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). About this sound Play  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" (About this sound Play ). 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.

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.

Byzantine music[edit]

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[edit]

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

About this sound play diatonic scale in 72-et 

About this sound contrast with just diatonic scale 

About this sound contrast with diatonic scale in 12-et 

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[edit]

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

Notation[edit]

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 down and up (16.6 cents), Half down arrow.png and Half up arrow.png for 1/6th down and up (33.3 cents), and Check arrow.gif and Check up arrow.gif for 1/4 up and down (50 cents).

They may be combined with the traditional sharp and flat symbols by being placed before them, for example: Half down arrow.png or Check up arrow.gif, but without the intervening space.

References[edit]

  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[edit]