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). Audio file "72-tet scale on C.mid" not found Each step represents a frequency ratio of 722, or ⁠16 + 2 / 3 cents, which divides the 100 cent 12 EDO "halftone" into 6 equal parts (100 cents ÷ ⁠16 + 2 / 3 = 6 steps, exactly) and is thus a "twelfth-tone" (Audio file "1 step in 72-et on C.mid" not found). 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 (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,[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 (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.[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 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.[citation needed] The ANS synthesizer uses 72 equal temperament.

Notation

The Maneri-Sims notation system designed for 72 EDO uses the accidentals and for 1/ 12  tone down and up (1 step = ⁠16 + 2 / 3 cents), File:Half down arrow.png and File:Half up arrow.png for  1 / 6 down and up (2 steps = ⁠33 + 1 / 3 cents), and File:Sims flagged arrow down.svg and File:Sims flagged arrow up.svg for septimal  1 / 4 up and down (3 steps = 50 cents = half a 12 EDO sharp). They may be combined with the traditional sharp and flat symbols (6 steps = 100 cents) by being placed before them, for example: File:Half down arrow.png or File:Sims flagged arrow up.svg, but without the intervening space. A  1 / 3 tone may be one of the following File:Sims flagged arrow up.svg, File:Sims flagged arrow down.svg, File:Half down arrow.png, or File:Half up arrow.png (4 steps = ⁠66 + 2 / 3 ) while 5 steps may be File:Half up arrow.pngFile:Sims flagged arrow up.svg, , or (⁠83 + 1 / 3 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 ⁠8 + 1 / 3 cents from its nearest 72 EDO note, since the step size between them is ⁠16 + 2 / 3 cents. 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.[citation needed]

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

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
12 tone Audio file "Major scale on C.mid" not found and 72 tone Audio file "Regular diatonic tunings 72-tone scale.mid" not found 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

References

  1. Hába, A. (1978) [1927]. Harmonické základy ctvrttónové soustavy [German translation Neue Harmonielehre des diatonischen, chromatischen Viertel-, Drittel-, Sechstel- und Zwölftel-tonsystems   English translation Harmonic Fundamentals of the Quarter-Tone System] (in čeština and Deutsch). Translated by Kistner, Fr. Leipzig (1927) / Wien, 1978: C.F.W. Siegel (1927) / Universal (1978).{{cite book}}: CS1 maint: location (link)
    Revised German edition:
    Hába, A. (2001) [1927, 1978]. Steinhard, Erich (ed.). Grundfragen der mikrotonalen Musik [Foundations of Microtonal Music] (in Deutsch). Vol. 3. Kistner, Fr. (original translation) (rev. ed.). München, DE: Musikedition Nymphenburg Filmkunst-Musikverlag.
  2. Wyschnegradsky, I. (1972). "L'ultrachromatisme et les espaces non octaviants". La Revue Musicale (290–291): 71–141.
  3. Jedrzejewski, Franck, ed. (1996) [1953]. La Loi de la Pansonorité [The Laws of Multitonal Music] (manuscript) (in français). Criton, Pascale (preface). Geneva, CH: Ed. Contrechamps. ISBN 978-2-940068-09-8.
  4. Jedrzejewski, Franck, ed. (2005) [1936]. Une philosophie dialectique de l'art musical [A Dialectical Philosophy of Musical Art] (manuscript) (in français). Paris, FR: Ed. L'Harmattan. ISBN 978-2-7475-8578-1.
  5. Chryssochoidis, G.; Delviniotis, D.; Kouroupetroglou, G. (11–13 July 2007). "A semi-automated tagging methodology for Orthodox ecclesiastic chant acoustic corpora" (PDF). Proceedings SMC'07. 4th Sound and Music Computing Conference. Lefkada, Greece. Archived (PDF) from the original on 15 August 2007. Retrieved 24 April 2008.

External links