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 is 13.79 cents.
Theory
87et 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 (see 87edo/13-limit detempering), and is the smallest equal temperament to do so. It is a zeta peak integer edo.
87et also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 29 are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they don't combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.
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 | ||
|---|---|---|---|---|---|
| 13-Limit | 31-Limit No-7s Extension | ||||
| 0 | 0.000 | 1/1 | P1 | D | |
| 1 | 13.793 | 126/125, 100/99, 91/90 | ^1 | ^D | |
| 2 | 27.586 | 81/80, 64/63, 49/48, 55/54, 65/64 | ^^1 | ^^D | |
| 3 | 41.379 | 50/49, 45/44, 40/39 | 39/38 | ^31 | ^3D/v3Eb |
| 4 | 55.172 | 28/27, 36/35, 33/32 | 34/33, 30/29, 32/31, 31/30 | vvm2 | vvEb |
| 5 | 68.966 | 25/24, 27/26, 26/25 | 24/23 | vm2 | vEb |
| 6 | 82.759 | 21/20, 22/21 | 20/19, 23/22 | m2 | Eb |
| 7 | 96.552 | 35/33 | 18/17, 19/18 | ^m2 | ^Eb |
| 8 | 110.345 | 16/15 | 17/16, 33/31, 31/29 | ^^m2 | ^^Eb |
| 9 | 124.138 | 15/14, 14/13 | 29/27 | vv~2 | ^3Eb |
| 10 | 137.931 | 13/12, 27/25 | 25/23 | v~2 | ^4Eb |
| 11 | 151.724 | 12/11, 35/32 | ^~2 | v4E | |
| 12 | 165.517 | 11/10 | 32/29, 34/31 | ^^~2 | v3E |
| 13 | 179.310 | 10/9 | vvM2 | vvE | |
| 14 | 193.103 | 28/25 | 19/17, 29/26 | vM2 | vE |
| 15 | 206.897 | 9/8 | 26/23 | M2 | E |
| 16 | 220.690 | 25/22 | 17/15, 33/29 | ^M2 | ^E |
| 17 | 234.483 | 8/7 | 31/27 | ^^M2 | ^^E |
| 18 | 248.276 | 15/13 | 22/19, 38/33, 23/20 | ^3M2/v3m3 | ^3E/v3F |
| 19 | 262.089 | 7/6 | 29/25, 36/31 | vvm3 | vvF |
| 20 | 275.862 | 75/64 | 27/23, 34/29 | vm3 | vF |
| 21 | 289.655 | 32/27, 33/28, 13/11 | m3 | F | |
| 22 | 303.448 | 25/21 | 19/16, 31/26 | ^m3 | ^F |
| 23 | 317.241 | 6/5 | ^^m3 | ^^F | |
| 24 | 331.034 | 40/33 | 23/19, 29/24 | vv~3 | ^3F |
| 25 | 344.828 | 11/9, 39/32 | v~3 | ^4F | |
| 26 | 358.621 | 27/22, 16/13 | 38/31 | ^~3 | v4F# |
| 27 | 372.414 | 26/21 | 31/25, 36/29 | ^^3 | v3F# |
| 28 | 386.207 | 5/4 | vvM3 | vvF# | |
| 29 | 400.000 | 44/35 | 34/27, 24/19, 29/23 | vM3 | vF# |
| 30 | 413.793 | 81/64, 14/11, 33/26 | 19/15 | M3 | F# |
| 31 | 427.586 | 32/25 | 23/18 | ^M3 | ^F# |
| 32 | 441.379 | 9/7, 35/27 | 22/17, 31/24, 40/31 | ^^M3 | ^^F# |
| 33 | 455.172 | 13/10 | 30/23 | ^3M3/v34 | ^3F#/v3G |
| 34 | 468.966 | 21/16 | 17/13, 25/19, 38/29 | vv4 | vvG |
| 35 | 482.759 | 33/25 | v4 | vG | |
| 36 | 496.552 | 4/3 | P4 | G | |
| 37 | 510.345 | 35/26 | 31/23 | ^4 | ^G |
| 38 | 524.138 | 27/20 | 23/17 | ^^4 | ^^G |
| 39 | 537.931 | 15/11 | 26/19, 34/25 | ^34 | ^3G |
| 40 | 551.724 | 11/8, 48/35 | ^44 | ^4G | |
| 41 | 565.517 | 18/13 | 32/23 | v4A4, vd5 | v4G#, vAb |
