87edo
The 87 equal temperament, often abbreviated 87-tET, 87-EDO, or 87-ET, is the scale derived by dividing the octave into 87 equally-sized steps, where each step represents a frequency ratio of 13.79 cents. It is solid as both a 13-limit (or 15 odd limit) and as a 5-limit system, and of course does well enough in any limit in between. It represents the 13-limit tonality diamond both uniquely and consistently, and is the smallest equal temperament to do so.
87et tempers out 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000 as well as the 29-comma, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.
87et is a particularly good tuning for rodan temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit POTE generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for clyde temperament is 0.04455 cents sharp of the 7-limit POTE generator.
Intervals
| # | Cents | Approximated Ratios | Ups and Downs Notation |
|---|---|---|---|
| 0 | 0.000 | 1/1 | D |
| 1 | 13.793 | 126/125, 100/99, 91/90 | ^D |
| 2 | 27.586 | 81/80, 64/63, 49/48, 55/54, 65/64 | ^^D |
| 3 | 41.379 | 50/49, 45/44 | ^3D/v3Eb |
| 4 | 55.172 | 28/27, 36/35, 33/32 | vvEb |
| 5 | 68.966 | 25/24, 27/26, 26/25 | vEb |
| 6 | 82.759 | 21/20, 22/21 | Eb |
| 7 | 96.552 | 35/33 | ^Eb |
| 8 | 110.345 | 16/15 | ^^Eb |
| 9 | 124.138 | 15/14, 14/13 | ^3Eb |
| 10 | 137.931 | 13/12 | ^4Eb |
| 11 | 151.724 | 12/11 | v4E |
| 12 | 165.517 | 11/10 | v3E |
| 13 | 179.310 | 10/9 | vvE |
| 14 | 193.103 | 28/25 | vE |
| 15 | 206.897 | 9/8 | E |
| 16 | 220.690 | 25/22 | ^E |
| 17 | 234.483 | 8/7 | ^^E |
| 18 | 248.276 | 15/13 | ^3E/v3F |
| 19 | 262.089 | 7/6 | vvF |
| 20 | 275.862 | 75/64 | vF |
| 21 | 289.655 | 33/28, 13/11 | F |
| 22 | 303.448 | 25/21 | ^F |
| 23 | 317.241 | 6/5 | ^^F |
| 24 | 331.034 | 63/52 | ^3F |
| 25 | 344.828 | 11/9, 39/32 | ^4F |
| 26 | 358.621 | 27/22, 16/13 | v4F# |
| 27 | 372.414 | 26/21 | v3F# |
| 28 | 386.207 | 5/4 | vvF# |
| 29 | 400.000 | 63/50, 44/35 | vF# |
| 30 | 413.793 | 14/11, 33/26 | F# |
| 31 | 427.586 | 32/25 | ^F# |
| 32 | 441.379 | 9/7 | ^^F# |
| 33 | 455.172 | 13/10 | ^3F#/v3G |
| 34 | 468.966 | 21/16 | vvG |
| 35 | 482.759 | 33/25 | vG |
| 36 | 496.552 | 4/3 | G |
| 37 | 510.345 | 75/56 | ^G |
| 38 | 524.138 | 27/20 | ^^G |
| 39 | 537.931 | 15/11 | ^3G |
| 40 | 551.724 | 11/8 | ^4G |
| 41 | 565.517 | 18/13 | v4G#, vAb |
| 42 | 579.310 | 7/5, 39/28 | v3G#, Ab |
| 43 | 593.103 | 45/32 | vvG#, ^Ab |
Rank two temperaments
| Periods per octave |
Generator | Cents | Associated ratio |
Temperament |
|---|---|---|---|---|
| 1 | 4\87 | 55.172 | 33/32 | Sensa |
| 1 | 10\87 | 137.931 | 13/12 | Quartemka |
| 1 | 14\87 | 193.103 | 28/25 | Luna / Hemithirds |
| 1 | 17\87 | 234.483 | 8/7 | Rodan |
| 1 | 23\87 | 317.241 | 6/5 | Hanson / Countercata / Metakleismic |
| 1 | 32\87 | 441.379 | 9/7 | Clyde |
| 1 | 38\87 | 524.138 | 65/48 | Widefourth |
| 1 | 40\87 | 551.724 | 11/8 | Emkay |
| 3 | 23\87 | 317.241 | 6/5 | Tritikleismic |
| 29 | 28\87 | 386.207 | 5/4 | Mystery |
87 can serve as a MOS in these:
13-limit detempering of 87et
See also: Detempering
Main article: 87edo/13-limit detempering