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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 [[cent|cents]].
{{Infobox ET}}
{{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]] (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]].


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 [[consistent|consistently]] (see [[87edo/13-limit detempering]]), and is the smallest equal temperament to do so.  
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.  


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.  
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]].  


87et [[tempering out|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.
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.


87et is a particularly good tuning for [[Gamelismic clan #Rodan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE tuning|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.
=== 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 [[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|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 ==
{| class="wikitable center-all right-2 left-3 left-4"
{| class="wikitable center-all right-2 left-3 left-4"
|-
! rowspan="2" | #
! rowspan="2" | #
! rowspan="2" | Cents
! rowspan="2" | Cents
! colspan="2" | Approximated Ratios
! colspan="2" | Approximated ratios
! colspan="2" rowspan="2" |[[Ups and Downs Notation]]
! colspan="2" rowspan="2" | [[Ups and downs notation]]
|-
|-
! 13-Limit
! 13-limit
! 31-Limit No-7s Extension
! 31-limit extension
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
|
|
Line 29: Line 38:
|-
|-
| 1
| 1
| 13.793
| 13.8
| [[126/125]], [[100/99]], [[91/90]]
| [[91/90]], [[100/99]], [[126/125]]
|
|
| ^1
| ^1
Line 36: Line 45:
|-
|-
| 2
| 2
| 27.586
| 27.6
| [[81/80]], [[64/63]], [[49/48]], [[55/54]], [[65/64]]
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
|
|
| ^^1
| ^^1
Line 43: Line 52:
|-
|-
| 3
| 3
| 41.379
| 41.4
| [[50/49]], [[45/44]], [[40/39]]
| [[40/39]], [[45/44]], [[50/49]]
| [[39/38]]
| [[39/38]]
| ^<sup>3</sup>1
| ^<sup>3</sup>1
Line 50: Line 59:
|-
|-
| 4
| 4
| 55.172
| 55.2
| [[28/27]], [[36/35]], [[33/32]]
| ''[[28/27]]'', [[33/32]], [[36/35]]
| [[34/33]], [[30/29]], [[32/31]], [[31/30]]
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
| vvm2
| vvm2
| vvEb
| vvEb
|-
|-
| 5
| 5
| 68.966
| 69.0
| [[25/24]], [[27/26]], [[26/25]]
| [[25/24]], [[26/25]], [[27/26]]
| [[24/23]]
| [[24/23]]
| vm2
| vm2
Line 64: Line 73:
|-
|-
| 6
| 6
| 82.759
| 82.8
| [[21/20]], [[22/21]]
| [[21/20]], [[22/21]]
| [[20/19]], [[23/22]]
| [[20/19]], [[23/22]]
Line 71: Line 80:
|-
|-
| 7
| 7
| 96.552
| 96.6
| [[35/33]]
| [[35/33]]
| [[18/17]], [[19/18]]
| [[18/17]], [[19/18]]
Line 78: Line 87:
|-
|-
| 8
| 8
| 110.345
| 110.3
| [[16/15]]
| [[16/15]]
| [[17/16]], [[33/31]], [[31/29]]
| [[17/16]], [[31/29]], [[33/31]]
| ^^m2
| ^^m2
| ^^Eb
| ^^Eb
|-
|-
| 9
| 9
| 124.138
| 124.1
| [[15/14]], [[14/13]]
| [[14/13]], [[15/14]]
| [[29/27]]
| [[29/27]]
| vv~2
| vv~2
Line 92: Line 101:
|-
|-
| 10
| 10
| 137.931
| 137.9
| [[13/12]], [[27/25]]
| [[13/12]], [[27/25]]
| [[25/23]]
| [[25/23]]
Line 99: Line 108:
|-
|-
| 11
| 11
| 151.724
| 151.7
| [[12/11]], [[35/32]]
| [[12/11]], [[35/32]]
|
|
Line 106: Line 115:
|-
|-
| 12
| 12
| 165.517
| 165.5
| [[11/10]]
| [[11/10]]
| [[32/29]], [[34/31]]
| [[32/29]], [[34/31]]
Line 113: Line 122:
|-
|-
| 13
| 13
| 179.310
| 179.3
| [[10/9]]
| [[10/9]]
|
|
Line 120: Line 129:
|-
|-
| 14
| 14
| 193.103
| 193.1
| [[28/25]]
| [[28/25]]
| [[19/17]], [[29/26]]
| [[19/17]], [[29/26]]
