87edo

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← 86edo 87edo 88edo →
Prime factorization 3 × 29
Step size 13.7931¢ 
Fifth 51\87 (703.448¢) (→17\29)
Semitones (A1:m2) 9:6 (124.1¢ : 82.76¢)
Consistency limit 15
Distinct consistency limit 13
Special properties

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.

Approximation of prime harmonics in 87edo
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)
Approximation of prime harmonics in 87edo (continued)
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.

15-odd-limit intervals in 87edo
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

Main article: 87edo/13-limit detempering

Rank-2 temperaments

Table of rank-2 temperaments by generator
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

Main article: List of MOS scales in 87edo

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

Gene Ward Smith
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