# 72 equal temperament

(Redirected from Maneri-Sims notation)

In music, 72 equal temperament, called twelfth-tone, 72-TET, 72-EDO, or 72-ET, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps (equal frequency ratios).   Each step represents a frequency ratio of 722, or ​16 23 cents, which divides the 100 cent "halftone" into 6 equal parts (100 ÷ ​16 23 = 6) and is thus a "twelfth-tone" ( ). Since 72 is divisible by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72, 72-EDO includes all those equal temperaments. Since it contains so many temperaments, 72-EDO contains at the same time tempered semitones, third-tones, quartertones and sixth-tones, which makes it a very versatile temperament.

This division of the octave has attracted much attention from tuning theorists, since on the one hand it subdivides the standard 12 equal temperament and on the other hand it accurately represents overtones up to the twelfth partial tone, and hence can be used for 11-limit music. It was theoreticized in the form of twelfth-tones by Alois Hába[1] and Ivan Wyschnegradsky,[2][3][4] who considered it as a good approach to the continuum of sound. 72-EDO is also cited among the divisions of the tone by Julián Carrillo, who preferred the sixteenth-tone as an approximation to continuous sound in discontinuous scales.

## History and use

### Byzantine music

The 72 equal temperament is used in Byzantine music theory,[5] dividing the octave into 72 equal moria, which itself derives from interpretations of the theories of Aristoxenos, who used something similar. Although the 72 equal temperament is based on irrational intervals (see above), as is the 12 tone equal temperament mostly commonly used in Western music (and which is contained as a subset within 72 equal temperament), 72 equal temperament, as a much finer division of the octave, is an excellent tuning for both representing the division of the octave according to the diatonic and the chromatic genera in which intervals are based on ratios between notes, and for representing with great accuracy many rational intervals as well as irrational intervals.

### Other history and use

A number of composers have made use of it, and these represent widely different points of view and types of musical practice. These include Alois Hába, Julián Carrillo, Ivan Wyschnegradsky and Iannis Xenakis.[citation needed]

Many other composers use it freely and intuitively, such as jazz musician Joe Maneri, and classically oriented composers such as Julia Werntz and others associated with the Boston Microtonal Society. Others, such as New York composer Joseph Pehrson are interested in it because it supports the use of miracle temperament, and still others simply because it approximates higher-limit just intonation, such as Ezra Sims and James Tenney. There was also an active Soviet school of 72 equal composers, with less familiar names: Evgeny Alexandrovich Murzin, Andrei Volkonsky, Nikolai Nikolsky, Eduard Artemiev, Alexander Nemtin, Andrei Eshpai, Gennady Gladkov, Pyotr Meshchianinov, and Stanislav Kreichi.[citation needed]

The ANS synthesizer uses 72 equal temperament.

## Notation

The Maneri-Sims notation system designed for 72-et uses the accidentals and for ​112-tone down and up (1 step = ​16 23 cents),   and   for ​16 down and up (2 steps = ​33 13 cents), and   and   for 14 up and down (3 steps = 50 cents).

They may be combined with the traditional sharp and flat symbols (6 steps = 100 cents) by being placed before them, for example:   or  , but without the intervening space. A ​13 tone may be one of the following  ,  ,  , or   (4 steps = ​66 23) while 5 steps may be   , , or (​83 13 cents).

## Interval size

Below are the sizes of some intervals (common and esoteric) in this tuning. For reference, differences of less than 5 cents are melodically imperceptible to most people:

Interval Name Size (steps) Size (cents) MIDI Just Ratio Just (cents) MIDI Error
octave 72 1200 2:1 1200 0
perfect fifth 42 700   3:2 701.96   −1.96
septendecimal tritone 36 600   17:12 603.00 −3.00
septimal tritone 35 583.33   7:5 582.51   +0.82
tridecimal tritone 34 566.67   18:13 563.38 +3.28
11th harmonic 33 550   11:8 551.32   −1.32
(15:11) augmented fourth 32 533.33   15:11 536.95   −3.62
perfect fourth 30 500   4:3 498.04   +1.96
septimal narrow fourth 28 466.66   21:16 470.78   −4.11
17:13 narrow fourth 17:13 464.43 +2.24
tridecimal major third 27 450   13:10 454.21   −4.21
septendecimal supermajor third 22:17 446.36 +3.64
septimal major third 26 433.33   9:7 435.08   −1.75
undecimal major third 25 416.67   14:11 417.51   −0.84
major third 23 383.33   5:4 386.31   −2.98
tridecimal neutral third 22 366.67   16:13 359.47 +7.19
neutral third 21 350   11:9 347.41   +2.59
septendecimal supraminor third 20 333.33   17:14 336.13 −2.80
minor third 19 316.67   6:5 315.64   +1.03
quasi-tempered minor third 18 300   25:21 301.85 -1.85
tridecimal minor third 17 283.33   13:11 289.21   −5.88
septimal minor third 16 266.67   7:6 266.87   −0.20
tridecimal ​54 tone 15 250   15:13 247.74 +2.26
septimal whole tone 14 233.33   8:7 231.17   +2.16
septendecimal whole tone 13 216.67   17:15 216.69 −0.02
whole tone, major tone 12 200   9:8 203.91   −3.91
whole tone, minor tone 11 183.33   10:9 182.40   +0.93
greater undecimal neutral second 10 166.67   11:10 165.00   +1.66
lesser undecimal neutral second 9 150   12:11 150.64   −0.64
greater tridecimal ​23 tone 8 133.33   13:12 138.57   −5.24
great limma 27:25 133.24   +0.09
lesser tridecimal ​23 tone 14:13 128.30   +5.04
septimal diatonic semitone 7 116.67   15:14 119.44   −2.78
diatonic semitone 16:15 111.73   +4.94
greater septendecimal semitone 6 100   17:16 104.95   -4.95
lesser septendecimal semitone 18:17 98.95   +1.05
septimal chromatic semitone 5 83.33   21:20 84.47   −1.13
chromatic semitone 4 66.67   25:24 70.67   −4.01
septimal third-tone 28:27 62.96   +3.71
septimal quarter tone 3 50   36:35 48.77   +1.23
septimal diesis 2 33.33   49:48 35.70   −2.36
undecimal comma 1 16.67   100:99 17.40 −0.73

Although 12-ET can be viewed as a subset of 72-ET, the closest matches to most commonly used intervals under 72-ET are distinct from the closest matches under 12-ET. For example, the major third of 12-ET, which is sharp, exists as the 24-step interval within 72-ET, but the 23-step interval is a much closer match to the 5:4 ratio of the just major third.

All intervals involving harmonics up through the 11th are matched very closely in this system; no intervals formed as the difference of any two of these intervals are tempered out by this tuning system. Thus 72-ET can be seen as offering an almost perfect approximation to 7-, 9-, and 11-limit music. When it comes to the higher harmonics, a number of intervals are still matched quite well, but some are tempered out. For instance, the comma 169:168 is tempered out, but other intervals involving the 13-th harmonic are distinguished.

Unlike tunings such as 31-ET and 41-ET, 72-ET contains many intervals which do not closely match any small-number (<16) harmonics in the harmonic series.

## Scale diagram

12-tone   and 72-tone   regular diatonic scales notated with the Maneri-Sims system

Because 72-EDO contains 12-EDO, the scale of 12-EDO is in 72-EDO. However, the true scale can be approximated better by other intervals.