Line 127: Line 136:
|-
|-
| 15
| 15
| 206.897
| 206.9
| [[9/8]]
| [[9/8]]
| [[26/23]]
| [[26/23]]
Line 134: Line 143:
|-
|-
| 16
| 16
| 220.690
| 220.7
| [[25/22]]
| [[25/22]]
| [[17/15]], [[33/29]]
| [[17/15]], [[33/29]]
Line 141: Line 150:
|-
|-
| 17
| 17
| 234.483
| 234.5
| [[8/7]]
| [[8/7]]
| [[31/27]]
| [[31/27]]
Line 148: Line 157:
|-
|-
| 18
| 18
| 248.276
| 248.3
| [[15/13]]
| [[15/13]]
| [[22/19]], [[38/33]], [[23/20]]
| [[22/19]], [[23/20]], [[38/33]]
| ^<sup>3</sup>M2/v<sup>3</sup>m3
| ^<sup>3</sup>M2/v<sup>3</sup>m3
| ^<sup>3</sup>E/v<sup>3</sup>F
| ^<sup>3</sup>E/v<sup>3</sup>F
|-
|-
| 19
| 19
| 262.089
| 262.1
| [[7/6]]
| [[7/6]]
| [[29/25]], [[36/31]]
| [[29/25]], [[36/31]]
Line 162: Line 171:
|-
|-
| 20
| 20
| 275.862
| 275.9
| [[75/64]]
| [[75/64]]
| [[27/23]], [[34/29]]
| [[20/17]], [[27/23]], [[34/29]]
| vm3
| vm3
| vF
| vF
|-
|-
| 21
| 21
| 289.655
| 289.7
| [[32/27]], [[33/28]], [[13/11]]
| [[13/11]], [[32/27]], [[33/28]]
|
|
| m3
| m3
Line 176: Line 185:
|-
|-
| 22
| 22
| 303.448
| 303.4
| [[25/21]]
| [[25/21]]
| [[19/16]], [[31/26]]
| [[19/16]], [[31/26]]
Line 183: Line 192:
|-
|-
| 23
| 23
| 317.241
| 317.2
| [[6/5]]
| [[6/5]]
|
|
Line 190: Line 199:
|-
|-
| 24
| 24
| 331.034
| 331.0
| [[40/33]]
| [[40/33]]
| [[23/19]], [[29/24]]
| [[23/19]], [[29/24]]
Line 197: Line 206:
|-
|-
| 25
| 25
| 344.828
| 344.8
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
Line 204: Line 213:
|-
|-
| 26
| 26
| 358.621
| 358.6
| [[27/22]], [[16/13]]
| [[16/13]], [[27/22]]
| [[38/31]]
| [[38/31]]
| ^~3
| ^~3
Line 211: Line 220:
|-
|-
| 27
| 27
| 372.414
| 372.4
| [[26/21]]
| [[26/21]]
| [[31/25]], [[36/29]]
| [[31/25]], [[36/29]]
Line 218: Line 227:
|-
|-
| 28
| 28
| 386.207
| 386.2
| [[5/4]]
| [[5/4]]
|
|
Line 225: Line 234:
|-
|-
| 29
| 29
| 400.000
| 400.0
| [[44/35]]
| [[44/35]]
| [[34/27]], [[24/19]], [[29/23]]
| [[24/19]], [[29/23]], [[34/27]]
| vM3
| vM3
| vF#
| vF#
|-
|-
| 30
| 30
| 413.793
| 413.8
| [[81/64]], [[14/11]], [[33/26]]
| [[14/11]], [[33/26]], [[81/64]]
| [[19/15]]
| [[19/15]]
| M3
| M3
Line 239: Line 248:
|-
|-
| 31
| 31
| 427.586
| 427.6
| [[32/25]]
| [[32/25]]
| [[23/18]]
| [[23/18]]
Line 246: Line 255:
|-
|-
| 32
| 32
| 441.379
| 441.4
| [[9/7]], [[35/27]]
| [[9/7]], [[35/27]]
| [[22/17]], [[31/24]], [[40/31]]
| [[22/17]], [[31/24]], [[40/31]]
Line 253: Line 262:
|-
|-
| 33
| 33
| 455.172
| 455.2
| [[13/10]]
| [[13/10]]
| [[30/23]]
| [[30/23]]
Line 260: Line 269:
|-
|-
| 34
| 34
| 468.966
| 469.0
| [[21/16]]
| [[21/16]]
| [[17/13]], [[25/19]], [[38/29]]
| [[17/13]], [[25/19]], [[38/29]]
Line 267: Line 276:
|-
|-
| 35
| 35
| 482.759
| 482.8
| [[33/25]]
| [[33/25]]
|
|
Line 274: Line 283:
|-
|-
| 36
| 36
| 496.552
| 496.6
| [[4/3]]
| [[4/3]]
|
|
Line 281: Line 290:
|-
|-
| 37
| 37
| 510.345
| 510.3
| [[35/26]]
| [[35/26]]
| [[31/23]]
| [[31/23]]
Line 288: Line 297:
|-
|-
| 38
| 38
| 524.138
| 524.1
| [[27/20]]
| [[27/20]]
| [[23/17]]
| [[23/17]]
Line 295: Line 304:
|-
|-
| 39
| 39
| 537.931
| 537.9
| [[15/11]]
| [[15/11]]
| [[26/19]], [[34/25]]
| [[26/19]], [[34/25]]
Line 302: Line 311:
|-
|-
| 40
| 40
| 551.724
| 551.7
| [[11/8]], [[48/35]]
| [[11/8]], [[48/35]]
|
|
Line 309: Line 318:
|-
|-
| 41
| 41
| 565.517
| 565.5
| [[18/13]]
| [[18/13]]
| [[32/23]]
| [[32/23]]
Line 316: Line 325:
|-
|-
| 42
| 42
| 579.310
| 579.3
| [[7/5]]
| [[7/5]]
| [[46/33]]
| [[46/33]]
Line 323: Line 332:
|-
|-
| 43
| 43
| 593.103
| 593.1
| [[45/32]]
| [[45/32]]
| [[24/17]], [[38/27]], [[31/22]]
| [[24/17]], [[31/22]], [[38/27]]
| vvA4, ^d5
| vvA4, ^d5
| vvG#, ^Ab
| vvG#, ^Ab
Line 337: Line 346:
|}
|}


