id AA15970; Fri, 14 Jun 91 13:54:24 met Return-Path: <@ruuinf.cs.ruu.nl:LISTSERV@AUVM.BITNET> Received: from accucx.cc.ruu.nl by ruuinf.cs.ruu.nl with SMTP (5.61+/IDA-1.2.8) id AA27253; Fri, 14 Jun 91 13:48:19 +0100 Received: from hearnvax.nic.surfnet.nl by accucx.cc.ruu.nl (5.65b/4.26) with SMTP id AA28254; Fri, 14 Jun 91 13:52:18 +0200 Received: from AUVM.AMERICAN.EDU by HEARNVAX.nic.SURFnet.nl; Fri, 14 Jun 91 13:52 MET Received: by AUVM (Mailer R2.07) id 0238; Fri, 14 Jun 91 07:35:02 EDT Reply-To: Electronic Music Discussion List From: TUTTEROW@UNCG.BITNET Sender: Electronic Music Discussion List Subject: Why divide by 12 and not 15? Date: Tue, 19 Mar 91 10:41:00 EST Allen. This may not be appropriate for EMUSIC but then again it might for those that play with different tuning systems. These statements are taken from "A History of Western Music" by Donald J. Grout, which is the standard History book for most college music schools. The first written record of the ocatave being divided into twelve parts comes from the Greeks, possibly from Aristoxenus and Pythagoras. Pythagoras discovered concords from the simple ratios among the divisions of a sounding string. String lengths in the ratio of 2:1 produced an octave, 3:2 the fifth, and 4:3 the fourth. All other intervals were considered discords. For the Greeks, the concept of note and intervals were dependent on the a distinction between two kinds of movement of the human voice: the continuous, in which the voice changes pitch in a constant sliding up and down without fixing a pitch; and the diastematic, in which pitches are sustained with discreet intervals perceptible within the the pitches. Intervals such as tones (whole step), semitones (half step), and ditones (thirds) were combined into a system of scales. These systems of scales WERE bases on twelve even divisions of the 2:1 ratio. Different people divided Pythagoras' concords differently, but the division by twelve was the only one that lasted throughout the two thousand years of Western Musical development. All through history music has experimented with dividing the octave into different divisions of microtones, but none has stayed with us. (An argument for Darwinian survival of the fittest chord?) This equal division provided us with the perfect fourth, perfect fifth, and perfect octave, but all other intervals were dissonant, even more than now. This problem was not solved until the late seventeenth and early eight -teenth century when our modern tuning system came into being. The perfect fourth and perfect fifth were tuned slightly flat with the octave being left perfect. This in turn brought the other interval the major and minor third and major and minor sixth closer to consonance, along with le lessoning the dissonance of the second and sixth. A good exercise for us in electronic music, especially if we are experimenting with different timbre qualities of patches would be to play with tuning our instruments differently in order to produce slightly different intervallic sounds. Then maybe experiment with different divisions of the octave and see what you can come up with. Write a tone poem based on a new scaling system and experiment. Twentieth century ears are not as attuned to consonance and dissonance as were earlier generations. Electronics can open up new ways of composition to take music into a brand new style having nothing to do with basic tonality as we know it. Experiment with tuning. What sounds bad today could be a hit ten years from now. I hope this wasn't to long and drawn out. Thanks. Kirk ========================================================================= Date: Wed, 20 Mar 91 10:50:14 GMT Reply-To: Electronic Music Discussion List Sender: Electronic Music Discussion List From: Nick Rothwell Subject: Re: Comments/opinions-Ensoniq Why 12 tones in the scale? The reason is, that for the better part of 4 or 5 thousand years, we had no set limit on notes. Even while Pythagoras was making his theoretical scales, you'd start at what ever pitch felt right, and as you sang, the notes would pretty much go where it sounded good. In the case, specifically, of Pythagoras' scale, it was built out of pure fifths: 3/2 ratios. But if you do this 12 times, then divide by 2 seven times, you end up with different answers! the ratio of the original pitch to the derived one is 1:1.013643...this is clearly not identity! This method did allow generating a scale of twelve tones. The reason for not going beyond 12 tones is that the next tone generated after the first 12 (and before the first 24) will always be 1.013643... above one of the already generated notes. (this will actually continue even after 24 notes, but each new note will come out 1.013643 above one already-generated note, and 1.0274726 above another, and on, and on...) This scale, however, is present in nature. The pythagorean scale is common in solo music: where a voice or fretless instrument is played with no accompaniement. The third and seventh are quite high, but this sounds OK with no other notes simultaneously present to compare to. On the other hand, when adjustable-pitch instruments play in consort (the renaissance word), Just intonation is used. (intonation=tuning. temperament= detuning for a purpose). In this tuning system, each interval is tuned against others to produce beatless chords. Although some have claimed this to be dull, I think they are listening to sine-wave voices. Anything more complex than that will result in sum and differnce tones between the fundamental and partials of each voice, yeilding notes which are also beatless, but not being directly produced. This is because the partials of most natural instruments (excepting bells and the piano, the latter of which could be argued as to its naturalness :-) are in integer multiples of the fundamental, and are therefore beatless. The whole thing of ringing chords in barbershop is based on just intonation. Helmholz observed that when the best performances occurred, they were played in Just intonation by artists who adjusted their pitch to match each other. These kind of performances are usually characterized in descriptions by an emphasis on the sound, tone, and quality of the sound, rather than on just the music. The temperaments came about when fixed-pitch instruments were used in different keys. This is accompanied by a cataclysmic change in musical style, but NOT an elimination of Just intonation for group playing and Pythagorean for solo work! It is to be noted that even the most modern instrument, wind or string, is still actively adjustable when being played! Any time a person has his hands on the strings, slides, his lips on the reed, or his diaphragm at the bottom (or top) of the wind column, pitch can be controlled. It is the education of the player into this control that has been neglected in recent years, more's the pity. Keyboard instruments, however, lack this kind of control. Even Synths (which can be tuned by microtonal tables) still are fairly fixed-the mod wheel bends the pitch of the whole instrument. (ever tried to tie aftertouch to pitch control? Can it even be done???) The problem with Just intonation for truly fixed-pitched instruments is that the first two whole-tone intervals are not the same! The first is larger than the second. There is a `major' and `minor' semitone: one whole interval is made up of two major semis, the other from one major and one minor. That means that a keyboard tuned to play just in C will not play just in D. There are lots of other problems, not the least of which is, `where are you after four modulations by a fifth? After 12?' Each modulation brings with it a shift in some notes, so that by 12 modulations (remember Pythagoras?) you're completely off! I won't go into any of those, since the major-semi minor-semi problem is basically insurmountable in Just fixed-pitch instruments. What to do? Well, the theorists decided that, if they took the error interval between 12 fifths and seven octaves, and divided it by a quarter and applied it to the fifth, shortening it, they could end up with a pure third and a pure octave. (the third is hit after four iterations, so the quarter adjustment brought that note down to what was needed for a pure third). This is called 1/4-comma Meantone, the mean being the note between the tonic and the fifth. It was in use for many years, and is still the tuning of choice for some music. The reason was because the scale produced by iterated fifths which are reduced by the fourth of the comma (that discrepancy between 12 fifths and 7 octaves) are not always right, and two of them are quite bad for using in some keys. These keys were called `wolf keys', for their howling! However, the great composers were aware of this (how could they not be?) and used chords on those notes as tension heighteners: how much more alive and meaningful is their music when played in 1/4-comma mean tone, than when played on equally tempered instruments! Equal temperament sacrifices the third, in order to make the fifth more pleasant. This is the `emotional' argument. Mathematically, it takes the comma and divides it into 12 parts, subtracting the 12-th comma from each fifth in the iteration. This way, the octave remains constant, the third ends up high, but not as bad as with Pythagorean, and the fifth is just a bit flat. Since this does divide the octave by equal ratios, hence the name equal temperament, you can now play in any key, and have it sound just as good, or just as bad, as any other. Grout's comments on the life and livelyhood of temperaments other than equal is some decades old. That, in my opinion, answers for his naivity in re: the survival of Just and Pythagorean, and the revival of other temperaments. There ARE other temperaments than ET12. ET32 approximates 1/4-comma mean- tone, and another one (helmholz called it Mercatorial, after Mercator. I don't know if this is the same Mercator whose map projection was the accepted norm until just recently) which divides the octave into 53 parts. This one allows playing in very-near-Just intonation in all keys, but tends to be dificult to put on a keyboard 8^) A recent scale development, which has been hullaballooed with all to many claims of `increasing universal harmony, mapping all chaotic systems, making it possible for a musician to play in any scale of any culture, and making it possible to communicate with whales and dolphins' is the Lucy scale. Lucy has decided that our two-dimensional representation of scales by frequency ratios is naive. He postulates a scale built on PI. His results are (amoung other claimed scales) a 22-note scale that makes it possible to play out-of-tune equally in all 12 of our normal scales, but is supposed to be extendible to other levels of harmony. I list this here out of general wish to be fair, but with the admonishment that I haven't heard the results of music played with or written for this scale, nor have I seen a justification of the 3-D view of sound. I am reserving judgement for that time. I hope this hasn't burned your mind out completely. I recommend getting your hands on Helmholz (On the Sensation of Tone), in the latest Dover reprint. The text is interesting, but the appendices, especially those from 20 on, are a fascinating historical, theoretical, and even practical tretise on tuning. raybro ========================================================================= Date: Thu, 21 Mar 91 15:29:02 EST Reply-To: Electronic Music Discussion List Sender: Electronic Music Discussion List From: Joe McMahon Subject: Re: tonalities In-Reply-To: Message of Thu, 21 Mar 91 14:53:05 EST from raybro's posting reminds me of a J. S. Bach story. It seems that Bach was a proponent of equal temperment over meantone and had a longstanding argument about this with an organ maker in the area. The organmaker had just completed a new organ (tuned meantone) and asked Bach to demonstrate it for the new owners. Bach said, "Why, certainly", and sat down and proceeded to play something in F# major, guaranteeing that he would hot all of the wolf tones. The organmaker got so angry he ran up and tore the wig off Bach's head. --- Joe M. ========================================================================= Date: Thu, 21 Mar 91 14:10:00 EDT Reply-To: Electronic Music Discussion List Sender: Electronic Music Discussion List From: "William R(ay) Brohinsky" Subject: Re: The mathematics of music. UNC/Greensboro's only E-music literate faculty member is a bassoon prof? That's funny, I was a bassoon major! Odd coincidences aside, I can't recommend any of the U's or colleges in Connecticut, although I'm not that familiar with Wesleyan. They are very much into ethnic music, so they might just have an E-music studio hidden away there, somewhere. I don't think I did a very good job on that listing, actually. I'm still in flux in my feelings about scale evolution---it seems to me that prior to the renaissance, scales were largely 8-toned in the West, with the building blocks of whole and half tones being the prime mover. At least, all the ancient Greek scales that were noticed by renaissance theorists and used (well, the names were used on the wrong modes, but what the hey?) were eight-toned. The name diatonic, though , infers that the two-toned ness was what differentiated these scales from the chromatic (which name refers to the `coloration' possible in a 12-toned scale). After the temperaments started being used (and 1800 years before, when Pythagoras had at his theories) the scales were 12-toned, and based on successions of fourths and/or fifths. The effort of all the tempering theorists (and the practicioner[sp]s like Werkemeister) was towards getting a 12 note scale that was homogeneously transposable on a fixed- pitch instrument, and the efforts of such instrument designers as Bohm and Sax were toward making flexible-pitch instruments into fixed pitch ones. All of this deserves more thought on my part before taking a mental dump on all of you. However, the last posting had the benifit of being shorter than helmholz... raybro . ^too much time on bix... ========================================================================= Date: Thu, 21 Mar 91 16:20:00 EDT Reply-To: Electronic Music Discussion List Sender: Electronic Music Discussion List From: "William R(ay) Brohinsky" Subject: Re: tonalities >>>>>>>>>>>>>>>original posting contains:<<<<<<<<<<<<<<<<<<<<<<<<< Subject: Re: Generating notes w/internal speaker >RAY, >THAT WAS A GREAT POSTING ON TUNINGS. A LOT OF IT STILL GOES OVER MY >HEAD BUT I'M GETTING THERE SLOWLY... > >DAN YUEN I HEARTILY agree! Thanks for taking the time to write so much!. As a newcomer to emusic and this kind of theory history I got really excited about the complexity of what you described. Especially I would be interested in an elaboration of this comment: >HOWEVER, THE GREAT COMPOSERS WERE AWARE OF THIS (HOW COULD THEY NOT >BE?) AND USED CHORDS ON THOSE NOTES AS TENSION HEIGHTENERS: HOW MUCH >MORE ALIVE AND MEANINGFUL IS THEIR MUSIC WHEN PLAYED IN 1/4-COMMA >MEAN TONE, THAN WHEN PLAYED ON EQUALLY TEMPERED INSTRUMENTS! Is this difference you are referring to, the difference between an orchestra playing vs a piano ? What is an example (instrumentation) of what you are saying and for what music? Thanks! >>>>>>>>>>>>>>>>>>end of post, start of reply<<<<<<<<<<<<<<<<<<<<<<<<< warning: I am not sure how long this will go on... Well, As far as instrumentation goes... 1/4-comma mean tone was used for a very long time in europe. It was fairly late in Bach's life when he finely found and fell in love with the Well-Temperament. Prior to that, from the middle-late 1500's, keyboards were being tuned in mean-tone temperaments. I'm not exactly sure when before that they were tuned in mean-tone (lapse in memory, or something) but prior to mean-tone they were pretty much tuned to just or maybe Pythagorean (which resulted in much theoretical writing on just how un-useful those tunings were for fixed-pitch instruments!) In actual fact, any instrument that can be tuned can be played in mean-tone. The differentiating characteristeric is the composer's music. A digression: When Carleen Hutchens was developing her [now famous] violin family, she tried a few experiments. One such was to take a viola, and re-make the bouts (the sides that hold the top to the back) to a height of 1/2", instead of the inch and three-quarters (or so) usually used. She figured that the body resonance (at Bb in the normal viola) might be sapping the strength of the viola at other pitches, and she thought this first experiment (making the air-cavity resonance very high and reducing the cavity volume) would make the viola very weak-sounding throughout the bottom range. Quite the contrary, if memory serves me right, the new viola sounded very even and loud throughtout it's range! This occasioned a few AB tests with a variety of listeners and players. All prefered the old viola. She claimed that she finally narrowed this unreasonable conclusion's cause to the fact that she'd used Mozart's viola concerto as the test piece. Her conjecture was that Mozart knew that the viola had a strong peak at Bb, and so he `played to [that] gallery' by putting the piece in Bb. He then contoured the music to use the strong tonic and weaker dominant as features of the instrument, rather than weaknesses. When played on the new viola, the resulting over-emphasis on the previously weaker parts of the scale resulted in a parody of viola-feeling. Although this is a very subjective sport (at the very least!), I feel she's hit the target on the bullseye. To return to my point: Because the wolf keys in mean-tone temperament are the odd ones (an instrument tuned mean-tone in C starts to get really bad on chords like E, which use the G#), those chords are used rarely in baroque music, and when they come, their effect is so tension-building that the arival at the tonic has a much heightened effect. This is true whether the music is played by an orchestra, a harpsichord, violin, bassoon, or voices. AS LONG AS THE TUNING IS ADHERED TO! It is not always possible to tell from concurrent musicological sources just what intonation or temperament was in use for a particular composer, but a sensitive observer can `read' the temperament from the chord useage. The scientist I work for has a few early-baroque pieces he's played recently that he remarked to me just this point: When played on modern instruments fixed at ET12, the pieces would sound stock, Bb arrangements, despite the quality of the players. When his little band (harpsichord, 2 baroque oboes, baroque bassoon) play them using 1/6th-comma mean-tone (which he just happens to like a bit more than 1/4-comma because the fifths are a bit purer) the music springs out at you. Baroque music is claimed by the theorists of the time to be a more emotional music. I've heard lots of performances of baroque music from the early 1960's which, although bombastic, were very un-differentiated and dull. Newer recordings of the same pieces with some attention paid to intonation/temperament result in wonderfully moving music, even for friends of mine who are relatively antithetical to baroque music. As for what music, English music from Byrd to Purcell (Holborne, Gibbons, Jenkins, North, Dowland), French and Italian music from the early to the high baroque (Vivaldi, Gabrielli, Costello, Lully, Couperin,etc.) Notice I mentioned Gabrielli: although his wind-band stuff was most likely played Just, his keyboard stuff is most likely intended for mean-tone. Early Bach, pre-dating the `Well Tempered Clavier' by a few years. OH, Yeah---There's an old Deutsche Grammophone recording of Ralph Kirkpatrick (from when he still had time to practice!) doing the Well-Tempered Clavier (Wolhe Tempierte Klavier-I think that's the German Spelling) on a clavichord tuned to Well-temperament. Find this at your local college library, and compare it to, say, Wanda Landowska (harpsichord) or any other 1950s harpsichord rendings ('scuse-renditions). Amazing difference. It's interesting to me as a transplanted-to-connecticutan, that UCONN's performance auditorium is named after Jorgensen. He was one of the first to give Temperament Concerts: he'd have two Grand Pianos and a harpsichord. One Grand would be modern ET12 (stretched, because of the non-harmonic overtones of steel strings under tension), one grand tuned in Well-Tempered, and the harpsichord in 1/4-comma Mean. He play pieces on both the modern ET12 and on which other temperament they were probably written for, contrasting the sound for audiences. He has left behind a book on tuning the temperaments, interesting reading even if you can't tell which end of a tuning hammer is used for driving the nails 8^) raybro From eijkhout@s41 (Victor Eijkhout) Fri Jul 5 13:38:58 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["495" "" "4" "July" "91" "00:03:07" "GMT" "Victor Eijkhout" "eijkhout@s41.csrd.uiuc.edu " nil "15" "Re: What's this scale?" "^From:" nil nil "7"]) Newsgroups: rec.music.makers Message-ID: <1991Jul4.000307.22885@csrd.uiuc.edu> References: <0094B0DA.71AD02A0@GOMEZ.PHYS.VIRGINIA.EDU> Organization: UIUC Center for Supercomputing Research and Development Lines: 15 From: eijkhout@s41.csrd.uiuc.edu (Victor Eijkhout) Subject: Re: What's this scale? Date: 4 Jul 91 00:03:07 GMT scott@GOMEZ.phys.virginia.edu writes: > While watching a Bugg's Bunny cartoon, I picked up this great >Persian sounding scale from the background music. It's as follows: >(in A) >A B C D# E F G# A >It's a harmonic minor with a sharp fourth. Is this Lydian harmonic minor? Some people call this the 'gypsy scale'. Given that there are too many scales in existence to give them names in any systematic way, this sort of evocative names are probably as good as any other. > -Scott Victor. From @ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi Tue Aug 27 15:45 MET 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Received: from ruuinf.cs.ruu.nl by alchemy.cs.ruu.nl with SMTP (15.11/15.6) id AA29443; Tue, 27 Aug 91 15:44:56 met Return-Path: <@ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi> Received: from att.att.com by ruuinf.cs.ruu.nl with SMTP (5.61+/IDA-1.2.8) id AA21496; Tue, 27 Aug 91 15:18:02 +0100 Message-Id: <9108271418.AA21496@ruuinf.cs.ruu.nl> From: milo%mvuxi@att.att.com To: piet@cs.ruu.nl Subject: Re: circle of fifths (& music theory) archive Date: Tue, 27 Aug 91 09:25 EDT "Circle of Fifths" - part 1 of 2 From: milo@cbnews.att.com (guy.f.klose) Subject: Re: African Gapped scale? Date: 26 Feb 91 22:53:31 GMT In article <1660029@hpsad.HP.COM>, smithj@hpsad.HP.COM (Jim Smith) writes: > I went ahead and tried a five-note equally-tempered scale on my synth last > night, and it sounds pretty good! Any note sounds good with any other note > in the scale. It would be a good choice for a scale for kids in a beginning > music (pre-music? :->) class to play on marimbas... Its pretty common for kid's musical toys to be based upon such "pleasing" scales, such as pentatonic scales (of which there are numerous varieties). That way they aren't discouraged by dissonance. As far as pentatonic scales go (though not the equally-tempered one described above), everyone probably knows that the black keys on a piano form a pentatonic scale. I think of this "short cut" when forming scales (I guess they would be considered major scales) of varying lengths. To form a pentatonic scale, chose any five consecutive notes from the circle of fifths: C G D A E B F#(Gb) Db Ab Eb Bb F C... (etc.) Note that the black keys from a piano are all adjacent in the circle of fifths. I would suppose that this can be generalized to form any length of scale. If you start on any given note (e.g., F) and count the next six notes, then you have a normal major scale (e.g., C-Major scale). As a side note, during the last heavy discussion of theory in this group, I started a file of "things I've noticed about the circle of fifths", but didn't really finish it well enough to post it. I have this strange feeling that many "secrets of Western Music" are wrapped up inside of the circle of fifths, but I've never seen an exhaustive study of it done in a theory book. Anyone else have any things they've noticed about it? Guy -- Guy Klose milo\@angate.att.com From: smithj@hpsad.HP.COM (Jim Smith) Subject: Re: African Gapped scale? Date: 27 Feb 91 23:26:06 GMT Re: Circle of fifths: The pentatonic scale generated by taking neighbors on the circle of fifths will not be equally-tempered, and, for me, always tends to evoke stereotypical 'Chinese' sound, due, I am sure, to my pre-school jams on the piano. :-) The equally-tempered pentatonic scale sounds quite different from this, since each step in the scale is somewhere between a whole step and a minor third. Yes, it's a bit 'dissonant,' but I like dissonance, especially in percussive parts (as was pointed out before). As for much of the secrets of harmony being wrapped up in the circle of fifths, I remember reading an interview with keyboardist/songwriter Bonnie Hayes, in which she said her guru/music teacher showed her the power of the circle of fifths. Don't remember his name, but I think he lived in Sebastopol, California... -Jim (Oh, yeah... you can also play 'Cowboy' music on the black keys!) :-) Date: Wed, 27 Feb 91 17:01:58 -0500 From: ogata@cs.UMD.EDU (Jefferson Ogata) To: milo@cbnews.att.com Subject: Re: African Gapped scale? In article <1991Feb26.225331.12189@cbnews.att.com> milo@cbnews.att.com (guy.f.klose) writes: >Its pretty common for kid's musical toys to be based upon such "pleasing" >scales, such as pentatonic scales (of which there are numerous varieties). >That way they aren't discouraged by dissonance. The scale he's talking about is not a scale to use if you don't want dissonance; in lower registers it will be much more dissonant than a 1 2 3 5 6 pentatonic. This is because it is not canceling beats very well, and the secondary beats are going to get subsonic at a much higher frequency than if you're sticking to notes from a beat-canceling scale like the regular modal scales. >I think of this "short cut" when forming scales (I guess they would >be considered major scales) of varying lengths. To form a pentatonic >scale, chose any five consecutive notes from the circle of fifths: > >C G D A E B F#(Gb) Db Ab Eb Bb F C... (etc.) Naturally this works because you've picked five notes...You can also use any other interval that doesn't divide an octave. >Note that the black keys from a piano are all adjacent in the circle of >fifths. I would suppose that this can be generalized to form any length >of scale. If you start on any given note (e.g., F) and count the next >six notes, then you have a normal major scale (e.g., C-Major scale). Well, if you start on F and consider the key to be F, you have F Lydian. But you also have C major, A minor, B Locrian, etc. This happens simply because the fifths precess one whole tone every other fifth. For the first six fifths you get two pieces of two whole-tone scales a fifth apart. Since standard mode scales contain exactly that, you wind up with a standard mode scale. Lots of other ways to do it too (whole whole half whole whole whole half)... -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From: djones@megatest.UUCP (Dave Jones) Subject: Re: African Gapped scale? Date: 1 Mar 91 00:13:02 GMT >From article <1660030@hpsad.HP.COM>, by smithj@hpsad.HP.COM (Jim Smith): > Re: Circle of fifths: The pentatonic scale generated by taking neighbors on > the circle of fifths will not be equally-tempered ... Huh? In my book, "equally tempered" means exactly the same thing as well tempered. It's the kind of temperment that's been used on pianos since the days of Bach. > and, for me, always tends > to evoke stereotypical 'Chinese' sound, due, I am sure, to my pre-school > jams on the piano. Ancient Chinese sacred music was indeed written in the pentatonic scale. :-) The equally-tempered pentatonic scale sounds quite different > from this, since each step in the scale is somewhere between a whole > step and a minor third. You mean the scale which is being called the "gapped scale", right? I think a better name for it might be the "symmetric pentatonic scale". > (Oh, yeah... you can also play 'Cowboy' music on the black keys!) :-) Also blues, jazz, gospel, pop, rock, and oh yes, classical. Ever hear "Humoresque"? I've got a book and practice record for jazz improvization built entirely around pentatonics. The pentatonic is used extensively in blues. Very often it is the minor pentatonic, often with an added tri-tone, making it the so-called "blues scale". From: djones@megatest.UUCP (Dave Jones) Subject: scales and fifths (Re: African Gapped scale?) Date: 27 Feb 91 23:22:28 GMT > I think of this "short cut" when forming scales (I guess they would > be considered major scales) of varying lengths. Major, minor, and other. See below. > To form a pentatonic scale, chose any five consecutive notes from > the circle of fifths: > > C G D A E B F#(Gb) Db Ab Eb Bb F C... (etc.) Two modes of this scale are generally given names. CDEGA is called the "major pentatonic" or simply "pentatonic". ACEGA is called the "minor pentatonic". What kind of names can we dream up for the others? > Note that the black keys from a piano are all adjacent in the circle of > fifths. I would suppose that this can be generalized to form any length > of scale. > > If you start on any given note (e.g., F) and count the next > six notes, then you have a normal major scale (e.g., C-Major scale). That's why Kenneth Russell, of "Lydian-Chromatic Concept of Tonal Orginization for Jazz Improvization" fame, chose to regard F-Lydian as the basic mode of the C-diatonic modes, not what we call C-major (Ionian mode). Funny thing. (Plate o' shrimp.) I was just experimenting with your concept the other day. To form a melodic minor scale, you pick five contiguous notes in the cycle of fifths, then add one one on each end, but leave a gap on both sides. Example: C-melodic minor B (E) A D G C F (Bb) Eb gap gap Are all popular scales like that? Not by a long shot. The harmonic minor scale is sort of splattered all over the cycle of fifths. It's not too hard to see why the fifths clump up more in the diatonic scheme than in the harmonic minor: The chords that the harmonic minor is built around don't contain as many perfect fifths. > As a side note, during the last heavy discussion of theory in this > group, I started a file of "things I've noticed about the circle of > fifths", but didn't really finish it well enough to post it. > ... but I've never seen an exhaustive > study of it done in a theory book. Anyone else have any things they've > noticed about it? Many. Why don't you finish up your list and post it? I'll be happy to add anything I can think of. I will say say much now: The importance of the fifth is an artifact of physics -- of how things, including our ears, vibrate. After the octave, it is the first note in the overtone series (harmonic series) of a fundamental frequency. When hear the sequence V-I, we hear the most basic release of tension. Try over-blowing a coke bottle. Now louder -- that's the octave. Louder yet! -- that's the fifth. Let it go back down. Ah. That's a five-one. When we hear the interval V/I in a chord we hear what is, after the octave, the most consonant way two notes can correspond. That's because 2 and 3 are the first prime numbers. For every three times the V vibrates, the I vibrates twice. Ah, again. That's a perfect five. From: milo@cbnews.att.com (guy.f.klose) Subject: Circle of Fifths Date: 5 Mar 91 17:45:04 GMT OK, so here's my file of notes on the Circle of Fifths (a.k.a. lots of other names). I'll just use the shorthand "cycle" from here on. First, some background...I've been playing trombone for twenty years, and took a couple of music theory classes in high school and college. My reasons for collecting these notes is that I very much enjoy the discussions of music theory in this newsgroup. It's obvious that there are several of you that know lots more about music theory (and jazz theory, in particular) than I do, so I'm very anxious to learn more from you as well. On the other hand, I suppose there are lots of you that have never had a theory class, and probably have never learned what the cycle is all about. Just about every music teacher I've ever run across has suggested that it's important to know the cycle "cold". These notes are just a simple summary of what I've learned about the cycle, with a couple of my own observations thrown in. Corrections and additions are welcome! It seems to me that the cycle has some "hidden" qualities, and perhaps has some applications that some of us haven't thought of yet. In fact, I would venture a silly, wild-assed guess that the cycle very compactly sums up an awful lot of Western music. Please excuse my sometimes basic approach. I've tried to assume only a rudimentary knowledge of music theory by the reader. Also, I've limited my thinking to "Western" music, which is the only basis from which I have studied. I would really love to hear other's observations about other types of music. First of all, here is a diagram of the circle: C F | G | Bb | D | | Eb---------------|-----------------A | | Ab | E | Db | B Gb/F# If the diagram didn't turn out on your terminal, trying drawing it on a separate piece of paper...the cycle goes C G D A E B F#(Gb) Db Ab Eb Bb F...starting with C at the top. The major axes are C and F#(Gb), and A and Eb...everyone (hopefully) knows the relationship used to build the circle. Start on C, and as you move clockwise, move up by an interval of a perfect fifth. Please note that F# is enharmonically the same as Gb (technically speaking they are not the "same" note, because they are not "spelled" the same way; they do, however, sound the same...hence the terminology "enharmonically the same"). Here are some of those ideas: 1. Key Signatures (sharps and flats): The circle can be used to summarize key signatures. Start by considering a virtual break in the circle, between Bb and F. You might even want to draw a line separating the two of them. Now draw numbers next to the notes as follows: C(0), G(1#), D(2#), A(3#), E(4#), B(5#), F#(6#) F(1b), Bb(2b), Eb(3b), Ab(4b), Db(5b), Gb(6b) Now to interpret those numbers...the key of C has no sharps or no flats. The key of G has 1 sharp, which is F# (to get the #, start at the "virtual break" and count sharps clockwise starting with F). The key of D has 2 sharps: F#, C#. The key of A has 3 sharps: F#, C#, G#. The key of E has 4 sharps: F#, C#, G#, D#. The key of B has 5 sharps: F#, C#, G#, D#, A#. The key of F# has 6 sharps: F#, C#, G#, D#, A#, E#. The key of F has 1 flat, which is Bb (to get the b, start at the "virtual break" and count flats counter-clockwise starting with Bb). The key of Bb has 2 flats: Bb, Eb. The key of Eb has 3 flats: Bb, Eb, Ab. The key of Ab has 4 flats: Bb, Eb, Ab, Db. The key of Db has 5 flats: Bb, Eb, Ab, Db, Gb. The key of Gb has 6 flats: Bb, Eb, Ab, Db, Gb, Cb. The key of Gb is enharmonically the same as the key of F#, but it is "spelled" differently. Theoretically, I suppose, you could have a key with seven sharps or seven flats, but it is certainly not a practical sort of key. It would be enharmonically the same as having a key a half-step higher or a half-step lower. Besides, these keys would be considered Cb and E#, which are notes we normally think of as B and F, respectively. 