72 equal temperament
Encyclopedia
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 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" .
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-et is also cited among the divisions of the tone by Julian 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
, Julian 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.
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 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 ratio
s between notes, and for representing with great accuracy many rational intervals as well as irrational intervals.
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.
and for 1/12th down and up, and for 1/6th down and up, and and for 1/4 up and down.
They may be combined with the traditional sharp and flat symbols by being placed before them, for example: or , but without the intervening space.
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
, which divides the 100 cent "halftone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
" into 6 equal parts (100/16.6 = 6) and is thus a "twelfth-tone" .
This division of the octave has attracted much attention from tuning theorists, since on the one hand it subdivides the standard 12 equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
and on the other hand it accurately represents overtones up to the twelfth partial tone, and hence can be used for 11-limit music
Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
. It was theoreticized in the form of twelfth-tones by Alois Hába
Alois Hába
Alois Hába was a Czech composer, musical theorist and teacher. He is primarily known for his microtonal compositions, especially using the quarter tone scale, though he used others such as sixth-tones and twelfth-tones....
and Ivan Wyschnegradsky, who considered it as a good approach to the continuum of sound. 72-et is also cited among the divisions of the tone by Julian Carrillo
Julián Carrillo
Julián Carrillo Trujillo was a Mexican composer, conductor, violinist and music theorist, famous for developing a theory of microtonal music which he dubbed "The Thirteenth Sound" .-Biography:...
, 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
Alois Hába
Alois Hába was a Czech composer, musical theorist and teacher. He is primarily known for his microtonal compositions, especially using the quarter tone scale, though he used others such as sixth-tones and twelfth-tones....
, Julian Carrillo
Julián Carrillo
Julián Carrillo Trujillo was a Mexican composer, conductor, violinist and music theorist, famous for developing a theory of microtonal music which he dubbed "The Thirteenth Sound" .-Biography:...
, Ivan Wyschnegradsky and Iannis Xenakis
Iannis Xenakis
Iannis Xenakis was a Romanian-born Greek ethnic, naturalized French composer, music theorist, and architect-engineer. He is commonly recognized as one of the most important post-war avant-garde composers...
.
Many other composers use it freely and intuitively, such as jazz musician Joe Maneri
Joe Maneri
Joseph Gabriel Esther "Joe" Maneri , was an American jazz composer, saxophone and clarinet player. Violinist Mat Maneri is his son....
, and classically-oriented composers such as Julia Werntz and others associated with the Boston Microtonal Society. Others, such as New York composer Joseph Pehrson
Joseph Pehrson
- Life :Pehrson comes from Detroit, Michigan. He studied at the University of Michigan and Eastman School of Music. . His teachers include Leslie Bassett, Joseph Schwantner, Otto Luening and Elie Siegmeister. From 1992 to 1993 he was composer-in-residence at the University of Akron. Since 1983 he...
are interested in it because it supports the use of miracle temperament
Miracle temperament
In music, miracle temperament is a regular temperament discovered by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones. Because 15:14 and 16:15 are equated, their ratio 225:224 \left is tempered out, and two secors give an 8:7...
, and still others simply because it approximates higher-limit just intonation, such as Ezra Sims
Ezra Sims
Ezra Sims is one of the pioneers in the field of microtonal composition. He invented a system of notation which was adopted by many microtonal composers after him, including Joseph Maneri....
and James Tenney
James Tenney
James Tenney was an American composer and influential music theorist.-Biography:Tenney was born in Silver City, New Mexico, and grew up in Arizona and Colorado. He attended the University of Denver, the Juilliard School of Music, Bennington College and the University of Illinois...
. There was also an active Soviet school of 72 equal composers, with less familiar names: Evgeny Alexandrovich Murzin, Andrei Volkonsky
Andrei Volkonsky
Prince Andrei Mikhaylovich Volkonsky also Andrey, André, Mikhailovich, Michailovich, Volkonski, Volkonskiy, etc. was a Russian composer of classical music, conductor and harpsichordist. He was a key figure in Early Music Revival in Russia.-Biography:...
, Nikolai Nikolsky, Eduard Artemiev, Alexander Nemtin, Andrei Eshpai
Andrei Eshpai
Andrei Yakovlevich Eshpai is an ethnic Mari composer.Eshpai was born at Kozmodemyansk, Mari El. A Red Army World War II veteran, he studied piano at Moscow Conservatory from 1948 to 1953 under Vladimir Sofronitsky, and composition under Nikolai Rakov, Nikolai Myaskovsky and Evgeny Golubev...
, Gennady Gladkov, Pyotr Meshchianinov, and Stanislav Kreichi.
Byzantine music
The 72 equal temperament is used in Byzantine musicByzantine music
Byzantine music is the music of the Byzantine Empire composed to Greek texts as ceremonial, festival, or church music. Greek and foreign historians agree that the ecclesiastical tones and in general the whole system of Byzantine music is closely related to the ancient Greek system...
