# 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 ${\sqrt[{96}]{2}}$, 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
As it can become 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 C$_4$–C$_5$, the notation ranges from C$_0$ to C$_8$. Thus, written D0 corresponds to sounding C↑↑$_4$ or note 2, and written A♭/G♯$_2$ corresponds to sounding E$_4$ 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 | play | 2:1 | 1200.00 | play | +00.00 |
| semidiminished octave | 92 | 1150 | play | 35:18 | 1151.23 | play | −01.23 |
| supermajor seventh | 91 | 1137.5 | | 27:14 | 1137.04 | play | +00.46 |
| major seventh | 87 | 1087.5 | | 15:80 | 1088.27 | play | −00.77 |
| neutral seventh, major tone | 84 | 1050 | play | 11:60 | 1049.36 | play | +00.64 |
| neutral seventh, minor tone | 83 | 1037.5 | | 20:11 | 1035.00 | play | +02.50 |
| large just minor seventh | 81 | 1012.5 | | 9:5 | 1017.60 | play | −05.10 |
| small just minor seventh | 80 | 1000 | play | 16:90 | 0996.09 | play | +03.91 |
| harmonic seventh | 78 | 0975 | | 7:4 | 0968.83 | play | +06.17 |
| supermajor sixth | 75 | 937.5 | | 12:7 | 933.13 | play | + 4.17 |
| major sixth | 71 | 0887.5 | | 5:3 | 0884.36 | play | +03.14 |
| neutral sixth | 68 | 0850 | play | 18:11 | 0852.59 | play | −02.59 |
| minor sixth | 65 | 0812.5 | | 8:5 | 0813.69 | play | −01.19 |
| subminor sixth | 61 | 0762.5 | | 14:90 | 0764.92 | play | −02.42 |
| perfect fifth | 56 | 0700 | play | 3:2 | 0701.96 | play | −01.96 |
| minor fifth | 52 | 0650 | play | 16:11 | 0648.68 | play | +01.32 |
| lesser septimal tritone | 47 | 0587.5 | | 7:5 | 0582.51 | play | +04.99 |
| major fourth | 44 | 0550 | play | 11:80 | 0551.32 | play | −01.32 |
| perfect fourth | 40 | 0500 | play | 4:3 | 0498.04 | play | +01.96 |
| tridecimal major third | 36 | 0450 | play | 13:10 | 0454.21 | play | −04.21 |
| septimal major third | 35 | 0437.5 | | 9:7 | 0435.08 | play | +02.42 |
| major third | 31 | 0387.5 | | 5:4 | 0386.31 | play | +01.19 |
| undecimal neutral third | 28 | 0350 | play | 011:9 | 0347.41 | play | +02.59 |
| superminor third | 27 | 0337.5 | | 017:14 | 0336.13 | play | +01.37 |
| 77th harmonic | 26 | 0325 | play | 077:64 | 0320.14 | play | +04.86 |
| minor third | 25 | 0312.5 | | 6:5 | 0315.64 | play | −03.14 |
| second septimal minor third | 24 | 0300 | play | 25:21 | 0301.85 | play | −01.85 |
| tridecimal minor third | 23 | 0287.5 | | 13:11 | 0289.21 | play | −01.71 |
| augmented second, just | 22 | 0275 | play | 75:64 | 0274.58 | play | +00.42 |
| septimal minor third | 21 | 0262.5 | | 7:6 | 0266.87 | play | −04.37 |
| tridecimal five-quarter tone | 20 | 0250 | play | 15:13 | 0247.74 | play | +02.26 |
| septimal whole tone | 18 | 0225 | | 8:7 | 0231.17 | play | −06.17 |
| major second, major tone | 16 | 0200 | play | 9:8 | 0203.91 | play | −03.91 |
| major second, minor tone | 15 | 0187.5 | | 10:90 | 0182.40 | play | +05.10 |
| neutral second, greater undecimal | 13 | 0162.5 | | 11:10 | 0165.00 | play | −02.50 |
| neutral second, lesser undecimal | 12 | 0150 | play | 12:11 | 0150.64 | play | −00.64 |
| greater tridecimal 2⁄3-tone | 11 | 0137.5 | | 13:12 | 0138.57 | play | −01.07 |
| septimal diatonic semitone | 10 | 0125 | play | 15:14 | 0119.44 | play | +05.56 |
| diatonic semitone, just | 09 | 0112.5 | | 16:15 | 0111.73 | play | +00.77 |
| undecimal minor second | 08 | 0100 | play | 128:121 | 0097.36 | play | −02.64 |
| septimal chromatic semitone | 07 | 087.5 | | 21:20 | 0084.47 | play | +03.03 |
| just chromatic semitone | 06 | 075 | play | 25:24 | 0070.67 | play | +04.33 |
| septimal minor second | 05 | 062.5 | | 28:27 | 0062.96 | play | −00.46 |
| undecimal quarter-tone | 04 | 0050 | play | 33:32 | 0053.27 | play | −03.27 |
| undecimal diesis | 03 | 0037.5 | | 45:44 | 0038.91 | play | −01.41 |
| septimal comma | 02 | 0025 | play | 64:63 | 0027.26 | play | −02.26 |
| septimal semicomma | 01 | 0012.5 | play | 126:125 | 0013.79 | play | −01.29 |
| unison | 00 | 0000 | play | 1:1 | 0000.00 | play | +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).