2. Scales I once noticed an interesting relationship between the cycle and scales. For example, start with F on the circle, and count the next six notes. Those notes make up a C major scale (or generalized, the modes of a C major scale). So, count any seven consecutive notes on the circle, and they will make up the modes of a major scale. For example, start with Db, and count the next six (Ab, Eb, Bb, F, C, G), rearrange them, and you are left with an Ab major scale. The modes are as follows: Ab Ionian: Ab Bb C Db Eb F G Bb Dorian: Bb C Db Eb F G Ab C Phrygian: C Db Eb F G Ab Bb Db Lydian: Db Eb F G Ab Bb C Eb Mixolydian: Eb F G Ab Bb C Db F Aeolian: F G Ab Bb C Db Eb G Locrian: G Ab Bb C Db Eb F I then started doing some thinking...what was so magic about a major scale? Scales don't have to have seven notes. They could be any number of notes. Think about a pentatonic scale. In my early music classes, I was always told: the black keys on a piano make up a pentatonic scale (C# D# F# G# A#; or, Db Eb Gb Ab Bb). These are all adjacent notes on the circle of fifths. So pick a random starting place like F, count the next four (C G D A) and play that scale. It has the same scalar sound as the black keys. Add one more to the scale, E. Clash city...not the same pleasing sound as the pentatonic scale (the E clashes with the F). Adding the next note (B), brings you back to the major scale, but it also adding a tritone (F vs. B) to the scalar sound. Tritones are also known as augmented fourths, or diminshed fifths (flat fifths in the jazz idiom...more on this later). I imagine any length scale can be derived from the circle. A pentatonic scale is five notes...I suppose you can have four note scales, or six note scales. Whatever you want. One observation here...of course a five note scale would not have to be built upon five adjacent notes from the circle...it could be any five notes. I would think, though, that kind of scale really departs from Western music. 3. Chord Progressions (ii-V-I) We've all heard it said that chord progressions down a major fifth are the strongest resolution in Western music (certainly our ears confirm this). It is also called an "authentic cadence" (either perfect or imperfect, depending upon the chord voicing). It is pretty easy to see that by moving counter-clockwise in the circle, the adjacent notes make up the movement of down a perfect fifth. So, to form an authentic cadence in the key of B, for example, resolve an F# chord to a B chord. When studying jazz, one is almost always told two things: memorize the circle of fifths, and learn ii-V-I progressions. Both are related. One is also told that the ii-V-I progressions permeates all of jazz. Granted, looking through something like the Real Book, you will see that lots of tunes have chord progressions based upon ii-V, or ii-V-I. For those of you not up on figured bass notation, consider a triad (three-note chord) built upon the tonic (or root note) of a key. It is a major triad (e.g., in the key of C, the triad is C E G). Number it with a roman number 1 (I), and make it a capital to signify major. A triad built upon the second note of the key will be minor (D F A in the key of C). Number it, but use lower case to signify minor (ii). Roman numerals are then used to express key-independence. You can have a I or ii or whatever in any key. Also, a small circle is used to indicate diminished (vii^o in the key of C is B D F, or diminished). Augmented is described with a +. The dominant (fifth) triad in a major key is written as V (also a major triad). Hence, the notation ii-V-I. So, then ii-V-I in the key of C would be a Dmi resolving to G resolving to C. Except in jazz, you usually extend the chords and at the very least add the seventh. ii7-V7-I becomes Dmi7 to G7 to CMA7 in the key of C. G7 is also called a "dominant seventh chord" which has an even stronger sounding need to resolve down a perfect fifth. In the key of C, the G7 is spelled G B D F, and the F (a tritone against the B) has the tendency to resolve to E, which is the third of the C major chord. Where does the circle of fifths come in? Well, the ii-V-I are all adjacent steps on the circle. If you know the circle well, then your mind can very quickly tell you where a ii-V-I chord is, and where it is leading. To come with a ii-V-I in any key, pick three adjacent steps. The progression is made up of those three chords, moving in a counter-clockwise motion. The first is played as a minor chord, the second is played as a dominant seventh chord, and the third is played as a major chord. "Extending the cycle" of the progression really just means that any chord can be labelled as a "temporary I chord" and you can progress to it with a ii-V, which are adjacent, of course. Here's an example of "extending the cycle"...first start with a basic chord progression, such as: | F / / / | / / / / | Gm7 / / / | C7 / / / | | Gm7 / / / | C7 / / / | F / / / | / / / / | Then, pick some key places, and consider those chords to be temporary I chords, and place a ii-V in front of it. I think good candidates for temporary I's are the Gm7 in bar 3 and the C7 in bar 6. Here's how it would come out: | F / / / | Am7 / D7 / | Gm7 / / / | C7 / / / | | Gm7 / D7 / | Gm7 / C7 / | F / / / | / / / / | One thing that is fun to do, is to open up a fake book to just about any jazz standard, and analyze it for the ii-V-I progressions. You'll be surprised how common it is, especially once you know about flat fifth substitutions. 4. Flat Fifths In the jazz idiom, I suppose since the advent of bebop, there has been talk of the "flat fifth" and the feel it gives to music. I'm just a novice with flat fifths, but here are some particular qualities... One use of a flat fifth is that when creating a chord progression, or making chord substitutions within an established progression, one can substitute for any dominant seventh chord, a dominant seventh chord built upon the flat fifth (a note that is a diminished fifth, or tritone away from the root of the original chord). One reason that this flat fifth substitution is so "acceptable", is that the original domininant seventh chord, and its flat fifth substitution (also a dominant seventh), share the third and the seventh. For example, a C7 (C E G Bb) and its b5 substitution Gb7 (Gb Bb Db Fb) share the third and the seventh, which are E (enharmonically the same as Fb) and Bb. Of course, the third of one is the seventh of the other. Sharing the third and the seventh is an okay concept, since they are considered the two most important tones of a jazz chord. That is, they help define the "quality" of the chord. A major chord is built with a major third from the root, and a minor chord is built with a minor third from the root. A major seventh chord is built with a major seventh interval from the root, and a dominant seventh or minor seventh chord is built with a minor seventh interval from the root. The flat fifth can be read from the circle of fifths. Any note has its flat fifth on the other side of the circle from it, e.g. F -- B (Cb), E -- Bb, Db -- G (Abb), etc. I've read that, in general, you can substitute a flat fifth for any chord (its just that for dominant chords, the reason is very obvious). So, a ii-V-I chord progression has a corresponding "alternate cycle" that's on the opposite side of the circle. I was just looking at the minor pentatonic scale. A minor pentatonic scale can be built with the tones 1 b3 5 6 b7 (C Eb G A Bb). These aren't exactly adjacent tones on the circle, but they come pretty close if you leave out the 4th (F) of the scale (more on this later) and if you leave out the ninth (D). One could argue that the ninth fits pretty well; certainly not as dissonant as the fourth. The blues scale is built by taking a minor pentatonic scale and adding the flat fifth to the scale. In C, this is C Eb Gb G A Bb. --- Finally, an issue not related to the cycle: At jazz camp this last summer, Jamey Aebersold offered a curious statement which he attributed to George Russell (New England Conservatory prof., author of "The Lydian/Chromatic Concept"). I'll throw out the statement here for discussion. Russell claims that the originators of Western Music probably made an mistake when they invented major scales. He said that they probably should have added a #4 to the major scale rather than the perfect fourth. One thing is that they make the distinction #4 rather than b5...the perfect fourth is left out, and the perfect fifth is left in. I'm not sure how this statement is justified...a #4 would really screw up the cycle. Also, 4 is a pretty dissonant tone in the scale. If you hear a beginning improviser hit lots of fourths, it sounds pretty bad. In defense of a #4, if you build a major chord (extended) out of thirds, the thirds alternate major and minor, until you hit the 4th (11th). Also, in jazz chords, you tend to see lots of alterations of the fourth, mostly raising it. I don't own the book "The Lydian/Chromatic Concept", but if any of you do, and if Russell discusses this issue in that book, I'd appreciate hearing more of his arguments. Thanks for reading...comments welcome... Guy -- Guy Klose milo\@angate.att.com From: djones@megatest.UUCP (Dave Jones) Subject: Re: Circle of Fifths Date: 7 Mar 91 22:20:00 GMT >From article <1991Mar5.174504.5927@cbnews.att.com>, by milo@cbnews.att.com (guy.f.klose): > In the jazz idiom, I suppose since the advent of bebop, there has been > talk of the "flat fifth" and the feel it gives to music. Much more commonly called a "tri-tone substitution". It predated Bebop. Did the classical cats use it? I dunno. I think of Coleman Hawkins's _Body and Soul_ from 1939 as a beginning of bebop, or at least a big hint. Listen to the way he plays it. There's a tritone substitution right at the start, just after the lyrics would have said, "My heart is sad and lonely...", if only saxophones spoke English. The chords in the first two and a half measures go something like this: Eb- Bb7(b9) | Eb-7 Ab7 | Dbmaj7 But he plays a fill in the melody as though it were like this: Eb- Bb7(b9) | Eb-7 D7 | Dbmaj7 (For those of you who are scoring at home, the first measure is in Eb harmonic minor, a nifty way to introduce the II-V-I in Db major which follows. See how the two sequences overlap? The II-V-I becomes a an extended cycle, VI II V I, with a functional substitution of VI7(b9) for the VI-7. Neat stuff.) > I was just looking at the minor pentatonic scale. A minor pentatonic scale > can be built with the tones 1 b3 5 6 b7 (C Eb G A Bb). These aren't exactly > adjacent tones on the circle ... The "minor pentatonic scale" in the traditional literature is built this way: 1 b3 4 5 b7 (C Eb F G Bb). Those *are* contiguous on the cycle. It's a mode of the major pentatonic. I've never seen the scale you list here given a name. > The blues scale is built by taking a minor pentatonic scale and adding the > flat fifth to the scale. In C, this is C Eb Gb G A Bb. Er..., sort of. You take the real minor pentatonic (the way I defined it above), and add the flat fifth: C Eb F Gb G Bb. That's what's called the "blues scale". Some R&B players really run it into the ground. :-( Take the major pentatonic scale and add a minor third, and you have the "major blues scale", a mode of the blues scale, which is also useful: Eb F Gb G Bb C = Eb major blues scale. > Russell claims that the originators of Western Music probably made an > mistake when they invented major scales. He said that they probably should > have added a #4 to the major scale rather than the perfect fourth. I've read, and eventually understood, _The Lydian Chromatic Concept of Tonal Organization for Jazz Improvization_. Maybe the hardest part is learning the title. Anyway, I think Russell is making a big to-do over nothing. I'll state it stronger: I'm going to go out on limb and say that his rationale does not hold water. He's out to lunch. The reason we use the scale we do is that you can use those same seven notes over the entire IImi V7 I sequence. True enough, you can only use the perfect fourth as a passing tone on the I. He substitutes the +4 on the I chord, because it is possible to hold that note. It sounds weird to most people though. As a passing note, I prefer the prefect IV, because it does not create a false impression that the V is the tonic rather than the I. It is a leading tone that begs to move to the III or the V. You *want* it to sound dissonant when used as a passing tone. He bases his argument around using notes in the cycle to build scales. That's the same mistake made with the "Pythagorean scale" millennia ago. The Pythagorean scale is built by taking successive just-tempered fifths, rather than successive well-tempered fifths as Russell would do. But there is much more than the cycle going on in note-selection! Pairwise, notes that are a fifth apart are musically important, and the movement of chords down a fifth is important, but those other intervals come close to notes in the cycle only by coincidence. That's the great insight that Bach and those cats had which makes possible music as we know it. It is impossible to overestimate the debt we owe the creators of the well- tempered scale. Second guessing them should be done with great circumspection. (Some violinists may *tune* their instruments to the Pythagorean scale, so that moving from string to string moves by just-tempered fifths, but they make constant intonation corrections as they play.) > I'm not sure how this statement is justified...a #4 would really screw up > the cycle. Not at all. For F-major, he starts with F - C - G - D, etc., the same notes we would identify with C major. In traditional theory, his F-Lydian scale is the Lydian mode of C major. That's where he gets the name "Lydian". > I don't own the book "The Lydian/Chromatic Concept", but if any of you do, > and if Russell discusses this issue in that book, I'd appreciate hearing > more of his arguments. I own it, but it's not here at the office. In my opinion, he doesn't make a very good case at all for dumping the traditional view and looking at things his way. (That's all it is you know: a way of looking at things. That's why he calls it a "concept".) His concept is based on observed coincidences, not physics and math. Because the the correspondences are only coincidences, they break down when he tries to extend them into a complete theory. He has to tweak with the scale to get in all of the chords that are generally used in jazz. So he is forced to use other scales that are not built by running the cycle. But he is loath to change more than one note in his beloved "Lydian" scale. The result is that for some chords he comes up with a mode of the harmonic MAJOR, an odd sounding scale that I very seldom hear used in jazz. Of course he picks some weird mode of the harmonic major and calls it the something-or-another-lydian scale. My advice: Forget about Russell. From: djones@megatest.UUCP (Dave Jones) Subject: Re: Circle of Fifths Date: 8 Mar 91 22:52:10 GMT >From article <1991Mar8.004756.22391@ida.liu.se>, by d87parfo@odalix.ida.liu.se (Par Fornland): > milo@cbnews.att.com (guy.f.klose) writes: >>the original domininant seventh chord, and its flat fifth substitution >>(also a dominant seventh), share the third and the seventh. > But: "Dm7b5 C#MAJ7 Cm" is also possible! Sounds great too. > There the 7th isn't the 3rd but the 4th of G7! How can one explain that? I know! I know! Call on me! I know! Let's look at the notes first. (Can we spell the C# as it Db? The key of C#, if there were such a thing, would have seven sharps. Five flats is way better. So, like okay...) In Dm7b5 (D half-diminishen) we have D F Ab C And, in the DbMAJ7, we have Db F Ab C Hmmm. Everything stays the same except that the root moves down a half step. How 'bout that? > Does is derive from G7sus4? Nope. It's a substitute for G7b9. This is the standard II V I sequence in C harmonic minor: Dm7b5 G7b9 Cm (The notes of C harmonic minor are C D Eb F G Ab B.) The key notes that give a 7b9 chord its color are (nacherly) the seven and the flat nine. Notice that the b9 (Ab), and the dom 7 (F), function in the DbMAJ7 as the five and three respectively. There's your rationale. How'd I do? While I've got you on the line, let me alert you to a sequence that uses the same kind of trick to change keys: Follow a dominant chord with a diminished chord up a half step. All the notes stay the same except for the bass, which moves *up* a half-step. The diminished chord is symmetric, so from there you can go almost anywhere. G7 G#dim > Isn't the 3rd (and the 5th) often omitted by jazzmusicians? I've heard > something about that. The fifth can be omited much more readily than the third. Maybe you're thinking of head-bangers. According to some postings a couple of months ago, sometimes they play only the one and five and call it a "power chord". Of course without a third, that's no chord at all. I remember from when I was a kid and my parents made me go to church, that about half the people there would be singing on perfect fifths. I don't think they knew they were doing it. I don't know what the connection is, unless head-bangers are kids who didn't get enough of that stuff in church. From: quayster@cynic.wimsey.bc.ca (Tony Chung) Subject: Re: Circle of Fifths Date: 9 Mar 91 21:27:53 GMT Hi Par, a little bit from your message, here: milo@cbnews.att.com (guy.f.klose) writes: >But: "Dm7b5 C#MAJ7 Cm" is also possible! Sounds great too. >There the 7th isn't the 3rd but the 4th of G7! How can one explain >that? Does is derive from G7sus4? What makes that progression sound strong is the fact you're using common tone bonding between chords, as well as voice leading. In a C# (we'll call it) Dbma7 chord, there's the tones Db F Ab C. The C is a common tone to the next chord (duh... C anything), and the Ab resolves downward to G, (also, the Db resolves downward to C). To be honest, this is bad theory practise. Usually in voice leading, semitone resolutions usually go up, don't they? A Db7 chord resolving to C would have a Cb note that resolves upwards to C. That creates a stronger cadence. If the final chord was a Cmi7, the Cb would ALSO resolve to Bb, which sounds even stronger. Then, if you add a #11 (G)... >Isn't the 3rd (and the 5th) often omitted by jazzmusicians? I've >heard something about that. No!! Never omit the third, OR the seventh in a chord... Those notes are what typifies the chord quality. Without them, the chord remains as ambiguous as a rap bassline (ugh). Usually the 3rd doesn't exist in a suspended voicing, or in some forms of dominant 11th chords. (personally, I never call an 11th with no 3rd an 11th chord, but a 7 sus 4 chord. don't ask me why... polychords all the way) Q:-) The fifth, on the other hand, is toast. Leave that to the guitar players who play power chords only (*hey!! I'm only kidding!!! no flames!!) :) >If you start on say C, and walk the circle, you'll find that you >get C G D A E B F# !!! You see F# not F. If however you would have >started on F for a C-scale you would, of course, get F instead of >F#. I think it's more logical to start on C than F, but I haven't >read anything about these matters. Of course, if you started on F for an F scale, you'd get a B natural, which is a #4, the same as the F# being a #4 in the C scale. You've just stumbled onto the intricasies of the Ima13(#11) chord family. In a major 7th chord, where upper extensions exist, you raise the 11th (4th) to avoid a minor ninth, or semitone interval between the 3rd and 4th from the root. It could have it's base in the cycle of fifths, or it could just have happened by accident. Of course, we all know the C G D... cycle is backwards to our natural way of hearing, don't we? We tend to hear cadences going V-I around the cycle. Try this: Take a chord shape (maj, min, dom) and go through the cycle backwards, and forwards. The one that sounds like V-I or II-V cadences should be the forward one. I'll leave it to your ears to figure this out! >>Thanks for reading...comments welcome... >>Guy >I've commented. Would someone please comment my comments! Likewise, if anybody (or their employers) has anything to say to me, either mail or post. You may be right, I may be crazy. -Tony 'Quays' ('keys' with a 'Q') +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- "I should be so lucky, | quayster@arkham.wimsey.bc.ca | ) lucky, lucky, lucky..." +----------------- Tony Quays -+ (____, -- a Stock Aitken Waterman gem From: quayster@cynic.wimsey.bc.ca (Tony Chung) Subject: Re: Circle of Fifths Date: 9 Mar 91 21:03:24 GMT djones@megatest.UUCP (Dave Jones) hunted and pecked: >From article <1991Mar5.174504.5927@cbnews.att.com>, by >milo@cbnews.att.com (guy. f.klose): > >> In the jazz idiom, I suppose since the advent of bebop, there has > >>been talk of the "flat fifth" and the feel it gives to music. > >Much more commonly called a "tri-tone substitution". > >It predated Bebop. Did the classical cats use it? I dunno. Sure did! They created a bunch of major chord substitutions called (by my theory teacher) "Mutant subdominant chords". The most common "Tri-tone" sub was the "Neopolitan 6th" chord -- a major triad on bII in first inversion. Any jazzers know this as a TTsub for V. Also, I believe there were German and French 6th chords as well, which were versions of dominant-sounding chords built on bVI. Those would be subs for IImi. I love the turn this discussion is taking. It's been 5 months since the last big theory post. Thanks to all of you on the net for sharing your knowledge. -Tony 'Quays' ('keys' with a 'Q') +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- "I should be so lucky, | quayster@arkham.wimsey.bc.ca | ) lucky, lucky, lucky..." +----------------- Tony Quays -+ (____, -- a Stock Aitken Waterman gem From: eethomas@cybaswan.UUCP (Andrew Thomas) Subject: Re: Circle of Fifths Date: 8 Mar 91 15:00:15 GMT In article <15580@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: >The "minor pentatonic scale" in the traditional literature is built >this way: 1 b3 4 5 b7 (C Eb F G Bb). Those *are* contiguous on the >cycle. It's a mode of the major pentatonic. I've never seen the scale >you list here given a name. > >(blues scale ) You take the real minor pentatonic (the way I defined it >above), and add the flat fifth: C Eb F Gb G Bb. That's what's called >the "blues scale". Some R&B players really run it into the ground. :-( >Take the major pentatonic scale and add a minor third, and you have >the "major blues scale", a mode of the blues scale, which is also >useful: Eb F Gb G Bb C = Eb major blues scale. I'm not a trained musician, but I have really enjoyed the above discussion. If you'll excuse my ignorance, I've got a query. The thing is, if I want to improvise in a minor key do I use the minor pentatonic/blues scale or the major. It seems like all the notes in the minor pentatonic scale are also in the natural minor so I guess that's what you use. But am I right? I believe it might be more complicated than that. In fact, now I think about it, I think I'm wrong. And if I'm improvising in a major key which of the scales should I use? I would greatly appreciate any comments on these scales, I think they're too primitive for most of my music books! Also comments on their general use would be very much appreciated. I know it's a terrible cliche to use them, but I find them quite effective for producing a "rock" feel. Please e-mail me direct, or put a message on the net if it would be of interest to other less musically-educated people. Thanks. Andrew Thomas. eethomas@pyr.swan.ac.uk I would appreciate any >From cbnews!att!philabs.Philips.Com!pjc Thu Mar 7 10:58:54 EST 1991 Date: Thu, 7 Mar 91 10:58:54 EST From: pjc@philabs.Philips.Com (Philip Cianci) To: milo@cbnews.att.com Subject: Re: Circle of Fifths it is always curious to find someone else who appreciates the cyclic nature of tonality, chord progressions etc. my associate and I began to develope a system based on these ideas a way back 1976 we call it the Matrix Music System I always find it surprising when someone else realizes the power of these concepts. we have taken it so far as to release software for IBM PC and Commodore 64 and I released one cassette of the material last year funny coincidence that I go back into the studio to begin recording the next generation of Matrix Music more power to you PJC From: d87parfo@odalix.ida.liu.se (Par Fornland) Subject: Re: Circle of Fifths Date: 9 Mar 91 19:29:42 GMT djones@megatest.UUCP (Dave Jones) writes: >From article <1991Mar8.004756.22391@ida.liu.se>, by d87parfo@odalix.ida.liu.se (Par Fornland): >> milo@cbnews.att.com (guy.f.klose) writes: >>>the original domininant seventh chord, and its flat fifth substitution >>>(also a dominant seventh), share the third and the seventh. >> But: "Dm7b5 C#MAJ7 Cm" is also possible! Sounds great too. >> There the 7th isn't the 3rd but the 4th of G7! How can one explain that? >> Does is derive from G7sus4? >Nope. It's a substitute for G7b9. This is the standard II V I sequence >in C harmonic minor: > Dm7b5 G7b9 Cm >(The notes of C harmonic minor are C D Eb F G Ab B.) >The key notes that give a 7b9 chord its color are (nacherly) the seven >and the flat nine. Notice that the b9 (Ab), and the dom 7 (F), function >in the DbMAJ7 as the five and three respectively. That goes for Db7 too, doesn't it? It's the MAJ7 that confuses me... Besides, Db7b5 sounds sometimes fit better than Db7. In Db7b5 you have Db F G B, that's 3 notes taken from the G7: G B D F. Db7b5b9 is Db F G B D, that's 4 out of 4 from the G7. So actually one could play the G7 and have the Db is the bass! (Wouldn't bet on that though...) I know very little about how do use 13-chords. I just love the sound of them, but can't fit them into my playing. Someone out there who knows something about them? -- !!!!!!!!! Par Fornland, Linkoping Institute of Technology, Sweden Bjornkarrsgatan 10c:10, 582 51 Linkoping, Sweden Tel. (Sweden)-013-260486 d87parfo@odalix.ida.liu.se !!!!!!!!! From: milo@cbnews.att.com (guy.f.klose) Subject: Re: Circle of Fifths Date: 11 Mar 91 17:06:42 GMT I can't believe the response I've received on this Circle of Fifths posting...thanks for all the appreciative comments that you've been sending to me. This is also making a lively discussion...I'd like to encourage everyone to keep up the comments, corrections, and new questions. >Isn't the 3rd (and the 5th) often omitted by jazzmusicians? I've >heard something about that. As other people have already pointed out, omitting 3rds is not common practice. The "3rd" defines a chord as major or minor, and a seventh defines other chord quality such as major-7th or dominant-7th. I have heard it said that the 3rd and the seventh are the two most important chord tones because of this quality definition. Other notes are left as "color" for the chord. What about the root? Isn't it important? Yes...but bass players can cover roots, so pianists don't need to double it. If you are orchestrating a large ensemble, you probably wouldn't leave out any chord tones, unless you were looking for a specific sound. You would probably be more concerned about which notes to double, thereby emphasizing them. Also, I'm not a pianist, but I've heard that when pianist are taught left-hand comping, for their own soloing, that they are first taught two-note voicings in the left hand, and those two notes are usually thirds and sevenths, assuming that they are playing with a bassist. Guy -- Guy Klose milo\@angate.att.com From: milo@cbnews.att.com (guy.f.klose) Subject: Re: Circle of Fifths Date: 11 Mar 91 17:12:22 GMT In article <2371@cybaswan.UUCP>, eethomas@cybaswan.UUCP (Andrew Thomas) writes: > > The thing is, if I want to improvise in a minor key do I > use the minor pentatonic/blues scale or the major... > > I would greatly appreciate any comments on these scales, I think they're > too primitive for most of my music books! Also comments on their general use > would be very much appreciated. I know it's a terrible cliche to use them, > but I find them quite effective for producing a "rock" feel. > You know, I wonder about this same sort of thing...when to apply blues scales. For example, when I listen to Cannonball Adderly, I hear a bluesy style... its almost as if he could take any sort of tune, and change its flavor to "blues" just by tasteful use of the blues scale and blue notes. I'd guess that the secret to avoiding the cliche is to avoid using the scale 100% of the time. But, how does one get that "Cannonball sound"? Guy -- Guy Klose milo\@angate.att.com From: quayster@arkham.wimsey.bc.ca (Tony Chung) Subject: Re: Circle of Fifths (Mar 11 version) Date: 11 Mar 91 10:07:26 GMT d87parfo@odalix.ida.liu.se (Par Fornland) wrote: >djones@megatest.UUCP (Dave Jones) writes: > >>The key notes that give a 7b9 chord its color are (nacherly) the seven >>and the flat nine. Notice that the b9 (Ab), and the dom 7 (F), function >>in the DbMAJ7 as the five and three respectively. > >That goes for Db7 too, doesn't it? It's the MAJ7 that confuses me... Don't you get it? The major 7th is the root of the chord you're going to!! It's so cool! You can hold the C all the way from Dmi7(b5) to C minor, using Dbma7 as the TTsub for G7! >Besides, Db7b5 sounds sometimes fit better than Db7. In Db7b5 you >have Db F G B, that's 3 notes taken from the G7: G B D F. >Db7b5b9 is Db F G B D, that's 4 out of 4 from the G7. >So actually one could play the G7 and have the Db is the bass! >(Wouldn't bet on that though...) Or one could play the Dbmi7(b5) chord and have the G in the bass, causing the total effect of the G+7 (b9) chord. One of the many versions of the V chord from minor. >I know very little about how do use 13-chords. I just love the >sound of them, but can't fit them into my playing. Someone out there >who knows something about them? If you like the sound of them, use them. Just note that you don't always need every extension between 7 and 13 (non-inclusive) in a 13th chord. It's common to omit the 11th and sometimes the 9th in those cases. If the V chord you're playing is in a minor key, you can use a b9 or #9 (or both) with your 13th chord. With 13th chords, the coolest scale is the 8-note dominant: root-b9-#9-3-#11 (b5)-5-13-b7-root so in the key of G (D7 altered) D Eb F F# G# A B C D on a D13(b9,#9,#11) chord! Another name for D------------^^^^^^^^ is D - demented %-) Of course there's more to 13th chords than the mickey mouse thing I've described... but in life, the things you cherish most are those you've paid for with your own sweat. Just play your tunes and don't worry about the theory behind them!! Then, when you find yourself getting into a rut, look for any book on harmony, and learn what it is you've just done. Theory makes so much more sense if it's been practised. -Tony 'Quays' ('keys' with a 'Q') +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- | quayster@arkham.wimsey.bc.ca | ) "Sig's keep getting shorter +----------------- Tony Quays -+ (____, every day..." --Myron Lewis From: djones@megatest.UUCP (Dave Jones) Subject: Re: Circle of Fifths Date: 12 Mar 91 01:40:43 GMT >From article <31179@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): > Dave: > you seem to have a lot of preconceptions and expectations about how music > should be constructed. From our discussions on scales to your remarks about > chords to your discussion about how Db should be signed to a bunch of other > things, I get the impression that you would like to standardize music to > your liking. This strikes me as a bit close-minded. I remark on this solely > as food for thought; it's not meant as personal criticism. Thanks for the belly-laugh. Judging by this thought-food, if you do decide to let go with a personal criticism, I sure don't want to get in its way. Whew! (Are you miffed at me for the little dig at "head-bangers" and their so-called "power chords"? Forgive me for I have sinned. I'll say twelve Hail Poisons.) Okay, if it's food for thought, I'll think about it. Am I closed minded? In some ways, yes. Some things you just figure out, then you don't have to worry about them again. But music is not one of them. If by "closed-minded" you mean, do I value some music and musical styles more than others, then I guess the answer is again yes. I've quit trying to enjoy bluegrass. It just wasn't worth the effort. Much as I tried to like it, I just could not. I can't even try to like "metal" because the attempt is so unpleasant to me. Is that closed minded, or is it facing the reality of my subjective likes and dislikes? Is it even possible to like everything? To me, enjoyment implies values. Maybe when I attain enlightenment, I'll throw off all judging, but I'm not there yet. I wonder if the enlightened have much fun. If by "closed-minded" you mean, am I not extremely eager to learn, then I am very open minded indeed. The food-metaphor is apt. Somebody smart once said that the purpose of an open mind is the same as that of an open mouth: to slam shut when something good enters. You misjudge my motives. It would be quite absurd of me to think that my postings here would have any chance to "standardize music". I'm posting this stuff partly to be helpful. I am gratified that more than one person has indicated appreciation for my postings. I'm also doing it to confirm for myself what I have learned. I don't feel comfortable that I know something until I can explain it to others. Besides, it's fun, and I've learned a few things in the exchanges. It should be obvious that I didn't make up any of this stuff. I read it in books, heard it on records, and learned it from teachers and fellow performers. Music is always changing and evolving, but it would be very vain and stupid to discard our musical heritage. The only justification for the theory is its application. It works. The object of the exercise is to make people happy by vibarating their ear drums. I'm getting to the point where I can actually do that. I got the biggest ovation I've ever received only last night. Thinking back on it, I did not feel the sense of pride I might have expected when I first undertook to learn jazz improvization eight years ago. I was very happy of course, but I knew without thinking about it that I was only doing the last step in a process that other people contributed to much much more than I. I think maybe that's why Dexter Gordon used to give his saxophone a bow before he took his. It was a way of honoring all those cats, including Adolf Sax. From: djones@megatest.UUCP (Dave Jones) Subject: Re: Circle of Fifths Date: 11 Mar 91 23:52:45 GMT >From article <2371@cybaswan.UUCP>, by eethomas@cybaswan.UUCP (Andrew Thomas): > I'm not a trained musician, but I have really enjoyed the above > discussion. If you'll excuse my ignorance, I've got a query. Why shouldn't we? Your ignorance is just as good as