84edo
84edo divides the octave into 84 equal parts of size 14.286 cents each.
Theory
Its patent val ⟨84 133 195 236 291] makes it an excellent orwell tuning and also a good one for compton, and the 84e val, ⟨84 133 195 236 290], is almost identical to the 11-limit POTE tuning for orwell. In the 13-limit it is the optimal patent val for the rank five temperament tempering out 144/143.
84edo is where the orwell temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the EDO, referencing the book 1984. The maximum evenness orwell in this temperament is a 31 note scale.
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Table of intervals
| Step | Size (Cents) | Orwell[31] Name | Associated ratio |
|---|---|---|---|
| 0 | 0.000 | unison, prime | 1/1 |
| 3 | 42.857 | second | |
| 6 | 85.714 | third | |
| 9 | 128.571 | ||
| 11 | 157.142 | ||
| 14 | 200.000 | ||
| 17 | 242.857 | ||
| 19 | 271.428 | eighth | 7/6 |
| 38 | fifteenth | 11/8 | |
| 57 | twenty-second | 5/3 |
Tempered commas
5-limit commas: 78732/78125, 531441/524288, 2109375/2097152
7-limit commas: 225/224, 1728/1715, 2430/2401, 6144/6125
11-limit commas: 441/440, 1344/1331, 1375/1372
84e: 99/98, 121/120, 176/175, 385/384, 540/539, 5632/5625
13-limit commas: 144/143, 351/350, 364/363, 625/625
84e: 275/273, 640/637, 351/350, 352/351, 625/624, 1001/1000
Music
- Ten by John Cage, 1991, for chamber ensemble. Ives Ensemble recording (YouTube)
- Two4 by John Cage, 1991, for violin and piano or shō. Harr & Miyata recording (YouTube)
- Two5 by John Cage, 1991, for tenor trombone and piano. Fulkerson & Denyer recording (YouTube).
- Two6 by John Cage, 1992, for violin and piano. Haar & Snijders recording (YouTube).