This page is by Rick Tagawa

Apparently Rick's site is no longer online. Sonic Arts is hosting this page as a service to interested readers. Several of the links here are also dead now.


"Ever since his own age a great controversy has raged about the teachings of Aristoxenus. Instead of using ratios, he divided the tetrachord into 30 parts, of which, in his diatonic syntonon, each tone has 12 parts, each semitone 6. . . ." [J. Murray Barbour, TUNING AND TEMPERAMENT A HISTORICAL SURVEY, Da Capo Press, New York, 1972, p22.]
Aristoxenos
Aristoxenus of Tarentum
FAQ Society of Ancient History

72 TONE EQUAL TEMPERAMENT (72-tET)

Temperament [G. Temperatur]. The term denotes those systems of tuning in which the intervals deviate from the "pure," i.e., acoustically correct intervals as used in the Pythagorean system and in Just intonation. . . . compromise[s] . . . which, instead of being perfect in the simple keys and intolerably wrong in the others, spread the . . . inaccuracy over all the tones and keys. [Apel, Willi, Harvard Dictionary of Music, Harvard University Press, Cambridge, MA 1964, p734]

Understanding Temperaments
Common Equal Tempered Scales
12 Tone Equal Temperament
24 Tone Equal Temperament
36 Tone Equal Temperament
Defects of Equal Temperaments Bar Chart
Definition of Tuning Terms
History of Tuning & Temperaments
Tuning Page
The Scales Page

Subsets of 72-tET

List of Musical Modes
BLACKJACK SCALE devised by Dave Keenan & Paul Erlich
CANASTA SCALE
Index of Dave Keenan MUSIC/MIRACLE
MIRACLE Generator and Scale by Joseph Monzo
Kathleen Schlesinger

The following is a selection from John H. Chalmers, Jr.book Divisions of the Tetrachord

. . . Martin Vogel in Bonn and Franz Richter Herf in Salzburg have been active in various microtonal systems, the latter especially in 72-tone equal temperament. [p2]

Aristoxenos described his genera in units of twelfths of a tone (Macran, Henry S. 1902. The harmonics of Aristoxenos. Oxford: The Clarendon Press), but later theorists, notably Cleonides, translated these units into a cipher consisting of 30 parts (moria) to the fourth (Barbera, Charles Andre. 1978. "Arithmetic and geometric divisions of the tetrachord," J. Music Theory 22:294-323.). . . . Two such 30-part tetrachords and a whole tone of twelve parts completed an octave of 72 parts. [pp18-19]

. . . neo-Aristoxenian tetrachords may be approximated in just intonation or realized in equal temperaments whose cardinalities are zero modulo 12. The zero modulo 12 temperaments provide opportunities to simulate many of the other genera in the Catalogs as their fourths are only two cents from 4/3 and other intervals of just intonation are often closely approximated. One may also use them to discover or invent new neo- Aristoxenian tetrachords. To articulate a single part difference, a temperament of 72 tones per octave is required. [p37] Chalmers, Jr., John H. Divisions of the Tetrachord, Frog Peak Music, 1993

John Chalmers' Articles & Reviews

NOTATION

Ezra Sims' 1/12th-tone Notation System
Graham Breed's Decimal Notation
Index of Dave Keenan MUSIC/MIRACLE
Dave Keenan's "mystery circle" of Graham Breed's Decimal Notation
Rick Tagawa's 2 Proposals (Cents Notation & Staff Notation)

The following is a section devoted to the music of Julian Carrillo and Augusto Novaro.

"Several experimentalists have managed to survive in the predominantly conservative musical culture of Mexico. Julian Carrillo (1875-1965), an early western- world experimenter with microtonal scales, is perhaps best known. The practically unknown Augusto Novaro (1891- 1960) designed and built string instruments, guitars and pianos with experimental intonational arrangements, and published his treatise on a "Natural System of Music" in Mexico City in 1951." [Garland, Peter, "Selected Studies for Player Piano," Soundings Book 4, spring- summer 1977, Berkeley, California, p1]

Jesus Bernal, Organista Mexicano, teaches acoustics @ National Conservatory, Mexico
Biography (part 1) of Julian Carrillo (in Spanish)
Biography (part 2) of Julian Carrillo (in English)
Julian Carrillo & the 13th Sound
Julian Carrillo 13th Sound & the Quarter Tone Guitar (in Spanish)
Julian Carrillo's Dos Bosquejos performed in March 1992 @ the Centre for Microtonal Music in London
Julian Carrillo's Preludio Impromptu & Lentamente from Suite Impromptu (1931) performed at Microthon May 20, 1999 New York
Julian Carrillo Cello Concertino (1927) in repertoire of Frances-Marie Uitti
Julian Carrillo Mass for Pope John XXIII sound recording
Biography of Augusto Novaro (in Spanish)

