87edo
| ← 86edo | 87edo | 88edo → |
The 87 equal divisions of the octave (87edo), or the 87(-tone) equal temperament (87tet, 87et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 87 equally-sized steps, where each step is about 13.8 cents.
Theory
87edo 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-odd-limit tonality diamond both uniquely and consistently (see 87edo/13-limit detempering), and is the smallest edo to do so. It is a zeta peak integer edo.
87edo 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, in addition to 245/243, 1029/1024, 3136/3125, and 5120/5103.
87edo 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.
Prime harmonics
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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 |
| … | … | … | … | … | … |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 15625/15552, 67108864/66430125 | [⟨87 138 202]] | -0.299 | 0.455 | 3.30 |
| 2.3.5.7 | 245/243, 1029/1024, 3136/3125 | [⟨87 138 202 244]] | +0.070 | 0.752 | 5.45 |
| 2.3.5.7.11 | 245/243, 385/384, 441/440, 3136/3125 | [⟨87 138 202 244 301]] | +0.033 | 0.676 | 4.90 |
| 2.3.5.7.11.13 | 196/195, 245/243, 352/351, 364/363, 625/624 | [⟨87 138 202 244 301 322]] | -0.011 | 0.625 | 4.53 |
| 2.3.5.7.11.13.17 | 154/153, 196/195, 245/243, 273/272, 364/363, 375/374 | [⟨87 138 202 244 301 322 356]] | -0.198 | 0.738 | 5.35 |
| 2.3.5.7.11.13.17.19 | 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363 | [⟨87 138 202 244 301 322 356 370]] | -0.348 | 0.796 | 5.77 |
13-limit detempering
Rank-2 temperaments
| Periods per octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio (Reduced) |
Temperament |
|---|---|---|---|---|
| 1 | 2\87 | 27.586 | 64/63 | Arch |
| 1 | 4\87 | 55.172 | 33/32 | Escapade / sensa / alphaquarter |
| 1 | 10\87 | 137.931 | 13/12 | Quartemka |
| 1 | 14\87 | 193.103 | 28/25 | Luna / didacus / hemithirds |
| 1 | 17\87 | 234.483 | 8/7 | Slendric / 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 | Emka / emkay |
| 3 | 18\87 (11\87) |
248.276 (151.724) |
15/13 (12/11) |
Hemimist |
| 3 | 23\87 (6\87) |
317.241 (82.759) |
6/5 (21/20) |
Tritikleismic |
| 3 | 28\87 (1\87) |
386.207 (13.793) |
5/4 (126/125) |
Mutt |
| 3 | 36\87 (7\87) |
496.552 (96.552) |
4/3 (18/17~19/18) |
Misty |
| 29 | 28\87 (1\87) |
386.207 (13.793) |
5/4 (121/120) |
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.