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.
Rank two temperaments[edit]
| 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:
270&87 <<24 -9 -66 12 27 ... ||
494&87 <<51 -1 -133 11 32 ... ||
13-limit detempering of 87et[edit]
[91/90, 49/48, 40/39, 28/27, 25/24, 21/20, 35/33, 16/15, 14/13, 13/12, 12/11, 11/10, 10/9, 28/25, 9/8, 25/22, 8/7, 15/13, 7/6, 75/64, 13/11, 25/21, 6/5, 40/33, 11/9, 16/13, 26/21, 5/4, 44/35, 14/11, 32/25, 9/7, 13/10, 21/16, 33/25, 4/3, 35/26, 27/20, 15/11, 11/8, 18/13, 7/5, 45/32, 64/45, 10/7, 13/9, 16/11, 22/15, 40/27, 52/35, 3/2, 50/33, 32/21, 20/13, 14/9, 25/16, 11/7, 35/22, 8/5, 21/13, 13/8, 18/11, 33/20, 5/3, 42/25, 22/13, 75/44, 12/7, 26/15, 7/4, 44/25, 16/9, 25/14, 9/5, 20/11, 11/6, 24/13, 13/7, 15/8, 66/35, 21/11, 25/13, 27/14, 39/20, 55/28, 99/50, 2]