72 TONE EQUAL TEMPERAMENT AS A PITCH CONTINUUM

. . . [Regarding] 72ET as representing the harmonic series, from a practical standpoint, you could consider 72ET to be a pitch continuum, where ANY interval is approximated well (with a maximum error of 8 1/3 cents), and this viewpoint is particularly appropriate to strings and such instruments, where the variable tuning is all by ear anyway. For fixed pitch instruments, it may be more reasonable to think of the 72 tones as only representing intervals that are "nearby" (the max error leads to ambiguity); in that view, the harmonic series is represented very well up to harmonic 12 (worst error 3.9c), but with 13 the error jumps to 7.2c, so from there on, the approximations are not clear representations of the harmonic relations... IMHO anyway. So one may need to be content with "only" pure octaves, fifths, thirds, sevenths, ninths, and elevenths! Not bad! From: Canright, David, Date: Sat, 21 Apr 2001 09:39:02 -0700

72-tET Approximations of Just Intervals

Below is a table of [the first 23]. . . intervals found in the harmonic series . . ., and their closest approximations by 72-tET. These are ordered by first appearance in the harmonic series (by numerator, then by denominator; . . .); Not all of the "octave complements" appear; of course, their tempered approximations have equal but opposite errors. Each entry includes the ratio, cents error relative to nearest tempered (so, for example, 3/2 is +2.0c relative to note #42), tempered note # (0-71), keyboard # (0-5), key # (0-11), where keyboards 1-5 are assumed progressively sharper relative to keyboard 0.

ratio cents error note key
1/1 +0.0¢ 0 0:0
3/2 +2.0¢ 42 0:7
4/3 -2.0¢ 30 0:5
5/3 +1.0¢ 53 5:8
5/4 +3.0¢ 23 5:3
6/5 -1.0¢ 19 1:3
7/4 +2.2¢ 58 4:9
7/5 -0.8¢ 35 5:5
7/6 +0.2¢ 16 4:2
8/5 -3.0¢ 49 1:8
8/7 -2.2¢ 14 2:2
9/5 +0.9¢ 61 1:10
9/7 +1.8¢ 26 2:4
9/8 +3.9¢ 12 0:2
10/7 +0.8¢ 37 1:6
10/9 -0.9¢ 11 5:1
11/6 -0.6¢ 63 3:10
11/7 -0.8¢ 47 5:7
11/8 +1.3¢ 33 3:5
11/9 -2.6¢ 21 3:3
11/10 -1.7¢ 10 4:1
12/7 -0.2¢ 56 2:9
12/11 +0.6¢ 9 3:1

Subject: table ordered by harmonic limit/Date: Wed, 2 May 2001 11:56:53 -0700/From: Canright, David

David Canright Homepage

I'd advise anyone wanting an ET to approximate some intervals with 11 and 13 as well as the lower primes to look at 41 or 43, if said person doesn't want 72. I also think of ETs as ETs, with all the usual properties, because if you don't need those, you might as well do JI to get JI. Ezra Sims has written several articles besides the one or two noted recently. I'm not wild about his notation, although it's not hard to figure out. Try Perspectives of New Music, vol. 29 no. 1 (1991). Paul Rapoport rapoport@mcmaster.ca Date: Sun, 12 Jun 94

Ezra Sims has been using 72ET since the early 70s. He uses it to approximate harmonics up into the 20s and even 30s. His main subset scale approximates the pitches 1/1 (33/32) 25/24 (17/16) 13/12 (35/32) 9/8 (37/32) 7/6 (19/16) 29/24 (39/32) 5/4 21/16 11/8 23/16 13/8 27/16 7/4 29/16 15/8 31/16 2/1 He says it's very similar to Franz Richter Herf's _Ecmelic Scale_. This info is out of the book _Mikrotoene III_, editor Horst-Peter Hesse, published Edition Helbling, Innsbruck, (c) 1990. Herf seems to have something of a following, this being the proceedings of their third annual conference, but the rest of the book is hard for me to follow because it's in German, and my German nearly isn't. But I hope this gives you some leads to follow anyway. -- pH <manynote@library.wustl.edu> Date: Fri, 10 Jun 94 13:53:25 -0700
Ezra Sims' 72-tET notation
EZRA SIMS AT SEVENTY by Julia Werntz (Sonneck Society for American Music)

Didn't Tenney compose a piece for 72-TET called "6 Harps" or something? If I remember correctly, each harp was in in 12-TET, but all were tuned 1/6 semitone apart from each other. He used a very small subset of the 72 pitches, I believe, to get close approximations of various just intervals. -greg higgs Date: Fri, 10 Jun 94 14:31:23 -0700
James Tenney biography @ Smith Archives Composer Profile
FROG PEAK MUSIC: A Composers' Collective

