In music, 72 equal temperament, called 72-tet, 72-edo, or 72-et, is the scale derived by dividing the octave into twelfth-tones, or in other words 72 equally large steps. Each step represents a frequency ratio of 2^1/72, or 16.667 cents. The cent is a logarithmic unit of measure used for musical intervals. ...

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 accuately represents overtones up to the twelfth partial tone, and hence can be used for 11-limit music. Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ...

A number of composers have made use of it, and these represent widely different points of view and types of musical practice. Many composers use it freely and intuitively, such as jazz musician Joe Maneri, and classically-oriented composers such 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. Other composers who have used it include Alois Haba, Julian Carrillo, Ivan Wyschnegradsky and Iannis Xenakis. 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. Joseph Gabriel Esther Maneri (born February 9, 1927, Brooklyn) is an American jazz composer, saxophone and clarinet player. ... In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, which serves as both the 15/14 and 16/15 semitones. ... Ezra Sims (born January 16, 1928 in Birmingham, Alabama) is one of the pioneers in the field of microtonal composition. ... James Tenney (August 10, 1934 in Silver City, NM) is an American composer and influential music theorist. ... Alois H ba (June 21, 1893 - November 18, 1973) was a Czech composer primarily known for his microtonal compositions, especially using the quarter tone scale, though he used others such as sixth-tones and twelfth-tones. ... Julián Carrillo, 1945. ... Ivan Alexandrovich Vïshnegradsky (1893-1979, also Wyschnegradsky) was a Russian composer primarily known for his microtonal compositions, including the quarter tone scale, though he used scales of up to 71 divisions. ... Iannis Xenakis (Ιάννης Ξενάκης) (May 29, 1922 Brăila - February 4, 2001 Paris) was a Greek composer and architect who spent much of his life in Paris. ...

It also is in use in connection with Byzantine music, which made theoretical use of it, dividing the octave into 72 equal moria, which itself derived from the theories of Aristoxenos, who used something similar. Aristoxenus of Tarentum (4th century BC) was a Greek peripatetic philosopher, and writer on music and rhythm. ...

Theoretical properties

In terms of tuning theory, the 72 equal harmonic system equates to the unison, or "tempers out", the small intervals 225/224, 243/242, 1029/1024, 385/384, 441/440, 540/539, as well as the Pythagorean comma and 15625/15552, among inumerable others; this gives it its own particular character in terms of functional harmony. It also means that 72 supports various temperaments which temper out some, but not all, of the above small intervals. When one ascends by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, one eventually reaches a note around seven octaves above the note one started on, which, when lowered to the same octave as the starting point, is 23. ...

It is important to notice, however, that it does not temper out the syntonic comma of 81/80, and is therefore not a meantone system. Instead, 81/80 becomes one step of the scale. Hence, common practice music needs to be adapted for it to be played in this harmonic system, though the option always remains to use only twelve of the 72 notes. The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ... Meantone temperament is a system of musical tuning. ... In music the common practice period is a long period in western musical history spanning from before the classical era proper to today, dated, on the outside, as 1600-1900. ...

Musical examples

• Night Piece by Ezra Sims, RealAudio excerpt

• In Full Cry by Joe Maneri, RealAudio excerpt

• Violin Cadenza by Julia Wertz, RealAudio excerpt

External Links

• Tonalsoft page on 72-tone equal-temperament / 72-edo
• Tonalsoft page on morion
• The Boston Microtonal Society official site