anybody's! :-) > The thing is, if I want to improvise in a minor key do I > use the minor pentatonic/blues scale or the major. Generally you will use several scales in a given tune. Playing all in one scale is likely to become monotonous. There are exceptions. Reggae, in which monotony seems to be part of the mystique, is very often played all in one major pentatonic scale. In selecting which pentatonic to play, the key of the tune is of less importance than the chord or chord sequence, and the amount of tension or repose that you want. Some chords will admit several pentatonic choices. Some don't work with any pentatonics very well. For example, a diminished chord does not fit with any pentatonic. More on this subject later... From: djones@megatest.UUCP (Dave Jones) Subject: Re: Circle of Fifths (Mar 11 version) Date: 12 Mar 91 03:02:18 GMT >From article <4a4my1w163w@arkham.wimsey.bc.ca>, by quayster@arkham.wimsey.bc.ca (Tony Chung): > With 13th chords, the coolest scale is the 8-note dominant: > root-b9-#9-3-#11 (b5)-5-13-b7-root so in the key of G (D7 altered) > D Eb F F# G# A B C D on a D13(b9,#9,#11) chord! > > Another name for D------------^^^^^^^^ is D - demented %-) I think I may have mentioned this already, but a sequence that has become very popular in jazz in the last ten or fifteen years contains a similar scale with the five augmented: D Eb F Gb Ab Bb C D The chord-symbol is D7+5+9 or just D7+9 for short. (Perhaps confusing, but that's convention.) It is used quite often as the dominant in a minor II V I: Ami7(b5) D7+9 Gmi The scale has a lot of different names, probably an indication that a lot of people discovered it through various paths. It is called, among other things, Ravel, Pomeroy, diminished whole tone, super locrian, and altered. Notice that it is the seventh mode of the Eb melodic minor scale. Jazz musicians often play modes of different melodic minors over the chords of the sequence above. Over the Ami7(b5), use mode VI of C-melodic minor (locrian #2), over the D7+9 use mode VII of the Eb melodic minor, and over Gmi, use mode I of G melodic minor. From: djones@megatest.UUCP (Dave Jones) Subject: Re: Circle of Fifths Date: 12 Mar 91 03:05:28 GMT >From article <1991Mar11.171222.5966@cbnews.att.com>, by milo@cbnews.att.com (guy.f.klose): > But, how does one get that "Cannonball sound"? The only foolproof way I've found is to visit the record store. From: milo@cbnews.att.com (guy.f.klose) Subject: New Theory Question: m3rd Substitutions Date: 12 Mar 91 22:36:36 GMT I've got a new theory question for you, the knowledgable public... I was playing in a small jazz group a couple of weeks ago, and took a closer look at one of the tunes we played. Unfortunately, I can't remember the name of the tune, or its chord progression. What I do remember is that it sort of followed the standard ii-V-I progression, with some tritone substitutions thrown in, but in a few places where I expected a certain chord, there was instead a chord that was a minor third higher. Almost as if it were a m3rd substitution. I asked the sax player about it, and he said that it was a pretty common substitution. Or, as he put it, I can go anywhere, you can substitute a T-T or a m3rd for any chord, and cadences are dominant 7th to tonic. I found his reasoning to be confusing. After some thinking, I realized that tritones, in effect, split an octave in half, and minor 3rds split a tritone in half. But one reason why tritone substitutions work is that they share thirds and sevenths with their original chords. It seems that m3rd substitutions wouldn't share much of anything, unless you are considering a minor chord to start with. Then its like changing the root of the chord. An example: original: C- C Eb G Bb D F subst: Eb7 Eb G Bb D F A Can anyone shed some light? Guy -- Guy Klose milo\@angate.att.com From: kmorgan@hpcuhe.cup.hp.com (Kevin Morgan) Subject: Re: Re: Circle of Fifths (Mar 11 version) Date: 13 Mar 91 20:17:55 GMT > As other people have already pointed out, omitting 3rds is not common > practice. The "3rd" defines a chord as major or minor, and a seventh > defines other chord quality such as major-7th or dominant-7th. I have > heard it said that the 3rd and the seventh are the two most important > chord tones because of this quality definition. Other notes are left > as "color" for the chord. Generally true. However another approach, used a fair amount in late 60's/early 70's fusion, is to omit the 3rd's in the basic progression. Then the lead player can define the quality via choice of major or minor 3rd. The comp'ers can then follow along for a bit, substituting in appropriate quality chords. After the lead player milks that mode, the comp'ers back off to the undefined progression again, allow the lead player to move into a new quality, and so on. Of course, this approach having the comp'er "follow along" with the lead player can be used for other than just 3rd's (use of 6 or b6, use of 7 or maj7, etc). But you need good ears, and probably a little planning in advance to get it going. I don't know, maybe this is jazz fundmentals? Check out John McGlaughlin's "Follow Your Heart", on an old Joe Farrell album from 1971 or so for a no-thirds example (also a good example of 11/8 time). -k From @ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi Tue Aug 27 17:54 MET 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Received: from ruuinf.cs.ruu.nl by alchemy.cs.ruu.nl with SMTP (15.11/15.6) id AA05985; Tue, 27 Aug 91 17:54:02 met Return-Path: <@ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi> Received: from att.att.com by ruuinf.cs.ruu.nl with SMTP (5.61+/IDA-1.2.8) id AA23584; Tue, 27 Aug 91 17:27:02 +0100 Message-Id: <9108271627.AA23584@ruuinf.cs.ruu.nl> From: milo%mvuxi@att.att.com To: piet@cs.ruu.nl Subject: Re: circle of fifths (& music theory) archive Date: Tue, 27 Aug 91 09:25 EDT "Circle of Fifths" - part 2 of 2 From: d87parfo@odalix.ida.liu.se (Par Fornland) Subject: Re: Circle of Fifths (Mar 11 version) Date: 14 Mar 91 01:20:44 GMT quayster@arkham.wimsey.bc.ca (Tony Chung) writes: >Or one could play the Dbmi7(b5) chord and have the G in the bass, >causing the total effect of the G+7 (b9) chord. One of the many >versions of the V chord from minor. Let's see... (Note: I'm not flaming, just discussing!) Dbmi7b5 = Db E G B ! That's the notes in G6b5 ! The G+7b9 is G B Eb F Ab ! How can you get G+7b9 to be G6b5? To get G+7b9 you'll have to take Db9 without the Db ! Or did I misunderstand completely? >With 13th chords, the coolest scale is the 8-note dominant: >root-b9-#9-3-#11 (b5)-5-13-b7-root so in the key of G (D7 altered) I see the "X7alt." (choose X) in lots of fakebooktunes. Does it mean: "Take any dominantchord you find proper" or are there a specific chord? >D Eb F F# G# A B C D on a D13(b9,#9,#11) chord! >Another name for D------------^^^^^^^^ is D - demented %-) Isn't this scale used also on Ddim/Fdim/G#dim/Bdim ? >... but in life, the things you cherish most are those >you've paid for with your own sweat. Just play your tunes and don't >worry about the theory behind them!! >Then, when you find yourself getting into a rut, look for any book >on harmony, and learn what it is you've just done. Theory makes so >much more sense if it's been practised. That's true. It is not easy if you don't have a hint on where to start! I had real trouble improvising, before I read an article about analyzing the chords, looking for sequences of chords on which one could use the same scale. I tried this on Blue Bossa, and found Cmi and Dbmaj to be the only scales really needed! After this things began to clear. (BTW: Listen to Michel Camillo's version of Blue Bossa! Stunning!) -- !!!!!!!!! Par Fornland, Linkoping Institute of Technology, Sweden Bjornkarrsgatan 10c:10, 582 51 Linkoping, Sweden Tel. (Sweden)-013-260486 d87parfo@odalix.ida.liu.se !!!!!!!!! From: djones@megatest.UUCP (Dave Jones) Subject: Space and laying out (Re: Circle of Fifths) Date: 15 Mar 91 01:19:07 GMT >From article <33800002@hpcuhe.cup.hp.com>, by kmorgan@hpcuhe.cup.hp.com (Kevin Morgan): > Of course, this approach having the > comp'er "follow along" with the lead player can be used for other > than just 3rd's (use of 6 or b6, use of 7 or maj7, etc). But > you need good ears, and probably a little planning in advance to get > it going. > > I don't know, maybe this is jazz fundmentals? I was watching Joe Henderson rehearse one time, and along the way he commented that he liked the way Jim McNeely often played triads on the piano, leaving himself free to select either a minor or major seventh. He also commented that for a long time he was recording without piano because he couldn't find anybody who would stay out of his way. (!) Another guy there, who had more balls than tact, asked something about "Herbie". Joe shrugged off the question and went back to rehearsing. I noticed that both during the rehearsal and at the concert that night, Jim often put his hands in his lap, turned on the piano bench, and just listened to Joe, Rufus Reed, and Tony Williams for rather long stretches. From: quayster@arkham.wimsey.bc.ca (Tony Chung) Subject: Re: Circle of Fifths (Mar 11 version) Date: 24 Mar 91 11:20:53 GMT d87parfo@odalix.ida.liu.se (Par Fornland) writes: > quayster@arkham.wimsey.bc.ca (Tony Chung) writes: [...really dumb mistake by moi deleted...] > > Let's see... (Note: I'm not flaming, just discussing!) > Dbmi7b5 = Db E G B ! That's the notes in G6b5 ! > The G+7b9 is G B Eb F Ab ! > How can you get G+7b9 to be G6b5? Well, uh... where I come from, (probably Jupiter) the E natural is the extremely #5. Yeah... that's the ticket.... :) > I see the "X7alt." (choose X) in lots of fakebooktunes. Does it mean: > "Take any dominantchord you find proper" or are there a specific chord? The "alt" means do whatever you want, within reason. The melody would suggest what notes you should add or change. For instance, if the melody had an "A" on the strong beats, and the chord was "G7alt", you wouldn't want to add a #9 nor a b9 on your G7 chord, because the "A" would clash. But you're right. Just about anything goes in the realm of fake books. (The other side of the coin, wich chould get me in trouble, is you can change the melody to suit YOUR harmony. Have heard John Coltrane/McCoy Tyner's "My Favourite Things"? the form is toast, and they used an Fma7, which threw it out of the Emi key for a few bars. > Isn't this scale used also on Ddim/Fdim/G#dim/Bdim ? You can use the 8-note dominant scale on any chord that has those altered notes in it. That is, anything but the #5. If you have a #5 chord, you'd use the "altered dominant" scale, which has already been discussed as being the 7th mode of the melodic minor. Now, the reason why your diminished chords work, is because the 8-note scales (there are 2 different types) are based on 2 diminished 7th chords, interlaced with each other. Those particular Dim chords you mentioned could be used with different bass notes to give you an X7b9 chord. Either E7, G7, Bb7, or Db7 (hey, that's ANOTHER DIM chord!) I hope this helps, this time! (sorry bout the goof) +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- | quayster@arkham.wimsey.bc.ca | ) "Sig's keep getting shorter +----------------- Tony Quays -+ (____, every day..." --Myron Lewis From: djones@megatest.UUCP (Dave Jones) Subject: Re: Circle of Fifths (Mar 11 version) Date: 26 Mar 91 01:29:06 GMT >From article , by quayster@arkham.wimsey.bc.ca (Tony Chung): > d87parfo@odalix.ida.liu.se (Par Fornland) writes: > >> I see the "X7alt." (choose X) in lots of fakebooktunes. Does it mean: >> "Take any dominantchord you find proper" or are there a specific chord? > > The "alt" means do whatever you want, within reason. The melody > would suggest what notes you should add or change. Well... Depends on what is "within reason". An "altered" chord is one in which neither the fifth nor the nineth occurs unaltered. So when you see C7alt, you can play a b5 or #5 (or both!), but NOT a natural perfect five. Ditto for the nine. The most popular choice may be C7+9, which has a raised five and a raised nine. (If the symbol looks anti-intuitive, think of C7+ as meaning C7-raised-five, and C7+9 adding a [raised] nine to that.) As I've pointed out already, the seventh mode of the melodic minor (A.K.A. "altered scale", "Super Locrian", etc., etc.) is indicated for playing over any altered chord. It contains all the suspects and no others: root b9 #9 maj-3 b5 #5 b7. From: bdt@cookie.uucp (Drew Turock) Subject: More Music Theory (Was:Re: Circle of Fifths (Mar 11 version)) Date: 30 Mar 91 22:22:25 GMT I've gotten a lot out of the Circle of Fifths discussion I can't thank you folks enough! To try to continue this education there's several questions I'd like to throw out to all of you. [Please excuse if I screw up this or that... I started to learn all of this about 2 years ago when I started taking guitar lessons from a fantastic jazz guitar player. Although I played basic guitar for many years, the education he gave me opened up a great new world. Unfortuately I recently lost my teacher to cancer and I'm now left with a lot of questions I wish I had the sense to ask when...] The first question is: How do musicians improvise over changes where the tempo of the song is very fast and the scales you use to improvise change with each 1 or 2 chord changes. There are some tunes that I can't see how to apply my two years of jazz/music theory education to and still keep up with the song! My guess is that one (or more) of the following is true: a) their brain is so quick that in the 1 second that a II/V/I flies by in the tune, they've processed the fact that they are in a II/V/I in Db (for example), they've played something interesting and are already thinking about the next chord sequence coming up that may be a II/V in Eb. b) They're playing a II/V/I in A for 2 bars, then a one bar II/V in D comes up, then a II/V in G. Do they ignore the 1 bar in the key of D and anticipate the II/V in G ? c) they can play so many notes in so little time that they play *any* notes (not necessarily tied to a specific scale) thru a number of changes but resolve the phrase using notes from a more "inside" scale. My sense is that from the Charlie Parker solos I've studied, he did some of this. [I've been told that you study scales etc. for a bunch of years and then just forget all of that and just play.... I equate that to item C ] On the Subject of Tritone substitutions... If I am improvising over an A7 (using an A mixolydian) and someone slips in an Eb7 as a tritone sub, doesn't this mean I'm now using the "wrong" scale for improvisation. I know that there really is no *wrong* scale, just some that sound a whole lot better than others but.. I was in the key of E. Now with the TT sub I went to the key of Ab. Is this because since (as I learned from this group) that the important tones are the 3rd and the 7th and since both of those are in both scales, it doesn't matter which scale I use (i.e. it depends on the sound you're looking for). IS ANYONE INTERESTED IN.... collectively analyzing a standard or two? We could post the changes to a tune and possibly see the different approaches we might take to improvising over it.... maybe a standard like How High The Moon or I Got Rhythym or a tune from Parker, Monk, Coltrane, etc..... whatever folks might be interested in. I was going to suggest "Valse Hot" since I recently picked up the Sonny Rollins Plus Four Album but from what I've learned in the last few weeks from you folks about II/V/I's in melodic minor, the changes don't look so bad to me any more. Drew -- ------------------------------------------------------------------------------- Drew Turock Bellcore, bellcore!bae!bdt Piscataway, NJ From: djones@megatest.UUCP (Dave Jones) Subject: Re: More Music Theory Date: 31 Mar 91 01:01:17 GMT >From article <1991Mar30.222225.28686@bellcore.bellcore.com), by bdt@cookie.uucp (Drew Turock): ) ) I've gotten a lot out of the Circle of Fifths discussion ) I can't thank you folks enough! Ah shucks. ) ... ) The first question is: How do musicians improvise over changes ) where the tempo of the song is very fast and the scales you use ) to improvise change with each 1 or 2 chord changes. ) There are some tunes that I can't see how to apply my two years ) of jazz/music theory education to and still keep up with the song! My opinion, which I try to apply when I play, is that the theory is used when preparing a song "in the woodshed", or when composing a song, not on stage. On stage, I hope that I am ready to play "by ear". In the shed you have all the time you need. When performance time comes, often you will not have the time to do all that symbolic processing as the changes whiz by. But by then you should have the tune and the changes internalized as sound rather than as symbols, allowing you to play what you hear, the way a singer sings. Of course there are exceptions. I know certain measures in certain tunes where a particular scale sounds really cool, so I tend use the scale rather mechanically. Also, you don't have to anticipate the chords if you know the tune well. The sound of the tune should be all the cue you need to know immediately what part of what measure you are in, and what the chord is. Of course, it takes work and experience to get to the point where you can leave all the symbols. In the mean time, there is nothing evil about thinking chord-symbols and scales as you play. You do what works for you. In any case, you should think in phrases, rather than in individual chords and notes. That will help you connect things up coherently. You already know to recognize formulas such as the II-V and the II-V-I as units. Look for other common formulas and for related patterns in general. One woodshed technique which I can not recommend highly enough when practicing a new tune is to play through the tune several times using bone-head formulas: First time, play only I-II-I-II in quarter notes on measures that keep the same chord for a whole measure. Make up simple patterns also for measures with two chords, etc.. Next time through, play I-II-III-V in quarter-notes, etc.. Keep making the formulas more complex. Go to eighth notes, then finally sixteenth note or even sixteenth note triplets. This will give you the sound of the changes. ) ... ) On the Subject of Tritone substitutions... ) If I am improvising over an A7 (using an A mixolydian) and someone ) slips in an Eb7 as a tritone sub, doesn't this mean I'm now using the ) "wrong" scale for improvisation. Not at all. The function of the dominant chord is to create harmonic tension, and as we have seen, the same tension-producing tritone interval occurs in both chords. You can keep right on playing the mixolydian scale over the original chord, or you can side-slip into the new one. Both techniques have their uses and their characteristic sound. ) I know that there really is no *wrong* scale ... Oh yeah there is. ) ... just some that sound a whole lot better than others but.. ) I was in the key of E. Now with the TT sub I went to the ) key of Ab. Is this because since (as I learned from this group) that ) the important tones are the 3rd and the 7th and since both of those are ) in both scales, it doesn't matter which scale I use (i.e. it depends on ) the sound you're looking for). That's right. ) IS ANYONE INTERESTED IN.... ) collectively analyzing a standard or two? We could post the changes ) to a tune and possibly see the different approaches we might take ) to improvising over it.... maybe a standard like How High The Moon ) or I Got Rhythym or a tune from Parker, Monk, Coltrane, etc..... ) whatever folks might be interested in. ) Sure. I always analyse a tune when I learn it, and usually I write my analysis down. I use the poop-sheets when I brush up on tunes I've gotten rusty on. I'll try to remember to bring some of them into the office to post. From: quayster@cynic.wimsey.bc.ca (Tony Chung) Subject: Re: More Music Theory (Was:Re: Circle of Fifths (Mar 11 version)) Date: 31 Mar 91 11:56:19 GMT Keywords Kill, your, jam, maniac, ! In answer to your first question, regarding soloing over II-V-I s at tempos beyond the comprehension of mortal musicians, there are two possible ideas I couls suggest. The first is that they know the tune so well, that they can find the one scale that would do the most good. For instance, in Autumn leaves, you could solo modally in G major all the way up until you hit the B7alt chord, where you could find a riff that worked or *pause* (silence is seldom seen in solos) :-) The second is that they know the tune so well that their ears tell them what sounds right and what doesn't. I've had a lot of success at singing lines before I play them (or while I'm playing them). As a piano player, I find singing with my lines brings my ear into the picture, and also improves my phrasing and articulation. When I run out of breath, I stop playing, breathe, then start again :-) Sax players may have some trouble singing while they solo... but they could mentally sing, I s'pose. Now, about Tritone subs, the first thing to consider is WHY your jamming partner would throw a TT where he did. Was it so he could set the next chord up chromatically, or so he could get a melodic bass line happening? Or was it just because he was bored. If the reason was the last, then play a standard A mixolydian and let him know who's boss :-). Otherwise, for TT subs the scale most recommended by jazz doctors is the Lydian b7 (4th mode of melodic minor). It's basically a major scale with a #4 and b7. This scale ties into the fact that TT subs belong to the chord family that Dick Groves calls "Family #6". This chord family considers that e all the chords that are used as chromatic dominant approach chords are IV chords from the melodic minor, functioning as a V eg TT sub. These chords are standard dominant (up to 13th) chords with a #11 replacing the natural 11th. The chords sound really neat. For instance, in a progression like in "Penny Lane" (Lennon/McCartney): F Dmi7 Bbma7 C7sus CDA chords could be inserted between each of them like this: ('/' = beat) F / / Eb7(#11) Dmi7 / Cmi7 B9(#11) Bbma7....... Now, I didn't put bar lines in there, but if you try the progression out, in time, it sounds really cool. Note that there is an added bonus, the Cmi7 - B9(#11) movement is a II-V utilizing the TT sub! I was shown this when a local jazzman played through a chart I wrote, and he said, "do this". I didn't understand it until I looked at it for a few hours, then came up with this conclusion. So... if there's any mistakes, let me know! Don't let me wander around this earth believing in falsities!! -TC -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- "If you drive, don't drink." -- Tony Chung quayster@cynic.wimsey.bc.ca quayster@arkham.wimsey.bc.ca -- +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- "I should be so lucky, | quayster@arkham.wimsey.bc.ca | ) lucky, lucky, lucky..." From: djones@megatest.UUCP (Dave Jones) Subject: Re: More Music Theory Date: 1 Apr 91 01:56:14 GMT >From article <1991Mar31.115619.6494@cynic.wimsey.bc.ca>, by quayster@cynic.wimsey.bc.ca (Tony Chung): > ... for TT subs the scale most > recommended by jazz doctors is the Lydian b7 (4th mode of melodic > minor). It's basically a major scale with a #4 and b7. I've heard myself doing that, but I didn't know it was a recognized thing. Notice that by playing the #4, you are playing the root note of the chord that you substituted for, rather than the major seven of the original dominant chord, which would sound bad if given any duration. For example take the standard progression E-7 A7 Dmaj and substitute the tritone, making it E-7 Eb7 Dmaj. The four of Eb is Ab. Over the Eb7, you tend to treat Ab as an "avoid-note", using it only in passing. But you can play A and it will sound fine. In fact, you can even play the A-mixolydian scale, "ignoring" the tritone sub. Using either mixolydian scale, Eb or A, just be careful not to hold the four (Ab or D). (Your ear will make you get off of it pronto, but the audience may whince before you do.) But let's go back to the Eb Lydian-dominant that Tony is recommending. If we play it off of the root A, rather than Eb, what have we got? It's our new old friend, the super-locrian or "altered" scale, which as we have seen fits nicely with any A7alt chord. Thus we see that the indicated scale for Eb7+11 is a *mode* of the indicated scale for A7alt! How about that?! I had never noticed this connection between tritone subs and altered dominants. Learn something every day. > This scale ties into the fact that TT subs belong to the chord family > that Dick Groves calls "Family #6". Tony, please elaborate. I am note familiar with this theory. From: quayster@cynic.wimsey.bc.ca (Tony Chung) Subject: Chord Families and Associate Scales Date: 5 Apr 91 11:25:28 GMT A long time ago Dave Jones forcefully pecked: >But let's go back to the Eb Lydian-dominant that Tony is recommending. >If we play it off of the root A, rather than Eb, what have we got? >It's our new old friend, the super-locrian or "altered" scale, which >as we have seen fits nicely with any A7alt chord. Thus we see that >the indicated scale for Eb7+11 is a *mode* of the indicated scale Well, of course it makes sense, because both are modes of the melodic minor. However, making sense and actually "making sense" aren't the same, are they? :-) >for A7alt! How about that?! I had never noticed this connection >between tritone subs and altered dominants. Learn something every >day. It escaped me that the Eb lydian-b7 was an A altered dominant! Is it really called a "ravel" scale? Bizarre! (What's even more bizarre is that the local radio station is playing The Flying Lizard's version of "Money" -- after 12 years, is there a resurgence?) Oh, BTW, Vancouver Canucks WON against the L.A. Kings. That's bizarre, too. (or TOO BIZARRE) >> This scale ties into the fact that TT subs belong to the chord >> family that Dick Grove calls "Family #6". > >Tony, please elaborate. I am note familiar with this theory. You should get the Grove series, "Modern Harmonic Relationships". I've posted it to every newsgroup I know related to music. It's a good way to standardize the classification of chords. -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- Basically, the thinking is this: Every chord fits into a distinct category that represents its own specific function in the context of a song. There are nine in all, each of them referring to a different function. #1 is called "I major". Every chord that functions as a I in major fits here. So, C, Cma7, C6, C6/9, and C6/9ma7 all belong to this family. The main rule is that if you add an 11th, it must be raised a half-step. So, you'll have additional chords like Cma9(#11), Cma7(b5), etc. Gears are spinning in Dave's mind as you read this, but I'll point out anyway that once you add the #11 it zooms the chord into the key a 4th below. So, these C chords will actually be IV chords in Gmajor, but they function as a I. There's another chord in this family, the Cma7(#5) chord. It's III in minor, but functions as a I. I think the only time you see this chord is in line progressions that go C C(#5) C6 C(#5) (Lucky Southern [Keith Jarrett], I think used this ... in D) Scale sources for #1 in C: C Ionian, C lydian (for the #11 chords) A melodic minor (for I(#5) chords) A harmonic minor (for the I(#5,b5) chords) #2 is the "II minor". You remember those II-V progressions in "Satin Doll" and a host of others? This family houses most of those II chords. Includes everything up to the minor 13th. (minor 7 chord, maj 9, perf. 11, maj 13). They say to avoid the 13th unless it's a melody note. In C, D F A C E G (B) Scale sources for #2 in C (Dmi7): D dorian (for "funky" style, Grove recommends the D and A blues scales) #3 is the "V13" family. Very boring chord family, as it's diatonic to a major key. The rule is that in a stack you avoid having both the 3rd and the 11th. Means you leave out the 11th or the 3rd at any given time. This is because of the minor 9th interval that occurs between them. (Ah... so that's where SUS chords come from; chord family #3!!) Scale sources for #3 in C (G7): G mixolydian, and that funky G blues scale :-) #4 is the "I in minor". Yuk. The minor (major 13th) chord is nowhere near the top of my list of favourites. Still, we cannot escape the fact it exists in this world. Every chord that functions as a I in minor (eg. Cmi, Cmi(add 9), Cmi6, and CmiMa7). All thirds-stacking is diatonic, and there are no alterations. Scale sources for #4 in Cminor [Cmi7(ma7)]: C melodic minor, C harmonic minor (also, depending on the chord, you can use the C natural minor, C dorian, and Bb ionian) #5 is the "II in minor". You can extend upon the basic minor7(b5) chord by adding the natural 9, 11 and b13. You'll notice the b13 is the same note as a #5 chord. This family gives a logical explanation for the existence of the mi7(#5,b5) chord (Not that you'd ever use it, but... :-) ). Scale sources for #5 in Cminor [Dmi7(b5)]: D locrian, D locrian #2 (aeloian b5) if you omit the 9th of the chord (as Grove recommends), you can use the C harmonic minor #6 is the chord family encompassing tritone subs. It is the dominant 13 (#11) chord that functions as a IV. It is often used as a chromatic dominant approach chord (approach a chord from a semitone above or below), or a dominant setup (IV sets up V, no matter how you slice it). Scale sources for #6 in Cminor [F13(#11)]: F lydian b7, F whole tone, funky F blues :-) #7 is the first altered-dominant chord family. There are really two different chord stacks for this family. The first is the V13(b9,#9) stack, and the second is the V13(b9,#9,#11) stack. The stack with the #11 in it is more common. What sets these chords apart from #6 is the altered ninth. Scale source for #7 in Cminor [G13(b9,#9,#11)]: G dominant 8-note scale. Scale source for the stack with the natural 11th: G Blues, F melodic minor (does this mode have a name?) #8 is the second of the altered-dominants. This stack includes the #5 (b13) in it, as well as the doubly-altered 9th and 11th. Scale sources for #8 in Cminor [G+7(b9,#9,#11)] G whole tone, G phrygian, G altered dominant, G spanish phrygian, C melodic minor, C blues #9 is the family that groups all diminished chords, or that which exists as a VII in minor, yet functions as a V without a root. (Classical theorists know this as the "incomplete dominant minor 9th." You don't add "extensions" to the 4-part diminished 7th stack, because stacking thirds only repeats notes. However, you can add notes to the stack: The ma9, P11, b13 and/or ma7. Note that this chord comes from VII in the harmonic minor, which is raised a semitone. Adding the ma7 gives you the bVII. Example, in Cminor, #9 stack is B dim7 (add C#, E, G, Bb). From a practical standpoint, adding notes to a diminished 7th chord doesn't help much. Scale source for #9 in Cminor (see above): B diminished 8-note scale -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- The reason Grove groups chords together in categories is to show that you can think through altered chords easily if you use polychords. I have by no means throroughly digested the information within Modern Harmonic Relationships, book 2. Every time I flip through its pages, I see something new, or understand something else that wasn't clear before. I've got a Jazz Theory final to prepare for next week, and this summary will help me remember what chords belong to what family. If I left anything out, or if anybody else has any additions I can make, (names of the modes of the minor scales would be nice :-] ), please E-mail me. If anybody has any really difficult questions, ask David Jones (only kidding, Dave! Put down that Jerusalem disk!) Thanks for giving me a way to study for my final, and answer your question at the same time. TNWT -Tony Chung If Modern Harmonic Relationships ain't at your music store, write to: Alfred Publishing Co., Inc. 15335 Morrison Street Post Office Box 5964 Sherman Oaks, California, U.S.A. 91413 and order it. Then tell them to send me a "recommender's fee" :-) -- -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- "If you drive, don't drink." -- Tony Chung quayster@cynic.wimsey.bc.ca quayster@arkham.wimsey.bc.ca From: djones@megatest.UUCP (Dave Jones) Subject: Minor mode names (Re: Chord Families and Associate Scales) Date: 5 Apr 91 17:46:33 GMT >From article <1991Apr5.112528.13559@cynic.wimsey.bc.ca>, by quayster@cynic.wimsey.bc.ca (Tony Chung): > If I left anything out, or if anybody else has any additions I can > make, (names of the modes of the minor scales would be nice :-] ), Here are the names that jazz people generally use for the modes of the melodic minor scale: I Melodic minor II Dorian flat two III Lydian augmented IV Lydian dominant V Hindu VI Locrian sharp two VII Super Locrian (a.k.a. altered, altered dominant, Pomeroy, Ravel, diminished whole-tone) I don't think the harmonic minor modes have any common names. Perhaps we should start a name-the-scales contest! From: djones@megatest.UUCP (Dave Jones) Subject: Re: Chord Families and Associate Scales Date: 5 Apr 91 18:46:56 GMT >From article <1991Apr5.112528.13559@cynic.wimsey.bc.ca), by quayster@cynic.wimsey.bc.ca (Tony Chung): ) Me sez: ) )) This scale ties into the fact that TT subs belong to the chord ) )) family that Dick Grove calls "Family #6". ) ) ) )Tony, please elaborate. I am note familiar with this theory. ) ) You should get the Grove series, "Modern Harmonic Relationships". ) I've posted it to every newsgroup I know related to music. It's a ) good way to standardize the classification of chords. Well, none of those chords is new to me. I guess I would have to read his book to figure out why he grouped them together the way he did. It would never have occured to me to put a major chord and a major chord with a sharp five in the same group. I mean, what's the point of it? I know, "Read the book." Anyway .... ) ... ) #7 is the first altered-dominant chord family. There are really two ) different chord stacks for this family. The first is the V13(b9,#9) ) stack, and the second is the V13(b9,#9,#11) stack. You should know that in jazz parlance, "altered" always implies a raised or lowered fifth. An "altered chord" means any chord in which the fifth is lowered or raised, and the ninth does not occur unless it is raised or lowered also. Maybe we could call category #7 the "modified dominant" chord family? ) The stack with ) the #11 in it is more common. What sets these chords apart from #6 ) is the altered ninth. ) ) Scale source for #7 in Cminor [G13(b9,#9,#11)]: ) G dominant 8-note scale. ) Scale source for the stack with the natural 11th: ) G Blues, F melodic minor (does this mode have a name?) Yeah. Jazz guys call it the "Hindu" scale. From: djones@megatest.UUCP (Dave Jones) Subject: Re: More Tritone subs and II V Is Date: 11 Apr 91 03:18:35 GMT Distribution: usa >From article <1991Apr9.233820.1559@morrow.stanford.edu), by AS.TLK@forsythe.stanford.edu (Timothy Kaufman): ) There is a great emphasis on II-V-I's in jazz (albeit in various ) contexts and key