I have been using 72 equal quite a bit. One thing that has helped me immensely is a 72TET 'Bingo' card I made up.
58 9 32 55 6 29 52 3 26
16 39 62 13 36 59 10 33 56
46 69 20 43 66 17 40 63 14
4 27 50 1 24 47 70 21 44
34 57 8 31 54 5 28 51 2
64 15 38 61 12 35 58 9 32
22 45 68 19 42 65 16 39 62
52 3 26 49 *0* 23 46 69 20
10 33 56 7 30 53 4 27 50
40 63 14 37 60 11 34 57 8
70 21 44 67 18 41 64 15 38
28 51 2 25 48 71 22 45 68
58 9 32 55 6 29 52 3 26
16 39 62 13 36 59 10 33 56
46 69 20 43 66 17 40 63 14
~3/2's go up and ~5/4's go right. On the card all intervals are uniformly spaced, that is a 7/6 for example is two right and one up. Some interval equivalences: 3/2 = 42, 9/8 = 12, 4/3 = 30, 16/9 = 60, 15/8 = 65, 5/4 = 23, 5/3 = 53, 10/9 = 11, 9/5 = 61, 6/5 = 19, 8/5 = 49, 16/15 = 7, 7/5 = 35, 7/4 = 58, 7/6 = 16, 14/9 = 46, 11/8 = 33.. etc. Many interesting symmetries and enharmonicities can be found.. --- james mccartney Date: Mon, 13 Jun 94 13:04:26 -0700

72-tET approximates all of Partch's consonant intervals (i.e., all the ratios in his 11-limit tonality diamond) consistently and with a maximum error of ~3.9 cents. This is much better than any ET with less than 118 notes. 144-tET cannot improve upon any of 72-tET's approximations. Partch obtained his full 43-tone system by transposing parts of the tonality diamond to centers other than 1/1. <snip> correct transcriptions of Harry Partch's music would never require them to leave 72-tET <snip> if 13- or higher-limit intervals were included, using 144-tET to approximate them could lead to dangerous inconsistencies, since 144-tET is only consistent through the 11-limit. 72-tET is consistent through the 17-limit, although the ~7.2- cent error in the 13:8 might cause one to deem it insufficiently accurate beyond the 11-limit. The simplest ET to approximate the 13-limit consistently with less than 4 cents error is 130-tET, and the simplest to do so for the 17-limit is 149-tET. Unfortunately, these are not multiples of 12. -Paul E. Subject: [tuning] Re: Partch lattice in 144-eq notation Date: Sun, 18 Apr 1999 18:36:28 -0400

The following article was contributed by Paul Erlich.

This table shows which 72-tET intervals best approximate the just ratios of the 17-limit within half an octave. "17- limit" here means the largest odd factor of either of the numbers forming the ratio does not exceed 17, as in Harry Partch's Tonality Diamonds. Intervals larger than half an octave are easily calculated: for example, 7:4 is an octave minus 8:7, so it's 72 - 14 = 58 steps.

The table shows that all 11-limit ratios are represented uniquely; that is, no 72-tET interval has to represent more than one of them. What the table does not show is that the maximum error of the 72-tET approximations of the 11-limit ratios is 4 cents. The lowest ET that improves on that is 118-tET where the largest error is 3 cents.

72-tET is consistent through the 17- limit. To explain what this means, let me show why 72-tET is not consistent in the 19-limit. In 72-tET, the best approximation of 13:8 is 50 steps, the best approximation of 19:8 is 90 steps, and the best approximation of 19:13 is 39 steps. That means that if you try to play the chord 8:13:19 in 72- tET, you cannot use the best approximation of all the intervals involved. That difficulty will never arise in a chord all of whose intervals are within the 17-limit, thus the term "consistency". 58-tET is also consistent through the 17-limit, but with worse approximations; the lowest ET that is consistent through the 19-limit is 80-tET, and the smallest that improves on 72-tET's 17-limit approximations is 94-tET, which is also consistent through the 23-limit.

72-tET steps Tones 11-limit ratio Remaining 17-limit ratios
0 0 1:1
6 1/2 18:17, 17:16
7 7/12 16:15, 15:14
8 2/3 14:13, 13:12
9 3/4 12:11
10 5/6 11:10
11 11/12 10:9
12 1 9:8
13 1-1/12 17:15
14 1-1/6 8:7
15 1-1/4 15:13
16 1-1/3 7:6
17 1-5/12 13:11, 20:17
19 1-7/12 6:5
20 1-2/3 17:14
21 1-3/4 11:9
22 1-5/6 16:13
23 1-11/12 5:4
25 2-1/12 14:11
26 2-1/6 9:7
27 2-1/4 13:10, 22:17
28 2-1/3 17:13
30 2-1/2 4:3
32 2-2/3 15:11
33 2-3/4 11:8
34 2-5/6 18:13
35 2-11/12 7:5
36 3 17:12, 24:17

Wed, 30 Jun 1999

Paul Erlich Harmonic Entropy (Farey Series N-79)
Paul Erlich Harmonic Entropy (Farey Series N-80)

Other Related Links:

Microtonal Groups World Wide
Other Selected Websites About Alternate Tuning
MICROTONAL BIBLIOGRAPHY (B. McLaren)
Schoenberg and the Further Possibilities of Music-Development
Stravinsky on the topic of microtonality
music related links you might find helpful
email
Hitometer

Updated: July 6, 2001