96 equal temperament


In music, 96 equal temperament, called 96-TET, 96-EDO, or 96-ET, is the tempered scale derived by dividing the octave into 96 equal steps. Each step represents a frequency ratio of, or 12.5 cents. Since 96 factors into 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96, it contains all of those temperaments. Most humans can only hear differences of 6 cents on notes that are played sequentially, and this amount varies according to the pitch, so the use of larger divisions of octave can be considered unnecessary. Smaller differences in pitch may be considered vibrato or stylistic devices.

History and use

96-EDO was first advocated by Julián Carrillo in 1924, with a 16th-tone piano. It was also advocated more recently by Pascale Criton and Vincent-Olivier Gagnon.

Notation

Since 96 = 24 × 4, quarter-tone notation can be used, and split into four parts.
One can split it into four parts like this:
C, C, C/C, C, C,..., C, C
Since it can get confusing with so many accidentals, Julián Carrillo proposed referring to notes by step number from C
Since the 16th-tone piano has a 97-key layout arranged in 8 conventional piano "octaves", music for it is usually notated according to the key the player has to strike. While the entire range of the instrument is only C4–C5, the notation ranges from C0 to C8. Thus, written D0 corresponds to sounding C4 or note 2, and written A♭/G♯2 corresponds to sounding E4 or note 32.

Interval size

Below are some intervals in 96-EDO and how well they approximate just intonation.
interval namesize size midijust ratiojust midierror
octave9612002:11200.000.00
semidiminished octave92115035:181151.23−1.23
supermajor seventh911137.527:141137.04+0.46
major seventh871087.515:81088.27−0.77
neutral seventh, major tone84105011:61049.36+0.64
neutral seventh, minor tone831037.520:111035.00+2.50
large just minor seventh811012.59:51017.60−5.10
small just minor seventh80100016:9996.09+3.91
subminor seventh789757:4968.83+6.17
supermajor sixth75937.512:7933.13+ 4.17
major sixth71887.55:3884.36+3.14
neutral sixth6885018:11852.59−2.59
minor sixth65812.58:5813.69−1.19
subminor sixth61762.514:9764.92−2.42
perfect fifth567003:2701.96−1.96
minor fifth5265016:11648.68+1.32
lesser septimal tritone47587.57:5582.51+4.99
major fourth4455011:8551.32−1.32
perfect fourth405004:3498.04+1.96
tridecimal major third3645013:10454.21−4.21
septimal major third35437.59:7435.08+2.42
major third31387.55:4386.31+1.19
undecimal neutral third2835011:9347.41+2.59
Superminor third27337.517:14336.13+1.37
77th harmonic2632577:64320.14+4.86
minor third25312.56:5315.64−3.14
Second septimal minor third2430025:21301.85−1.85
Tridecimal minor third23287.513:11289.21−1.71
augmented second, just2227575:64274.58+0.42
septimal minor third21262.57:6266.87−4.37
tridecimal five-quarter tone2025015:13247.74+2.26
septimal whole tone182258:7231.17−6.17
major second, major tone162009:8203.91−3.91
major second, minor tone15187.510:9182.40+5.10
neutral second, greater undecimal13162.511:10165.00−2.50
neutral second, lesser undecimal1215012:11150.64−0.64
Greater tridecimal ⅔-tone11137.513:12138.57−1.07
Septimal diatonic semitone1012515:14119.44+5.56
diatonic semitone, just9112.516:15111.73+0.77
Undecimal minor second 8100128:12197.36−2.64
Septimal chromatic semitone787.521:2084.47+3.03
Just chromatic semitone67525:2470.67+4.33
Septimal minor second562.528:2762.96−0.46
Undecimal quarter-tone 45033:3253.27−3.27
Undecimal diesis337.545:4438.91−1.41
septimal comma22564:6327.26−2.26
septimal semicomma112.5126:12513.79−1.29
unison001:10.000.00

Moving from 12-EDO to 96-EDO allows the better approximation of a number of intervals, such as the minor third and major sixth.

Scale diagram

Modes

96-EDO contains all of the 12-EDO modes. However, it contains better approximations to some intervals.