72 equal temperament

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). {{errorTemplate:Main other|Audio file "72-tet scale on C.mid" not found}}Template:Category handler Each step represents a frequency ratio of Template:Radic, or Template:Nobr which divides the 100 cent 12 EDO "halftone" into 6 equal parts (100 cents ÷ Template:Nobr = 6 steps, exactly) and is thus a "twelfth-tone" ({{errorTemplate:Main other|Audio file "1 step in 72-et on C.mid" not found}}Template:Category handler). 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 and Ivan Wyschnegradsky, 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 (96 EDO) as an approximation to continuous sound in discontinuous scales.

History and use

Byzantine music

The 72 equal temperament is used in Byzantine music theory, 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 (12 EDO) 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 ancient Greek 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.

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 EDO 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.

The ANS synthesizer uses 72 equal temperament.

Notation

The Maneri-Sims notation system designed for 72 EDO uses the accidentals Template:Music and Template:Music for Template:Nobr down and up (1 step Template:Nobr Template:Music and Template:Music for Template:Nobr and up (2 steps Template:Nobr and Template:Music and Template:Music for [[septimal quarter tone|septimal Template:Sfrac]] up and down (3 steps Template:Nobr Template:Nobr

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

Interval size

File:72ed2.svg

Just intervals approximated in 72 EDO. Note that any pitch must be within 8.3 cents of its nearest 72 EDO note.

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, and approaching the limits of feasible tuning accuracy for acoustic instruments. Note that it is not possible for any pitch to be further than Template:Nobr from its nearest 72 EDO note, since the step size between them is Template:Nobr Hence for the sake of comparison, pitch errors of about 8 cents are (for this fine a tuning) poorly matched, whereas the practical limit for tuning any acoustical instrument is at best about 2 cents, which would be very good match in the table – this even applies to electronic instruments if they produce notes that show any audible trace of vibrato.

