96edo
| ← 95edo | 96edo | 97edo → |
The 96 equal divisions of the octave (96edo), or the 96-tone equal temperament (96tet), 96 equal temperament (96et) when viewed from a regular temperament perspective, divides the octave into 96 equal parts of exactly 12.5 cents each.
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.
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 (%) | +0.0 | -15.6 | +9.5 | +49.4 | -10.5 | -24.2 | -39.6 | +19.9 | -26.2 | -36.6 | +39.7 | |
| Steps (reduced) |
96 (0) |
152 (56) |
223 (31) |
270 (78) |
332 (44) |
355 (67) |
392 (8) |
408 (24) |
434 (50) |
466 (82) |
476 (92) | |
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 |
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
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"