centers). This is not without reason--for whatever ) psychological reason, the resolution a fifth down sounds very ) "solid" to most ears, at least Western ears. ALL ears. Eastern music is just as much based on that resolution as Western. It's physics. From: djones@megatest.UUCP (Dave Jones) Subject: Re: More Tritone subs and II V Is Date: 11 Apr 91 03:15:42 GMT Distribution: usa >From article <1991Apr9.205911.1832@bellcore.bellcore.com>, by bdt@cookie.uucp (Drew Turock): > Start Editorial: ... > START New Subject: > > So far I know of: > > II-7 V7 IMaj7 - progression for Major > II-7b5 V7b9 I-7 - progression for Melodic Minor The second of these is the II V I of the Harmonic Minor, not the Melodic. A II-V in the melodic minor is much the same as a II-V in the same (melodic) major key. The only differences are that in the minor, the II dorian has a flat two, and the V mixolydian has a flat six (the "Hindu" scale). For example, melodic major... Bmi7 E7 Amaj7 and the melodic minor... Bmi7(b9) E7(b6) Ami(maj7) Usually you will see this less specific sequence instead: Bmi7 E7 Ami You can treat that a melodic minor, or as A major on the II-V with a functional change on the I. You can overlap sequences to expedite key changes. For example... seq 1 seq 3 ________________ ______________ Amaj7 | Bmi7 E7 | Ami D7 | Gmaj7 ------------- seq 2 Lookie at that. We've made a very slick change from A major to G major by way of A minor. The trick is "mulitple function". The Bmi7 E7 serves as a II-V in A following the Amaj7. It also serves as a II V in A minor preceding the Ami chord. The Ami D7 serves as a II-V in G preceding the Gmaj7. The trick in soloing over sequences like this is not to "take sides". You have to play chord notes and other notes common to the different functions of the ambiguous parts. Do not use connecting notes that would tie the chords to one function or another. That ruins the pun. In practice there is often no one scale that fits a whole minor cadence, and if there is one, you may not want to use it anyway. The example which has come up repeatedly is... B-7b5 E7alt A- Over the B, jazz guys tend to play the locrian sharp two, and over the E they play the super locrian. Those are modes of *different* melodic minor scales, neither being A minor. The fact that they are modes of melodic minor scales is only of practical importance because it means you get several scales "for free" when you learn one melodic minor. I don't think it means that those chords, as they function in the sequence, are related to the respective melodic minors. It's just a kind of a trick to remember the scales. From: knid%midiline.uucp@elroy.jpl.nasa.gov (Vermicious Knid) Subject: Re: More Tritone subs and II V Is Date: 12 Apr 91 04:29:55 GMT Tritone subs - rule of thumb is they can ONLY be used on dominant chords. This is the only type of chord that the true tritone (i.e., the 3rd and flat 7th) is present. A tritone sub is generally considered a "self-contained" substitution; that is, it takes the place of the dominant chord in the progression, rather than being added to the progression. Example: a II-V-I in C major. When moving from the Dmin7 to the G7 (whichever form you choose), the natural tritone is F natural and B natural; the flat 7th and 3rd, respectively. To perform the tritone sub, the 3rd and 7th change functions, i.e., the F now becomes the 3rd and the B becomes the seventh, which outlines the chord of Db7, which is the tritone sub of G7. Thus, the progression now becomes Dmin7 - Db7 - C. The use of the tritone sub to create approach chords is much the same as the above, however, an approach chord must be placed before a definitive chord (II-V-I) and is used *in addition* to the already present dominant chord, thus taking time away from the dominant chord. Using the sub as an approach chord and assuming the original progression consisted of four counts of II and four counts of V leading to the tonic, we get: Dmin7 G7 Db7 C / / / / | / / / / | And, yes, of course there's such thing as a II7 (maj or min) - V7 - I7. That's what the blues is....you're pretty much using ALL dominant chords. Regarding "Anthropology", the G7b9 CAN be thought of as a momentary V - I in melodic minor, but given the context of the key (Bb), the key is to think of the G7b9 as a altered dominant VI chord, which helps to justify the B natural as a flat 9 of the key signature, rather than a move to another key area. I think you'll find the I - VI - II - V is a fairly common progression. The choice of using a major or minor VI depends largely on the style of the tune; '20's tunes almost without exception used the major VI7 (think of the "Charleston" progression), for example. From: quayster@cynic.wimsey.bc.ca (Tony Chung) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 12 Apr 91 15:16:28 GMT To David Jones, thanks for naming those scales, and yes, maybe we should hold a "name the scales" contest. I think I know the answers to the I, II and V in harmonic minor: I = "harmonic minor" (not very original) :-) II = "locrian sharp 6" (because that's what it looks like) :-) V = "Spanish phrygian" (at least that's what the Improv teach says) Now, in my post, I made a most grevious error. I said that over a Cma7(#5) chord you use the A melodic minor, and Cma7(#5,b5) chord you use the A harmonic minor. They should be reversed. Sorry if someone else posted this correction earlier; I haven't been here of late! :-) -Tony Chung -- -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- "If you drive, don't drink." -- Tony Chung quayster@cynic.wimsey.bc.ca quayster@arkham.wimsey.bc.ca From: wayne@inmet.inmet.com Subject: Re: More Tritone subs and II V Is Date: 12 Apr 91 17:51:00 GMT > As an example, here are the changes to Jerome Kern's Yesterdays > from 2 sources. Looking at bars 7 and 8 from the Real Book > there is: > > | B-7b5 / / / | E7 / / / | > > but from another source there is: > > | C-7 / F9 / | B-7 / E9 / | > > In this case is F9 a TTsub for B-7b5? No. F9 is the TTsub of B9, which is the V chord of E9. The person who arranged this tune spiced it up by adding a few chords. Here is one way of looking at it. B-7b5 / / / | E7 / / / | original B9 / / / | E9 / / / | changed the form of resolution to extended dominant (V of V), added tensions. F9 / / / | E9 / / / | changed the dominant to a tritone sub. C-7 / F9 / | B-7 / E9 / | added related IIs. Notice the chromatic motion of the II-V's -- it's a nice effect. > Another example(?) is in Bars 7 and 8 of Wes Montgomery's "Four on Six" > The bars look like this: | A-7 / D7 / | Eb-9 / Ab7 / | > > Note that the second bar changes are a flatted fifth from the first! > Ab7 is D7s TT sub, so is Eb-9 being used as a TT sub of A-7 or is this > an example of "cycle extension" (where the "real" changes are D7 to Ab7 > and for added fun he's just sticking the II chord in front of each). The "real" changes were probably the plain ol' vanilla A-7 D7 | Gmaj7 (you left out the target chord, so I assume it is Gmaj7). The writer/ arranger again spiced up the changes by inserting a II-V of the tritone sub of the dominant chord. This is not uncommon. D7 and Ab7 relate to each other only incidently, in that they have the same target of resolution. > On the subject of II-V-I progressions again, > I've now learned (through this group) > the standard II V I for Melodic Minor. I'm just wondering if there > are a number of other II V I progressions for things like > Harmonic minor etc. Are there only a few or are they too > numerous to state? Actually, there are two: one for target chord of major and one for minor. There are of course some slight variations. There is no "II-V" whose target is a m7b5 chord (because there is no V chord). > as for Melodic Minor, I think I've seen variations on this such as: > II7+9 V(TT sub) I-7. For example in the tune Yesterdays you could go > from: > | E-7b5 / A7b9 / | D-7 / / / | > to: > | E7+9 / Eb9 / | D-7 / / / | This is actually an extended dominant, where you go V of V -> V -> I. In the above example, the progression goes V of V -> subV -> I. This is not a II-V resolution. > Is there such a thing as a II7 V- I7? > For example the first two bars of Anthropology (I Got Rhythym changes) are: > > | BbMaj7 G7b9 | C-7 F7 | > > I know the second bar is really a II-V in Bb but where does the G7b9 > come from?? (is the G7b9 C-7 a V I in C melodic minor?) Yes, the G7b9 C-7 is a V -> I in C melodic minor. So in answer to your first question: II7 V- I7 is not a standard progression. I would even say it is a wrong analysis of this situation because of the harmonic rhythm: the I chord has to be on a strong beat, like the first beat of a measure or the the first beat of series of measures. > As an aside to all of this, are there any books etc, that lay some of this out > that anyone would recommend? Sometimes it seems like the more I learn the more > I don't know...(just like the rest of life.. :) I recommend the following book: JAZZ HARMONY by Andrew Jaffe. I also recommend you hear these changes and be familiar with them so that you can identify them by ear. > Drew I hope I'm not too technical in my explanations; I sort of take my understanding of theory for granted and sometimes I don't know my audience. If you have any questions, please feel free to ask. Wayne Wylupski wayne@inmet.com From: sf@sco.COM (Steve Finney) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 15 Apr 91 00:07:26 GMT In article <1991Apr12.151628.21590@cynic.wimsey.bc.ca> quayster@cynic.wimsey.bc.ca (Tony Chung) writes: >To David Jones, thanks for naming those scales, and yes, maybe >we should hold a "name the scales" contest. I think I know the >answers to the I, II and V in harmonic minor: > > I = "harmonic minor" (not very original) :-) > II = "locrian sharp 6" (because that's what it looks like) :-) > V = "Spanish phrygian" (at least that's what the Improv teach says) > Or go with middle eastern mode names. The V is _really_ common in middle eastern and balkan musics (greek, bulgarian, ...), and is referred to as "hejaz" (alternatively "hijaz"). Actually, it's a little trickier, since there are various forms of hejaz. The minor starting on the 4 is "nikris". sf -- From: dbb@tc.fluke.COM (Dave Bartley) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 16 Apr 91 15:21:12 GMT In <1991Apr12.151628.21590@cynic.wimsey.bc.ca> quayster@cynic.wimsey.bc.ca (Tony Chung) writes: >To David Jones, thanks for naming those scales, and yes, maybe >we should hold a "name the scales" contest. I think I know the >answers to the I, II and V in harmonic minor: >... > V = "Spanish phrygian" (at least that's what the Improv teach says) This is also called Hijaz, named for the Arabic scale with the same notes. (For the 3 Mustaphas 3 fans in the audience, there are also scales called Sabah and Isfa'ani...). There's also an Arabic name for the IV hm mode, which escapes me at the moment. -- Dave Bartley / John Fluke Mfg. Co. / P O Box 9090 / Everett, WA 98206-9090 USA dbb@tc.fluke.COM ....!{uw-beaver,sun,uunet}!fluke!dbb (Tel. +1 206 356 5781) From: jaz@icd.ab.com (Jack A. Zucker) Subject: Re: More Tritone subs and II V Is Date: 16 Apr 91 16:50:24 GMT In article , knid%midiline.uucp@elroy.jpl.nasa.gov (Vermicious Knid) writes: |> Tritone subs - rule of thumb is they can ONLY be used on dominant chords. This is misleading. Rules of thumb don't always apply well to chord progressions. One of the most common substitutions for a I (Maj 7) chord is to use the tri-tone root half diminished chord in place of the one chord. For example take the wizard of oz: Original (or close to it) | C | Emi7 C7 | etc... | With substitution: | F#min7b5 B7 alt | Emin7 etc | ^^^^^^^^ tri-tone In general, there are a minimum of 144 correct substitutions for every chord, with many substitutions implying a different chord-scale relationship. I.E. Dmin7 based on a Cmaj scale is different than a Dmin7 based on a Bb scale. This is the principal of the Dodecaphonic Harmonic System. -jaz | Jack A Zucker {cwjcc,pyramid,decvax,uunet}!jaz@icd.ab.com | | Allen-Bradley Company, Inc. or ICCGCC::ZUCKER | | 747 Alpha Drive | Highland Hts., OH 44143 phone: (216) 646-4668 FAX: (216) 646-4484 | From: djones@megatest.UUCP (Dave Jones) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 15 Apr 91 21:31:39 GMT In article <1991Apr12.151628.21590@cynic.wimsey.bc.ca) quayster@cynic.wimsey.bc.ca (Tony Chung) writes: )To David Jones, thanks for naming those scales, and yes, maybe )we should hold a "name the scales" contest. I think I know the )answers to the I, II and V in harmonic minor: ) ) I = "harmonic minor" (not very original) :-) ) II = "locrian sharp 6" (because that's what it looks like) :-) ) V = "Spanish phrygian" (at least that's what the Improv teach says) ) Whoa! This one blew by me the first time around. Ask your improv teach how come a "phrygian" got a major third, eh? (Demand a refund.) From: cbolton@csd475a.erim.org (Chris Bolton) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 18 Apr 91 03:43:05 GMT In article <17335@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: > )To David Jones, thanks for naming those scales, and yes, maybe > )we should hold a "name the scales" contest. I think I know the > )answers to the I, II and V in harmonic minor: > ) > ) I = "harmonic minor" (not very original) :-) > ) II = "locrian sharp 6" (because that's what it looks like) :-) > ) V = "Spanish phrygian" (at least that's what the Improv teach says) > ) > Whoa! This one blew by me the first time around. Ask your improv teach how > come a "phrygian" got a major third, eh? (Demand a refund.) If my memory serves me correctly, my improv. teacher said a Spanish phrygian was the same as a phrygian but with a major third as well. That is, it contains both a major AND minor third. -Chris Bolton cbolton@csd460a.erim.org From: will@ogre.cica.indiana.edu (William Sadler) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 18 Apr 91 16:59:54 GMT In <17335@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: >) I = "harmonic minor" (not very original) :-) >) II = "locrian sharp 6" (because that's what it looks like) :-) >) V = "Spanish phrygian" (at least that's what the Improv teach says) >) >Whoa! This one blew by me the first time around. Ask your improv teach how >come a "phrygian" got a major third, eh? (Demand a refund.) If you have a REAL book, and I know no one does since they are illegal, look at the chart for Chick Corea's "Spain." They have a scale notated in that they call Spanish Phyrgian with a major and a minor third. The scale built on V in harmonic minor closely resembles an octatonic scale (half-whole) without the flat 3. Will -- *************************************************************************** * _______________\|/_ Will Sadler will@cica.indiana.edu * * Laser 44888 /|\ sadler@iubacs.bitnet * *************************************************************************** From: mjs@hpfcso.FC.HP.COM (Marc Sabatella) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 19 Apr 91 15:12:48 GMT >The scale built on V in harmonic minor closely resembles an octatonic >scale (half-whole) without the flat 3. Octatonic? Haven't heard that one before. I love it! The concept of a phrygian with a major third (with or without a minor third) doesn't bother me. I think of the minor second as the defining characteristic of phrygian anyhow. From: djones@megatest.UUCP (Dave Jones) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 19 Apr 91 23:58:30 GMT >From article , by cbolton@csd475a.erim.org (Chris Bolton): > If my memory serves me correctly, my improv. teacher said a Spanish > phrygian was the same as a phrygian but with a major third as well. That > is, it contains both a major AND minor third. That makes more sense. That makes it the third mode of the so-called "major bebop scale". When played in eighth-notes, all the odd-numbered modes of that scale sound similar, because there are eight notes in the scale. Everything stays "lined up". In the Spanish phrygian as you define it, the major third is only a passing tone between the minor third, which occurs on a down-beat, and the four, which does same. (In the major bebop scale, the same note occurs between the five and the six.) From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Re: Minor mode names (Re: Chord Families and Associate Scales) Date: 24 Apr 91 07:28:08 GMT In article <17911@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: |> From article , by cbolton@csd475a.erim.org (Chris Bolton): |> |> > If my memory serves me correctly, my improv. teacher said a Spanish |> > phrygian was the same as a phrygian but with a major third as well. That |> > is, it contains both a major AND minor third. |> |> That makes more sense. That makes it the third mode of the so-called |> "major bebop scale". When played in eighth-notes, all the odd-numbered |> modes of that scale sound similar, because there are eight notes in the |> scale. Everything stays "lined up". In the Spanish phrygian as you define |> it, the major third is only a passing tone between the minor third, which |> occurs on a down-beat, and the four, which does same. (In the major bebop |> scale, the same note occurs between the five and the six.) This also makes the scale handy for playing over the I minor II I minor II chord progressive common in "Spanish" music. Also found in songs like Jefferson Airplane's "Go Ask Alice". In this usage, the major third is not a passing tone at all, but the dominant sense of the scale is major, flat 2, 6, and 7, and the minor third is actually being used strictly as the second of the lydian-sounding minor II. Try: G (1 meas.) Ab/G (1 meas.) [repeat]. with the Spanish phrygian in G over it all. And listen to "Go Ask Alice" to hear how Grace is putting the Spanish phrygian very nicely over this progression (in G I think, with following bass note, rather than the fixed G). -- Jefferson Ogata ogata@cs.umd.edu University of Maryland Department of Computer Science "Sure. Understanding today's complex world of the future *is* a little like having bees live in your head." From: milo@cbnews.cb.att.com (guy.f.klose) Subject: Bebop Scales (Re: Minor mode names) Date: 26 Apr 91 21:17:31 GMT In article <17911@prometheus.megatest.UUCP>, djones@megatest.UUCP (Dave Jones) writes: > That makes more sense. That makes it the third mode of the so-called > "major bebop scale". When played in eighth-notes, all the odd-numbered > modes of that scale sound similar, because there are eight notes in the > scale. Everything stays "lined up". In the Spanish phrygian as you define > it, the major third is only a passing tone between the minor third, which > occurs on a down-beat, and the four, which does same. (In the major bebop > scale, the same note occurs between the five and the six.) Hi Dave: I'd love to hear a more detailed explanation of the bebop scale from you, and I'm sure others would too, if you have the inclination to do so. I've read David Baker's description in one of his books, but I'm not really sure what he meant. It seems that a passing tone is added to a scale, in order to make linear phrases come out more evenly, but I'm not sure why that particular passing tone it added. Thanks... Guy -- Guy Klose milo@mvuxi.att.com From: bdt@cookie.bae.bellcore.com (Drew Turock) Subject: Re: Re: More Tritone subs and II V Is Date: 2 May 91 13:07:35 GMT >> Wayne Wylupski Writes: me>> as for Melodic Minor, I think I've seen variations on this such as: me>> II7+9 V(TT sub) I-7. For example in the tune Yesterdays you could go me>> from: me>> | E-7b5 / A7b9 / | D-7 / / / | me>> to: me>> | E7+9 / Eb9 / | D-7 / / / | >This is actually an extended dominant, where you go V of V -> V -> I. >In the above example, the progression goes V of V -> subV -> I. >This is not a II-V resolution. I think I've seen mention of this before... is there any background on where one uses a V of V type of technique that you could offer? >I recommend the following book: JAZZ HARMONY by Andrew Jaffe. > Would you know the publisher on this? >I hope I'm not too technical in my explanations; I sort of take my >understanding of theory for granted and sometimes I don't know my audience. >If you have any questions, please feel free to ask. > >Wayne Wylupski Thanks. It was great for me. I knew enough to understand what you wrote but not so much that I didn't learn from what you wrote. Thanks also to the others who posted comments on this. I printed them all out, took them home, and tried them out! (sorry for the delay on this posting, I'm just now catching up on the last few weeks...) Drew ------------------------------------------------------------------------------- Drew Turock Bellcore, bellcore!bae!bdt Piscataway, NJ From @ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi Tue Aug 27 17:54 MET 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Received: from ruuinf.cs.ruu.nl by alchemy.cs.ruu.nl with SMTP (15.11/15.6) id AA05989; Tue, 27 Aug 91 17:54:14 met Return-Path: <@ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi> Received: from att.att.com by ruuinf.cs.ruu.nl with SMTP (5.61+/IDA-1.2.8) id AA23590; Tue, 27 Aug 91 17:27:14 +0100 Message-Id: <9108271627.AA23590@ruuinf.cs.ruu.nl> From: milo%mvuxi@att.att.com To: piet@cs.ruu.nl Subject: Re: circle of fifths (& music theory) archive Date: Tue, 27 Aug 91 09:25 EDT "Music Theory" - part 1 of 2 From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Re: African Gapped scale? Date: 24 Feb 91 19:45:13 GMT In article quayster@cynic.wimsey.bc.ca (Tony Chung) writes: >In theory class last year we discussed the African "gapped" scale, >which consists mostly of notes not on a keyboard instrument. The >scale takes a standard 12-semitone octave, and divides it into >5 equal parts. Would anyone happen to know either how to calculate >the frequencies for the 5 notes, or the frequencies themselves? > >I want to start getting into more electronic music; this information >would help me out. My computer would be able to determine the pitches >for me (if I program the right frequencies), and I could tune 5 notes >on my synth to the computer! Don't know this scale personally, it may not be actually divided into exactly equal parts, but: If the octave is really divided up into 5 equal parts then you may get the frequencies as follows (assuming you are using A440 as your starting point) (^ is "to the power of") (x_y is "the pitch falling between notes x and y"): A = 440 * 2 ^ (0/5) == 440 * 2 ^ 0 == 440 * 1 == 440 B_C = 440 * 2 ^ (1/5) == 440 * 2 ^ .2 Db_D = B_C * 2 ^ (1/5) == 440 * 2 ^ (2/5) == 440 * 2 ^ .4 E_F = Db_D * 2 ^ (1/5) == 440 * 2 ^ (3/5) == 440 * 2 ^ .6 Gb_G = E_F * 2 ^ (1/5) == 440 * 2 ^ (4/5) == 440 * 2 ^ .8 A = Gb_G * 2 ^ (1/5) == 440 * 2 ^ (5/5) == 440 * 2 == 880 The point is that you get the next pitch going up by multiplying the frequency by the fifth root of 2. You may go down by dividing the frequency by the fifth root of 2. Every five times you do this you will hit a pitch some integral number of octaves from your original. To get other octaves multiply or divide by 2. To root the scale at a pitch other than A replace 440 with the frequency of that pitch. In cents: A is 0 cents from A B_C is 40 cents sharp from B Db_D is 20 cents flat from D E_F is 20 cents sharp from E Gb_G is 40 cents flat from G Hope this helps. -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From: curt@cynic.wimsey.bc.ca (Curt Sampson) Subject: Re: African Gapped scale? Date: 25 Feb 91 11:32:59 GMT quayster@cynic.wimsey.bc.ca (Tony Chung) writes: > In theory class last year we discussed the African "gapped" scale, > which consists mostly of notes not on a keyboard instrument. The > scale takes a standard 12-semitone octave, and divides it into > 5 equal parts. Would anyone happen to know either how to calculate > the frequencies for the 5 notes, or the frequencies themselves? Well, that "standard 12-semitone octave" is an octave. All octaves are the same, since an octave is defined as a doubling of frequency. (e.g., 1000 to 2000 Hz is an octave, and 2000 to 4000 Hz is another octave.) The standard 12-tone equal temperment scale is an octave divided into 12 equal parts. Given the frequency of a note, you can calculate the next semitone by multipling the original frequency by the 12th root of 2. The equation for this is: f = fb * 2^(s/12) where f is the frequency of the note you want, fb is the base frequency (usually A = 440 Hz) and s is the number of semitones above the base frequency the original note is. Thus, given a root of 440 Hz for A, A# and C will be: A# 440 * 2^(1/12) = 466.16 Hz. C 440 * 2^(4/12) = 554.37 Hz. Now, if we divide the ocatave into five equal parts, rather than twelve, we take the fifth root of two as our increment instead. We'll use A = 440 as our base again and calculate a five part equally-tempered scale: A 440 * 2^(0/5) = 440.00 Hz B 440 * 2^(1/5) = 505.43 Hz C 440 * 2^(2/5) = 580.58 Hz D 440 * 2^(3/5) = 666.92 Hz E 440 * 2^(4/5) = 766.08 Hz A 440 * 2^(5/5) = 880.00 Hz (one octave up from the original) So there's your equally-tempered 5 note scale. cjs curt@cynic.wimsey.bc.ca | "Sometimes it's like a party you go to where curt@cynic.uucp | there are no lights and everyone is doing {uunet|ubc-cs}!van-bc!cynic!curt | animal impressions." -Phillip Evans on usenet From: smithj@hpsad.HP.COM (Jim Smith) Subject: Re: African Gapped scale? Date: 25 Feb 91 05:01:17 GMT I know nothing about the scale you mention, but if you want to divide an octave into five equal parts, each note's frequency will be the fifth root of two higher than the frequency of the note before it. This is the basis of equal temperament. I don't have a calculator here, or I'd tell you what the fifth root of two is. (Another way to express the fifth root of two is: two to the one-fifth power.) To get the next note, just multiply its frequency by (two to the one-fifth power). While we're on the subject of scales, does anyone know what the scale is that Foday Muso Sosa (sp?), the african harp player who plays with Herbie Hancock, uses? -jim Smith smithj@hpsad.HP.COM From: akg@mentor.cc.purdue.edu (Mike Hughey) Subject: Re: African Gapped scale? Date: 26 Feb 91 00:33:20 GMT quayster@cynic.wimsey.bc.ca (Tony Chung) writes: >In theory class last year we discussed the African "gapped" scale, >which consists mostly of notes not on a keyboard instrument. The >scale takes a standard 12-semitone octave, and divides it into >5 equal parts. Would anyone happen to know either how to calculate >the frequencies for the 5 notes, or the frequencies themselves? Actually, according to the "Harvard Dictionary of Music", most of the African "gapped" scales are so called because they are "chasmatonic", implying larger intervals in some places than in others. A couple of examples given in the "Dictionary" are C-D-Eb-G-Bb-C and A-C-Eb-F-G-A. Of course, it goes on to describe actual intonation. These are not the actual notes according to our equal-temperament scale, merely approximations. In any event, the idea is that it is NOT equal parts. The "Dictionary" goes on to mention that some Ugandan scales may actually be "isotonic" (which would be 5 equal parts), but sounded uncertain, as if they may not be actually completely isotonic. It also mentions that scales are frequently formed with a lower 4th and upper 5th, like C-G-C. In other words, African tuning is a very complex, typically tribal thing, in which you are likely to be merely making an approximation unless you are instructed by an African who knows the music. I would recommend either getting a recording of African music which you like, if you wish to be truly authentic, and copying the scale, or, if you don't care so much about authenticity, try playing with tunings around the scales listed above until you find one you like (I did that on one of my pieces once, and I was happy with it. Music is a very subjective thing...). Of course, an isotonic 5-tone scale can also produce interesting results! Mike Hughey akg@mentor.cc.purdue.edu From: smithj@hpsad.HP.COM (Jim Smith) Subject: Re: African Gapped scale? Date: 25 Feb 91 21:41:22 GMT I went ahead and tried a five-note equally-tempered scale on my synth last night, and it sounds pretty good! Any note sounds good with any other note in the scale. It would be a good choice for a scale for kids in a beginning music (pre-music? :->) class to play on marimbas... From: djones@megatest.UUCP (Dave Jones) Subject: Re: African Gapped scale? Date: 26 Feb 91 00:31:54 GMT >From article , by quayster@cynic.wimsey.bc.ca (Tony Chung): > In theory class last year we discussed the African "gapped" scale, > which consists mostly of notes not on a keyboard instrument. The > scale takes a standard 12-semitone octave, and divides it into > 5 equal parts. Would anyone happen to know either how to calculate > the frequencies for the 5 notes, or the frequencies themselves? Each note in such a scale would have a frequency of 2**(1/5) times that of the previous one. (Only two of these notes are a great distance from ones on the keyboard.) Are you quite sure that these few notes comprise the whole scale? Symmetric scales are seldom used exclusively in a tune, because they very quickly start to sound purposeless, due to the lack of a the lack of a tonal center. (Ravel and those cats made extensive use of symmetric scales, for temporary sections with a "suspended" sound, just as jazz musicians do today, but they have all those other notes in their bags.) I ran a little computer program to see how closely the notes of this scale match up the small integral ratios. The news is not too good. scale nearest just error degree ratio temper actual in cents I 1/1 = 1.000000 1.000000 0.000000 II 8/7 = 1.142857 1.148698 8.825906 III 4/3 = 1.333333 1.319508 -18.044999 IV 3/2 = 1.500000 1.515717 18.044999 V 7/4 = 1.750000 1.741101 -8.825906 The most important ratio to have is the one we call the perfect fifth, or 3/2. The fourth of this scale comes closest, but is sharp by 18 cents, or 18 logrithmic percent of a 12-tone half step. That would definitely be noticeable to all but the profoundly tone deaf. There is nothing remotely near a major third (5/4) or to a minor third (6/5). The 7/5 is missing, making it impossible to get a dominant seven sound. Notice the very discordant ratios of 8/7 and 7/4, which do not have a close relatives on the keyboard. The nearest notes on the keyboard are respectively the major second and the minor seventh, but they are off quite a bit. Without actually having heard it played, I have doubts as to the usefulness of this scale. From: djones@megatest.UUCP (Dave Jones) Subject: Re: I need to know the frequency of the notes on a piano. Date: 26 Feb 91 01:01:55 GMT lpdjb@brahms.amd.com (Jerry Bemis): > I know 'A' is 440 Hz. Can someone fill in the rest for me? > > respond to me at lpdjb@brahms.amd.com I don't normally read this group. I'm sending this directly, as requested, but also posting it, because it may be of general interest. In general, to objtain successive notes of the well-tempered scale, you just multiply by the twelfth root of two. However be warned that tuning a real piano is not that easy. There are two gotchas that I know of. One is that because strings are not perfectly elastic, notes on even the finest, longest grand pianos do not have overtones that are well in tune with the fundamental. The big grands are better, but by no means perfect, particularly in the higher registers. Most people prefer to hear chords whose overtones on the low notes are in tune with the high notes. For that reason, piano tuners always "stretch the octaves", making an piano octave just a little bit larger than a true octave. They stretch more at the ends than in the midrange. You can find charts of stretched octave frequencies in books. Try the library. The charts are averages derived from pianos tuned by professional piano tuners. In double blind tests they have proven to be about as good as the real thing. The other gotcha is that the notes that use two or three strings need to be tuned with the strings slightly out of tune with each other. That makes the accoustic coupling between the strings and the piano less efficient, keeping the piano from losing sound to heat. I don't understand why mistuning diminishes heat loss more than sound generation, so I may not have explained that just right. Anyway, suffice it say that you need to mistune the strings just so, or the piano will not have good sustain. From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Re: African Gapped scale? Date: 26 Feb 91 20:47:05 GMT In article <15398@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: >From article , by quayster@cynic.wimsey.bc.ca (Tony Chung): >> In theory class last year we discussed the African "gapped" scale, >> which consists mostly of notes not on a keyboard instrument. The >> scale takes a standard 12-semitone octave, and divides it into >> 5 equal parts. Would anyone happen to know either how to calculate >> the frequencies for the 5 notes, or the frequencies themselves? >> > >Each note in such a scale would have a frequency of 2**(1/5) times that >of the previous one. (Only two of these notes are a great distance from >ones on the keyboard.) Are you quite sure that these few notes comprise >the whole scale? Symmetric scales are seldom used exclusively in a tune, >because they very quickly start to sound purposeless, due to the lack of a >the lack of a tonal center. (Ravel and those cats made extensive use >of symmetric scales, for temporary sections with a "suspended" sound, just >as jazz musicians do today, but they have all those other notes in their bags.) > >I ran a little computer program to see how closely the notes of this >scale match up the small integral ratios. The news is not too good. > >scale nearest just error >degree ratio temper actual in cents >I 1/1 = 1.000000 1.000000 0.000000 >II 8/7 = 1.142857 1.148698 8.825906 >III 4/3 = 1.333333 1.319508 -18.044999 >IV 3/2 = 1.500000 1.515717 18.044999 >V 7/4 = 1.750000 1.741101 -8.825906 Here you calculate cents error from perfect intervals, but the poster was requiring synthesizer tunings, which are usually referenced from an equal- tempered scale. Here is the equal-tempered result: position cents sharp from tonic I 0 II 240 (1200/5) III 480 IV 720 V 960 Since 12-tone equal-tempered semitones land on 100's of cents, we can see that II is 40 cents sharp from the II of a major scale, II is 80 cents sharp from the III of a major scale, IV is 20 cents sharp from the V of a major scale, and V is 60 cents sharp from the VI of a major scale. I don't understand why you call the news "not too good". This news is fine. It's just what the doctor ordered. >The most important ratio to have is the one we call the perfect fifth, or 3/2. >The fourth of this scale comes closest, but is sharp by 18 cents, or >18 logrithmic percent of a 12-tone half step. That would definitely be >noticeable to all but the profoundly tone deaf. There is nothing >remotely near a major third (5/4) or to a minor third (6/5). The 7/5 is >missing, making it impossible to get a dominant seven sound. Notice the >very discordant ratios of 8/7 and 