Interval name	Size (steps)	Size (cents)	MIDI audio	Just ratio	Just (cents)	MIDI audio	Error
octave	72	1200		2:1	1200		0
harmonic seventh	58	966.67		7:4	968.83		−2.16
perfect fifth	42	700	{{errorTemplate:Main other|Audio file "Perfect fifth on C.mid" not found}}Template:Category handler	3:2	701.96	{{errorTemplate:Main other|Audio file "Just perfect fifth on C.mid" not found}}Template:Category handler	−1.96
septendecimal tritone	36	600	{{errorTemplate:Main other|Audio file "Tritone on C.mid" not found}}Template:Category handler	17:12	603.00		−3.00
septimal tritone	35	583.33	{{errorTemplate:Main other|Audio file "35 steps in 72-et on C.mid" not found}}Template:Category handler	7:5	582.51	{{errorTemplate:Main other|Audio file "Lesser septimal tritone on C.mid" not found}}Template:Category handler	+0.82
tridecimal tritone	34	566.67	{{errorTemplate:Main other|Audio file "34 steps in 72-et on C.mid" not found}}Template:Category handler	18:13	563.38		+3.28
11th harmonic	33	550	{{errorTemplate:Main other|Audio file "Eleven quarter tones on C.mid" not found}}Template:Category handler	11:8	551.32	{{errorTemplate:Main other|Audio file "Eleventh harmonic on C.mid" not found}}Template:Category handler	−1.32
(15:11) augmented fourth	32	533.33	{{errorTemplate:Main other|Audio file "32 steps in 72-et on C.mid" not found}}Template:Category handler	15:11	536.95	{{errorTemplate:Main other|Audio file "Undecimal augmented fourth on C.mid" not found}}Template:Category handler	−3.62
perfect fourth	30	500	{{errorTemplate:Main other|Audio file "Perfect fourth on C.mid" not found}}Template:Category handler	4:3	498.04	{{errorTemplate:Main other|Audio file "Just perfect fourth on C.mid" not found}}Template:Category handler	+1.96
septimal narrow fourth	28	466.66	{{errorTemplate:Main other|Audio file "28 steps in 72-et on C.mid" not found}}Template:Category handler	21:16	470.78	{{errorTemplate:Main other|Audio file "Twenty-first harmonic on C.mid" not found}}Template:Category handler	−4.11
17:13 narrow fourth				17:13	464.43		+2.24
tridecimal major third	27	450	{{errorTemplate:Main other|Audio file "Nine quarter tones on C.mid" not found}}Template:Category handler	13:10	454.21	{{errorTemplate:Main other|Audio file "Tridecimal major third on C.mid" not found}}Template:Category handler	−4.21
septendecimal supermajor third				22:17	446.36		+3.64
septimal major third	26	433.33	{{errorTemplate:Main other|Audio file "26 steps in 72-et on C.mid" not found}}Template:Category handler	9:7	435.08	{{errorTemplate:Main other|Audio file "Septimal major third on C.mid" not found}}Template:Category handler	−1.75
undecimal major third	25	416.67	{{errorTemplate:Main other|Audio file "25 steps in 72-et on C.mid" not found}}Template:Category handler	14:11	417.51	{{errorTemplate:Main other|Audio file "Undecimal major third on C.mid" not found}}Template:Category handler	−0.84
quasi-tempered major third	24	400	{{errorTemplate:Main other|Audio file "Major third on C.mid" not found}}Template:Category handler	5:4	386.31	{{errorTemplate:Main other|Audio file "Just major third on C.mid" not found}}Template:Category handler	13.69
major third	23	383.33	{{errorTemplate:Main other|Audio file "23 steps in 72-et on C.mid" not found}}Template:Category handler	5:4	386.31	{{errorTemplate:Main other|Audio file "Just major third on C.mid" not found}}Template:Category handler	−2.98
tridecimal neutral third	22	366.67	{{errorTemplate:Main other|Audio file "22 steps in 72-et on C.mid" not found}}Template:Category handler	16:13	359.47		+7.19
neutral third	21	350	{{errorTemplate:Main other|Audio file "Neutral third on C.mid" not found}}Template:Category handler	11:9	347.41	{{errorTemplate:Main other|Audio file "Neutral third on C.mid" not found}}Template:Category handler	+2.59
septendecimal supraminor third	20	333.33	{{errorTemplate:Main other|Audio file "20 steps in 72-et on C.mid" not found}}Template:Category handler	17:14	336.13		−2.80
minor third	19	316.67	{{errorTemplate:Main other|Audio file "19 steps in 72-et on C.mid" not found}}Template:Category handler	6:5	315.64	{{errorTemplate:Main other|Audio file "Just minor third on C.mid" not found}}Template:Category handler	+1.03
quasi-tempered minor third	18	300	{{errorTemplate:Main other|Audio file "Minor third on C.mid" not found}}Template:Category handler	25:21	301.85		−1.85
tridecimal minor third	17	283.33	{{errorTemplate:Main other|Audio file "17 steps in 72-et on C.mid" not found}}Template:Category handler	13:11	289.21	{{errorTemplate:Main other|Audio file "Tridecimal minor third on C.mid" not found}}Template:Category handler	−5.88
septimal minor third	16	266.67	{{errorTemplate:Main other|Audio file "16 steps in 72-et on C.mid" not found}}Template:Category handler	7:6	266.87	{{errorTemplate:Main other|Audio file "Septimal minor third on C.mid" not found}}Template:Category handler	−0.20
tridecimal Template:Nobr	15	250	{{errorTemplate:Main other|Audio file "Five quarter tones on C.mid" not found}}Template:Category handler	15:13	247.74		+2.26
septimal whole tone	14	233.33	{{errorTemplate:Main other|Audio file "14 steps in 72-et on C.mid" not found}}Template:Category handler	8:7	231.17	{{errorTemplate:Main other|Audio file "Septimal major second on C.mid" not found}}Template:Category handler	+2.16
septendecimal whole tone	13	216.67	{{errorTemplate:Main other|Audio file "13 steps in 72-et on C.mid" not found}}Template:Category handler	17:15	216.69		−0.02
whole tone, major tone	12	200	{{errorTemplate:Main other|Audio file "Major second on C.mid" not found}}Template:Category handler	9:8	203.91	{{errorTemplate:Main other|Audio file "Major tone on C.mid" not found}}Template:Category handler	−3.91
whole tone, minor tone	11	183.33	{{errorTemplate:Main other|Audio file "11 steps in 72-et on C.mid" not found}}Template:Category handler	10:9	182.40	{{errorTemplate:Main other|Audio file "Minor tone on C.mid" not found}}Template:Category handler	+0.93
greater undecimal neutral second	10	166.67	{{errorTemplate:Main other|Audio file "10 steps in 72-et on C.mid" not found}}Template:Category handler	11:10	165.00	{{errorTemplate:Main other|Audio file "Greater undecimal neutral second on C.mid" not found}}Template:Category handler	+1.66
lesser undecimal neutral second	9	150	{{errorTemplate:Main other|Audio file "Neutral second on C.mid" not found}}Template:Category handler	12:11	150.64	{{errorTemplate:Main other|Audio file "Neutral second on C.mid" not found}}Template:Category handler	−0.64
greater tridecimal Template:Nobr	8	133.33	{{errorTemplate:Main other|Audio file "8 steps in 72-et on C.mid" not found}}Template:Category handler	13:12	138.57	{{errorTemplate:Main other|Audio file "Greater tridecimal two-third tone on C.mid" not found}}Template:Category handler	−5.24
great limma				27:25	133.24	{{errorTemplate:Main other|Audio file "Semitone Maximus on C.mid" not found}}Template:Category handler	+0.09
lesser tridecimal Template:Nobr				14:13	128.30	{{errorTemplate:Main other|Audio file "Lesser tridecimal two-third tone on C.mid" not found}}Template:Category handler	+5.04
septimal diatonic semitone	7	116.67	{{errorTemplate:Main other|Audio file "7 steps in 72-et on C.mid" not found}}Template:Category handler	15:14	119.44	{{errorTemplate:Main other|Audio file "Septimal diatonic semitone on C.mid" not found}}Template:Category handler	−2.78
diatonic semitone				16:15	111.73	{{errorTemplate:Main other|Audio file "Just diatonic semitone on C.mid" not found}}Template:Category handler	+4.94
greater septendecimal semitone	6	100	{{errorTemplate:Main other|Audio file "Minor second on C.mid" not found}}Template:Category handler	17:16	104.95	{{errorTemplate:Main other|Audio file "Just major semitone on C.mid" not found}}Template:Category handler	−4.95
lesser septendecimal semitone				18:17	98.95	{{errorTemplate:Main other|Audio file "Just minor semitone on C.mid" not found}}Template:Category handler	+1.05
septimal chromatic semitone	5	83.33	{{errorTemplate:Main other|Audio file "5 steps in 72-et on C.mid" not found}}Template:Category handler	21:20	84.47	{{errorTemplate:Main other|Audio file "Septimal chromatic semitone on C.mid" not found}}Template:Category handler	−1.13
chromatic semitone	4	66.67	{{errorTemplate:Main other|Audio file "4 steps in 72-et on C.mid" not found}}Template:Category handler	25:24	70.67	{{errorTemplate:Main other|Audio file "Just chromatic semitone on C.mid" not found}}Template:Category handler	−4.01
septimal third-tone				28:27	62.96	{{errorTemplate:Main other|Audio file "Septimal minor second on C.mid" not found}}Template:Category handler	+3.71
septimal quarter tone	3	50	{{errorTemplate:Main other|Audio file "Quarter tone on C.mid" not found}}Template:Category handler	36:35	48.77	{{errorTemplate:Main other|Audio file "Septimal quarter tone on C.mid" not found}}Template:Category handler	+1.23
septimal diesis	2	33.33	{{errorTemplate:Main other|Audio file "1 step in 36-et on C.mid" not found}}Template:Category handler	49:48	35.70	{{errorTemplate:Main other|Audio file "Septimal diesis on C.mid" not found}}Template:Category handler	−2.36
undecimal comma	1	16.67	{{errorTemplate:Main other|Audio file "1 step in 72-et on C.mid" not found}}Template:Category handler	100:99	17.40		−0.73