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
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
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
Diatonic genus
In ancient Greek music theory, the diatonic genus is the division of the tetrachord from which the modern diatonic scale evolved. The distinguishing characteristic of the diatonic genus is that its largest interval is about the size of a major second...
and the chromatic
Chromatic genus
In Ancient Greek music theory, the chromatic genus is a genus of the tetrachord characterized by an upper interval of a minor third. The two middle notes of the tetrachord were movable while the two outer notes were immovable...
genera in which intervals are based on ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s 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 Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones... |
42 | 700 | 3:2 | 701.96 | −1.96 | ||
| septimal tritone Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7... |
35 | 583.33 | 7:5 | 582.51 | +0.82 | ||
| tridecimal tritone Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... |
34 | 566.67 | 18:13 | 563.38 | +3.28 | ||
| tritone Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... , 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 Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there... |
30 | 500 | 4:3 | 498.04 | +1.96 | ||
| septimal fourth | 28 | 466.66 | 21:16 | 470.78 | −4.11 | ||
| tridecimal major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
27 | 450 | 13:10 | 454.21 | −4.21 | ||
| septimal major third Septimal major third In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal... |
26 | 433.33 | 9:7 | 435.08 | −1.75 | ||
| undecimal major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
25 | 416.67 | 14:11 | 417.51 | −0.84 | ||
| major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
23 | 383.33 | 5:4 | 386.31 | −2.98 | ||
| tridecimal neutral third Neutral third A neutral third is a musical interval wider than a minor third but narrower than a major third . Three distinct intervals may be termed neutral thirds:... |
22 | 366.67 | 16:13 | 359.47 | +7.19 | ||
| neutral third Neutral third A neutral third is a musical interval wider than a minor third but narrower than a major third . Three distinct intervals may be termed neutral thirds:... |
21 | 350 | 11:9 | 347.41 | +2.59 | ||
| minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major... |
19 | 316.67 | 6:5 | 315.64 | +1.03 | ||
| tridecimal minor third | 17 | 283.33 | 13:11 | 289.21 | −5.88 | ||
| septimal minor third Septimal minor third In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5... |
16 | 266.67 | 7:6 | 266.87 | −0.20 | ||
| tridecimal 5/4 tone | 15 | 250 | 15:13 | 247.74 | +2.26 | ||
| septimal whole tone Septimal whole tone In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies. It is about 231 cents wide in just intonation. Although 24 equal temperament does not match this interval particularly well, its... |
14 | 233.33 | 8:7 | 231.17 | +2.16 | ||
| 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 Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... |
10 | 166.67 | 11:10 | 165.00 | +1.66 | ||
| lesser undecimal neutral second Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... |
9 | 150 | 12:11 | 150.64 | −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 | 14:13 | 128.30 | +5.04 | ||
| septimal diatonic semitone | 7 | 116.67 | 15:14 | 119.44 | −2.78 | ||
| diatonic semitone Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... |
7 | 116.67 | 16:15 | 111.73 | +4.94 | ||
| septimal chromatic semitone Septimal chromatic semitone In music, a septimal chromatic semitone or minor semitone is the interval 21:20 . It is about 84.47 cents. The septimal chromatic semitone may be derived from the harmonic series as the interval between the twentieth and twenty-first harmonics.... |
5 | 83.33 | 21:20 | 84.47 | −1.13 | ||
| chromatic semitone Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... |
4 | 66.67 | 25:24 | 70.67 | −4.01 | ||
| septimal third-tone Septimal third-tone A septimal 1/3-tone is an interval with the ratio of 28:27, which is the difference between the perfect fourth and the supermajor third. It is about 62.96 cents wide. The septimal 1/3-tone can be viewed either as a musical interval in its own right, or as a comma; if it is tempered out in a given... |
4 | 66.67 | 28:27 | 62.96 | +3.71 | ||
| septimal quarter tone Septimal quarter tone A septimal quarter-tone is an interval with the ratio of 36:35 , which is the difference between the septimal minor third and the Just minor third , or about 48.77 cents wide. The name derives from the interval being the 7-limit approximation of a quarter tone... |
3 | 50 | 36:35 | 48.77 | +1.23 | ||
| septimal diesis Septimal diesis In music, septimal diesis is an interval with the ratio of 49:48 , which is the difference between the septimal whole tone and the septimal minor third. It is about 35.7 cents wide, which is narrower than a quarter-tone but wider than the septimal comma... |
2 | 33.33 | 49:48 | 35.70 | −2.36 | ||
| undecimal comma Comma (music) In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning... |
1 | 16.67 | 100:99 | 17.40 | −0.73 |
Although 12-ET
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
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
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
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
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
and 41-ET
41 equal temperament
In music, 41 equal temperament, often abbreviated 41-tET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally-sized steps . Each step represents a frequency ratio of 21/41, or 29.27 cents , an interval close in size to the septimal comma. 41-ET can be seen as a...
, 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 accidentalsand for 1/12th down and up, and for 1/6th down and up, and and for 1/4 up and down.
They may be combined with the traditional sharp and flat symbols by being placed before them, for example: or , but without the intervening space.
