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 C₄–C₅, the notation ranges from C₀ to C₈. Thus, written D0 corresponds to sounding C↑↑₄ 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
harmonic 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 2⁄3-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	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	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).

See also

• Musical temperament
• Equal temperament

Further reading

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