87edo
← 86edo | 87edo | 88edo → |
87 equal divisions of the octave (abbreviated 87edo or 87ed2), also called 87-tone equal temperament (87tet) or 87 equal temperament (87et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 87 equal parts of about 13.8 ¢ each. Each step represents a frequency ratio of 21/87, or the 87th root of 2.
Theory
87edo is solid as both a 13-limit (or 15-odd-limit) and as a 5-limit system, and does well enough in any limit in between. It is the smallest edo that is distinctly consistent in the 13-odd-limit tonality diamond, and also the smallest edo with relative interval errors of no greater than 25% on all of the first 16 harmonics of the harmonic series. It is also 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 do not combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.
The equal temperament tempers out 15625/15552 (kleisma), [26 -12 -3⟩ (misty comma), and [46 -29⟩ (29-comma) in the 5-limit, in addition to 245/243, 1029/1024, 3136/3125, and 5120/5103 in the 7-limit. In the 13-limit, notably 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000.
87edo is a particularly good tuning for rodan, the 41 & 46 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
In higher limits it excels as a subgroup temperament, especially as an incomplete 71-limit temperament with 128/127 and 129/128 (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of Ringer 87, thus tempering S116 through S137 by patent val and corresponding to the gravity of the fact that 87edo is a circle of 126/125's, meaning (126/125)87 only very slightly exceeds the octave.
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.00 | +1.49 | -0.11 | -3.31 | +0.41 | +0.85 | +5.39 | +5.94 | +6.21 | +4.91 | -0.21 | -3.07 |
Relative (%) | +0.0 | +10.8 | -0.8 | -24.0 | +2.9 | +6.2 | +39.1 | +43.0 | +45.0 | +35.6 | -1.5 | -22.2 | |
Steps (reduced) |
87 (0) |
138 (51) |
202 (28) |
244 (70) |
301 (40) |
322 (61) |
356 (8) |
370 (22) |
394 (46) |
423 (75) |
431 (83) |
453 (18) |
Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -1.48 | -1.17 | -3.44 | -4.54 | +2.90 | +0.36 | +3.45 | -0.39 | +6.69 | -5.92 | +5.13 | -5.36 |
Relative (%) | -10.7 | -8.5 | -24.9 | -32.9 | +21.0 | +2.6 | +25.0 | -2.8 | +48.5 | -42.9 | +37.2 | -38.9 | |
Steps (reduced) |
466 (31) |
472 (37) |
483 (48) |
498 (63) |
512 (77) |
516 (81) |
528 (6) |
535 (13) |
539 (17) |
548 (26) |
555 (33) |
563 (41) |
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 | 91/90, 100/99, 126/125 | ^1 | ^D | |
2 | 27.586 | 49/48, 55/54, 64/63, 65/64, 81/80 | ^^1 | ^^D | |
3 | 41.379 | 40/39, 45/44, 50/49 | 39/38 | ^31 | ^3D/v3Eb |
4 | 55.172 | 28/27, 33/32, 36/35 | 30/29, 31/30, 32/31, 34/33 | vvm2 | vvEb |
5 | 68.966 | 25/24, 26/25, 27/26 | 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, 31/29, 33/31 | ^^m2 | ^^Eb |
9 | 124.138 | 14/13, 15/14 | 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, 23/20, 38/33 | ^3M2/v3m3 | ^3E/v3F |
19 | 262.089 | 7/6 | 29/25, 36/31 | vvm3 | vvF |
20 | 275.862 | 75/64 | 20/17, 27/23, 34/29 | vm3 | vF |
21 | 289.655 | 13/11, 32/27, 33/28 | 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 | 16/13, 27/22 | 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 | 24/19, 29/23, 34/27 | vM3 | vF# |
30 | 413.793 | 14/11, 33/26, 81/64 | 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, 31/22, 38/27 | vvA4, ^d5 | vvG#, ^Ab |
… | … | … | … | … | … |
Approximation to JI
Interval mappings
The following table shows how 15-odd-limit intervals are represented in 87edo. Prime harmonics are in bold.
As 87edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.
Interval and complement | Error (abs, ¢) | Error (rel, %) |
---|---|---|
1/1, 2/1 | 0.000 | 0.0 |
5/4, 8/5 | 0.107 | 0.8 |
11/8, 16/11 | 0.406 | 2.9 |
13/11, 22/13 | 0.445 | 3.2 |
11/10, 20/11 | 0.513 | 3.7 |
15/13, 26/15 | 0.535 | 3.9 |
13/12, 24/13 | 0.642 | 4.7 |
13/8, 16/13 | 0.852 | 6.2 |
13/10, 20/13 | 0.958 | 6.9 |
15/11, 22/15 | 0.980 | 7.1 |
11/6, 12/11 | 1.087 | 7.9 |
15/8, 16/15 | 1.386 | 10.1 |
3/2, 4/3 | 1.493 | 10.8 |
5/3, 6/5 | 1.600 | 11.6 |
13/9, 18/13 | 2.135 | 15.5 |
11/9, 18/11 | 2.580 | 18.7 |
9/8, 16/9 | 2.987 | 21.7 |
9/5, 10/9 | 3.093 | 22.4 |
7/5, 10/7 | 3.202 | 23.2 |
7/4, 8/7 | 3.309 | 24.0 |
11/7, 14/11 | 3.715 | 26.9 |
13/7, 14/13 | 4.160 | 30.2 |
15/14, 28/15 | 4.695 | 34.0 |
7/6, 12/7 | 4.802 | 34.8 |
9/7, 14/9 | 6.295 | 45.6 |
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 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
---|---|---|---|---|
1 | 2\87 | 27.586 | 64/63 | Arch |
1 | 4\87 | 55.172 | 33/32 | Escapade / escaped / 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
87 can serve as a MOS in these:
Scales
MOS scales
Harmonic scale
87edo accurately approximates the mode 8 of harmonic series, and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently.
(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 12)
Overtones | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
JI Ratios | 1/1 | 13/12 | 7/6 | 5/4 | 4/3 | 17/12 | 3/2 | 19/12 | 5/3 | 7/4 | 11/6 | 23/12 | 2/1 |
… in cents | 0.0 | 138.6 | 266.9 | 386.3 | 498.0 | 603.0 | 702.0 | 795.6 | 884.4 | 968.8 | 1049.4 | 1126.3 | 1200.0 |
Degrees in 87edo | 0 | 10 | 19 | 28 | 36 | 44 | 51 | 58 | 64 | 70 | 76 | 82 | 87 |
… in cents | 0.0 | 137.9 | 262.1 | 386.2 | 496.6 | 606.9 | 703.4 | 800.0 | 882.8 | 965.5 | 1048.3 | 1131.0 | 1200.0 |
- The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.
- 13, 15, 16, 18, 20, and 22 are close matches.
- 14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
Other scales
Music
- Pianodactyl (archived 2010) – SoundCloud | detail | play – rodan[26] in 87edo tuning