7/4, which do not have a close relatives >on the keyboard. The nearest notes on the keyboard are respectively the >major second and the minor seventh, but they are off quite a bit. > >Without actually having heard it played, I have doubts as to the usefulness >of this scale. Your idea of "important" seems very absolutist. As far as I know, there is no "most important" ratio to have, since many of the scales in the world are not geared towards integral ratios. The integral ratios camp comes from Western Christian monk music and is only one representative of the many scales currently in use on the planet. You seem to be saying that a scale is only useful if it cancels beats. Perhaps you are unaware that a lot of music does not contain sustained combinations of notes, so beats do not arise. This is true of most percussive music; the tunings are often not so critical, because there is no time for combined notes to generate subsonic beats. The same is true of much vocal music; when you have a solo vocal, no beats are going to happen. This scale sounds very useful me, and I intend to try it out when I have some time. -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From: pd2@dolphin.cis.ufl.edu (Philip Duvalsaint) Subject: Re: African Gapped scale? Date: 27 Feb 91 03:21:07 GMT Could it be that these notes sound discordant because you have been taught what is harmonious? What I mean is, perhaps, if someone was raised in an environment wherein the 5 tone scale was commonplace, would he/she learn to accept this as "normal". Just a thought. Phil pd2@lightning.cis.ufl.edu From: quayster@cynic.wimsey.bc.ca (Tony Chung) Subject: Re: African Gapped scale? Date: 27 Feb 91 07:58:44 GMT Well, I plugged all the info on "fifth roots of two" into a simple basic program that calculates the notes of my isotonic pentatonic scale from any starting frequency, and plays a random succession of notes in various octaves on my pc speaker. Needless to say, it sounds like garbage. I can't wait until I get my synth back. I'm wondering if any people with true perfect pitch have tried this scale, and if the sounds aggravate them to the ends of the earth. After reading the now-exhausted discussion on perfect pitch, how non-musical tones really upset some PPers, I'd like to know if the "almost-but-not-quite" notes of the isotonic scale drives them batty. I should think that if I was that sensitive to pitch colour, it would make me rather insane!! -Quays +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- "Stop war | quayster@arkham.wimsey.bc.ca | ) save babies..." +----------------- Tony Quays -+ (____, --A dream I had, Feb 19, 1991 From: chb@valideast.COM (Charlie Berg) Subject: Re: Mark Levine on salsa Date: 26 Feb 91 20:09:58 GMT In article <9102181838.AA03943@sp24.csrd.uiuc.edu> hsu@csrd.uiuc.edu (William Tsun-Yuk Hsu) writes: >Some quick notes on salsa, adapted from Mark Levine's _The Jazz Piano >Book_. Disclaimer: I've done some listening but this is my only source >for the technical side of salsa. I'm sure I'll get lots of corrections >etc... > >The rhythmic basis for salsa is the clave. The forward clave (or 3 & 2): > >1&2&3&4&|1&2&3&4&|| >x x x x x > Most salseros just call it "2-3" or "3-2" now. The concept of "forward" & "reverse" have pretty much disappeared. >The reverse clave (or 2 & 3): > >1&2&3&4&|1&2&3&4&|| > x x x x x > >The African or "Rumba" clave: > >1&2&3&4&|1&2&3&4&|| >x x x x x > >(I don't know if there's a reverse rumba clave.) > Also known as Cuban clave, which makes sense, since the rumba is a Cuban song form. There is also Brazilian clave which is.. 1&2&3&4&|1&2&3&4&|| x x x x x (Just think about the rhythm to any Bossa). > >A bass pattern (tumbao): > >1&2&3&4&|1&2&3&4&|1&2&3&4&|1&2&3&4&| >x x x x x x x x x etc. ^ | | | I think you made a typo...this first beat is not usually played in the tumbao. > >I'm sure you salsa experts are laughing to death by now, so how about >some corrections/comments. Jones? Eric Majani? > >Bill Do I qualify as an expert? I played timbales in salsa bands in Philly in the mid-70s...this is before the big merengue & bomba fads. Mostly, things were still driven by New York salsa. "Tito suena los timbales... Ran kan kan, kan kan" Charlie Berg From: paul@kuhub.cc.ukans.edu Subject: Re: African Gapped scale? Date: 27 Feb 91 15:19:38 GMT In article <15398@prometheus.megatest.UUCP>, djones@megatest.UUCP (Dave Jones) writes: > From article , by quayster@cynic.wimsey.bc.ca (Tony Chung): >> In theory class last year we discussed the African "gapped" scale, >> which consists mostly of notes not on a keyboard instrument. The >> scale takes a standard 12-semitone octave, and divides it into >> 5 equal parts. Would anyone happen to know either how to calculate >> the frequencies for the 5 notes, or the frequencies themselves? >> > I ran a little computer program to see how closely the notes of this > scale match up the small integral ratios. The news is not too good. > > > scale nearest just error > degree ratio temper actual in cents > I 1/1 = 1.000000 1.000000 0.000000 > II 8/7 = 1.142857 1.148698 8.825906 > III 4/3 = 1.333333 1.319508 -18.044999 > IV 3/2 = 1.500000 1.515717 18.044999 > V 7/4 = 1.750000 1.741101 -8.825906 > > The most important ratio to have is the one we call the perfect fifth, or 3/2. > The fourth of this scale comes closest, but is sharp by 18 cents, or > 18 logrithmic percent of a 12-tone half step. That would definitely be > noticeable to all but the profoundly tone deaf. There is nothing > remotely near a major third (5/4) or to a minor third (6/5). The 7/5 is > missing, making it impossible to get a dominant seven sound. Notice the > very discordant ratios of 8/7 and 7/4, which do not have a close relatives > on the keyboard. The nearest notes on the keyboard are respectively the > major second and the minor seventh, but they are off quite a bit. > > Without actually having heard it played, I have doubts as to the usefulness > of this scale. 7/4 is a harmonic "7th". It is so consonant that, when included in a major 7th chord, the seventh fades into the harmonics of the major chord quite quickly. It sounds great. I'll have to try 8/7 and see what it sounds like. Congratulations regarding the scholarship behind the computer program! From: djones@megatest.UUCP (Dave Jones) Subject: More on temperment Date: 28 Feb 91 00:36:33 GMT In conjunction with the discussion of scale-temperment, I thought I would mention that I have been experimenting with intentionally playing the saxophone "out of tune". The idea is not just to play what sounds good on inspiration alone, but to use theory to look for oportunities to improve the intonation. Luckily, I've got a couple of horns that I can play in very good tune with the well-tempered scale, so I know theoritically what my point of departure is. My best success to date I think is on the end of "I Remember Cliford", a poingant tribute to the late Cliford Brown. It ends with a double phrase, the first part of which has a major third, the second having a minor third. To simulate just-temperment on this phrase, I play the major third purposefully flat and the minor third purposefully sharp. The first time I performed this tune in public (on the soprano), the first substantive comment I heard after I finished (with that phrase, recall), was, "Your intonation was perfect!" I chuckled to myself. How could I explain how perfectly gratifying that remark was? Just a heartfelt thanks had to do. Of course singers bend things all over the place, as do many instrumentalists. Helen Merril sends shivers up my spine sometimes. The first horn player I ever noticed doing this, long before I was nearly accomplished enough to attempt it myself, was Dexter Gordon. More recently, maybe three years ago, an ex-roomate of mine and I were trying to cop a lick from the bridge of Dexter's 'Round Midnight. The e.r. was using a piano, and I was using a tenor sax -- same thing Dexter was on. After a really long period of frustration, we became convinced that one note was "in a crack". So I cheated: Rigged the C.D. to loop on that one note, and put the electronic tuner on it. Sure nuff. In a crack. A quick session with the calculator proved that he was playing very close to just-temperment in the implied key -- a tritone substitution. Wow. From: djones@megatest.UUCP (Dave Jones) Subject: Re: African Gapped scale? Date: 27 Feb 91 23:56:19 GMT >From article <28821.27cb79aa@kuhub.cc.ukans.edu>, by paul@kuhub.cc.ukans.edu: > 7/4 is a harmonic "7th". It is so consonant that, when included in a > major 7th chord, the seventh fades into the harmonics of the major > chord quite quickly. It sounds great. It was stupid of me to call 7/4 "dissonant" without having heard it. I had never heard of a "harmonic seventh", but in theory it would seem perfectly reasonable. To clear up a possible misconception, let's reiterate that the "harmonic seventh" you refer to is not to be found on a standard piano. In particuar, it is not the major seven, which is simply a fifth above the major third, (3/2)*(5/4). It comes closer to the minor seventh, but my little program says that the well-tempered minor seven misses it by 31 cents -- way off. > I'll have to try 8/7 and see what it sounds like. Let me know how it comes out. > Congratulations regarding the scholarship behind the computer program! Ah shucks. From: paul@kuhub.cc.ukans.edu Subject: Re: African Gapped scale? (Jazzy!) Date: 28 Feb 91 15:12:39 GMT >> scale nearest just error >> degree ratio temper actual in cents >> I 1/1 = 1.000000 1.000000 0.000000 >> II 8/7 = 1.142857 1.148698 8.825906 >> III 4/3 = 1.333333 1.319508 -18.044999 >> IV 3/2 = 1.500000 1.515717 18.044999 >> V 7/4 = 1.750000 1.741101 -8.825906 >> The fourth of this scale comes closest, but is sharp by 18 cents, or >> 18 logrithmic percent of a 12-tone half step. That would definitely be >> noticeable to all but the profoundly tone deaf. There is nothing >> remotely near a major third (5/4) or to a minor third (6/5). The 7/5 is >> missing, making it impossible to get a dominant seven sound. Notice the >> very discordant ratios of 8/7 and 7/4, which do not have a close relatives >> on the keyboard. The nearest notes on the keyboard are respectively the >> major second and the minor seventh, but they are off quite a bit. Thinking more about it (and I hope to play it tonight) this scale is going to sound pretty "jazzy". Degree 2 is slightly more than a quarter tone above the Major 2nd (less than 1/4 tone down from a minor 3rd). With the addition of pitches close to the 4th, 5th, and very close to the harmonic seventh this is going to make a wonderful "jazz" scale! From: djones@megatest.UUCP (Dave Jones) Subject: Re: African Gapped scale? Date: 27 Feb 91 21:03:18 GMT >From article <30744@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): > [re. my chart ...] > Here you calculate cents error from perfect intervals, but the poster was > requiring synthesizer tunings, which are usually referenced from an equal- > tempered scale. I didn't write out the chart specifically to use tuning a synth. I don't have any experience with that. I was just commenting on the scale itself. > Since 12-tone equal-tempered semitones land on 100's of cents, we can see > that II is 40 cents sharp from the II of a major scale, II is 80 cents > sharp from the III of a major scale... That's a typo. You meant to say the III is 80 cents sharp from the III of the well-tempered scale. But why not say it is 20 cents flat from the IV of the well-tempered scale? Compare that to the -18 in my chart, showing how much it is flat of just-temperment. Recall that I said that the II and V of the scale were not very close to any piano tunings, but the III and IV were close to the piano perfect IV and V, but badly out of tune. > I don't understand why you call the news "not too good". I explained it. I guess what you mean is that you don't agree. Fine. > This news is > fine. It's just what the doctor ordered. > >>The most important ratio to have is the one we call the perfect fifth, >> or 3/2. >> ... >> >>Without actually having heard it played, I have doubts as to the usefulness >>of this scale. > > Your idea of "important" seems very absolutist. Based on physics. How absolute is that? What other basis would you propose for choosing scale tones? Random number generators? > As far as I know, there > is no "most important" ratio to have, since many of the scales in the > world are not geared towards integral ratios. Please name and describe a few. Seriously, I would really like to know of some. > The integral ratios camp > comes from Western Christian monk music and is only one representative > of the many scales currently in use on the planet. Baloney. I once heard on PBS series called "What Is Music?" that every culture that ever developed music has discovered and used the pentatonic scale. I believed it at the time. Is that not true? Certainly many have, and not just in the West. Ancient Chinese sacred music is written entirely in the pentatonic scale. It was once illegal in China to write secular music in that scale because it was considered too perfect to trivialize! The perceived perfection is due to physics: how sound moves; how our ears are built. [ ... stuff about things he says I don't understand omited. ] > This scale sounds very useful me ... When you say it "sounds" useful, do you mean you have heard it? > ... and I intend to try it out when I have > some time. Good. Let us know how it comes out. From: djones@megatest.UUCP (Dave Jones) Subject: Re: African Gapped scale? Date: 27 Feb 91 23:40:15 GMT >From article Thinking more about it (and I hope to play it tonight) this scale is > going to sound pretty "jazzy". Degree 2 is slightly more than a > quarter tone above the Major 2nd (less than 1/4 tone down from a minor > 3rd). With the addition of pitches close to the 4th, 5th, and very close to > the harmonic seventh this is going to make a wonderful "jazz" scale! Okay, I've got my keyboard back now, and I programmed the tunings in as Jefferson Ogata pointed out. The hardest thing to do is figure out what key I'm playing in, with this isotonic pentatonic. Each scale step sounds like a major second, but could be a minor third. Each "third" sounds close to a perfect fourth. I based my scale on G, but I use the fingering of F pentatonic. To say it's in the key of F is incorrect, because the F is 40 cents flat! We've almost hit a quarter tone in this thing! If you play some really quick lines, you get the feeling of Africa, or at least "Walking in your footsteps" by the Police. It would be a cool key to solo in. I'll wait until my band does a song in G, (or Ab, which I programmed the black notes for), then flip the tuning to my user-programmed scale and wank out! Thanks to all who responded with formulas and tuning. Now I know what to do if I want to divide an octave into 7 equal parts. Or even 8. How about 4? This is soooooo cool. Maybe 15? 117...? -Tony 'Quays' ('keys' with a 'Q') +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- "I should be so lucky, | quayster@arkham.wimsey.bc.ca | ) lucky, lucky, lucky..." +----------------- Tony Quays -+ (____, -- a Stock Aitken Waterman gem From: richgi@microsoft.UUCP (Richard GILLMANN) Subject: Re: I need to know the frequency of the notes on a piano. Date: 28 Feb 91 01:36:11 GMT In article <8bmI1si00WBN41zUse@andrew.cmu.edu> go0b+@andrew.cmu.edu (Gregory P. Otto) writes: | Alright, here's the general formula for the frequency for a note (in Hz): | f = 2^(N/12 + M) [that's "two to the power of ((N/12) + M)"] | where N is the number of half steps above the C in this octave | M is the number of octaves above the bottom octave Actually, piano tuners do not follow the equal tempered scale that you suggest. The very lowest notes are tuned a little flat and the very highest notes on the piano are tuned a little sharp. These extremes of the piano range just sound a little better that way. Something to do with the harmonics, I believe. From: smithj@hpsad.HP.COM (Jim Smith) Subject: Re: Re: African Gapped scale? (Jazzy!) Date: 3 Mar 91 16:24:07 GMT Re: Dave Jones' response to my comment that the 'black keys' scale is not equally tempered. Maybe I misunderstood what equal/well temperament is. I always thought it meant that the ratio between any two adjacent notes in a scale is constant. Since the 'black keys' pentatonic scale has both (well tempered) major seconds and major thirds in it, I didn't think this scale *by itself* could be considered equally tempered, even though it is derived from an equally tempered chromatic scale. I'd be interested in hearing a definitive answer on this.. And, in case it wasn't clear, I was KIDDING about the 'black keys' scale and chinese and cowboy music. I LIKE the black keys etude.. "It rained all night the day I left, the weather it was dry, the sun so hot I froze to death, Susanna don't you cry..." -Jim From: paul@kuhub.cc.ukans.edu Subject: African gapped scale tried, 8/7 tried Date: 4 Mar 91 16:53:44 GMT In article <28837.27ccc987@kuhub.cc.ukans.edu>, paul@kuhub.cc.ukans.edu writes: >>> scale nearest just error >>> degree ratio temper actual in cents >>> I 1/1 = 1.000000 1.000000 0.000000 >>> II 8/7 = 1.142857 1.148698 8.825906 >>> III 4/3 = 1.333333 1.319508 -18.044999 >>> IV 3/2 = 1.500000 1.515717 18.044999 >>> V 7/4 = 1.750000 1.741101 -8.825906 > Thinking more about it (and I hope to play it tonight) this scale is > going to sound pretty "jazzy". Degree 2 is slightly more than a > quarter tone above the Major 2nd (less than 1/4 tone down from a minor > 3rd). With the addition of pitches close to the 4th, 5th, and very close to > the harmonic seventh this is going to make a wonderful "jazz" scale! I tried the scale and it is "dominated" by the major 4th, i.e. harmonies generated by playing one note and its neighbor two notes above it sound as though they belong to an implied IV chord. This scale seems to sound "best" (not blatantly "out of tune") when realized with a brass or flute sound. Separately I programmed a "pure" 8/7 ratio as a second degree in a equally tempered scale, making it replace the "D" note in the key of C. Running up the scale this new "D" jives well with the equal tempered "E" above it, but makes the "C" below it sound a bit flat. From: wags@cimage.com (Bill Wagner) Subject: Re: I need to know the frequency of the notes on a piano. Date: 4 Mar 91 18:12:49 GMT In article <7}?&+7@warwick.ac.uk> csuaw@warwick.ac.uk (Patrick Clark) writes: >In article <8bmI1si00WBN41zUse@andrew.cmu.edu> go0b+@andrew.cmu.edu (Gregory P. Otto) writes: >>Alright, here's the general formula for the frequency for a note (in Hz): >> >>f = 2^(N/12 + M) [that's "two to the power of ((N/12) + M)"] >> >>where N is the number of half steps above the C in this octave >> M is the number of octaves above the bottom octave >> >>You may need to play around with M to get the right octave. >> > Is this right ? I always thought that the frequency of a 'pitch' >varied on where you played ? isnt there something called the `concert hall A' >(or something). > Also this formula implies that the interval between two notes is >according to a strict law ( this may be true on a keyboard instrument but >it certainly isnt so on other instruments ) I think to show this you play a >melodic minor scale on an instrument like a violin and compare it with a >scale played on the piano ? > I can try to clarify. For starters, we all know about A-440Hz. I think we also know that one octave higher is 880Hz, one octave lower 220Hz. Now, tunings break into two different realms. Most instruments today (including the piano) user "equal temperment" tuning. Basically, divide the octave into twelve equal divisions, one for each half-step. I believe the above formula works correctly for this method. (I didn't try it though.) The other method is harmonic tuning (I'm not sure I remember the term correctly.) In this method, the instrument is tuned by using the proper values for the overtone sequence. An octave is double (or half) the frequency. A perfect fifth + one octave is 3 times the frequency. After that I forget. Eventually, working up the overtone sequence, the entire scale can be generated. The problem: it is only in tune for one key & scale type. Shifting keys puts you WAY out of tune. In equal temperment, everything is a bit out of tune, but not so as you would notice. (See later exception). The only exception I've experienced is with guitar ensembles. When I work with two or more other guitarists in a guitar only piece, we tune to a fingering of the tonic chord of the piece. This gives us a good harmonic tuning. If you tune in ET, some harmonic relationships don't function properly. -- Bill Wagner USPS net: Cimage Corporation Internet: wags@cimage.com 3885 Research Park Dr. AT&Tnet: (313)-761-6523 Ann Arbor MI 48108 FaxNet: (313)-761-6551 From: djones@megatest.UUCP (Dave Jones) Subject: Re: I need to know the frequency of the notes on a piano. Date: 4 Mar 91 21:18:11 GMT >From article <1991Mar4.181249.28467@cimage.com>, by wags@cimage.com (Bill Wagner): > ... Most instruments today > (including the piano) user "equal temperment" tuning. Basically, > divide the octave into twelve equal divisions, one for each half-step. You must have come in late on this thread. It ain't that easy. The piano is intentionally mistuned from the theoretical equal temperment in a couple of ways. See previous articles. From: djones@megatest.UUCP (Dave Jones) Subject: Scales (Re: African Gapped scale?) Date: 4 Mar 91 19:31:57 GMT >From article <30950@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): ... > |> Please name and describe a few [scales not based on integer ratios]. > |> Seriously, I would really like to > |> know of some. > > Various Malaysian, Gamelan, Javanese, and Indian scales are simple > examples. You left out the "describe" part. I am seriously interested. Please elucidate. > ... (BTW, I've heard that the Locrian mode > was not used in Western church music because of the presence of a > tritone in the V position.) > I've heard that story also. I even heard that the tri-tone V was *illegal* in the Catholic church of the middle ages. Because it could resolve in either of two directions, it was identified with the horns of the devil. I don't know if there is an ounce of truth to this rumor, but knowing the kind of thought that went down in those days, it sounds credible. From: djones@megatest.UUCP (Dave Jones) Subject: Re: Re: African Gapped scale? (Jazzy!) Date: 4 Mar 91 19:42:31 GMT >From article <1660031@hpsad.HP.COM>, by smithj@hpsad.HP.COM (Jim Smith): > Re: Dave Jones' response to my comment that the 'black keys' scale is not > equally tempered. We got a misunderstanding going here. I said that in the popular jargon, if you say "equal tempered", you're talking about the way pianos have been tuned since Johann S. B. and those cats whacked the octave up into twelve equal parts. (Or close to it.. Refer to the thread on piano-tuning and "stretching octaves".) If you want to call the scale under consideration, which whacks the octave into five equal parts, "equal tempered", go ahead I guess. I suggested "symmetric pentatonic" as an alternative that might avoid misunderstanding. Obviously I failed. From: djones@megatest.UUCP (Dave Jones) Subject: Re: I need to know the frequency of the notes on a piano. Date: 5 Mar 91 19:14:23 GMT >From article <1421@gvgspd.GVG.TEK.COM>, by mrk@gvgspd.GVG.TEK.COM (Michael R. Kesti): > In equal temperment, how far out of tune is everything? And > when a piano tuner uses "stretch" tuning, how far does he stretch? I've got a listing somewhere at home for the average of a lot of tunings done by "master" piano tuners. I'll try to see if I can dig it up. The book says that in double blind studies, the average tunings were judged by laymen to be as good as "masterful" tunings. In the mean time, here is the "unstreached" equal tempered diatonic scale. It's not difficult to calculate the frequencies. It's a little tricky to figure out what ratios the notes stand in for and what the error is. I wrote a little computer program to do the calculations. Here are the results, where a "cent" is one percent (logrithmic) of an equal tempered half step, 2**(1/12). scale ratio just equal error degree to root tempered tempered in cents I 1/1 1.000000 1.000000 0.000000 II 9/8 1.125000 1.122462 -3.910001 III 5/4 1.250000 1.259921 +13.686287 IV 4/3 1.333333 1.334840 +1.955002 V 3/2 1.500000 1.498307 -1.955001 VI 5/3 1.666667 1.681793 +15.641287 VII 15/8 1.875000 1.887749 +11.731286 VIII 2/1 2.000000 2.000000 0.000000 So there you are. Notice that the IV and V are very good, as is the II, but the other notes are very *sharp*. If you'll calculate the interval between the VI and the VIII, you will of course find that the minor third is *flat* from 6/5 by the same amount that the major six is sharp, 15.64 cents. Other minor intervals will also be flat. (I'm figuring this out as I go along. What fun!) So we see that the perfect intervals and the major II are almost right, while other major intervals are sharp, and minor intervals other than the -VII are flat. Gosh, do I ever feel edified! In fact, we can list the minor intervals: -VII 16/9 +3.91 -VI 8/5 -13.69 -III 6/5 -15.64 -II 16/15 -11.73 The only other interval is the tri-tone. What's the deal on that one? Left as an exercise for the reader, and for me when I have more time. I do know that its equal tempered value is the square root of two, which is close to 7/5, but I don't know if that has anything to do with the price of tea in China. Take it from there. From: simons@tetrauk.UUCP (Simon Shaw) Subject: Re: Re: African Gapped scale? (Jazzy!) Date: 6 Mar 91 16:25:47 GMT In article <1660031@hpsad.HP.COM> smithj@hpsad.HP.COM (Jim Smith) writes: >Re: Dave Jones' response to my comment that the 'black keys' scale is not >equally tempered. No Jim, it isn't, in the sense that it has different intervals. However, it _is_ based on an equal temperament 12-note scale (e.g. on a piano). The derivation of the underlying set of notes will affect the sound of the scale. Thus it is important to look separately at the tuning of the instrument and the scale one chooses to play on it. >Maybe I misunderstood what equal/well temperament is. I always thought it >meant that the ratio between any two adjacent notes in a scale is constant. Yes, this definition is correct. Incidentally, the five-note equal temperament scale and the western pentatonic scale are not so far apart, hence the jazzy sound. Only the addition of blue notes at 4+ and 7+ would be needed to complete the effect. -- Simon Shaw ; simons@tetrauk.uucp From: daver@felix.UUCP (Dave Richards) Subject: Re: I need to know the frequency of the notes on a piano. Date: 12 Mar 91 03:16:44 GMT Patrick Clark writes: - Is this right ? I always thought that the frequency of a 'pitch' -varied on where you played ? isnt there something called the `concert hall A' -(or something). - Also this formula implies that the interval between two notes is -according to a strict law ( this may be true on a keyboard instrument but -it certainly isnt so on other instruments ) I think to show this you play a -melodic minor scale on an instrument like a violin and compare it with a -scale played on the piano ? Don't know about the pitch of 'A' varying with location, but it has varied over time. It doesn't make much difference, as long as all the instruments that are playing together are designed for the same concert pitch. You would think that with the equal-tempered scale, any instrument that can be tuned can be tuned to play correctly in any pitch, but that's not true. On an instrument with permanent fingerings built in, such as the holes on a flute for example, those holes are only the right distances apart from each other when the distance to the mouthpiece is correct. If you re-tune the pitch by moving the mouthpiece in or out, now the finger holes are in the wrong place. Jim Smith writes: -Re: Dave Jones' response to my comment that the 'black keys' scale is not -equally tempered. - -Maybe I misunderstood what equal/well temperament is. I always thought it -meant that the ratio between any two adjacent notes in a scale is constant. -Since the 'black keys' pentatonic scale has both (well tempered) major seconds -and major thirds in it, I didn't think this scale *by itself* could be -considered equally tempered, even though it is derived from an equally -tempered chromatic scale. I'd be interested in hearing a definitive -answer on this.. Jim is right. The 'black keys' do not represent a true pentatonic scale (as- suming pentatonic means 5-note equal-temper). But he is not correct when he says the black-key scale has major thirds in it. It only has major seconds and minor thirds. But it seems kind of funny to refer to them using 7-note- scale terms like "second" and "third", because those terms have no meaning in this context. Here's a table for the black-key scale and a true ET pen- tatonic scale assuming both scales start at 100Hz: "Black keys" pentatonic 1 100.0 100.0 2 112.25 114.87 3 133.5 131.95 4 149.8 151.57 5 168.2 174.11 1' 200.0 200.0 Note that some black keys are below the ET pentatonic values, and one is above because of the uneven spacing. Bill Wagner writes: -I can try to clarify. For starters, we all know about A-440Hz. I think -we also know that one octave higher is 880Hz, one octave lower 220Hz. - -Now, tunings break into two different realms. Most instruments today -(including the piano) use "equal temperment" tuning. Basically, -divide the octave into twelve equal divisions, one for each half-step. It is potentially confusing to describe equal-temperment as "equal divis- ions". A safer phrase is "Equal Ratios" -The other method is harmonic tuning (I'm not sure I remember the term -correctly.) In this method, the instrument is tuned by using the proper -values for the overtone sequence. An octave is double (or half) the -frequency. A perfect fifth + one octave is 3 times the frequency. -After that I forget. Eventually, working up the overtone sequence, -the entire scale can be generated. The scale obtained this way will have truly equal intervals in the absolute sense, or equal-divisions if you prefer. This is the basis for Just Inton- ation (JI). I believe the sequence or overtones is: 1, 1', 5', 1'', 3'', 5'', 7'', 1''', 2''', 3''', 4''', 5''', 6''', 7''', 8''' where the apostrophes represent successive octaves. These numbers don't cor- respond to the numbers of the equal-temper scale degrees, because you get one extra interval in the octave. This means that several of them are "in the cracks", or between the equal-temper values. Consider two scales based on a starting pitch of 100 Hz: A. Western diatonic ("8-note") Scale (based on 12 Equal-Ratio half-steps). B. 8-note scale based on the overtone series (Equal-Intervals). Do Re Mi Fa Sol La Ti Do A. 100.0 112.25 126.0 133.5 149.8 168.2 188.8 200.0 B. 100.0 112.5 125.0 137.5 150.0 162.5 175.0 187.5 200.0 How B compares to A: beats/s 0. 