96edo
| ← 95edo | 96edo | 97edo → |
96 equal divisions of the octave (abbreviated 96edo or 96ed2), also called 96-tone equal temperament (96tet) or 96 equal temperament (96et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 96 equal parts of exactly 12.500 ¢ each. Each step represents a frequency ratio of 21/96, or the 96th root of 2.
Theory
As a 5-limit system, 96edo can be characterized by the fact that it tempers out both the Pythagorean comma, 531441/524288, Würschmidt's comma, 393216/390625, the unicorn comma, 1594323/1562500, and the kwazy comma, [-53 10 16⟩. It therefore has the same familiar 700-cent fifth as 12edo, and has a best major third of 387.5 cents, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the würschmidt family of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the unicorn family. It supports sitcom temperament.
In the 7-limit, 96 has two possible mappings for 7/4, a sharp one of 975 cents from the patent val, and a flat one of 962.5 cents from 96d. Using the sharp mapping, 96 tempers out 225/224 and supports 7-limit würschmidt temperament, and using the flat mapping it tempers out 126/125 and supports worschmidt temperament. We can also dispense with 7 altogether, and use it as a no-sevens system, where it tempers out 243/242 in the 11-limit and 676/675 in the 13-limit. If we include 7, then the sharp mapping tempers out 99/98 and 176/175 in the 11-limit, and 169/168 in the 13-limit, and this provides the optimal patent val for the interpental temperament. With the flat 7 it tempers out 385/384 in the 11-limit and 196/195 and 364/363 in the 13-limit, and serves for the various temperaments of the unicorn family.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -1.96 | +1.19 | +6.17 | -1.32 | -3.03 | -4.96 | +2.49 | -3.27 | -4.58 | +4.96 |
| relative (%) | +0 | -16 | +9 | +49 | -11 | -24 | -40 | +20 | -26 | -37 | +40 | |
| Steps (reduced) |
96 (0) |
152 (56) |
223 (31) |
270 (78) |
332 (44) |
355 (67) |
392 (8) |
408 (24) |
434 (50) |
466 (82) |
476 (92) | |
As a tuning standard
A step of 96edo is known as a triamu (third MIDI-resolution unit, 3mu, 23 = 8 equal divisions of the 12edo semitone). The internal data structure of the 3mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and three other bits which are not used.
Subsets and supersets
Since 96 factors into 25 × 3, 96edo has subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, and 48.
Intervals
| Steps | Cents | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 12.5 | ^D, v7Eb | |
| 2 | 25 | ^^D, v6Eb | 64/63, 65/64, 66/65 |
| 3 | 37.5 | ^3D, v5Eb | 45/44 |
| 4 | 50 | ^4D, v4Eb | 33/32, 65/63, 77/75 |
| 5 | 62.5 | ^5D, v3Eb | 27/26, 80/77 |
| 6 | 75 | ^6D, vvEb | 25/24 |
| 7 | 87.5 | ^7D, vEb | 21/20 |
| 8 | 100 | D#, Eb | 55/52 |
| 9 | 112.5 | ^D#, v7E | 16/15 |
| 10 | 125 | ^^D#, v6E | |
| 11 | 137.5 | ^3D#, v5E | 13/12 |
| 12 | 150 | ^4D#, v4E | 12/11 |
| 13 | 162.5 | ^5D#, v3E | 11/10 |
| 14 | 175 | ^6D#, vvE | 72/65 |
| 15 | 187.5 | ^7D#, vE | |
| 16 | 200 | E | 9/8, 28/25 |
| 17 | 212.5 | ^E, v7F | 44/39 |
| 18 | 225 | ^^E, v6F | 25/22 |
| 19 | 237.5 | ^3E, v5F | 55/48, 63/55 |
| 20 | 250 | ^4E, v4F | 15/13, 52/45 |
| 21 | 262.5 | ^5E, v3F | 64/55 |
| 22 | 275 | ^6E, vvF | 75/64 |
| 23 | 287.5 | ^7E, vF | 13/11 |
| 24 | 300 | F | 25/21 |
| 25 | 312.5 | ^F, v7Gb | 6/5 |
| 26 | 325 | ^^F, v6Gb | 65/54, 77/64 |
| 27 | 337.5 | ^3F, v5Gb | 39/32, 40/33 |
| 28 | 350 | ^4F, v4Gb | 11/9, 27/22 |
| 29 | 362.5 | ^5F, v3Gb | 16/13 |
| 30 | 375 | ^6F, vvGb | |
| 31 | 387.5 | ^7F, vGb | 5/4 |
