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
| 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
See detempering.
In this table, "Difference in Cents" indicates whether 87 is flat (negative) or sharp (positive) of the detempered interval. For example, 15 steps, at 206.89655 cents, corresponds to 9/8 and is 3.0 cents sharp.
| Steps of 87 | Cents value | Detempered Interval | Difference in Cents |
| 1 | 13.79310 | 91/90 | -5.3 |
| 2 | 27.58621 | 49/48 | -8.1 |
| 3 | 41.37931 | 40/39 | -2.5 |
| 4 | 55.17241 | 28/27 | -7.8 |
| 5 | 68.96552 | 25/24 | -1.7 |
| 6 | 82.75862 | 21/20 | -1.7 |
| 7 | 96.55172 | 35/33 | -5.3 |
| 8 | 110.34483 | 16/15 | -1.4 |
| 9 | 124.13793 | 14/13 | -4.2 |
| 10 | 137.93103 | 13/12 | -0.6 |
| 11 | 151.72414 | 12/11 | 1.1 |
| 12 | 165.51724 | 11/10 | 0.5 |
| 13 | 179.31035 | 10/9 | -3.1 |
| 14 | 193.10345 | 28/25 | -3.1 |
| 15 | 206.89655 | 9/8 | 3.0 |
| 16 | 220.68966 | 25/22 | -0.6 |
| 17 | 234.48276 | 8/7 | 3.3 |
| 18 | 248.27586 | 15/13 | 0.5 |
| 19 | 262.06897 | 7/6 | -4.8 |
| 20 | 275.86207 | 75/64 | 1.3 |
| 21 | 289.65517 | 13/11 | 0.4 |
| 22 | 303.44828 | 25/21 | 1.6 |
| 23 | 317.24138 | 6/5 | 1.6 |
| 24 | 331.03448 | 40/33 | -2.0 |
| 25 | 344.82759 | 11/9 | -2.6 |
| 26 | 358.62069 | 16/13 | -0.9 |
| 27 | 372.41379 | 26/21 | 2.7 |
| 28 | 386.20690 | 5/4 | -0.1 |
| 29 | 400.00000 | 44/35 | 3.8 |
| 30 | 413.79310 | 14/11 | -3.7 |
| 31 | 427.58621 | 32/25 | 0.2 |
| 32 | 441.37931 | 9/7 | 6.3 |
| 33 | 455.17241 | 13/10 | 1.0 |
| 34 | 468.96552 | 21/16 | -1.8 |
| 35 | 482.75862 | 33/25 | 2.1 |
| 36 | 496.55172 | 4/3 | -1.5 |
| 37 | 510.34483 | 35/26 | -4.3 |
| 38 | 524.13793 | 27/20 | 4.6 |
| 39 | 537.93103 | 15/11 | 1.0 |
| 40 | 551.72414 | 11/8 | 0.4 |
| 41 | 565.51724 | 18/13 | 2.1 |
| 42 | 579.31035 | 7/5 | -3.2 |
| 43 | 593.10345 | 45/32 | 2.9 |
| 44 | 606.89655 | 64/45 | -2.9 |
| 45 | 620.68966 | 10/7 | 3.2 |
| 46 | 634.48276 | 13/9 | -2.1 |
| 47 | 648.27586 | 16/11 | -0.4 |
| 48 | 662.06897 | 22/15 | -1.0 |
| 49 | 675.86207 | 40/27 | -4.6 |
| 50 | 689.65517 | 52/35 | 4.3 |
| 51 | 703.44828 | 3/2 | 1.5 |
| 52 | 717.24138 | 50/33 | -2.1 |
| 53 | 731.03448 | 32/21 | 1.8 |
| 54 | 744.82759 | 20/13 | -1.0 |
| 55 | 758.62069 | 14/9 | -6.3 |
| 56 | 772.41379 | 25/16 | -0.2 |
| 57 | 786.20690 | 11/7 | 3.7 |
| 58 | 800.00000 | 35/22 | -3.8 |
| 59 | 813.79310 | 8/5 | 0.1 |
| 60 | 827.58621 | 21/13 | -2.7 |
| 61 | 841.37931 | 13/8 | 0.9 |
| 62 | 855.17241 | 18/11 | 2.6 |
| 63 | 868.96552 | 33/20 | 2.0 |
| 64 | 882.75862 | 5/3 | -1.6 |
| 65 | 896.55172 | 42/25 | -1.6 |
| 66 | 910.34483 | 22/13 | -0.4 |
| 67 | 924.13793 | 75/44 | 0.9 |
| 68 | 937.93103 | 12/7 | 4.8 |
| 69 | 951.72414 | 26/15 | -0.5 |
| 70 | 965.51724 | 7/4 | -3.3 |
| 71 | 979.31035 | 44/25 | 0.6 |
| 72 | 993.10345 | 16/9 | -3.0 |
| 73 | 1006.89655 | 25/14 | 3.1 |
| 74 | 1020.68966 | 9/5 | 3.1 |
| 75 | 1034.48276 | 20/11 | -0.5 |
| 76 | 1048.27586 | 11/6 | -1.1 |
| 77 | 1062.06897 | 24/13 | 0.6 |
| 78 | 1075.86207 | 13/7 | 4.2 |
| 79 | 1089.65517 | 15/8 | 1.4 |
| 80 | 1103.44828 | 66/35 | 5.3 |
| 81 | 1117.24138 | 21/11 | -2.2 |
| 82 | 1131.03448 | 25/13 | -1.1 |
| 83 | 1144.82759 | 27/14 | 7.8 |
| 84 | 1158.62069 | 39/20 | 2.5 |
| 85 | 1172.41379 | 55/28 | 3.6 |
| 86 | 1186.20690 | 99/50 | 3.6 |
| 87 | 1200.00000 | 2/1 | 0.0 |