0.25 1.0 4.0 0.2 5.7 6.8 1.3 0. % err 0. +0.22 -0.79 +3.0 +0.13 -3.3 +4.04 -0.69 0. cents 0. +3.75 -14.12 +50.4 +2.25 -60.0 +68.0 -12.25 0. Dave From: milo@cbnews.att.com (guy.f.klose) Subject: Re: I need to know the frequency of the notes on a piano. Date: 12 Mar 91 22:09:24 GMT There are a few things that really confuse me about Dave Richards' followup that I would like clarified: In article <158490@felix.UUCP>, daver@felix.UUCP (Dave Richards) writes: > > Jim is right. The 'black keys' do not represent a true pentatonic scale (as- > suming pentatonic means 5-note equal-temper). But he is not correct when he > says the black-key scale has major thirds in it. It only has major seconds > and minor thirds. But it seems kind of funny to refer to them using 7-note- > scale terms like "second" and "third", because those terms have no meaning in > this context. Here's a table for the black-key scale and a true ET pen- > tatonic scale assuming both scales start at 100Hz: I've never really heard of this notion of "true" pentatonic scale. It seems to me that I've never even heard about equally tempering five notes within an octave until just about two weeks ago, so I don't know how the term "true" enters into it. It also seems that a couple of postings have had a discrepancy about the use of the term "equal tempering". One person spoke of it in terms of the equal tempering of twelve notes within an octave, and then choosing a pentatonic scale from those notes, while someone else spoke of equally tempering five notes within an octave. Those ol' apples and oranges. I think that a pentatonic scale refers to a five-note scale, with no assumption of tempering at all. Further, from our perspective of Western music, the tempering is implied to be twelve notes within an octave (usually "well", as in slightly adjusted), and given this perspective, there is a large, but finite, number of pentatonic scales. The one that is most common to my studies is the [1 2 3 5 6] or [1 2 4 5 6] scales. The "black keys" can be represented by either of these scales. Maybe I missed it, but did someone give an example of a culture that has scales based upon five equally-tempered notes within an octave? Do those cultures use the same concept of octave that we use? They certainly wouldn't have access to the "strongest" sound in our Western music, the V to I resolution, so do they have different concepts of resolution? Guy -- Guy Klose milo\@angate.att.com From: jgk@osc.COM (Joe Keane) Subject: Re: Origins of Equal Temperament(was Re: Mersenne) Date: 13 Mar 91 00:32:49 GMT In article <30870@usc> alves@calvin.usc.edu (William Alves) writes: >The theoretical possibility of temperament (irrational ratios in tuning) >goes at least all the way back to Aristoxeneus (3rd cent BC), but the >question was about the *mathematical details* of equal temperament. Appa- >rently, fretted instrument players were commonly using equal temperament >back to the fifteenth century. The most common method of deriving equal >temperament practically was to tune to an 18/17 semitone (about 99 cents). The `rule of 18' is an empirical rule which says that each fret should be about 1/18 closer to the nut than the previous one. People knew empirically that this procedure didn't exactly make an octave, and it's easy to show mathematically that (18^17)^12 != 2. I believe that they compensated by making each interval some fixed amount bigger than the `rule of 18' would suggest. Given this, i'd say they were shooting for equal temperament. Although they may not have had the mathematical concept of the twelfth root of two, they could approximate that ratio to the necessary accuracy. [stuff about correct mathematical explanation deleted] >Marpurg insisted that J.S. Bach had used equal temperament and accused >Kirnberger, a student of Bach's, of abandoning his teachings. Quite the >contrary, said Kirnberger, Bach advocated unequal temperaments and C.P.E. >Bach wrote a testimonial backing up Kirnberger in this regard. It's unfortunate that the advocates of equal temperament cited Bach as a strong supporter. Even today we hear people claiming that his "Well-Tempered Clavier" is a demonstration of equal temperament. This is not true. >Though of course we do not have a mention of any tuning system from Bach's >own mouth, the fact that he wrote preludes and fugues in every key does not >mean he advocated or used equal temperament. In fact, he calls it the WELL- >Tempered Clavier because the temperament is good enough to be played in >every key, but, I believe, unequal enough that the different keys still >retain some differences in the intervals, and, hence, the mood and >affectation. In unequal temperament, each key has slightly different intervals in the various positions. This is similar to the seven modes but at a higher level of detail. I think that people who were used to unequal temperament learned to associate these differences with the actual key. Therefore when they heard music in equal temperament they still heard those differences which weren't actually there. Those of us who grew up on equal temperament have no idea what they were talking about. This is all confused by the fact that modern instruments have a mixture of temperaments. Here are the ones i can think of off the top of my head: free form: voice, fretless strings (e.g. violin) fixed equal temparement: keyed strings (e.g. piano) equal temperament but flexible: fretted strings (e.g. guitar) harmonic but flexible: simple wind instruments (e.g. bugle) none of the above: valved wind instruments (e.g. trumpet) If you know how a trumpet is really tuned, then you know more than most people. One thing i can tell you is that it's not equal temperament. -- Joe Keane, amateur mathematician From @ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi Tue Aug 27 17:54 MET 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Received: from ruuinf.cs.ruu.nl by alchemy.cs.ruu.nl with SMTP (15.11/15.6) id AA05993; Tue, 27 Aug 91 17:54:25 met Return-Path: <@ruuinf.cs.ruu.nl,@att.att.com:milo@mvuxi> Received: from att.att.com by ruuinf.cs.ruu.nl with SMTP (5.61+/IDA-1.2.8) id AA23601; Tue, 27 Aug 91 17:27:26 +0100 Message-Id: <9108271627.AA23601@ruuinf.cs.ruu.nl> From: milo%mvuxi@att.att.com To: piet@cs.ruu.nl Subject: Re: circle of fifths (& music theory) archive Date: Tue, 27 Aug 91 09:25 EDT "Music Theory" - part 2 of 2 From: djones@megatest.UUCP (Dave Jones) Subject: scales and temperment (Re: Circle of Fifths) Date: 13 Mar 91 23:31:55 GMT >From article <31358@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): > ... I just don't want to see peoples ideas discarded before > they've been heard. If anything miffed me, it was when you said that the > five-tone equal-tempered scale didn't sound very "useful". > wasn't it? If not, I will eat a 7-11 burrito, *and* enjoy ... Got 59 cents handy? What I said, word-for-word, was, "Without actually having heard it played, I have doubts as to the usefulness of this scale." I specifically said I didn't know how it sounded, only that I had doubts. I still haven't heard it played, - (I don't have a fancy synth) - but based on some more discussion, particularly the posting about the "harmonic seventh", I have stated that my doubts are somewhat allayed. > It seemed that you were basing > your idea of "useful" on how closely the scale approximated a jazz scale, or > how well/poorly it canceled beats, when the intended use of the scale may > have been music completely unrelated to these attributes. You are really beginning to test my "well-temperment". My definition is purely practical. To me, "useful" is what can be used to entertain an audience. I have developed some theories on the subject. If you can't handle that, ask yourself why not. I don't think just any random collection of tones is very "useful". I think that some scales are more useful than others, even within the context of well-tempered music. If you disagree, what criteria do you use to choose scales? (I asked that question once before and received no answer.) You keep saying that in other cultures there are scales based not on the integer ratios -- you call it "canceling beats". I am very skeptical that there is any such scale in common use, but I am not qualified to say it is not so. I would very much like to have an example of such a thing explained. I asked you to name and explain one such scale. You named several, calling them "simple examples", but omited any definition or explanation. Are you going to make me go to the library? Well, I'm overdue for a visit anyway. From: djones@megatest.UUCP (Dave Jones) Subject: Pentatonics Date: 14 Mar 91 00:21:23 GMT A posting or two ago someone asked which pentatonic to use. I attempted a partial answer, but quickly became overwhelmed. For jazz and blues, I recommend two excellent books: _Modern Concepts in Jazz Improvization_ by David Baker, and _Jazz Improvization and Pentatonic_ by Adelhard Roidinger. The former contains many very useful formulae. The latter comes with a demo/practice tape. I'll just throw in one little tidbit to give you a taste of what it's like. Baker lists the following turnarounds to use over the last two measures of a blues in F. (He writes out sixteenth-note riffs, something I can't easily do in ascii, so I'll just use the letter 'P' to indicate major-pentatonic, and let you devise your own riffs.) 1. FP AbP | GP GbP 2. FP AbP | DbP GbP 3. FP EbP | DbP CP 4. FP EbP | AbP GbP 5. FP DP | GP CP 6. AP AbP | DbP CP From: djones@megatest.UUCP (Dave Jones) Subject: Re: I need to know the frequency of the notes on a piano. Date: 13 Mar 91 22:47:12 GMT >From article <1991Mar12.220924.22435@cbnews.att.com>, by milo@cbnews.att.com (guy.f.klose): ... > Maybe I missed it, but did someone give an example of a culture that has > scales based upon five equally-tempered notes within an octave? Yep. He said it was African and that he learned about it in a college class. > Do those cultures use the same concept of octave that we use? Undoubtedly. They have the same laws of physics and the same ears. > They certainly > wouldn't have access to the "strongest" sound in our Western music, > the V to I resolution, so do they have different concepts of resolution? As I showed, that scale *does* have the V to I resolution, although it is rather badly out of tune from the theoretical 3-to-2 ratio. I would imagine that good singers would automatically compenstate. From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Re: scales and temperment (Re: Circle of Fifths) Date: 14 Mar 91 08:24:57 GMT In article <15664@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: |> From article <31358@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): |> |> > wasn't it? If not, I will eat a 7-11 burrito, *and* enjoy ... |> |> Got 59 cents handy? What I said, word-for-word, was, "Without actually |> having heard it played, I have doubts as to the usefulness of this scale." Well, I'm skeptical as to whether I should actually eat a burrito here, but maybe tomorrow if I'm hungry at the right time. |> You are really beginning to test my "well-temperment". Well, at least you're considering non-standard-Western scaling then! ;-) Anyway, please bear in mind that I'm feeling friendly when I type this stuff, so if I say anything that actually sounds nasty, I'm not being clear. |> My definition is purely practical. To me, "useful" is what can be used to |> entertain an audience. I have developed some theories on the subject. |> If you can't handle that, ask yourself why not. I think that the human brain is pleasantly excited by so many things that trying to develop a system of rules to discern what they are is absurd. I can handle it if you want to have your rules; I just don't think it's fair to export them to anyone else. This, for me, is the proper place for empiricism. |> I think that some scales |> are more useful than others, even within the context of well-tempered music. |> If you disagree, what criteria do you use to choose scales? (I asked that |> question once before and received no answer.) Sorry, I must have missed the question before. There are two criteria I use offhand, and a guideline. The criteria are: do I really feel like reprogramming the scale tables in my synths? and Does it sound good? in that order. The guideline is: in the range I'm playing in, when I combine sustained notes of frequencies X and Y, Y > X, is either Y - X < ~40 or is 2X - Y < ~40? I will violate this guideline if it still sounds good. In particular, lately I have been thinking about how to combine pitches so that the beat provides an implied bass note. I will also ignore this guideline if I am never combining sustained notes anyway. |> You keep saying that in other cultures there are scales based not on |> the integer ratios -- you call it "canceling beats". Actually I would like to discern between these two things (c.f. below about Chinese scales). Note that 16/15, while being a fairly low integer ratio (wait'll you see the ones in the Hindu scale!) doesn't cancel beats very well; if you are under 300 Hz your beat is subsonic. |> I am very |> skeptical that there is any such scale in common use, but I am not qualified |> to say it is not so. I would very much like to have an example of such a |> thing explained. I asked you to name and explain one such scale. You named |> several, calling them "simple examples", but omited any definition or |> explanation. I'll go ahead and post the stuff about scales I have typed up. It's long, but I hope it has some value for interested people. The stuff about Chinese scales may interest you; it seems that Chinese scales, while having their origins in integer ratios, actually classically didn't cancel beats because they were blowing into stopped tubes to generate their fifths. The tubes were cut to generate fifths by cutting new tubes to be 2/3 the length of old ones, but this actually produces a small fifth because it does not take into consideration the error introduced by blowing into the tubes. According to the book I found on the subject, the average fifth in a Chinese scale has been measured at 678 cents. But all that will be in the next posting. -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Some scales of various cultures (long) Date: 14 Mar 91 09:35:22 GMT As there has been some interest in non-(twelve-tone equal-tempered) scales lately, I decided to dig out some numbers. I have used a variety of scales myself, but some of the pitches I got out of books I don't own, and others were already programmed into synths. Not having a frequency counter, it would be difficult for me to get the actual pitches or deviations for these scales, so I dug up some books in the music library here at U of MD. The topics are in the following order: equal-tempered and Just twelve- tone scales, Pythagorean twelve-tone and Hindu 22-tone scales, Chinese and Japanese scales, Balinese and Javanese scales, scales of the Shona people of Zimbabwe. I will try to follow up this post with a bit more info in a few days. I have used several of the scales in discussion; I recorded some music in Just scales last year. I wish I could provide a table for this interesting 19-tone tuning on my Proteus/1, but I don't know what the values are (I'll see if I can find out). Some basic scale theory: The octave is the standard for measurement of intervals by Western scholars. An octave is the interval attained by multiplying the frequency of the lower pitch by exactly 2. Octaves are divided into cents. A cent is 1/1200 of an octave, or 1/100 of an equal- tempered semitone. To raise a pitch by one cent, multiply its frequency by 2 ** (1/1200), where ** denotes "to the power of". To raise a pitch by one semitone (== 100 cents) multiply its frequency by 2 ** (100/1200) (== 2 ** (1/12)). In general, to raise a pitch by n cents, multiply its frequency by 2 ** (n/1200). To determine how many cents frequency Y is above frequency X, take 1200 * log2 (Y/X). If you don't have a calculator with log2, take 1200 * (logx (Y/X)) / (logx 2) where logx is the log you do have on your calculator. The tables I give are all measured in cents. Where I give a table for a scale, I give a version where every pitch is in cents above the tonic. When tuning a synthesizer to one of these scales, pick a tonic for reference. For each note that you decide to map to one of the pitches in question, note how many semitones above the tonic it is. Now subtract 100 cents from the target interval for each semitone. For example: suppose the tonic is A, I need to map a pitch 340 cents above the tonic to the C. I subtract 300 cents and tune C 40 cents sharp. To map a pitch 685 cents above the tonic to the E, I subtract 700 cents and tune E 15 cents flat. The "standard" of the octave, while common, is not universal; check out the stuff on the Shona mbira tunings at the end. When two sustained notes are played together, a third implied tone arises, called a beat or interference beat. The frequency of the beat tone is equal to the difference in frequency between the two sounding tones. Frequently this beat lands in subsonic frequencies (i.e. < ~20 Hz), and people have traditionally avoided such beats because they are often disturbing to human physiology (perhaps it is a sign of an earthquake that triggers an emotional response). For an example of the problem of beats, play two notes together a semitone apart on a piano in a low octave. You may be able to perceive a low-frequency tone, and the overall sound will probably be unpleasant. Now play the same two notes in a high octave. The beat will no longer sound offensive, since it is no longer a subsonic; higher notes on an equal-tempered instrument are farther apart frequency-wise than lower notes. Intervals that are separated by a factor of low integer ratios (e.g. the Just perfect fifth 3/2) have beats that are easy to keep out of subsonic frequencies. For example, the 3/2 ratio always has a beat that is exactly one octave lower than the low tone. The 2/1 or octave ratio has a beat that is exactly equal to the low tone. This is why octaves sound so good; the beat reinforces the chord. Note that subsonics are not always unpleasant; some frequencies are very soothing; other frequencies sound good by themselves. Subsonics between ~1 and ~20 Hz often sound annoying when added to music. Here is a table from _The Gamelan Music of Java and Bali_ by Donald A. Lentz (1965, University of Nebraska Press LCCCN 65-10545) pp. 24-25. I have chopped this table into bits, as it is in a wide format in the book. The table is based on a tonic of C and gives a lot of cent values. The cent values are given with no fractional parts. Comments are mine. The equal-tempered twelve-tone (Tempered) scale is the standard for modern Western music. The Just (perfect) scale is the ancient beat-canceling integral-ratio scale commonly used until the eighteenth century in Western music. It is still used by some performers of ancient music for the sake of authenticity and in certain contexts by other performers. The actual pitches of the Just scale depend on which note is taken as the tonic of the scale. The ratio and cent values, however, remain constant regardless of the tonic. Lentz apparently made this table in C so he could provide note names. -I- -II- -III- Cents Equal Just above Tem- (Using Funda- pered C as a mental Tonic) Name Interval Ratio Interval Name 0 C Unison 1/1 Unison C 100 Db Half step 200 D Whole step 204 9/8 Whole step D 300 Db Minor 3rd 386 5/4 Major 3rd E 400 E Major 3rd 498 4/3 Perfect 4th F 500 F Perfect 4th 600 F# Aug. 4th 700 G Perf. 5th 702 3/2 Perf. 5th G 800 G# Aug. 5th 884 5/3 Maj. 6th A 900 A Maj. 6th 1000 A# Aug. 6th 1088 15/8 Maj. 7th B 1100 B Maj. 7th 1200 C Perf. 8ve 2/1 Perf. 8ve C The Pythagorean scale is derived from perfect fifths alone. The intonation of the Renaissance period used eight ascending fifths and three descending fifths. When 12 perfect 3/2 fifths have been ascended and the octave has been corrected back to the original register, the resulting ratio is 531441/524288 (== (3/2) ** 12 / 2 ** 7). This interval is known as a comma and is equal to 24 cents. The Hindu scale is based both on ascending fifths and ascending fourths. The eleven pitches from ascending fifths and the eleven pitches from ascending fourths are combined to produce a 22-tone scale. There are quite a few details about this scale that I won't try to summarize here. The Pramana is the distance between Srutis 4 and 5, which are the two whole-tone intervals of the Just scale: 9/8 (between I and II) and 10/9 (between II and III). -I- -IV- -V- Cents Pythagorean Hindu above Funda- mental Cycle Sruti No. Interval Name Ratio No. Name 0 0 Unison C 1/1 1 Sa 22 81/80 Pramana 24 12 Comma 90 Minor 2nd (Limma) 256/243 2 Ri 112 16/15 3 Ri 114 7 Aug. Prime C# 182 10/9 4 Ri 204 2 Maj. 2nd D 9/8 5 Ri 294 -3 Min. 3rd Eb 32/27 6 Ga 316 6/5 7 Ga 318 9 Aug. 2nd D# 386 5/4 8 Ga 408 4 Maj. 3rd E 81/64 9 Ga 498 -1 Perf. 4th F 4/3 10 Ma 520 27/20 11 Ma 590 45/32 12 Ma 600 6 Aug. 4th F# 64/45 13 Pa 702 1 Perf. 5th G 3/2 14 Fa 792 128/81 15 Dha 814 8/5 16 Dha 816 8 Aug. 5th G# 884 5/3 17 Dha 906 3 Maj. 6th A 27/16 18 Dha 996 -2 Min. 7th Bb 16/9 19 Ni 1018 9/5 20 Ni 1020 10 Aug. 6th A# 1088 15/8 21 Ni 1110 5 Maj. 7th B 243/128 22 Ni I am skipping Columns VI and VII of the table, which give blown fifths and string-length division values. The blown fifths are Chinese intervals arrived at by cutting bamboo tubes of particular lengths. These tubes produce small fifths that range between 670 and 680 cents. A study of these tubes by Dr. E. M. von Hornbostel claimed that the average interval is 678 cents. This produces a cycle of 23 fifths, and the comma after 23 fifths is 6 cents. Here are some quotes from pp. 27-28 of Lentz. "Two theoretical systems evolved in China, one derived from the Cyclic Pentatonic and the other from the division of string lengths. They are found combined in the highest form of Ch'in music. The Cyclic Pentatonic, arrived at mathematically, is important in Chinese musical thought. The conception of the twelve liis (tones) dates back to the Han dynasty. They are formed by building empirical fifths in a manner similar to that used in Pythagorean tuning. Methods of arriving at these fifths included the use of twelve tubes. Levis indicates that a stopped bamboo tube 230 millimeters long, 8.12 millimeters in diameter, and vibrating at 366 vibrations per second was the Yellow Bell (Huang Chong), the standard established by the Bureau of Weights and Measures in 239 B.C. Tube number two was made two-thirds the length of the reference tube; number three was made equal to two-thirds of number two and then doubled to bring the tone within the ambit of an octave. This process was repeated for each of the twelve tubes. The fifths produced by these tubes were small compared to Western fifths. Various musicologists place them between 670 and 680 cents as compared to the Just fifth of 702 cents." "In Java musicians and gamelan makers, when queried about the small fifth, explained it as coming from nature but gave no specific examples. At dawn one morning in a small village in Java, a bird was singing with a call of an octave and a small fifth. This I recorded. The same song was frequently heard later and again recorded for confirmation of interval. It is a fascinating bit of fancy that the fifth in the bird call when measured on the stroboscope varied only two or three cents from the one of 678 cents. There are also indications that the small fifth might have been a standard in Sumeria and Egypt." "In music for the ch'in, a zither-type instrument, the seven strings are tuned to the Cyclic Pentatonic. Each string has frets or nodes dividing it into the following lengths: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 4/5, 1/6, 5/6, 1/8, and 7/8. The resultant cent values are listed in column VII of Chart IV. Each string employs the principle of Just intonation, but many microtonic intervals result when this theory of string length division is combined with the above Cyclic Pentatonic procedure, which is used to tune the seven open strings. There is no counterpart in Western Tempered music. Many Oriental stringed instruments use frets which are movable. The accuracy of placing the fret, which is done by ear, can greatly affect the pitch and thus produce noticeable deviances in a system using natural intervals of microtonic size. The tones of the present-day scales of Japanese koto and gekkin music evolved with variance from the Ch'in principle. "Western musicians think of the fifth as being an interval of 700 to 702 cents. The deviation of only two cents betwen Just, Pythagorean, and Tempered fifths is so small that these sizes are accepted as being the true fifth. This is not the case in Oriental music, even though some Western musicologists try to explain their fifths as anomalies from the Western norm. The conception of the fifths is in many cases very different. For the most part they are smaller than the Western fifth. In Chinese music, another common theoretical fifth of 693 cents, as contrasted with the Cyclic fifth of 678 cents, results from combining three of the characteristic large seconds derived from string-length division (see column VII in Chart IV). Each of the large seconds has a value of 231 cents and a ratio of 8/7." "Fifths of varying sizes are produced on different pipes when the end-correction factor is not considered, thus not fitting a theoretical system. These convert to a standard when duplicated. This procedure of duplication is found in China along with the theoretical fifth, and although one cannot find positive documentation of it for the gamelans of Java and Bali, it is highly possible that it became a factor in the varying sizes of the fifth there too." On p. 33 the gamelan scale is described: "Three basic tones and two or four secondary tones are the background of the gamelan tonal system. The main tone, called dong in Bali, is supported by two tones, one a fifth above (called dang) and a the other a fifth below (called dung). The secondary tones are a fifth above (d`eng) and a fifth below (ding) the supporting tones. By bringing the five tones within an octave, the following scale results: dong, d`eng, dung, dang, ding. "...For convenience the tone names of Western notation will be used, with C arbitrarily chosen as a starting tone. But it should be recalled that the Oriental fifths are variable in size and in all probability will not correspond to a Western fifth. This results in a scale named C D F G Bb. "In the Balinese-Javanese five-tone scale, a large interval, approximately a minor third, which will vary in size from one gamelan to the next, occurs between the second and third and the fourth and fifth degrees." Using a fifth of 678 cents, we can generate one example of this scale: Degree Cents above tonic I 0 == unison II 156 == two fifths minus one octave III 522 == octave minus one fifth IV 678 == one fifth V 1044 == two octaves minus two fifths In _Musics of Vietnam_ by Pham Duy, Edited by Dale R. Whiteside (1975, Southern Illinois University Press) Duy explains that theoretically the Khmer scale of South Vietnam is divided into seven equally-spaced tones. This would make cent values as follows: Degree Cents above tonic I 0 II 171 III 342 IV 514 V 685 VI 857 VII 1028 However, Duy notes that in practice the intervals in the Khmer scale are not exactly equal. In _The Soul of Mbira_ by Paul F. Berliner (1978, 1981, University of California Press), Berliner describes a number of tunings used for the mbira, which is a kalimba-like African instrument common in Zimbabwe. Apparently each region has its own tuning, and different instrument makers tune their instruments differently. The prevailing theory of Shona mbira tuning is that "mbira makers and players use a distinctive, well defined scale, with only slight variation in different parts of the country....It can be described as a seven-note scale, with all the intervals equal." (p. 66) However, Berliner found in a sample of tunings that the variation was very large, varying between 37 to 286 cents between adjacent scale degrees, and not equal at all. The various mbira players select their instruments based on a variety of factors, including tuning, which they refer to collectively as the "chuning" of the instrument. "...I asked several musicians who owned these mbira...to select from a set of fifty-four forks tuned 4 c.p.s. apart (from 212 c.p.s. to 424 c.p.s.) the individual forks which each thought matched the tuning of the keys on his respective instrument. The fact that they sometimes said that the pitch of an mbira key fell between two tuning forks demonstrated that the musicians could discern fine variations in tuning." Here is one of the tunings Berliner gives: Between Mbira interval in cents C and D 185 D and E 204 E and F 204 F and G 163 G and A 158 A and B 137 B and C 251 This gives the following table: Degree Cents above tonic I 0 II 185 III 389 IV 593 V 756 VI 914 VII 1051 oct. 1302 Note that the octave is not a factor of two in this tuning. Apparently the "octaves" are highly variable in mbira tunings. Berliner gives some tables of the variations from a "true" octave he found in various mbiras. The octave also is not exact in Scottish and Irish bagpipe music, since the high overtone used for the octave on the canter pipe is somewhat off. The high notes of a bagpipe melody tend to come out weaker and a little bit flat. This concludes the summary of information I collected; I hope to follow this up with some other examples in a few days. Happy tuning. -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From: djones@megatest.UUCP (Dave Jones) Subject: Re: scales and temperment (Re: Circle of Fifths) Date: 15 Mar 91 00:22:50 GMT >From article <31436@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): > (wait'll you see the ones in the Hindu scale!) It may interest you to know that the term "Hindu" is used by jazz guys to designate the fifth mode of the melodic minor scale. (Or then it may not.) It's like the mixolydian, but with a lowered VI. > I'll go ahead and post the stuff about scales I have typed up. It's > long, but I hope it has some value for interested people. The stuff > about Chinese scales may interest you ... Oh yeah. I heard a nifty Japanse scale on TV the other night. A woman in a kimono was playing an instrument with lots of strings and moveable bridges. Presumably it was authentic stuff. Okay, I'll admit it: It was a PBS kid's show. The scale was G# A C# D# E. Notice that these notes belong to the Phrygian mode of E, a minor mode, but the minor third (B) is missing. Also notice the major third between A and C#. You can look at it as an inversion of A-Lydian. From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Re: scales and temperment (Re: Circle of Fifths) Date: 15 Mar 91 10:36:45 GMT In article <15680@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: |> From article <31436@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): |> |> It may interest you to know that the term "Hindu" is used by jazz guys |> to designate the fifth mode of the melodic minor scale. (Or then it |> may not.) It's like the mixolydian, but with a lowered VI. Really? I wonder why...maybe Ravi used it in his jam sessions... =) Sort of inverted logic of the melodic minor; major III but minor VI and VII. |> > I'll go ahead and post the stuff about scales I have typed up. It's |> > long, but I hope it has some value for interested people. The stuff |> > about Chinese scales may interest you ... |> |> Oh yeah. |> I heard a nifty Japanse scale on TV the other night. A woman in a kimono was |> playing an instrument with lots of strings and moveable bridges. Presumably |> it was authentic stuff. Okay, I'll admit it: It was a PBS kid's show. |> The scale was G# A C# D# E. Notice that these notes belong to the |> Phrygian mode of E, a minor mode, but the minor third (B) is missing. |> Also notice the major third between A and C#. You can look at it as |> an inversion of A-Lydian. May have been a kayagum. What was the name of the show? Did you ever see Laurie Anderson's _Home of the Brave_? Or hear _Mister Heartbreak_? The oriental instrument in "Kokoku" is a kayagum, if I'm not mistaken. I think this instrument is a lot like the ch'in that Lentz talks about in the quotes I give in the long scale posting. If it were tuned like the ch'in Lentz talks about, it might have microtonal differences that would be hard to pick up in a PBS kids' show. Not having heard the music, given those notes my first way of looking at it would be A-Lydian. Guess I'm more of a major-head? Hell, I can't keep the major thirds out of my so-called blues scales... ;-) -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From: alves@calvin.usc.edu (William Alves) Subject: Re: Info on some scales (long) Date: 15 Mar 91 18:55:33 GMT I'm sure that a lot of people are grateful for Jefferson Ogata for his posting of tuning systems. Such tables are a good place to start for people looking to experiment with non-Western tunings. However, I want to clarify a couple of things. Tables such as these some- times give the impression that these cultures have some sort of stan- dard, when, in fact, very few of them do. The mbira tunings from Berliner show this. As Mr. Ogata says, Berliner points out that mbira tunings vary widely from one player to another. The only time when they NEED to be the same is in the infrequent occasions when ensembles of mbira players play together. In this case, Berliner states elsewhere, the players tune to the mbira of the most senior player. Likewise, it has long been known that the gamelan of Indonesia are not tuned to any standard. Instead, each gamelan of instruments is tuned to itself. The "tuning systems" of pelog and slendro are really "families" of tuning systems that share some similarities. Alexander Ellis was the first to make this critical error when he tried to fit all the gamelan in Java to an equal-tempered 5-tone slendro, and a 7-tone pelog of un- equal tones. Kunst (in Java) and McPhee (in Bali) found that no standard really exists, but made the mistake of assuming that the octave was constant. See Mantle Hood's "Slendro and Pelog Redefined" in _Selected Reports in Ethnomusicology_ I/i 1966. In both India and China there seems to have been an obsession with coming up with theoretical tuning systems. In China, especially, this was related to mystical associations with the mathematics of sound (hardly unknown in the West), but these tuning systems were, as far as I'm aware, related only very incidentally to practice. Studies of Indian tunings, for example, find virtually no trace of a 22-tone equal tem- pered system in practice. Instead, the drone tones are tuned to a pure fourth or fifth, and the tuning of the rest of the solo instrument, as in the case of the mbira player, is up to the soloist. I think it's most important to understand when and why tuning systems are used. First of all, they're most important in instruments with a single sounding body per note (keyboard instruments, harps, mallet instruments [including the gamelan], the mbira, zithers, and so on). Secondly, they only have to be standardized when either two of these instruments play together, when instruments have to be able to play with any other instruments (as in the West, but not in Indonesia), or when a composer explicitly asks for a certain tuning system (as in Partch). I think the ultimate goal of studying other tuning systems would be to understand WHY the person who tunes an instrument tunes it that way. Are they interested in beatless intervals? A certain frequency of beats? Different keys sounding the same? Sounding different? Are there extra- musical considerations? Unfortunately, there has been very little of this sort of research that I am aware of. So go and explore these tunings, by all means. Just keep these other things in mind as you do. Bill Alves From: jimh@ultra.com (Jim Hurley) Subject: Re: Info on some scales Date: 15 Mar 91 18:04:51 GMT Jefferson Ogata has posted a fairly interesting set of tuning tables. It is hard to find such exotic information and Western researchers seem obsessed with measurements of exotic instruments and don't get into the spirit of how they are played. Here are some other references: Tuning In, by Scott Wilkinson has a very comprehensive coverage of the history of temperament and covers things up to the present time. On the Sensations of Tone, the classic, has a section at the end that has a very elaborate list of intervals in fractional cents with their names and some historic info. -- Jim Hurley --> jimh@ultra.com ...!ames!ultra!jimh (408) 922-0100 Ultra Network Technologies / 101 Daggett Drive / San Jose CA 95134 From: quayster@arkham.wimsey.bc.ca (Tony Chung) Subject: Re: Big discussion on right and wrong (WAS: C of 5ths) Date: 16 Mar 91 10:43:23 GMT ogata@cs.umd.edu (Jefferson Ogata) thoughtfully worded >I think it wise to be careful to qualify negative >value judgements with your own experience and influence. What you >have already decided about music may apply only to you. E.g. this >C# vs. Db thing: it may be easier for you as a sax player to read >Db, but as a keyboard player, I find it easier to read C#. What you fail to consider, my dear friend, is that you, as Jefferson Ogata, prefer to read C#, whereas I, as Tony Chung (or Quays, depending on the mood) prefer to read Db. That is, when I prefer to read :) (Yes, I'm a quayboard player, too, just don't read much, is all) -Tony 'Quays' ('keys' with a 'Q') +- Tony Chung -----------------+ \ ^ | quayster@cynic.wimsey.bc.ca | -- o- | quayster@arkham.wimsey.bc.ca | ) "Sig's keep getting shorter +----------------- Tony Quays -+ (____, every day..." --Myron Lewis From: djones@megatest.UUCP (Dave Jones) Subject: Re: Big discussion on right and wrong (WAS: C of 5ths) Date: 17 Mar 91 00:37:14 GMT >From article <1aewy2w164w@arkham.wimsey.bc.ca>, by quayster@arkham.wimsey.bc.ca (Tony Chung): > [another guy] prefer[s] to read C#, whereas I, as Tony Chung (or Quays, > depending on the mood) prefer to read Db. There is a nice symmetry to the key-signatures, arranged in cycle-of-fifths order: sharps.. 6 5 4 3 2 1 KEY F# B E A D G C F Bb Eb Ab Db Gb flats.. 1 2 3 4 5 6 You glue the F# and the Gb togther, and you get the cycle. Of course you don't have to glue it into a cycle; you can wrap it around into a helix, coming up with key-signatures of 7 sharps, 8 sharps (one double sharp), nine sharps, thirty-six sharps, four hundred sharps. I just don't see the point in it. Happily, the practice is not common, at least not in jazz. The only place I ever saw key signatures of more than six flats or sharps was in a legit etude and exercises book. But I frequently see chords written that way in fake books. It always goofs me up for a few seconds. Quick! What are the notes of the mixolydian scale for A#7? Brrrrzzzt! Sorry. Time's up. The rhythm section is already in the next measure. From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Re: Big discussion on right and wrong (WAS: C of 5ths) Date: 17 Mar 91 13:58:07 GMT In article <15708@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: |> From article <1aewy2w164w@arkham.wimsey.bc.ca>, by quayster@arkham.wimsey.bc.ca (Tony Chung): |> |> > [another guy] prefer[s] to read C#, whereas I, as Tony Chung (or Quays, |> > depending on the mood) prefer to read Db. |> |> There is a nice symmetry to the key-signatures, arranged in cycle-of-fifths |> order: |> |> sharps.. 