96 equal temperament

Template:Short description 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, CTemplate:Music, CTemplate:MusicTemplate:Music/CTemplate:MusicTemplate:MusicTemplate:Music, CTemplate:MusicTemplate:Music, CTemplate:Music, ..., CTemplate:Music, 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 CTemplate:MusicTemplate:Music₄ 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	Template:Audio	2:1	1200.00	Template:Audio	Template:00.00
semidiminished octave	92	1150	Template:Audio	35:18	1151.23	Template:Audio	−Template:01.23
supermajor seventh	91	1137.5		27:14	1137.04	Template:Audio	+Template:00.46
major seventh	87	1087.5		15:8Template:0	1088.27	Template:Audio	−Template:00.77
neutral seventh, major tone	84	1050	Template:Audio	11:6Template:0	1049.36	Template:Audio	+Template:00.64
neutral seventh, minor tone	83	1037.5		20:11	1035.00	Template:Audio	+Template:02.50
large just minor seventh	81	1012.5		9:5	1017.60	Template:Audio	−Template:05.10
small just minor seventh	80	1000	Template:Audio	16:9Template:0	Template:0996.09	Template:Audio	+Template:03.91
harmonic seventh	78	Template:0975		7:4	Template:0968.83	Template:Audio	+Template:06.17
supermajor sixth	75	937.5		12:7	933.13	Template:Audio	+ 4.17
major sixth	71	Template:0887.5		5:3	Template:0884.36	Template:Audio	+Template:03.14
neutral sixth	68	Template:0850	Template:Audio	18:11	Template:0852.59	Template:Audio	−Template:02.59
minor sixth	65	Template:0812.5		8:5	Template:0813.69	Template:Audio	−Template:01.19
subminor sixth	61	Template:0762.5		14:9Template:0	Template:0764.92	Template:Audio	−Template:02.42
perfect fifth	56	Template:0700	Template:Audio	3:2	Template:0701.96	Template:Audio	−Template:01.96
minor fifth	52	Template:0650	Template:Audio	16:11	Template:0648.68	Template:Audio	+Template:01.32
lesser septimal tritone	47	Template:0587.5		7:5	Template:0582.51	Template:Audio	+Template:04.99
major fourth	44	Template:0550	Template:Audio	11:8Template:0	Template:0551.32	Template:Audio	−Template:01.32
perfect fourth	40	Template:0500	Template:Audio	4:3	Template:0498.04	Template:Audio	+Template:01.96
tridecimal major third	36	Template:0450	Template:Audio	13:10	Template:0454.21	Template:Audio	−Template:04.21
septimal major third	35	Template:0437.5		9:7	Template:0435.08	Template:Audio	+Template:02.42
major third	31	Template:0387.5		5:4	Template:0386.31	Template:Audio	+Template:01.19
undecimal neutral third	28	Template:0350	Template:Audio	Template:011:9	Template:0347.41	Template:Audio	+Template:02.59
superminor third	27	Template:0337.5		Template:017:14	Template:0336.13	Template:Audio	+Template:01.37
77th harmonic	26	Template:0325	Template:Audio	Template:077:64	Template:0320.14	Template:Audio	+Template:04.86
minor third	25	Template:0312.5		6:5	Template:0315.64	Template:Audio	−Template:03.14
second septimal minor third	24	Template:0300	Template:Audio	25:21	Template:0301.85	Template:Audio	−Template:01.85
tridecimal minor third	23	Template:0287.5		13:11	Template:0289.21	Template:Audio	−Template:01.71
augmented second, just	22	Template:0275	Template:Audio	75:64	Template:0274.58	Template:Audio	+Template:00.42
septimal minor third	21	Template:0262.5		7:6	Template:0266.87	Template:Audio	−Template:04.37
tridecimal five-quarter tone	20	Template:0250	Template:Audio	15:13	Template:0247.74	Template:Audio	+Template:02.26
septimal whole tone	18	Template:0225		8:7	Template:0231.17	Template:Audio	−Template:06.17
major second, major tone	16	Template:0200	Template:Audio	9:8	Template:0203.91	Template:Audio	−Template:03.91
major second, minor tone	15	Template:0187.5		10:9Template:0	Template:0182.40	Template:Audio	+Template:05.10
neutral second, greater undecimal	13	Template:0162.5		11:10	Template:0165.00	Template:Audio	−Template:02.50
neutral second, lesser undecimal	12	Template:0150	Template:Audio	12:11	Template:0150.64	Template:Audio	−Template:00.64
greater tridecimal Template:Frac-tone	11	Template:0137.5		13:12	Template:0138.57	Template:Audio	−Template:01.07
septimal diatonic semitone	10	Template:0125	Template:Audio	15:14	Template:0119.44	Template:Audio	+Template:05.56
diatonic semitone, just	Template:09	Template:0112.5		16:15	Template:0111.73	Template:Audio	+Template:00.77
undecimal minor second	Template:08	Template:0100	Template:Audio	128:121	Template:097.36	Template:Audio	−Template:02.64
septimal chromatic semitone	Template:07	Template:087.5		21:20	Template:084.47	Template:Audio	+Template:03.03
just chromatic semitone	Template:06	Template:075	Template:Audio	25:24	Template:070.67	Template:Audio	+Template:04.33
septimal minor second	Template:05	Template:062.5		28:27	Template:062.96	Template:Audio	−Template:00.46
undecimal quarter-tone	Template:04	Template:050	Template:Audio	33:32	Template:053.27	Template:Audio	−Template:03.27
undecimal diesis	Template:03	Template:037.5		45:44	Template:038.91	Template:Audio	−Template:01.41
septimal comma	Template:02	Template:025	Template:Audio	64:63	Template:027.26	Template:Audio	−Template:02.26
septimal semicomma	Template:01	Template:012.5	Template:Audio	126:125	Template:013.79	Template:Audio	−Template:01.29
unison	Template:00	Template:00	Template:Audio	1:1	Template:00.00	Template:Audio	Template: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