96 equal temperament

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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[edit]

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[edit]

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/Chalf sharp, Chalf sharp, Chalf sharp, ..., 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[edit]

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

Modes[edit]

96-EDO contains all of the 12-EDO modes. However, it contains better approximations to some intervals (such as the minor third).

See also[edit]

References[edit]

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

Further reading[edit]

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