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 , 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 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 0.00 semidiminished octave 92 1150 35:18 1151.23 −1.23 supermajor seventh 91 1137.5 27:14 1137.04 +0.46 major seventh 87 1087.5 15:8 1088.27 −0.77 neutral seventh, major tone 84 1050 11:6 1049.36 +0.64 neutral seventh, minor tone 83 1037.5 20:11 1035.00 +2.50 large just minor seventh 81 1012.5 9:5 1017.60 −5.10 small just minor seventh 80 1000 16:9 996.09 +3.91harmonic seventh 78 975 7:4 968.83 +6.17 supermajor sixth 75 937.5 12:7 933.13 + 4.17 major sixth 71 887.5 5:3 884.36 +3.14 neutral sixth 68 850 18:11 852.59 −2.59 minor sixth 65 812.5 8:5 813.69 −1.19 subminor sixth 61 762.5 14:9 764.92 −2.42 perfect fifth 56 700 3:2 701.96 −1.96 minor fifth 52 650 16:11 648.68 +1.32 lesser septimal tritone 47 587.5 7:5 582.51 +4.99 major fourth 44 550 11:8 551.32 −1.32 perfect fourth 40 500 4:3 498.04 +1.96 tridecimal major third 36 450 13:10 454.21 −4.21 septimal major third 35 437.5 9:7 435.08 +2.42 major third 31 387.5 5:4 386.31 +1.19 undecimal neutral third 28 350 11:9 347.41 +2.59 superminor third 27 337.5 17:14 336.13 +1.37 77th harmonic 26 325 77:64 320.14 +4.86 minor third 25 312.5 6:5 315.64 −3.14 second septimal minor third24 300 25:21 301.85 −1.85 tridecimal minor third 23 287.5 13:11 289.21 −1.71 augmented second, just 22 275 75:64 274.58 +0.42 septimal minor third 21 262.5 7:6 266.87 −4.37 tridecimal five-quarter tone 20 250 15:13 247.74 +2.26 septimal whole tone 18 225 8:7 231.17 −6.17 major second, major tone 16 200 9:8 203.91 −3.91 major second, minor tone 15187.5 10:9 182.40 +5.10 neutral second, greater undecimal 13 162.5 11:10 165.00 −2.50 neutral second, lesser undecimal 12 150 12:11 150.64 −0.64 greater tridecimal -tone 11 137.5 13:12 138.57 −1.07 septimal diatonic semitone 10 125 15:14 119.44 +5.56 diatonic semitone, just 9 112.5 16:15 111.73 +0.77 undecimal minor second 8 100 128:121 97.36 −2.64 septimal chromatic semitone 7 87.5 21:20 84.47 +3.03 just chromatic semitone 6 75 25:24 70.67 +4.33 septimal minor second 5 62.5 28:27 62.96 −0.46 undecimal quarter-tone4 50 33:32 53.27 −3.27undecimal diesis 3 37.5 45:44 38.91 −1.41septimal comma 2 25 64:63 27.26 −2.26septimal semicomma 1 12.5 126:125 13.79 −1.29 unison 0 0 1:1 0.00 0.00

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).

96 equal temperament

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