• {{errorTemplate:Main other|Audio file "72-et diatonic scale on C.mid" not found}}Template:Category handler
• {{errorTemplate:Main other|Audio file "Just diatonic scale on C.mid" not found}}Template:Category handler
• {{errorTemplate:Main other|Audio file "Diatonic scale on C.mid" not found}}Template:Category handler

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

12 EDO has a very good approximation for the perfect fifth (third harmonic), especially for such a small number of steps per octave, but compared to the equally-tempered versions in 12 EDO, the just major third (fifth harmonic) is off by about a sixth of a step, the seventh harmonic is off by about a third of a step, and the eleventh harmonic is off by about half of a step. This suggests that if each step of 12 EDO were divided in six, the fifth, seventh, and eleventh harmonics would now be well-approximated, while 12 EDO‑s excellent approximation of the third harmonic would be retained. Indeed, all intervals involving harmonics up through the 11th are matched very closely in 72 EDO; no intervals formed as the difference of any two of these intervals are tempered out by this tuning system. Thus, 72 EDO 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 13th harmonic are distinguished.

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

Scale diagram

File:Regular diatonic tunings 72-tone versus 12-tone.png

Audio file "Regular diatonic tunings 72-tone scale.mid" not found}}Template:Category handler 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.

See also

• Musical temperament
• Equal temperament

External links

• — symbols for Maneri-Sims notation and others