87edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|87}}
{{ED intro}}


== Theory ==
== 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]], the smallest edo that is [[purely consistent]]{{idiosyncratic}} in the [[15-odd-limit]] (meaning no greater than 25% [[relative interval error]]s on all of the first 16 [[harmonic]]s of the [[harmonic series]]). It is also a [[zeta peak integer edo]].
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]], the smallest edo that is [[purely consistent]]{{idiosyncratic}} in the [[15-odd-limit]] (maintains [[relative interval error]]s of no greater than 25% on all of the first 16 [[harmonic]]s of the [[harmonic series]]). It is also a [[zeta peak integer edo]]. Since {{nowrap|87 {{=}} 3 × 29}}, 87edo shares the same perfect fifth with [[29edo]].  


87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]] and [[29/1|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 [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.  
87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]], and [[29/1|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 [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.  


The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]), {{monzo| 26 -12 -3 }} ([[misty comma]]), and {{monzo| 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]].  
It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]), {{monzo| 26 -12 -3 }} ([[misty comma]]), and {{monzo| 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 {{nowrap|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 [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
87edo is a particularly good tuning for [[rodan]], the {{nowrap|41 & 46}} temperament. The 8/7 generator of 17\87 is a remarkable 0.00061{{c}} sharper than the 13-limit [[CWE tuning|CWE generator]]. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.01479{{c}} sharp of the 13-limit CWE generator.


=== Prime harmonics ===
=== 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 scale|Ringer]] 87, thus tempering [[Square superparticular|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]])<sup>87</sup> only very slightly exceeds the octave.
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 scale|Ringer]] 87, thus tempering [[S-expression|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]])<sup>87</sup> only very slightly exceeds the octave.
 
{{Harmonics in equal|87|columns=12}}
{{Harmonics in equal|87|columns=12}}
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
=== Subsets and supersets ===
87edo contains [[3edo]] and [[29edo]] as subset edos.


== Intervals ==
== Intervals ==
Line 347: Line 349:
=== Interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals|87}}
{{Q-odd-limit intervals|87}}
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[483zpi]]
| 87.0139255957575
| 13.7908960178956
| 8.869041
| 1.439474
| 18.061741
| 87edo
| 1199.80795355692
| 16
| 14
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 541: Line 512:
87 can serve as a mos in these:
87 can serve as a mos in these:


* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]]) {{multival| 24 -9 -66 12 27 … }}
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])  
* [[Breed|87 & 494]] {{multival| 51 -1 -133 11 32 … }}
* [[Breed|87 & 494]]  


== Scales ==
== Scales ==
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{{main|List of MOS scales in 87edo}}
{{main|List of MOS scales in 87edo}}


=== Harmonic scale ===
=== Harmonic scales ===
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.  
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.  


Line 610: Line 581:
|}
|}


* The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.  
The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.  


==== (Mode 12) ====
==== (Mode 12) ====
Line 691: Line 662:
|}
|}


* The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.  
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.
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 ===
=== Other scales ===
* [[Sequar5m]]
* [[Sequar5m]]
== Instruments ==
* [[Lumatone mapping for 87edo]]
* [[Skip fretting system 87 2 17]]


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025)
; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning
* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning

Latest revision as of 00:25, 16 August 2025

← 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

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, the smallest edo that is purely consistent[idiosyncratic term] in the 15-odd-limit (maintains 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. Since 87 = 3 × 29, 87edo shares the same perfect fifth with 29edo.

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.

It 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 0.00061 ¢ sharper than the 13-limit CWE generator. Also, the 32\87 generator for clyde temperament is 0.01479 ¢ sharp of the 13-limit CWE 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)

Subsets and supersets

87edo contains 3edo and 29edo as subset edos.

Intervals

# Cents Approximated ratios Ups and downs notation
13-limit 31-limit extension
0 0.0 1/1 P1 D
1 13.8 91/90, 100/99, 126/125 ^1 ^D
2 27.6 49/48, 55/54, 64/63, 65/64, 81/80 ^^1 ^^D
3 41.4 40/39, 45/44, 50/49 39/38 ^31 ^3D/v3Eb
4 55.2 28/27, 33/32, 36/35 30/29, 31/30, 32/31, 34/33 vvm2 vvEb
5 69.0 25/24, 26/25, 27/26 24/23 vm2 vEb
6 82.8 21/20, 22/21 20/19, 23/22 m2 Eb
7 96.6 35/33 18/17, 19/18 ^m2 ^Eb
8 110.3 16/15 17/16, 31/29, 33/31 ^^m2 ^^Eb
9 124.1 14/13, 15/14 29/27 vv~2 ^3Eb
10 137.9 13/12, 27/25 25/23 v~2 ^4Eb
11 151.7 12/11, 35/32 ^~2 v4E
12 165.5 11/10 32/29, 34/31 ^^~2 v3E
13 179.3 10/9 vvM2 vvE
14 193.1 28/25 19/17, 29/26 vM2 vE
15 206.9 9/8 26/23 M2 E
16 220.7 25/22 17/15, 33/29 ^M2 ^E
17 234.5 8/7 31/27 ^^M2 ^^E
18 248.3 15/13 22/19, 23/20, 38/33 ^3M2/v3m3 ^3E/v3F
19 262.1 7/6 29/25, 36/31 vvm3 vvF
20 275.9 75/64 20/17, 27/23, 34/29 vm3 vF
21 289.7 13/11, 32/27, 33/28 m3 F
22 303.4 25/21 19/16, 31/26 ^m3 ^F
23 317.2 6/5 ^^m3 ^^F
24 331.0 40/33 23/19, 29/24 vv~3 ^3F
25 344.8 11/9, 39/32 v~3 ^4F
26 358.6 16/13, 27/22 38/31 ^~3 v4F#
27 372.4 26/21 31/25, 36/29 ^^3 v3F#
28 386.2 5/4 vvM3 vvF#
29 400.0 44/35 24/19, 29/23, 34/27 vM3 vF#
30 413.8 14/11, 33/26, 81/64 19/15 M3 F#
31 427.6 32/25 23/18 ^M3 ^F#
32 441.4 9/7, 35/27 22/17, 31/24, 40/31 ^^M3 ^^F#
33 455.2 13/10 30/23 ^3M3/v34 ^3F#/v3G
34 469.0 21/16 17/13, 25/19, 38/29 vv4 vvG
35 482.8 33/25 v4 vG
36 496.6 4/3 P4 G
37 510.3 35/26 31/23 ^4 ^G
38 524.1 27/20 23/17 ^^4 ^^G
39 537.9 15/11 26/19, 34/25 ^34 ^3G
40 551.7 11/8, 48/35 ^44 ^4G
41 565.5 18/13 32/23 v4A4, vd5 v4G#, vAb
42 579.3 7/5 46/33 v3A4, d5 v3G#, Ab
43 593.1 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 distinct

87 can serve as a mos in these:

Scales

Mos scales

Main article: List of MOS scales in 87edo

Harmonic scales

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

Instruments

Music

Bryan Deister
Gene Ward Smith
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