72 equal temperament

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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). <phonos file="72-tet scale on C.mid">Play</phonos> Each step represents a frequency ratio of 21/72, or 16.67 cents, which divides the 100 cent "halftone" into 6 equal parts (100/16.6 = 6) and is thus a "twelfth-tone" (Audio file "1 step in 72-et on C.mid" not found). 72 being divisible by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, and 72, 72-tet includes those equal temperaments.

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] who considered it as a good approach to the continuum of sound. 72-et 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.

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]

Byzantine music

The 72 equal temperament is used in Byzantine music theory,[3] 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.

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

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.

Theoretical properties

72 equal temperament contains at the same time tempered semitones, third-tones, quartertones and sixth-tones, which makes it a very versatile temperament.

Notation

The Maneri-Sims notation system designed for 72-et uses the accidentals Maneri-Sims down arrow.gif and Maneri-Sims up arrow.gif for 1/12th-tone down and up (1 step = 16.6 cents), Half down arrow.png and Half up arrow.png for 1/6th down and up (2 steps = 33.3 cents), and x14px and x14px for 1/4 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: Half down arrow.png♭ or x14px♭, but without the intervening space. A 1/3rd tone may be one of the following ↑x14px, ↓x14px, Half down arrow.png♯, or Half up arrow.png♭ (4 steps = 66.6) while 5 steps may be Half up arrow.pngx14px or ↑♭ (83.3 cents).

References

  1. ↑ A. Hába: "Harmonické základy ctvrttónové soustavy". German translation: "Neue Harmonielehre des diatonischen, chromatischen Viertel-, Drittel-, Sechstel- und Zwölftel-tonsystems" by the author. Fr. Kistner & C.F.W. Siegel, Leipzig, 1927. Universal, Wien, 1978. Revised by Erich Steinhard, "Grundfragen der mikrotonalen Musik"; Bd. 3, Musikedition Nymphenburg 2001, Filmkunst-Musikverlag, München, 251 pages.
  2. ↑ I. Wyschnegradsky: "L'ultrachromatisme et les espaces non octaviants", La Revue Musicale # 290-291, pp. 71-141, Ed. Richard-Masse, Paris, 1972; La Loi de la Pansonorité (Manuscript, 1953), Ed. Contrechamps, Geneva, 1996. Preface by Pascale Criton, edited by Franck Jedrzejewski. ISBN 978-2-940068-09-8; Une philosophie dialectique de l'art musical (Manuscript, 1936), Ed. L'Harmattan, Paris, 2005, edited by Franck Jedrzejewski. ISBN 978-2-7475-8578-1.
  3. ↑ [1] G. Chryssochoidis, D. Delviniotis and G. Kouroupetroglou, "A semi-automated tagging methodology for Orthodox Ecclesiastic Chant Acoustic corpora", Proceedings SMC'07, 4th Sound and Music Computing Conference, Lefkada, Greece (11–13 July 2007).

External links

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