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:
13-limit detempering of 87et
See detempering.
In this table, "Difference in Cents" indicates whether the 87-interval 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.
todo: Align cent precision of size and difference!
| Steps of 87 |
Size in Cents |
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 |