6 5 4 3 2 1 |> KEY F# B E A D G C F Bb Eb Ab Db Gb |> flats.. 1 2 3 4 5 6 There is a happy *asymmetry* to most music. How about: sharps.. 7 6 5 4 3 2 1 KEY C# F# B E A D G C F Bb Eb Ab Db flats.. 1 2 3 4 5 Gee, this reflects the number of semitones in the perfect fourth and fifth intervals, which, as we all know, are more perfect than any other intervals. =) I like to see things a bit off center sometimes. And we might try: sharps.. 5 4 3 2 1 KEY B E A D G C F Bb Eb Ab Db Gb Cb flats.. 1 2 3 4 5 6 7 I, for one, like to view the circle of fifths as extending from B# to Dbb. This means that instead of just gluing two scales at F#/Gb, you get to glue *every* scale to another, and the degrees extend to 12 on both sides. Doesn't this seem more symmetric to you? |> You glue the F# and the Gb togther, and you get the cycle. Of course you |> don't have to glue it into a cycle; you can wrap it around into a helix, |> coming up with key-signatures of 7 sharps, 8 sharps (one double sharp), |> nine sharps, thirty-six sharps, four hundred sharps. You just did wrap it into a helix, when you glued F# and Gb together. |> I just don't see the |> point in it. Then you didn't read what I said about C#. There are two points in it: - seven sharps is easier to remember than five flats, because everything is sharped; - C# can be played in C instantly by ignoring the key signature and dropping one sharp from every accidental. This can be handy. |> Happily, the practice is not common, at least not in jazz. |> The only place I ever saw key signatures of more than six flats |> or sharps was in a legit etude and exercises book. But I frequently see chords |> written that way in fake books. It always goofs me up for a few seconds. As you are no doubt aware, frequently notes are named to reflect a scale degree that might be named in more than one way, but is meant to be read in a particular way. For this reason, you will find many double sharps, double flats, and namings like A# in the music of Johann Sebastian Bach, for example, who liked to indicate that he was modulating up a fourth from Gb by writing in Cb; this means you get to just add one flat to the scale, instead of suddenly switching from 6 flats to five sharps, so you don't have to cancel the entire key signature of Gb. It also reflects the modulation as a perfect fourth, rather than an augmented third, which is what switching to B would imply. A# mixolydian goes to D# major; this is an easy one: from all naturals, sharp everything and add an extra sharp to F and C, refecting the signature of D major. Eb major may be easier, but not necessarily if the song is written in F#. I prefer not to find Eb's in the middle of a piece that's got six sharps in the signature! -- Jefferson Ogata ogata@cs.umd.edu University Of Maryland Department of Computer Science From: clemens@ut-emx.uucp (J. Christopher Clemens) Subject: Re: Circle of Fifths Date: 17 Mar 91 21:37:57 GMT In article <15628@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: >Somebody smart once said that the purpose of an open mind is the >same as that of an open mouth: to slam shut when something good enters. "The only reason I know for having an open mind is the same as that for having an open mouth; that I may close it again on something solid. --G. K. Chesterton I think its from "Orthodoxy" but I'm going by memory here. G. K. C. weighed several hundred pounds when he died, which makes the quote somewhat ironic (a fact which was not lost on him). Sorry this is not directly related to music. Chris Clemens U.T. Austin Astronomy Department From: milo@cbnews.att.com (guy.f.klose) Subject: Re: Big discussion on right and wrong (WAS: C of 5ths) Date: 18 Mar 91 17:16:11 GMT In article <31583@mimsy.umd.edu>, ogata@leviathan.cs.umd.edu (Jefferson Ogata) writes: > > I, for one, like to view the circle of fifths as extending from B# to > Dbb. This means that instead of just gluing two scales at F#/Gb, you > get to glue *every* scale to another, and the degrees extend to 12 on > both sides. Doesn't this seem more symmetric to you? > Then you didn't read what I said about C#. There are two points in it: > - seven sharps is easier to remember than five flats, because everything > is sharped; This one is definitely a value judgement...band instruments in concert keys (flutes, trombones, tubas, etc.) would see Db long before they would C#. Personally, I find E# and B# a lot less easy to remember, regardless of the key signature. On the other hand, I've never practiced in C#, since trombonists rarely face it (granted, I'm not into orchestral music). > - C# can be played in C instantly by ignoring the key signature and > dropping one sharp from every accidental. This can be handy. I can think of only one instance where I was faced with the key of C#...in a pit band for the musical "Lil' Abner". While reading it the first time, the musical director said "this is ridicluous, ignore all the sharps, read it as written". He went on to say "some composers are under the mistaken impression that sharp keys sound brighter". Personally, I won't argue either way, other than if you have a group of people who are reading C# for the first time, the music will definitely not sound bright. His instructions might have also been kind of callous, especially if the composer was trying to get a certain sound, like maybe a half-step modulation later on, or something like that. Another bit of trivia, I've played several Basie tunes (riff tunes, actually) that are written in the key of Db..."One O'Clock Jump" and "Splanky" come to mind. Both are Db blues, and actually, trombone-wise, are good blues keys, since the blues scale sits in a pretty good set of positions. Can't speak for any other instruments, though. Guy -- Guy Klose milo\@angate.att.com From: djones@megatest.UUCP (Dave Jones) Subject: Open minds (Re: Circle of Fifths) Date: 18 Mar 91 23:21:51 GMT This is not about music. So hit whatever key you hit to proceed if you are not interested. >From article <45711@ut-emx.uucp>, by clemens@ut-emx.uucp (J. Christopher Clemens): > In article <15628@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: >>Somebody smart once said that the purpose of an open mind is the >>same as that of an open mouth: to slam shut when something good enters. > > "The only reason I know for having an open mind is the same as that > for having an open mouth; that I may close it again on something solid. > > --G. K. Chesterton > Thanks! I'm trusting your sources and making an entry into my largest book of quotations, which doesn't have this in its 1088 pages. By coincidence, I saw a panel discussion on TV on this subject just the day after I posted the above. No, it didn't mention G.K. Chesterton. It featured William F. Buckley, a journalist, and a professor from Dartmoth (I think). W.F. and the j. were ganging up on the p. saying, that except in "politically correct" indoctrination courses, colleges now teach that any critical value-judgement is the mark of a "closed mind". The collegiate ideal of teaching students to answer difficult questions is abandoned. Study at the university no longer begins with the Socratic questions. In the political indoctrination courses, it begins with the answers. Students are routinely graded according to how accurately they parrot the professors' party line, a practice that would have been shameful twenty years ago. In other courses, they learn that the questions of value and worth are not to be asked. As an example they said that in a literature course, one does not dare to venture the opinion that Shakespeare is better literature than Terry and the Pirates comics. They claimed that the idea of a "marketplace of ideas", where honorable people can hold and argue strongly differing opinions, is virtually extinct. Much as I felt sorry for the prof. being two-to-oned like that, I had to admit he made a pretty lousy rebutal. I got the feeling that he knew they were right and just wished they wouldn't blab it all around. From: djones@megatest.UUCP (Dave Jones) Subject: Re: Big discussion on right and wrong (WAS: C of 5ths) Date: 18 Mar 91 23:37:06 GMT >From article <1991Mar18.171611.9523@cbnews.att.com>, by milo@cbnews.att.com (guy.f.klose): > Another bit of trivia, I've played several Basie tunes (riff tunes, > actually) that are written in the key of Db..."One O'Clock Jump" and > "Splanky" come to mind. Both are Db blues, and actually, trombone-wise, > are good blues keys, since the blues scale sits in a pretty good set of > positions. Can't speak for any other instruments, though. "Five flats" as it is commonly called, is a pretty common key in jazz. I don't know why. Maybe pianists like to hit the black keys for a change of pace. :-) If I had a fake book handy, I could easily find a dozen examples or more. "Body and Soul" and "'Round Midnight" come to mind. And of course, because jazz tunes change keys quite a bit, you tend to hit all the keys. "Joy Spring" has the first and last eight measures in F, (except for a couple of tri-tone substitutions), but the second eight is in - guess what? - five flats. The third eight or "bridge" changes keys several times. These key changes are infrequently noted in the key signature; usually they are handled with accidentals. From: mu298ai@sdcc6.ucsd.edu (mu298ai) Subject: Re: Slonimsky (was Coltrane, My Favorite Things) Date: 24 Mar 91 15:18:32 GMT In article <16211@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: >Certainly the "Giant Steps" trick has that flavor about it: Divide the >cycle of fifths into three equal parts, and embellish around each corner >of the triangle. His "sheets-of-sound" runs also come to mind. Could you explain what you mean by the embellishments around each corner of the triangle? In what way, what modes/scales/patterns (if any), did he use? Did he choose certain corners more often than others? Sue . From: marcoz@enquirer.scandal.cs.cmu.edu (Marco Zagha) Subject: Re: Coltrane changes on Body and Soul Date: 1 Apr 91 19:47:17 GMT In article <16484@prometheus.megatest.UUCP>, djones@megatest.UUCP (Dave Jones) writes: > Recall that I said Dexter Gordon used the Coltrane changes on the > bridge to Body and Soul? Somebody said Coltrane did too. Last night > I got out my "Coltrane's Sound" CD and listened to both B&S cuts, > the one from the original 33 and an alternate take. Neither used the > Coltrane changes on the bridge, although the piano did add some passing > chords between the standard ones. In each case the intro was quite > similar to the way Dexter did it. I just listened to it and he *does* use the changes. Perhaps you were looking for this Coltrane turnaround, C Eb Ab Db C when he uses the Countdown type turnaround instead: D- Eb Ab B E G C == Marco Internet: marcoz@cs.cmu.edu Uucp: ...!seismo!cs.cmu.edu!marcoz Bitnet: marcoz%cs.cmu.edu@cmuccvma CSnet: marcoz%cs.cmu.edu@relay.cs.net From: ogata@leviathan.cs.umd.edu (Jefferson Ogata) Subject: Re: OverTone Series Listing - ? Date: 24 Apr 91 10:24:05 GMT In article <17912@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: |> From article <71931@eerie.acsu.Buffalo.EDU>, by v097pba8@ubvmsd.cc.buffalo.edu (Ken F Morton): |> > |> > Could someone send me a listing of the components of the overtone |> > series? thanks... |> |> Pick a frequency. Any frequency. Now multiply it by 2, 3, 4, 5, 6, ... |> "Viola!", as they say in France. Only the first few matter, though. You have accurately described the harmonic series, not the "overtone series". An overtone series is any succession of progressively higher frequencies that sound with the fundamental. The harmonic series is one example. The poster did not specify which overtone series he was looking for, so it may be that's what he wanted. The overtone series for a high string on a piano is somewhat different from that of a low string; the overtone series for a tympani head is very weird; and so on... I thought it was a very strange request. Maybe he is actually asking for all the models of a series of equipment manufactured by a small company called "OverTone"...?????? -- Jefferson Ogata ogata@cs.umd.edu University of Maryland Department of Computer Science "Sure. Understanding today's complex world of the future *is* a little like having bees live in your head." From: djones@megatest.UUCP (Dave Jones) Subject: Re: OverTone Series Listing - ? Date: 29 Apr 91 22:23:24 GMT >From article <33453@mimsy.umd.edu>, by ogata@leviathan.cs.umd.edu (Jefferson Ogata): > In article <17912@prometheus.megatest.UUCP> djones@megatest.UUCP (Dave Jones) writes: > |> From article <71931@eerie.acsu.Buffalo.EDU>, by v097pba8@ubvmsd.cc.buffalo.edu (Ken F Morton): > |> > > |> > Could someone send me a listing of the components of the overtone > |> > series? thanks... > |> > |> Pick a frequency. Any frequency. Now multiply it by 2, 3, 4, 5, 6, ... > |> "Viola!", as they say in France. Only the first few matter, though. > > You have accurately described the harmonic series, not the "overtone > series". An overtone series is any succession of progressively higher > frequencies that sound with the fundamental. The harmonic series is one > example. When I was a mathematician, "the harmonic series" was definided as the following: 1 + 1/2 + 1/3 + 1/4 + 1/5 ... or more generally, "a harmonic series" was any series of the form 1 + 1/(2**a) + 1/(3**a) + 1/(4**a) + ... The salient feature of such a series is that it converges when the norm of 'a' is less than 1 and diverges when it is more than 1. > The poster did not specify which overtone series he was looking for, so > it may be that's what he wanted. The overtone series for a high string > on a piano is somewhat different from that of a low string; 'Struth. That's why they have to "stretch" the octaves when they tune an accoustical piano. But there is a theoretical overtone series, which does not take physical non-linearities into account. BTW, didja know that clarinets don't produce even numbered overtones? That's why their register key pops them up an octave and a fifth rather than an octave like on the saxophone, and why they can sound, well, nasal. > I thought it was a very strange request. Maybe he is actually asking for > all the models of a series of equipment manufactured by a small company > called "OverTone"...?????? Thinking back on it, I suspect he wants to know what notes in the well-tempered scale come closest to the notes in (ahem) the overtone series. If so, he wanted something like this: 1 2 3 4 5 6 7(*) 8 9 10 11(**) C C' G' C'' E'' G'' ?? C''' D''' E''' ?? The question-mark notes don't come close to anything in the well-tempered scale. (*) Bb'' comes closest but is very badly out of tune. Sharp by 31 cents. (**) This overtone is almost exactly midway between F''' and Gb'''. From: hsu@csrd.uiuc.edu (William Tsun-Yuk Hsu) Subject: Mark Levine on salsa Date: 18 Feb 91 18:38:44 GMT Some quick notes on salsa, adapted from Mark Levine's _The Jazz Piano Book_. Disclaimer: I've done some listening but this is my only source for the technical side of salsa. I'm sure I'll get lots of corrections etc... The rhythmic basis for salsa is the clave. The forward clave (or 3 & 2): 1&2&3&4&|1&2&3&4&|| x x x x x The reverse clave (or 2 & 3): 1&2&3&4&|1&2&3&4&|| x x x x x The African or "Rumba" clave: 1&2&3&4&|1&2&3&4&|| x x x x x (I don't know if there's a reverse rumba clave.) The chapter is pretty brief on rhythmic patterns played by instruments other than the piano. A typical 2&3 pattern called cascara for drums or cymbals: 1&2&3&4&|1&2&3&4&|| x x xx x x xx x x A bass pattern (tumbao): 1&2&3&4&|1&2&3&4&|1&2&3&4&|1&2&3&4&| x x x x x x x x x etc. Salsa pianists usually play repeated figures called montunos or guajeos. The chapter gives a number of examples (for typical harmonic progressions), but I haven't been able to figure out exactly what the rhythmic rules are, and what you can change. I guess: 1) the montuno should work with the chosen clave and 2) large chunks of the pattern should be off the beat, to give that neat syncopated effect. A basic pattern (these can be 4 or 8 bars long too): 1&2&3&4&|1&2&3&4&|| x xx x x x x x x More examples: 1&2&3&4&|1&2&3&4&|| x xxx xx xxxx xx 1&2&3&4&|1&2&3&4&|| x xx xx xxxxxxx Solos are mostly rhythmic (since the harmony is usually simple), lots of triplets, syncopations. I've heard some wild vocal patterns with repeating groups of 7 and other great tricks. I'm sure you salsa experts are laughing to death by now, so how about some corrections/comments. Jones? Eric Majani? Bill From djh@neuromancer (Dallas J. Hodgson) Thu Sep 5 15:30:14 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["2775" "" "5" "September" "91" "00:38:55" "GMT" "Dallas J. Hodgson" "djh@neuromancer.metaphor.com " nil "63" "Re: Blues scale question" "^From:" nil nil "9"]) Newsgroups: rec.music.makers Message-ID: <1229@cronos.metaphor.com> References: <1991Sep4.131423.24103@noose.ecn.purdue.edu> Organization: Metaphor Computer Systems, Mountain View, CA Lines: 63 From: djh@neuromancer.metaphor.com (Dallas J. Hodgson) Subject: Re: Blues scale question Date: 5 Sep 91 00:38:55 GMT In article <1991Sep4.131423.24103@noose.ecn.purdue.edu> blairk@gus19.ecn.purdue.edu (Kim B Blair) writes: >Here is a question on music theory. After years of playing, I >am finally taking the time to learn some blues lead. Thus the >motivation for the question. According to a method book, a >standard 12 bar blues pattern in E includes the chords E, A, and B >(or some version of their 7ths and 9ths). Now, consider the >following: > >E major scale: E F# G# A B C# D# E. OK, this is right. >Notes in the chord progression (7ths and 9ths in parenthesis). > >E: E G# B (D F#) >A: A C# E (G B) >B: B D# F# (A C#) E9, A9, B9 - fancy, but OK - see later on >Now, the five note "blues scale" in E. (I have heard this called a >pentatonic scale, but I don't think that is a correct use of >pentatonic.) > >E G A B D E This would be the 5-note basic blues E (minor) pentatonic, as derived from E (natural) Minor: E F# G A B C D E (note similarity). Since it has 5 notes, it qualifies as a pentatonic scale. Compare to major pentatonic: E G# A B D#. Note how the 2nd & 6th major scale degrees get left out. >Questions: > >1. Why do the notes in the scale not conform to the notes in the > chord progression it is played over? >2. Why does it sound ok? Seems as though the half step intervals > would (A-A#, D-D#) sound wrong. >3. What defines the blues scale? i.e (root, flat-third, ...) >4. Related to 3, are there similar definitions for minor keys? The closest fit here would be using all 7th chords (no 9ths) over a major pentatonic scale; you can I7, IV7, V7 and have a perfect match. If you wander outside of this, you're getting subjective. Often blues backing chords are just root-fifth, which work over major or minor pentatonics. Blues soloing is fairly forgiving over what chords are playing underneath, so rely on your ear to decide if something sounds good or not. The original scale grew up around the tonic,subdominant/dominant progression, and it's those chords which fit best. >p.s. Electronic mail is great! You can ask all sorts of questions > and retain your anonymity! Damn straight. -- +----------------------------------------------------------------------------+ | Dallas J. Hodgson | "These days, you have to be pretty | | Metaphor Computer Systems | technical before you can even | | Mountain View, Ca. | aspire to crudeness." | | USENET : djh@metaphor.com | - William Gibson | +============================================================================+ | "The views I express are my own, and not necessarily those of my employer" | +----------------------------------------------------------------------------+ From ljnelson@amherst Mon Sep 23 11:14:29 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] [nil nil nil nil nil nil nil nil nil nil nil nil "^From:" nil nil nil]) Newsgroups: rec.music.makers Message-ID: <13871.28d65c58@amherst.bitnet> References: <13801.28ce3a86@amherst.bitnet> <20509@prometheus.megatest.UUCP> Organization: Amherst College, Amherst, MA. Lines: 130 From: ljnelson@amherst.bitnet Subject: Re: Blues scale question (Long) Date: 17 Sep 91 19:49:44 GMT In article <20509@prometheus.megatest.UUCP>, djones@megatest.UUCP (Dave Jones) writes: > ljn>> ... In numeric terms, the Chicago blues scale (at least that's what I've >> heard it called...) is 1, b3, 4, b5, 5, b7. > dj> That has pretty much won the battle for the title "blues scale". Mode II of > it is called the "major blues scale", and contains both the minor and major > third: > > 1 2 b3 3 5 6 > > It is a very good scale for blues, but unlike the pure "blues scale" it > can not be used, ad nauseum, over the whole I-IV-I-V-I blues form. You have > to change when the chords change. (I consider that a plus.) > This is quite true, and I thank you for putting it into words successfully. The following excerpt... >> Over dominant chords, I >> use just about every note in some way [notes in parentheses are those which >> sound ugly if repeatedly hammered upon :)]: >> 1, 2, b3, 3, 4, b5, 5, (6) [IMHO], b7, (7) [passing ONLY!]. ..which I wrote in response to the original question, illustrates the way that I learned blues by ear--I combined the Chicago scale with the "major blues" scale into one big scale that I alter to fit the chord at hand. If you look at the scale immediately above, (1, 2, b3, 3, 4 etc.) you will see that the Chicago scale is merged in with the major blues scale. This works depending on whether the blues that you're playing over has a major or a minor "feel" to it--don't know how to describe this more technically. Your "major blues scale" on its own sounds pretty dixielandish [:)], and over older style tunes works very well. The other thing that I guess is important at this juncture is that I kind of lied about what I played earlier. The original question to which I was responding concerned the very fundamentals of blues, asked by what I guessed was a guitar or bass player who (presumably) had not had much experience in the area. I definitely agree with you that the "straight" blues scale is boring on its own if it is not varied. Hence the reason that I blend it with the major blues scale to yield a very rich plethora of notes (excuse the corny prose). Understood, of course, is that not every note in this scale is going to work over every chord; hence I tried to give the addressee a sense of which were the not-so-strong or not-guitar-oriented notes (particularly over the tonic chord). Oh, BTW when I referred to "dominant chords" in my previous posting, I meant, as you rightly corrected, dominant-7 chords that are used as the tonic--e.g. for an F blues progression, F7 is the tonic, obviously, but is also a dominant seventh chord; hence my lazy term "dominant chord". I hope that didn't generate too much confusion. > Hmmmmm. I play the major 6 over dominant seven chords more often > that I play the minor 7. I think it is one of the best, sweetest notes around. Well, 1/2 agreement. It all depends on the style of the blues being played (I'm limiting myself rather simplistically to two styles to prove a point: electric [B.B. King, for example]/contemporary, and old-school [Acoustic stuff >from 50's and earlier]). In a contemporary setting--let's stay with the key of F just for consistency :)--your root/tonic chord is going to be F7 in some voicing or another, and an Eb is going to be very significant in defining the quality of the chord. If you hammer on that note (the b7--and I don't *really* mean hammer; anything sounds bad if hammered), you're not only emphasizing the quality and the color of that chord, but you're setting up tension that wants to resolve. Since your western listener is allegedly versed in the blues tradition, s/he is going to expect the IV 7 chord for the next change, which means that the expected resolution from the Eb (b7 in the F scale) is to the D (3 in the Bb scale). Now the 6 of the F scale, a D, while a pretty note, I don't think is as inherently "bluesy" as the b7, if for no other reason than it doesn't really want to resolve anywhere (except, possibly, to the b7 of the same scale--and I certainly am not arguing its validity as a passing note [I assume that's what those are called :)]). Again, please keep in mind that the original scope of the original posting to which I replied was limited to two basic styles. Miles Davis' _All_Blues_ for example is definitely a blues, but would people define it as basic? I refer to that tune to point out that in modal styles such as this the 6 is indispensable. But then in a modal style, the focus is on each individual scale played over each individual chord, as opposed to one or more modifications on a stock "blues" scale that fit over the entire chorus. Hence the 6 in *this* style is perfect because it doesn't *need* to resolve anywhere but to notes within "its" scale. That is, it doesn't advance the overall progression to the IV (or whatever) because that isn't the focus. Summary: The b7 is better in "basic" contemporary blues than the 6 (IMHO), but the 6 has its indispensable place in modally influenced blues. > Particularly when the so-called "dominant" chord is really functioning as > the tonic. That mistake is documented somewhere above. I got lazy; whenever I referred to a "dominant" chord I meant merely a dominant 7 on any degree of the tonic scale. Hence, using my lazy (and incorrect) terminology, I meant to refer to a particular style of blues, where each degree in the I-IV-V progression is a dominant 7--as opposed, for example, to a style of blues where each degree of the progression is a *minor* 7. Whatever. It was wrong, and I think I confused too many people. :) > Many jazz books list the 4 as an "avoid note" over the dominant seven chord. Yeah, and I disagree entirely. I've been taught that in jazz, anyway, one of the most common scales used over a dominant 7 chord (there; got the terminology right :)) is a lydian b7 (the lydian, of course, raises the 4). But, if used correctly, the 4 *and* the #4 are both great as tension-setters. I know; I know--in classical theory, the 4 is one of the most unstable notes in the scale. But I guess that's the point. If you hang on that note for a while, or hit it on quarter notes for the first bar, and then (somehow--via a riff, for example) resolve to the b3 of your tonic scale, you have duplicated one of the most commonly used blues licks of all time. > I will say that the four does seem to do better in dominant seven chords that > serve as the tonic than it does in dom-7 chords that lead to a chord a > fourth up. Huh? Aren't these cases the same thing? Your first change in a blues prog. is >from the I7 to the IV7. Isn't that a case of a dominant seventh chord "leading to a chord a fourth up"? > Different strokes, I guess. One man's blue-note is another man's avoid-note. Assuredly, and I'm not discounting this view. THis is just fun, and we can all learn a little something. :) Thanks for your interest and corrections. I've played blues pretty consistently for at least 11 years, and only now, as a sophomore in college, am I in a course that's teaching me what I've been doing all along. Hence the flaws in the terms. LAiRD -- ljnelson@amherst.bitnet I am the Keymaster. From ljnelson@amherst Mon Nov 18 12:10:27 1991 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["4110" "" "17" "November" "91" "17:12:44" "GMT" "ljnelson@amherst.edu" "ljnelson@amherst.edu" nil "82" "Re: Modal composition in pop/rock" "^From:" nil nil "11"]) Newsgroups: rec.music.synth Message-ID: <16076.2926a30c@amherst.edu> References: <1991Nov15.174212.11441@rice.edu> <1991Nov16.120851.4355@crl.dec.com> Organization: Amherst College, Amherst, MA. Lines: 82 From: ljnelson@amherst.edu Subject: Re: Modal composition in pop/rock Date: 17 Nov 91 17:12:44 GMT In article <1991Nov16.120851.4355@crl.dec.com>, rwk@crl.dec.com (Bob Kerns) writes: > all, it's EASIER. Just use the same notes, but pick a different > note of the scale as the root. So if you're playing in C major, > and want a wistful interlude, just switch your tune up a note > in the C scale, and you've got yourself C Dorian (well, I remember > that one, anyway!). Play your interlude, and then switch back. > Or for a subtle variation, switch up to the fourth or fifth. Close; that gives you D dorian. C dorian is C D Eb F G A Bb C. D dorian is D E F G A B C D. So really if you shift your piece up the way you say then you ARE switching keys AND modes. Remember that a mode is named after the pitch it starts on, and not the scale that it was derived from. > "Scarborough Fair" is a familar example in Dorian mode. But You can actually play in the dorian mode (the CORRECT dorian mode--see above) over any minor or minor seventh chord. So over a C minor seventh you would play C dorian; over a D minor seventh, D dorian and so on. This should be an automatic response because it doesn't matter how that minor chord is used; you can always play the dorian of the key that the chord is in over it. > But the BEST thing about playing in alternate modes, is that, > because the sonic space is different, you'll find yourself being > tempted into new and different territory. You'll find yourself > playing with a wider range of contrasts, and putting new meanings > and interpretations on old structures, phrases, riffs, etc. In > short, playing modally is the best cure for a stale imagination. Okay, so the following example is jazz, but where do you think rock came from? Check out Miles Davis' _Kind_of_Blue_ (1958, Columbia/CBS) for the most famous and best-implemented examples of modal improvisation. All pieces on the album are modal, and extended quite a bit, to boot. > One bit of practical advice: All of the modes are easy, except > for any in which the interval between the first and fifth note > is a tritone. (For example, if you're playing with an unaltered > C major scale, the one which begins on B has this characteristic.) B Locrian, if anyone's interested. You would play this over a B-7 (b5) chord, which are not really common in rock. Nevertheless, they are found. > It *IS* possible to do meaningful music in these modes, but it Miles calleth. :) > definitely a challenge, and I'd be hardpressed to produce > approachable, commercial, mainstream-type stuff. You'll find > the one that starts on the third note (E in our C-major-scale That would be E Phrygian (yecch) for 200 points, Bob. :) Note that you would actually NEVER play this over a C major chord unless you wanted to send small furry animals scurrying for cover. You have to keep in mind that "a mode is named after the pitch it actually starts *on* and not the scale *from* which it was diatonically derived." (Jaffe, Andrew, _Jazz_Theory_, p. 6). This is where Bob is leading you astray unintentionally. You would play E phrygian over some chord in the key of *E* and *not* in the key of C. Confidentially, I have no idea what you would play phrygian over...I know it's used in "Flamenco Sketches" on _Kind_of_Blue_, but I'll be damned if I know what Bill Evans is playing. Intuition tells me that if you saw a possible application for it, it would definitely *not* be in rock. > example) takes a bit of getting used to as well, because it has To say the least. > a half-step above the root and a whole step below, instead of > the more familiar reverse situation, but playing with it a bit > will get you used to it. > > Go for it, dude! Can ya dig it, man. LAiRD -- _______________________________________________________________________________ LAiRD j. NeLSoN '94 | What key is this in? What key is it in? (Kenny Kirkland) ljnelson@amherst.edu | Don't ask me; I'm just improvising. (Neil Peart) (413) 542-3374 | When the red light's on, everybody gon' be quiet. (Miles) ------------------------------------------------------------------------------- From andy@oda (Andy Spiceley) Mon Jan 20 16:43:58 1992 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["7645" "" "16" "January" "92" "14:10:27" "GMT" "Andy Spiceley" "andy@oda.icl.co.uk " nil "142" "Microtonality forum report" "^From:" nil nil "1"]) Newsgroups: rec.music.synth Message-ID: <9201161410.AA11342@oda.icl.co.uk> Lines: 142 From: andy@oda.icl.co.uk (Andy Spiceley) Subject: Microtonality forum report Date: 16 Jan 92 14:10:27 GMT Newsgroups: rec.music.synth, rec.music.classical Subject: report on SPNM Microtonality forum (LONG) Keywords: alternative tunings, quartertones, gamuts, temperament Date: 16th January 1992 From: andy@oda.icl.co.uk (Andy Spiceley) Lines: 131 SPNM Composers Forum on Microtonality: report Brookmans Park, Hertfordshire, UK, 16-19 December 1991 Each year the Society for the Promotion of New Music holds a composers forum. This year the topic was Microtonality, at the instigation of the percussionist and composer James Wood. The forum included lectures, instrumental demonstrations, discussions, open rehearsals, and culminated in a concert of ten specially written microtonal works, played by the Guildhall New Music Ensemble. (The Guildhall is one of London's music colleges or conservatoires, and the ensemble is formed from the best students. James Wood directs the ensemble and has worked with the students in developing facility in microtonal playing). In all some 60-70 people took part, including 40 composers, together with instrumentalists, theorists, and - believe it or not - some disinterested (in the neutral sense) concert-goers. So, what is microtonality - or rather, what aspects of microtonality were considered? Well, the forum encompassed consideration of other divisions of the semitone, and of the octave: unconventional (in Western music terms) scales and modes, use of the "natural" intervals of the overtone series (in progressively smaller intervals) and almost any other alternative to the 12-note tempered scale that has become so ubiquitous and which holds Western music in its straitjacket. In an historical survey, we reviewed -amongst many others - the work of Huygens, the Dutch mathematician, astronomer etc. who devised a 31-note equal tempered scale back in the late 17th century, giving rise only in this century (thanks to Fokker) to the building of a 31 note-octave organ, and a school of Dutch composers working in the 31-note scale. (And an enterprising series of publications from the 'Diapason Press' which has reprinted several key works on tuning systems and temperament by Werckmeister, Bosanquet, Huygens & others.) The composers Jonathan Harvey, George Benjamin, Michael Finnissy, Justin Connolly, Nicola Lefanu and James Wood gave talks about their own music and about other music which exploited microtones: from 15th and 16th century European music, ethnic musics of Indonesia, Africa, through Ives, Haba and Harry Partch, to Tristan Murail. Harvey and Benjamin and Murail have all worked at IRCAM, and exploited the facilities there for generating