| 32 | 400 | F#, Gb | 63/50 |
| 33 | 412.5 | ^F#, v7G | 33/26, 80/63 |
| 34 | 425 | ^^F#, v6G | 32/25 |
| 35 | 437.5 | ^3F#, v5G | |
| 36 | 450 | ^4F#, v4G | 13/10 |
| 37 | 462.5 | ^5F#, v3G | 55/42, 72/55 |
| 38 | 475 | ^6F#, vvG | 21/16 |
| 39 | 487.5 | ^7F#, vG | |
| 40 | 500 | G | 4/3 |
| 41 | 512.5 | ^G, v7Ab | |
| 42 | 525 | ^^G, v6Ab | 65/48 |
| 43 | 537.5 | ^3G, v5Ab | 15/11 |
| 44 | 550 | ^4G, v4Ab | 11/8 |
| 45 | 562.5 | ^5G, v3Ab | 18/13 |
| 46 | 575 | ^6G, vvAb | |
| 47 | 587.5 | ^7G, vAb | 7/5, 45/32 |
| 48 | 600 | G#, Ab | 55/39, 78/55 |
| 49 | 612.5 | ^G#, v7A | 10/7, 64/45 |
| 50 | 625 | ^^G#, v6A | 63/44 |
| 51 | 637.5 | ^3G#, v5A | 13/9, 75/52 |
| 52 | 650 | ^4G#, v4A | 16/11 |
| 53 | 662.5 | ^5G#, v3A | 22/15 |
| 54 | 675 | ^6G#, vvA | 65/44 |
| 55 | 687.5 | ^7G#, vA | |
| 56 | 700 | A | 3/2 |
| 57 | 712.5 | ^A, v7Bb | |
| 58 | 725 | ^^A, v6Bb | 32/21 |
| 59 | 737.5 | ^3A, v5Bb | 55/36 |
| 60 | 750 | ^4A, v4Bb | 20/13, 77/50 |
| 61 | 762.5 | ^5A, v3Bb | 81/52 |
| 62 | 775 | ^6A, vvBb | 25/16 |
| 63 | 787.5 | ^7A, vBb | 52/33, 63/40 |
| 64 | 800 | A#, Bb | |
| 65 | 812.5 | ^A#, v7B | 8/5 |
| 66 | 825 | ^^A#, v6B | |
| 67 | 837.5 | ^3A#, v5B | 13/8 |
| 68 | 850 | ^4A#, v4B | 18/11, 44/27 |
| 69 | 862.5 | ^5A#, v3B | 33/20, 64/39 |
| 70 | 875 | ^6A#, vvB | |
| 71 | 887.5 | ^7A#, vB | 5/3 |
| 72 | 900 | B | 42/25 |
| 73 | 912.5 | ^B, v7C | 22/13 |
| 74 | 925 | ^^B, v6C | 75/44 |
| 75 | 937.5 | ^3B, v5C | 55/32 |
| 76 | 950 | ^4B, v4C | 26/15, 45/26 |
| 77 | 962.5 | ^5B, v3C | |
| 78 | 975 | ^6B, vvC | 44/25 |
| 79 | 987.5 | ^7B, vC | 39/22 |
| 80 | 1000 | C | 16/9, 25/14 |
| 81 | 1012.5 | ^C, v7Db | |
| 82 | 1025 | ^^C, v6Db | 65/36 |
| 83 | 1037.5 | ^3C, v5Db | 20/11 |
| 84 | 1050 | ^4C, v4Db | 11/6 |
| 85 | 1062.5 | ^5C, v3Db | 24/13 |
| 86 | 1075 | ^6C, vvDb | |
| 87 | 1087.5 | ^7C, vDb | 15/8 |
| 88 | 1100 | C#, Db | |
| 89 | 1112.5 | ^C#, v7D | 40/21 |
| 90 | 1125 | ^^C#, v6D | 48/25 |
| 91 | 1137.5 | ^3C#, v5D | 52/27, 77/40 |
| 92 | 1150 | ^4C#, v4D | 64/33 |
| 93 | 1162.5 | ^5C#, v3D | |
| 94 | 1175 | ^6C#, vvD | 63/32, 65/33 |
| 95 | 1187.5 | ^7C#, vD | |
| 96 | 1200 | D | 2/1 |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 3293216/390625, 531441/524288 | [⟨96 152 223]] | +0.240 | 0.732 | 5.86 |
| 2.3.5.11 | 243/242, 5632/5625, 131769/131072 | [⟨96 152 223 332]] | +0.276 | 0.637 | 5.10 |
Rank-2 temperaments
| Periods per 8ve |
Generator | Names |
|---|---|---|
| 1 | 5\96 | Unicorn |
| 1 | 21\96 | Spog |
| 1 | 29\96 | Interpental / Submajor |
| 1 | 31\96 | Würschmidt |
| 2 | 13\96 | Kwazy / Bisupermajor |
| 2 | 25\96 | Vines |
| 12 | 1\96 | Compton / Sitcom |
| 24 | 1\96 | Hours |
Scales
History
96 equal divisions of the octave was first used by the Mexican composer and theorist Julián Carrillo. It has subsequently been used by a number of other composers.
Carrillo
- Main article: Julián Carrillo
Other composers
Works for the Sauter's 1/16tone microtone piano by the composers Ernest Helmuth Flammer, Marc Kilchenmann, Bernfried E. G. Pröve, Martin Imholz, Franck Cristoph Yeznikian, Werner Grimmel, and Alain Bancquart, are recompilated on this CD: 'The Carrillo tone piano' .
- Mohajeri, Shaahin
- Marie, Jean-Etienne
- Criton, Pascale
- Martin Salinas, J.A. 'Autumn' conic bellophone & mixed quintet.mp3 / Pictures of the 96edo conic bellophone
- Haas, Georg Friedrich, "flow and friction"
Music
- Butterfly Skin (2021)
- Dream of a memory, memory of a dream (2021)
- The Persistence of Memory (2021)
See also
- Equal multiplications of MIDI-resolution units