| 42 | 579.310 | 7/5 | 46/33 | v3A4, d5 | v3G#, Ab |
| 43 | 593.103 | 45/32 | 24/17, 38/27, 31/22 | vvA4, ^d5 | vvG#, ^Ab |
| … | … | … | … | … | … |
Just approximation
Selected just intervals
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | prime 23 | prime 29 | prime 31 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.00 | +1.49 | -0.11 | -3.31 | +0.41 | +0.85 | +5.39 | +5.94 | +6.21 | +4.91 | -0.21 |
| relative (%) | 0.0 | +10.8 | -0.8 | -24.0 | +2.9 | +6.2 | +39.1 | +43.0 | +45.0 | +35.6 | -1.5 | |
Temperament Measures
The following table shows TE temperament measures (RMS normalized by the rank) of 87et.
| 3-limit | 5-limit | 7-limit | 11-limit | 13-limit | 17-limit | 19-limit | 23-limit | 29-limit | 31-limit | ||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Octave stretch (¢) | -0.471 | -0.299 | +0.070 | +0.033 | -0.011 | -0.198 | -0.348 | -0.462 | -0.517 | -0.466 | |
| Error | absolute (¢) | 0.471 | 0.455 | 0.752 | 0.676 | 0.625 | 0.738 | 0.796 | 0.816 | 0.792 | 0.772 |
| relative (%) | 3.42 | 3.30 | 5.45 | 4.90 | 4.53 | 5.35 | 5.77 | 5.92 | 5.74 | 5.60 | |
13-limit detempering of 87et
- See also: Detempering
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 | 26\87 | 358.621 | 16/13 | Restles |
| 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 |
| 3 | 36\87 | 496.552 | 4/3 | Misty |
| 29 | 28\87 | 386.207 | 5/4 | Mystery |
87 can serve as a MOS in these:
Scales
Harmonic Scale
87edo accurately approximates the mode 8 of harmonic series, and the only intervals not distinct are 14/13 and 15/14. It does mode 16 fairly decent, with the only anomaly at 28/27 (4 steps) and 29/28 (5 steps).
Mode 8
| Overtones | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| JI Ratios | 1/1 | 9/8 | 5/4 | 11/8 | 3/2 | 13/8 | 7/4 | 15/8 | 2/1 |
| … in cents | 0.0 | 203.9 | 386.3 | 551.3 | 702.0 | 840.5 | 968.8 | 1088.3 | 1200.0 |
| Degrees in 87edo | 0 | 15 | 28 | 40 | 51 | 61 | 70 | 79 | 87 |
| … in cents | 0.0 | 206.9 | 386.2 | 551.7 | 703.5 | 841.4 | 965.5 | 1089.7 | 1200.0 |
- The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.
Mode 16
| Odd overtones | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 |
| JI Ratios | 17/16 | 19/16 | 21/16 | 23/16 | 25/16 | 27/16 | 29/16 | 31/16 |
| … in cents | 105.0 | 297.5 | 470.8 | 628.3 | 772.6 | 905.9 | 1029.6 | 1145.0 |
| Degrees in 87edo | 8 | 22 | 34 | 46 | 56 | 66 | 75 | 83 |
| … in cents | 110.3 | 303.4 | 469.0 | 634.5 | 772.4 | 910.3 | 1034.5 | 1144.8 |
- The scale in adjacent steps is 8, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 4, 5, 4, 4, 4.
- 25 and 31 are close matches.
- 21 is a little bit flat, but still decent.
- The others (17, 19, 23, 27 and 29) are extremely sharp, but the intervals between them are close.