# 96 equal temperament

In music, 96 equal temperament, called 96-TET, 96-EDO ("Equal Division of the Octave"), or 96-ET, is the tempered scale derived by dividing the octave into 96 equal steps (equal frequency ratios). Each step represents a frequency ratio of 962, 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 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.[1]

## 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 (e.g. 0, 1, 2, 3, 4, ..., 95, 0)

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 name size (steps) size (cents) midi just ratio just (cents) midi error (cents)
octave 96 1200   2:1 1200.00   +00.00
semidiminished octave 92 1150   35:18 1151.23   01.23
supermajor seventh 91 1137.5 27:14 1137.04   +00.46
major seventh 87 1087.5 15:80 1088.27   00.77
neutral seventh, major tone 84 1050   11:60 1049.36   +00.64
neutral seventh, minor tone 83 1037.5 20:11 1035.00   +02.50
large just minor seventh 81 1012.5 9:5 1017.60   05.10
small just minor seventh 80 1000   16:90 0996.09   +03.91
subminor seventh 78 0975 7:4 0968.83   +06.17
supermajor sixth 75 937.5 12:7 933.13   + 4.17
major sixth 71 0887.5 5:3 0884.36   +03.14
neutral sixth 68 0850   18:11 0852.59   02.59
minor sixth 65 0812.5 8:5 0813.69   01.19
subminor sixth 61 0762.5 14:90 0764.92   02.42
perfect fifth 56 0700   3:2 0701.96   01.96
minor fifth 52 0650   16:11 0648.68   +01.32
lesser septimal tritone 47 0587.5 7:5 0582.51   +04.99
major fourth 44 0550   11:80 0551.32   01.32
perfect fourth 40 0500   4:3 0498.04   +01.96
tridecimal major third 36 0450   13:10 0454.21   04.21
septimal major third 35 0437.5 9:7 0435.08   +02.42
major third 31 0387.5 5:4 0386.31   +01.19
undecimal neutral third 28 0350   011:9 0347.41   +02.59
Superminor third 27 0337.5 017:14 0336.13   +01.37
77th harmonic 26 0325   077:64 0320.14   +04.86
minor third 25 0312.5 6:5 0315.64   03.14
Second septimal minor third 24 0300   25:21 0301.85   01.85
Tridecimal minor third 23 0287.5 13:11 0289.21   01.71
augmented second, just 22 0275   75:64 0274.58   +00.42
septimal minor third 21 0262.5 7:6 0266.87   04.37
tridecimal five-quarter tone 20 0250   15:13 0247.74   +02.26
septimal whole tone 18 0225 8:7 0231.17   06.17
major second, major tone 16 0200   9:8 0203.91   03.91
major second, minor tone 15 0187.5 10:90 0182.40   +05.10
neutral second, greater undecimal 13 0162.5 11:10 0165.00   02.50
neutral second, lesser undecimal 12 0150   12:11 0150.64   00.64
Greater tridecimal ⅔-tone 11 0137.5 13:12 0138.57   01.07
Septimal diatonic semitone 10 0125   15:14 0119.44   +05.56
diatonic semitone, just 09 0112.5 16:15 0111.73   +00.77
Undecimal minor second (121st subharmonic) 08 0100   128:121 0097.36   02.64
Septimal chromatic semitone 07 087.5 21:20 0084.47   +03.03
Just chromatic semitone 06 075   25:24 0070.67   +04.33
Septimal minor second 05 062.5 28:27 0062.96   00.46
Undecimal quarter-tone (33rd harmonic) 04 0050   33:32 0053.27   03.27
Undecimal diesis 03 0037.5 45:44 0038.91   01.41
septimal comma 02 0025   64:63 0027.26   02.26
septimal semicomma 01 0012.5   126:125 0013.79   01.29
unison 00 0000   1:1 0000.00   +00.00

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 (such as the minor third).

## References

1. ^ Monzo, Joe (2005). "Equal-Temperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo. Retrieved 26 February 2019.

## Further reading

• Sonido 13, Julián Carillo's theory of 96-EDO