sounds using tunings derived from the natural harmonic series, in combination with timbres derived from acoustic instruments (Harvey's 'Vivos Voco, Mortuos Plango', and Benjamin's 'Antara', for example). In instrumental workshops, eminent intrumentalists demonstrated the ease (or otherwise) with which existing wind instruments could play on a quartertone scale. (I was impressed with specially designed alto and bass flutes built for Kate Lukas by a Dutch maker which, by ingenious provision of "open-hole" mechanism, permit quartertones.) Also present at the forum were a quarter-tone metallaphone and quarter-tone glockenspiel, instruments specially built for the newly-formed Centre for Microtonal Music. These were exploited in several of the works written by composers specially for the forum, rehearsed in a few extremely concentrated open rehearsals, and performed at the end of the forum. For rec.music.synth: as well as the emphasis on playing techniques and new acoustic instruments, special attention was paid to the role of electronics. Yamaha loaned, free of charge, an SY77 and two TG77 modules, which have extensive alternative tuning possibilities: the tuning potential of the Proteus and DX11 were also demonstrated, though the only instruments heard in the concert were the TG77s (operated through two KX88s to give 7 octaves of quarter-tones), and two samplers: uncompleted fragments for two violins (using very specialised intervals involving small fractions of a tone) by Patrick Ozzard-Low were realised - very convincingly - on a Roland S750; and my own work used the EPS16+ to produce the effect of Steinways with 10 and 16 note tempered scales, and quarter-tone string pads, in conjunction with live instruments. I was quite taken with the SY77 tuning capabilities. As on earlier Yamaha instruments, several temperaments are built in, including quarter and eighthtones: I particularly liked the effect of the natural harmonic series laid out on adjacent white keys, which seems to be new for the SY77. We did not spend a lot of time comparing the potential of the various approaches of the different manufacturers, but it is worth noting that not all machines permit (as does the EPS) the use of several different tuning systems simultaneously for different voices. (Indeed the EPS permits eight different tables in the eight layers of each voice...) There also is a tremendous difference in the ease with which different tunings may be set up (and saved and edited) on the various synths. There is also some variation in the accuracy of the resulting tunings: I suspect, though have not investigated, that the EPS is not quite accurate enough, when using the "extrapolate pitch table" command. (I would recommend Scott Wilkinson's book "Tuning In" as an excellent introduction to the history of alternative tuning systems, and a useful, though now rather dated, survey of microtuning capabilities of commercial synthesisers.) Some interesting psychological effects were demonstrated by tuning the SY77 so that the black keys played an equal-tempered 5 note scale. When well known pentatonic tunes were played, most of us found that our brains tried hard to hear the pitches as "just out of tune" normal pentatonic scales, even when the equal-temperament was demonstrated by simply playing the same tune upone degree of the scale! I also found that the timbre of a normal piano seems altered when heard in different tunings - something to do with the disturbed natural resonance of the strings perhaps? In summary, I was struck by the many aspects of music which were implicated in what may seem like an esoteric interest: instrument design (both acoustic and electronic), instrumental technique, compositional technique, psychoacoustics, acoustics, aesthetics and musicology were all significant areas of exploration. Some (such as psychoacoustics) were only briefly touched on & need much more time and study: others, such as instrumental technique, received - necessarily - more in-depth consideration. I believe we all came away stimulated, excited, and with a buzz of new ideas to work on, and having made useful new contacts with other microtonal enthusiasts. As a follow-up, there will be a weekend festival (March 7-9) at the Barbican in London, which will include performances of works by Harvey, Murail, Scelsi, Xenakis and others, Harry Partch films, open discussions and more. And of course there will be more works written, by those composers present and others, to exploit the potential of the new instruments, and the skills of the Guildhall students and their peers and mentors. [If anyone would like more information on any aspect of the Forum, the SPNM, or any of the works or publications I mentioned, please email me: andy@oda.icl.co.uk: I picked up details of a number of publications and contacts which I would be happy to pass on, swap etc!] END Regards, Andy Spiceley ICL Bracknell email andy@oda.icl.co.uk phone +44 344 424842 x2616 From midkiff@public (Neil E. Midkiff midkiff@btr.com) Mon Jun 15 12:47:01 1992 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["32863" "" "13" "June" "92" "04:30:56" "GMT" "Neil E. Midkiff midkiff@btr.com" "midkiff@public.BTR.COM " nil "629" "Tuning and Temperament, was Re: tuning information needed" "^From:" nil nil "6"]) Newsgroups: rec.music.early Summary: Lots of detailed information on tuning and temperament Keywords: meantone tuning Message-ID: <7016@public.BTR.COM> References: <1066@zogwarg.etl.army.mil> Organization: BTR Public Access UNIX, MtnView CA. Contact: Customer Service cs@BTR.COM Lines: 629 From: midkiff@public.BTR.COM (Neil E. Midkiff midkiff@btr.com) Subject: Tuning and Temperament, was Re: tuning information needed Date: 13 Jun 92 04:30:56 GMT In article <1066@zogwarg.etl.army.mil> mike@zogwarg.etl.army.mil (Mike McDonnell) writes: >I would like to have the details needed to set a meantone bearing on >my harpsichord. What I specifically need is the intervals for both >quarter-comma and sixth-comma meantone tuning. > >I can accept information in any form such as Hz or cents difference >from equal temperament. The tuner I use takes its tables in the form >of cents deviation from equal temperament, so that is the best form >for me to get the information in. The most difficult information to >work with is "beats", though I think I can convert that to cents. Fortunately, I've been gathering just that information! Your article prompted me to prepare it for posting. Here it is. ------------------------------------------------------------------------- TUNING AND TEMPERAMENT - An Overview with Some Real Numbers Neil Midkiff (midkiff@btr.com) 6/11/92 This document is intended for musicians who wish to understand some of the mathematical details involved in commonly-encountered systems of tuning keyboard instruments. It does not purport to give a complete historical picture of temperament, nor does it replace tuning recipes based on beats. Non-Western tunings and tunings with more than 12 notes per octave are also specifically excluded from consideration. This information should be especially useful with electronic tuning aids which are calibrated in cents of deviation from the equal-tempered scale. With some additional information it can be used to generate beat charts for tuning by ear, or to compute the data for microtuning of synthesizers such as the Yamaha TG77/SY77 and -99 which allow the user to specify individual note pitches. (Contact me at the address above if more information is desired.) As always, I'd appreciate comments and corrections if any of this is unclear or inaccurate. I'm sure that others will be able to help flesh out the bits of historical data I include. There may be a little repetition since this was assembled from a set of individual documents created from Microsoft Excel spreadsheets. (Spreadsheets turn out to be useful "calculators" for this sort of work!) I hope, however, that others will find this as useful as I would have found it several months ago before I began tracking all this down to put it into one place. --------------------------------------------------------------------------- Let's start with a few definitions. Engineers and scientists deal with musical tones in terms of frequency, with the units of hertz (Hz), equivalent to vibrations per second. Musicians also have agreed (well, mostly) on a standard frequency of 440 Hz for the A above middle C. But because our sense of pitch is related to the logarithm of frequency, it's easier to deal with these logarithmic numbers in discussing tuning. This allows us to add and subtract intervals, rather than multiplying and dividing ratios of frequencies, and has the advantage that musical intervals like octaves have the same size in pitch units, no matter where they are on the keyboard (how high or low the pitch, that is). Musicians usually talk about tuning in units of cents, or hundredths of an equal-temperament semitone. An octave is a 2:1 frequency ratio, and is divided in equal temperament into 12 semitones, or 1200 cents. Cents are computed by taking 1200 times the logarithm (base 2) of the frequency ratio. EQUAL TEMPERAMENT Equal temperament, the system used today for keyboard instruments, makes all semitones equal in size; the frequency ratio is the twelfth root of 2 (about 1.0594631), or 100 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 523.25 1200.00 0.00 B3 493.88 1100.00 0.00 A#3 466.16 1000.00 0.00 A3 440.00 900.00 0.00 G#3 415.30 800.00 0.00 G3 392.00 700.00 0.00 F#3 369.99 600.00 0.00 F3 349.23 500.00 0.00 Note: Because I originally computed these E3 329.63 400.00 0.00 tables for use on a MIDI synthesizer, the D#3 311.13 300.00 0.00 note names use the Yamaha octave numbers D3 293.66 200.00 0.00 so that C3 equals middle C, and C4 is one C#3 277.18 100.00 0.00 octave above it. C3 261.63 0.00 0.00 The "problem" with equal temperament is that it is only an approximation of the ideal intervals our ears hear as pure. We tend to hear consonances between frequencies whose ratios are equal to the ratio of small integers. Thus the octave is 2:1, the fifth is 3:2, the fourth is 4:3, the major third is 5:4, the minor third is 6:5 and so on. The trouble is that these intervals can't be combined in ways that "come out even" with the scales we see on our keyboards. The Pythagorean Comma To take the simplest example, start at the lowest C on the piano and go upward by fifths to G, D, A, and so forth. When you come back to C after twelve fifths, you'll have covered the seven complete octaves on the keyboard. But this is only possible because we've agreed to use equally- tempered intervals on the piano, where a fifth is 700 cents, or 7/12 of an octave. A pure 3:2 fifth turns out to be 1200*log2(3/2), or 701.96 cents. That is, we make fifths on the piano about 2 cents flatter (narrower) than pure, so that twelve of them equal seven octaves. The total discrepancy if we used pure fifths would be (3/2)**12/2**7, or about 23.46 cents. This interval is called the Pythagorean comma, and the amount by which equal- tempered fifths are flattened is one-twelfth of it. The following table of intervals may look a little redundant since all the lines are the same. But I'm using it here to prepare for the discussion of other tunings. The note names at the left refer to the lower note of an interval. So the first entry means that the minor second from B3 up to C4 is 100 cents, which is 4.96 cents flat with respect to a 17:16 frequency ratio. (This is one of several choices of integer ratios for the minor second, but I'll use it throughout for consistency). minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 100.00 -4.96 200.00 -3.91 300.00 -15.64 A#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 A3 100.00 -4.96 200.00 -3.91 300.00 -15.64 G#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 G3 100.00 -4.96 200.00 -3.91 300.00 -15.64 F#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 F3 100.00 -4.96 200.00 -3.91 300.00 -15.64 E3 100.00 -4.96 200.00 -3.91 300.00 -15.64 D#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 D3 100.00 -4.96 200.00 -3.91 300.00 -15.64 C#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 C3 100.00 -4.96 200.00 -3.91 300.00 -15.64 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 400.00 13.69 500.00 1.96 700.00 -1.96 A#3 400.00 13.69 500.00 1.96 700.00 -1.96 A3 400.00 13.69 500.00 1.96 700.00 -1.96 G#3 400.00 13.69 500.00 1.96 700.00 -1.96 G3 400.00 13.69 500.00 1.96 700.00 -1.96 F#3 400.00 13.69 500.00 1.96 700.00 -1.96 F3 400.00 13.69 500.00 1.96 700.00 -1.96 E3 400.00 13.69 500.00 1.96 700.00 -1.96 D#3 400.00 13.69 500.00 1.96 700.00 -1.96 D3 400.00 13.69 500.00 1.96 700.00 -1.96 C#3 400.00 13.69 500.00 1.96 700.00 -1.96 C3 400.00 13.69 500.00 1.96 700.00 -1.96 As the table makes evident, fourths and fifths are not badly compromised by equal temperament, but major and minor thirds are much farther out of pure tuning. We have learned to accept these slightly dissonant thirds as the price we have to pay for a tuning system which allows modulating into different keys without encountering even worse out-of-tuneness. Historically, other tradeoffs have been made in an attempt to have greater consonance in the key of C and those "near" it (with few sharps or flats in the key signature), at the expense of unusable intervals in remote keys. The so-called "wolf" interval is found between G# and Eb in the tunings described below, though another place around the circle of fifths could equally well be chosen. The Syntonic Comma Before we describe some of the tunings, we need to define another kind of comma, the syntonic comma. It's most quickly defined in the key of C as the difference between the major 2nd between F and G and the major 2nd between Eb and F. That is, it's (fifth - fourth) minus (fourth - minor 3rd). In frequency terms it's ((3/2)/(4/3)) divided by ((4/3)/(6/5)), or 9/8 divided by 10/9. And this ratio is the key to its real definition, and to its importance. In the harmonic series based on the C three octaves below middle C (that is, C0 on my synthesizer), C1 (one octave up) is twice the fundamental frequency, G1 is close to three times the fundamental, C2 is four times, E2 is close to five times, G2 is close to 6 times, A#2 is about seven times, C3 is eight times, D3 is close to 9 times, and E3 is close to 10 times. If we tuned pure intervals based on C, then we'd make all the "close to" intervals exact. Then the major second from C3 to D3 would be 9:8, and the major second from D3 to E3 would be 10:9, and the difference is once again the syntonic comma. 9/8 divided by 10/9 is 81/80; in pitch this works out to about 21.51 cents. MEANTONE TUNINGS The various varieties of meantone tunings are attempts to average out the differences in these major seconds. In a quarter-comma meantone, all but one of the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-fourth of the syntonic comma, or 5.38 cents. So fifths are tuned to 696.58 cents, except for the wolf which must be seven octaves minus eleven flattened fifths, or 737.64 cents. Here the wolf is shown inverted as the fourth of 462.36 cents, between G# and Eb. All other fourths are 503.42 cents. Hz cents adjustments from A=440 equal temperament (cents) C4 526.36 1200.00 10.26 B3 491.93 1082.89 -6.84 A#3 470.79 1006.84 17.11 A3 440.00 889.74 0.00 G#3 411.22 772.63 -17.11 G3 393.55 696.58 6.84 F#3 367.81 579.47 -10.26 F3 352.00 503.42 13.69 E3 328.98 386.31 -3.42 D#3 314.84 310.26 20.53 D3 294.25 193.16 3.42 C#3 275.00 76.05 -13.69 C3 263.18 0.00 10.26 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 117.11 12.15 193.16 -10.75 310.26 -5.38 A#3 76.05 -28.91 193.16 -10.75 269.21 -46.43 A3 117.11 12.15 193.16 -10.75 310.26 -5.38 G#3 117.11 12.15 234.22 30.31 310.26 -5.38 G3 76.05 -28.91 193.16 -10.75 310.26 -5.38 F#3 117.11 12.15 193.16 -10.75 310.26 -5.38 F3 76.05 -28.91 193.16 -10.75 269.21 -46.43 E3 117.11 12.15 193.16 -10.75 310.26 -5.38 D#3 76.05 -28.91 193.16 -10.75 269.21 -46.43 D3 117.11 12.15 193.16 -10.75 310.26 -5.38 C#3 117.11 12.15 234.22 30.31 310.26 -5.38 C3 76.05 -28.91 193.16 -10.75 310.26 -5.38 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 427.37 41.06 503.42 5.38 696.58 -5.38 A#3 386.31 0.00 503.42 5.38 696.58 -5.38 A3 386.31 0.00 503.42 5.38 696.58 -5.38 G#3 427.37 41.06 503.42 5.38 737.64 35.68 G3 386.31 0.00 503.42 5.38 696.58 -5.38 F#3 427.37 41.06 503.42 5.38 696.58 -5.38 F3 386.31 0.00 503.42 5.38 696.58 -5.38 E3 386.31 0.00 503.42 5.38 696.58 -5.38 D#3 386.31 0.00 462.36 -35.68 696.58 -5.38 D3 386.31 0.00 503.42 5.38 696.58 -5.38 C#3 427.37 41.06 503.42 5.38 696.58 -5.38 C3 386.31 0.00 503.42 5.38 696.58 -5.38 In summary: Quarter-comma meantone gives pure major thirds in most cases (except for four wide ones) and evens out the major seconds (except for two wide ones from C# to D# and G# to A#) as half the pure major third. This is accomplished at the expense of minor seconds of two widely different sizes, a very sour "wolf" fifth, and even worse mistunings for the wide major thirds and three narrow minor thirds. Usually when "meantone" is mentioned without further specifics, this is the variety that is meant. In a fifth-comma meantone, the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-fifth of the syntonic comma, or 4.30 cents. So fifths are tuned to 697.65 cents, except for the wolf which must be 725.81 cents. Here the wolf is shown inverted as the fourth of 474.19 cents, between G# and Eb. All other fourths are 502.35 cents. Hz cents adjustments from A=440 equal temperament (cents) C4 525.38 1200.00 7.04 B3 492.55 1088.27 -4.69 A#3 469.33 1004.69 11.73 A3 440.00 892.96 0.00 G#3 412.50 781.23 -11.73 G3 393.06 697.65 4.69 F#3 368.49 585.92 -7.04 F3 351.13 502.35 9.39 E3 329.18 390.61 -2.35 D#3 313.67 307.04 14.08 D3 294.06 195.31 2.35 C#3 275.68 83.58 -9.39 C3 262.69 0.00 7.04 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 111.73 6.78 195.31 -8.60 307.04 -8.60 A#3 83.58 -21.38 195.31 -8.60 278.88 -36.76 A3 111.73 6.78 195.31 -8.60 307.04 -8.60 G#3 111.73 6.78 223.46 19.55 307.04 -8.60 G3 83.58 -21.38 195.31 -8.60 307.04 -8.60 F#3 111.73 6.78 195.31 -8.60 307.04 -8.60 F3 83.58 -21.38 195.31 -8.60 278.88 -36.76 E3 111.73 6.78 195.31 -8.60 307.04 -8.60 D#3 83.58 -21.38 195.31 -8.60 278.88 -36.76 D3 111.73 6.78 195.31 -8.60 307.04 -8.60 C#3 111.73 6.78 223.46 19.55 307.04 -8.60 C3 83.58 -21.38 195.31 -8.60 307.04 -8.60 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 418.77 32.46 502.35 4.30 697.65 -4.30 A#3 390.61 4.30 502.35 4.30 697.65 -4.30 A3 390.61 4.30 502.35 4.30 697.65 -4.30 G#3 418.77 32.46 502.35 4.30 725.81 23.85 G3 390.61 4.30 502.35 4.30 697.65 -4.30 F#3 418.77 32.46 502.35 4.30 697.65 -4.30 F3 390.61 4.30 502.35 4.30 697.65 -4.30 E3 390.61 4.30 502.35 4.30 697.65 -4.30 D#3 390.61 4.30 474.19 -23.85 697.65 -4.30 D3 390.61 4.30 502.35 4.30 697.65 -4.30 C#3 418.77 32.46 502.35 4.30 697.65 -4.30 C3 390.61 4.30 502.35 4.30 697.65 -4.30 Once again, we have achieved major seconds that are half the size of the major thirds in most cases, so this is also a meantone tuning. The difference is that we've allowed the "nice" major thirds to expand from pure to one-fifth comma wider than pure. This reduces most of the other deviations from pure intervals; only the "nice" minor thirds are a little farther from pure than in quarter-comma. The wolves are howling less loudly! In a sixth-comma meantone, the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-sixth of the syntonic comma, or 3.58 cents. So fifths are tuned to 698.37 cents, except for the wolf which must be 717.92 cents. Here the wolf is shown inverted as the fourth of 482.08 cents between G# and Eb. All other fourths are 501.63 cents. Hz cents adjustments from A=440 equal temperament (cents) C4 524.73 1200.00 4.89 B3 492.95 1091.85 -3.26 A#3 468.36 1003.26 8.15 A3 440.00 895.11 0.00 G#3 413.35 786.96 -8.15 G3 392.73 698.37 3.26 F#3 368.95 590.22 -4.89 F3 350.55 501.63 6.52 E3 329.32 393.48 -1.63 D#3 312.89 304.89 9.78 D3 293.94 196.74 1.63 C#3 276.14 88.59 -6.52 C3 262.37 0.00 4.89 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 108.15 3.19 196.74 -7.17 304.89 -10.75 A#3 88.59 -16.36 196.74 -7.17 285.34 -30.30 A3 108.15 3.19 196.74 -7.17 304.89 -10.75 G#3 108.15 3.19 216.29 12.38 304.89 -10.75 G3 88.59 -16.36 196.74 -7.17 304.89 -10.75 F#3 108.15 3.19 196.74 -7.17 304.89 -10.75 F3 88.59 -16.36 196.74 -7.17 285.34 -30.30 E3 108.15 3.19 196.74 -7.17 304.89 -10.75 D#3 88.59 -16.36 196.74 -7.17 285.34 -30.30 D3 108.15 3.19 196.74 -7.17 304.89 -10.75 C#3 108.15 3.19 216.29 12.38 304.89 -10.75 C3 88.59 -16.36 196.74 -7.17 304.89 -10.75 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 413.04 26.73 501.63 3.58 698.37 -3.58 A#3 393.48 7.17 501.63 3.58 698.37 -3.58 A3 393.48 7.17 501.63 3.58 698.37 -3.58 G#3 413.04 26.73 501.63 3.58 717.92 15.97 G3 393.48 7.17 501.63 3.58 698.37 -3.58 F#3 413.04 26.73 501.63 3.58 698.37 -3.58 F3 393.48 7.17 501.63 3.58 698.37 -3.58 E3 393.48 7.17 501.63 3.58 698.37 -3.58 D#3 393.48 7.17 482.08 -15.97 698.37 -3.58 D3 393.48 7.17 501.63 3.58 698.37 -3.58 C#3 413.04 26.73 501.63 3.58 698.37 -3.58 C3 393.48 7.17 501.63 3.58 698.37 -3.58 By this point, I think the trend should be clear. The wolves are almost tamed; the major thirds aren't too wide, but they're getting farther off, and the minor thirds are going flat. The logical extension of this trend is a one-twelfth (Pythagorean, not syntonic) comma meantone, which is precisely equal temperament. WELL-TEMPERED TUNINGS Other systems of tuning, which get lumped under the heading of "well- tempered" tunings, don't follow the meantone pattern of having a single wolf fifth. Instead, they distribute the discrepancies more-or-less evenly through some of the remote-from-C-major intervals. In this way, they attempt to preserve the consonances of purer intervals in the usual keys, while smoothing over the consequences of modulating into remote keys. All keys are usable, but the sizes of the intervals between various degrees of the scale are not exactly the same in each key. This is often thought of as an advantage, since it lends a distinctive character to the different keys. The tuning Bach had in mind for Das Wohltemperierte Klavier was almost certainly *not* equal temperament (though it had been invented prior to Bach's time) but one of the many forms of well-tempered tunings. Werckmeister III A well-tempered tuning adopted for organs in the time of Bach is known as Werckmeister III. In this tuning, the fifths C-G-D-A and B-F# are each tempered by 1/4 of the Pythagorean comma, or 5.87 cents. Pure fifths of 3:2 are 701.96 cents; the tempered ones are then 696.09 cents. Pure fourths of 4:3 are 498.04 cents; the tempered ones are then 503.91 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 526.81 1200.00 11.73 B3 495.00 1092.18 3.91 A#3 468.27 996.09 7.82 A3 440.00 888.27 0.00 G#3 416.24 792.18 3.91 G3 393.77 696.09 7.82 F#3 369.99 588.27 0.00 F3 351.21 498.04 9.78 E3 330.00 390.22 1.96 D#3 312.18 294.13 5.87 (Yes, it's true: F# and A are the same D3 294.33 192.18 3.91 as in equal temperament, and the others C#3 277.50 90.22 1.96 are all *sharper*. This is a coincidence C3 263.40 0.00 11.73 due to using A=440 as a standard pitch.) minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 107.82 2.86 198.04 -5.87 300.00 -15.64 A#3 96.09 -8.87 203.91 0.00 294.13 -21.51 A3 107.82 2.86 203.91 0.00 311.73 -3.91 G#3 96.09 -8.87 203.91 0.00 300.00 -15.64 G3 96.09 -8.87 192.18 -11.73 300.00 -15.64 F#3 107.82 2.86 203.91 0.00 300.00 -15.64 F3 90.22 -14.73 198.04 -5.87 294.13 -21.51 E3 107.82 2.86 198.04 -5.87 305.87 -9.77 D#3 96.09 -8.87 203.91 0.00 294.13 -21.51 D3 101.96 -3.00 198.04 -5.87 305.87 -9.77 C#3 101.96 -3.00 203.91 0.00 300.00 -15.64 C3 90.22 -14.73 192.18 -11.73 294.13 -21.51 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 401.96 15.65 498.04 0.00 696.09 -5.87 A#3 396.09 9.78 498.04 0.00 701.96 0.00 A3 401.96 15.65 503.91 5.87 701.96 0.00 G#3 407.82 21.51 498.04 0.00 701.96 0.00 G3 396.09 9.78 503.91 5.87 696.09 -5.87 F#3 407.82 21.51 503.91 5.87 701.96 0.00 F3 390.22 3.91 498.04 0.00 701.96 0.00 E3 401.96 15.65 498.04 0.00 701.96 0.00 D#3 401.96 15.65 498.04 0.00 701.96 0.00 D3 396.09 9.78 503.91 5.87 696.09 -5.87 C#3 407.82 21.51 498.04 0.00 701.96 0.00 C3 390.22 3.91 498.04 0.00 696.09 -5.87 The result is that major thirds are stretched by 2, 5, 8, or 11/12 of the Pythagorean comma, and minor thirds flattened by 2, 5, 8, or 11/12 of it. 2 5 8 11 Pure major third is 386.31 cents 390.22 396.09 401.96 407.82 Pure minor third is 315.64 cents 311.73 305.87 300.00 294.13 Vallotti Another popular well-tempered tuning is the Vallotti tuning, historically accurate for music of Mozart's time. In the Vallotti tuning, the fifths F-C- G-D-A-E-B are each tempered by 1/6 of the Pythagorean comma, or 3.91 cents. Pure fifths of 3:2 are 701.96 cents; the tempered ones are then 698.04 cents. Pure fourths of 4:3 are 498.04 cents; the tempered ones are then 501.96 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 525.03 1200.00 5.87 B3 492.77 1090.22 -3.91 A#3 467.75 1000.00 5.87 A3 440.00 894.13 0.00 G#3 415.77 796.09 1.96 G3 392.88 698.04 3.91 F#3 369.58 592.18 -1.96 F3 350.81 501.96 7.82 E3 329.26 392.18 -1.96 D#3 311.83 298.04 3.91 D3 294.00 196.09 1.96 C#3 277.18 94.13 0.00 C3 262.51 0.00 5.87 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 109.78 4.82 203.91 0.00 305.87 -9.77 A#3 90.22 -14.73 200.00 -3.91 294.13 -21.51 A3 105.87 0.91 196.09 -7.82 305.87 -9.77 G#3 98.04 -6.91 203.91 0.00 294.13 -21.51 G3 98.04 -6.91 196.09 -7.82 301.96 -13.68 F#3 105.87 0.91 203.91 0.00 301.96 -13.68 F3 90.22 -14.73 196.09 -7.82 294.13 -21.51 E3 109.78 4.82 200.00 -3.91 305.87 -9.77 D#3 94.13 -10.82 203.91 0.00 294.13 -21.51 D3 101.96 -3.00 196.09 -7.82 305.87 -9.77 C#3 101.96 -3.00 203.91 0.00 298.04 -17.60 C3 94.13 -10.82 196.09 -7.82 298.04 -17.60 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 407.82 21.51 501.96 3.91 701.96 0.00 A#3 396.09 9.78 498.04 0.00 701.96 0.00 A3 400.00 13.69 501.96 3.91 698.04 -3.91 G#3 403.91 17.60 498.04 0.00 701.96 0.00 G3 392.18 5.87 501.96 3.91 698.04 -3.91 F#3 407.82 21.51 498.04 0.00 701.96 0.00 F3 392.18 5.87 498.04 0.00 698.04 -3.91 E3 403.91 17.60 501.96 3.91 698.04 -3.91 D#3 400.00 13.69 498.04 0.00 701.96 0.00 D3 396.09 9.78 501.96 3.91 698.04 -3.91 C#3 407.82 21.51 498.04 0.00 701.96 0.00 C3 392.18 5.87 501.96 3.91 698.04 -3.91 The result is that major thirds are stretched by 3, 5, 7, 9, or 11/12 of the Pythagorean comma, and minor thirds flattened by 5, 7, 9, or 11/12 of it. 3 5 7 9 11 Pure major third is 386.31 cents 392.18 396.09 400.00 403.91 407.82 Pure minor third is 315.64 cents 305.87 301.96 298.05 294.14 The Fisk-Vogel tunings at Stanford Finally, a pair of tunings which have gotten some discussion on the net: the tunings devised by Charles Fisk and Harald Vogel for the Fisk organ in Memorial Church at Stanford University, which can be switched by a lever above the music desk from mean-tone to well-tempered; the lever activates an alternate set of tracker mechanisms for the sharp keys, so that there are 17 pipes per octave. Stanford's Fisk organ uses a modified meantone tuning which shares the natural keys with the well-tempered tuning. This means that the six intervals F-C-G-D-A-E-B are identical in both systems; they're flattened by one-fifth the Pythagorean comma, or 4.69 cents. The five intervals F-Bb-Eb and G#-C#-F#-B are flattened by one-fourth the syntonic comma, or 5.38 cents. So natural fifths are tuned to 697.26 cents and "other" fifths are tuned to 696.58 cents, except the wolf which must be seven octaves minus six natural fifths minus five accidental fifths, or 733.53 cents. Here the wolf is shown inverted as the fourth of 466.47 cents, between G# and Eb. All natural fourths are 502.74 cents, and "other" fourths are 503.42 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 525.74 1200.00 8.21 B3 492.32 1086.31 -5.47 A#3 470.05 1006.16 14.37 A3 440.00 891.79 0.00 G#3 411.55 776.05 -15.74 G3 393.24 697.26 5.47 F#3 368.10 582.89 -8.90 F3 351.44 502.74 10.95 E3 329.11 389.05 -2.74 D#3 314.34 309.58 17.79 D3 294.13 194.53 2.74 C#3 275.22 79.47 -12.32 C3 262.87 0.00 8.21 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 113.69 8.73 193.16 -10.75 308.21 -7.43 A#3 80.16 -24.80 193.84 -10.07 273.31 -42.33 A3 114.37 9.41 194.53 -9.38 308.21 -7.43 G#3 115.74 10.78 230.11 26.20 310.26 -5.38 G3 78.79 -26.17 194.53 -9.38 308.90 -6.74 F#3 114.37 9.41 193.16 -10.75 308.90 -6.74 F3 80.16 -24.80 194.53 -9.38 273.31 -42.33 E3 113.69 8.73 193.84 -10.07 308.21 -7.43 D#3 79.47 -25.48 193.16 -10.75 273.31 -42.33 D3 115.05 10.10 194.53 -9.38 308.21 -7.43 C#3 115.05 10.10 230.11 26.20 309.58 -6.06 C3 79.47 -25.48 194.53 -9.38 309.58 -6.06 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 423.27 36.96 502.74 4.69 696.58 -5.38 A#3 388.37 2.06 503.42 5.38 696.58 -5.38 A3 387.68 1.37 502.74 4.69 697.26 -4.69 G#3 423.95 37.64 503.42 5.38 733.53 31.57 G3 389.05 2.74 502.74 4.69 697.26 -4.69 F#3 423.27 36.96 503.42 5.38 696.58 -5.38 F3 389.05 2.74 503.42 5.38 697.26 -4.69 E3 387.00 0.69 502.74 4.69 697.26 -4.69 D#3 387.68 1.37 466.47 -31.57 696.58 -5.38 D3 388.37 2.06 502.74 4.69 697.26 -4.69 C#3 423.27 36.96 503.42 5.38 696.58 -5.38 C3 389.05 2.74 502.74 4.69 697.26 -4.69 These should be compared with quarter-comma meantone; the differences are hardly ever more than a couple of cents. The nifty thing is that by switching only the sharps, you get the well-tempered tuning described next. Stanford's Fisk organ uses a well-tempered tuning which shares the natural keys with the modified meantone tuning. This means that the six intervals F- C-G-D-A-E-B are identical in both systems; they're flattened by one-fifth the Pythagorean comma. G#-C# is similarly flattened in the well-tempered tuning. The three intervals Bb-Eb-G# and F#-B are pure; the two intervals F-Bb and C#-F# are sharpened (widened) by one-fifth Pythagorean comma. So natural fifths are tuned to 697.26 cents, and wide fifths are tuned to 706.65 cents, compared with pure fifths of 701.96 cents. All natural fourths are 502.74 cents, and wide fourths are 493.35 cents, compared with pure fourths of 498.04 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 525.74 1200.00 8.21 B3 492.32 1086.31 -5.47 A#3 467.32 996.09 4.30 A3 440.00 891.79 0.00 G#3 415.40 792.18 0.39 G3 393.24 697.26 5.47 F#3 369.24 588.27 -3.52 F3 351.44 502.74 10.95 E3 329.11 389.05 -2.74 D#3 311.55 294.13 2.35 D3 294.13 194.53 2.74 C#3 277.68 94.92 3.13 C3 262.87 0.00 8.21 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 113.69 8.73 208.60 4.69 308.21 -7.43 A#3 90.22 -14.73 203.91 0.00 298.83 -16.81 A3 104.30 -0.65 194.53 -9.38 308.21 -7.43 G#3 99.61 -5.35 203.91 0.00 294.13 -21.51 G3 94.92 -10.04 194.53 -9.38 298.83 -16.81 F#3 108.99 4.04 203.91 0.00 303.52 -12.12 F3 85.53 -19.42 194.53 -9.38 289.44 -26.20 E3 113.69 8.73 199.22 -4.69 308.21 -7.43 D#3 94.92 -10.04 208.60 4.69 294.13 -21.51 D3 99.61 -5.35 194.53 -9.38 308.21 -7.43 C#3 99.61 -5.35 199.22 -4.69 294.13 -21.51 C3 94.92 -10.04 194.53 -9.38 294.13 -21.51 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 407.82 21.51 502.74 4.69 701.96 0.00 A#3 398.44 12.13 498.04 0.00 706.65 4.69 A3 403.13 16.82 502.74 4.69 697.26 -4.69 G#3 407.82 21.51 502.74 4.69 701.96 0.00 G3 389.05 2.74 502.74 4.69 697.26 -4.69 F#3 407.82 21.51 498.04 0.00 706.65 4.69 F3 389.05 2.74 493.35 -4.69 697.26 -4.69 E3 403.13 16.82 502.74 4.69 697.26 -4.69 D#3 403.13 16.82 498.04 0.00 701.96 0.00 D3 393.74 7.43 502.74 4.69 697.26 -4.69 C#3 407.82 21.51 493.35 -4.69 697.26 -4.69 C3 389.05 2.74 502.74 4.69 697.26 -4.69 ------------------------------------------------------------------------- version 1.0 (C) Copyright 1992 by Neil Midkiff. This document may be freely distributed for educational or other non-profit use; please retain this notice and do not distribute it in altered form. Commercial use or publication in whole or part requires my consent. In other words, it's freeware, but not public domain. From alves@calvin (William Alves) Wed Nov 25 16:58:47 1992 X-VM-v5-Data: ([nil nil nil nil nil nil nil nil nil] ["2650" "" "24" "November" "1992" "14:26:27" "-0800" "William Alves" "alves@calvin.usc.edu " nil "49" "Re: frequencies <=> tuning" "^From:" nil nil "11"]) Newsgroups: comp.music Organization: University of Southern California, Los Angeles, CA Lines: 49 Distribution: inet Message-ID: <1eua6jINNf5d@calvin.usc.edu> References: <4f4XQo200iUyE1QFkk@andrew.cmu.edu> NNTP-Posting-Host: calvin.usc.edu From: alves@calvin.usc.edu (William Alves) Subject: Re: frequencies <=> tuning Date: 24 Nov 1992 14:26:27 -0800 In article <4f4XQo200iUyE1QFkk@andrew.cmu.edu> Jeffrey C Kunins writes: >Speaking of the discrepancies between tempered scales and the scale of >true intonation, does anyone know if there IS a way to mathematically >derive the "true intonation" frequencies for, say G major ? I know that >many synthesizers have built in micro-tuning for tunings such as "true >major", "true minor", and other more-exotic true tunings. Does anyone >know how those are derived? Well, first of all, tuning systems that use interval ratios that are reducible to fractions with relatively small numerators and denominators are called "just" tuning systems, or just intonation. Tuning systems that include irrational frequency ratios are called temperaments. The simplest just system (though some people don't count it as a just tuning) is "Pythagorean" tuning. It is based on the ratio 3/2 for the perfect fifth. Thus if C=262 Hz, then G=262*3/2=392 hz. D=(392*3/2)/2 (have to subtract an octave)=294 hz and so on. You will find, however, that when you get to B#, it is about 24 cents sharp of C. This interval is called the Pytha- gorean comma. Also, if you go the other way around the circle of fifths, you will find that the flat notes do not have the same frequency as their sharp enharmonics. Other just tuning systems use not only 3/2 but also such intervals as 5/4 for a major third, 6/5 for a minor third, or even such intervals as 7/6 or 11/8, though these intervals lie pretty far outside the nearest equally-tempered scale degree. A very common just tuning system is this one (obviously it could be transposed to start on any scale degree): 1/1 16/15 9/8 6/5 5/4 4/3 *** 3/2 8/5 5/3 16/9 15/8 2/1 C C# D Eb E F F# G Ab A Bb B C *** There are a number of ways to derive the tritone. For that matter, there are a number of ways to derive many of these pitches, depending on which triads or other chords are most important to you. I'm not sure if the Yamaha "Pure Major" is this tuning system, but it's something pretty close. There are many, many other tuning systems. A class of tuning systems used in the Baroque, for example, tempered the fifths but kept pure major thirds (i.e. 5/4 thirds) in the most commonly used keys. Such a tuning system is usually called a "mean tone" temperament. There are several good historical surveys of tuning systems, one of the most comprehensive (despite his annoying prejudice in favor of equal temperament) is J. Murray Barbour's _Tuning and Temperament: A Historical Survey_ (Michigan State College Press, 1953). Bill Alves