== Just approximation ==
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|87}}


=== Selected just intervals ===
== Regular temperament properties ==
{| class="wikitable center-all"
{| class="wikitable center-4 center-5 center-6"
! colspan="2" |
|-
! prime 2
! rowspan="2" | [[Subgroup]]
! prime 3
! rowspan="2" | [[Comma list]]
! prime 5
! rowspan="2" | [[Mapping]]
! prime 7
! rowspan="2" | Optimal<br>8ve stretch (¢)
! prime 11
! colspan="2" | Tuning error
! prime 13
! prime 17
! prime 19
! prime 23
! prime 29
! prime 31
|-
|-
! rowspan="2" | Error
! [[TE error|Absolute]] (¢)
! absolute (¢)
! [[TE simple badness|Relative]] (%)
| 0.00
| +1.49
| -0.11
| -3.31
| +0.41
| +0.85
| +5.39
| +5.94
| +6.21
| +4.91
| -0.21
|-
|-
! [[Relative error|relative]] (%)
| 2.3.5
| 0.0
| 15625/15552, 67108864/66430125
| +10.8
| {{mapping| 87 138 202 }}
| -0.8
| −0.299
| -24.0
| 0.455
| +2.9
| 3.30
| +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.
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 11-limit
! 13-limit
! 17-limit
! 19-limit
! 23-limit
! 29-limit
! 31-limit
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5.7
| -0.471
| 245/243, 1029/1024, 3136/3125
| -0.299
| {{mapping| 87 138 202 244 }}
| +0.070
| +0.070
| 0.752
| 5.45
|-
| 2.3.5.7.11
| 245/243, 385/384, 441/440, 3136/3125
| {{mapping| 87 138 202 244 301 }}
| +0.033
| +0.033
| -0.011
| 0.676
| -0.198
| 4.90
| -0.348
| -0.462
| -0.517
| -0.466
|-
|-
! rowspan="2" |Error
| 2.3.5.7.11.13
! [[TE error|absolute]] (¢)
| 196/195, 245/243, 352/351, 364/363, 625/624
| 0.471
| {{mapping| 87 138 202 244 301 322 }}
| 0.455
| −0.011
| 0.752
| 0.676
| 0.625
| 0.625
| 4.53
|-
| 2.3.5.7.11.13.17
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| {{mapping| 87 138 202 244 301 322 356 }}
| −0.198
| 0.738
| 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
| {{mapping| 87 138 202 244 301 322 356 370 }}
| −0.348
| 0.796
| 0.796
| 0.816
| 0.792
| 0.772
|-
! [[TE simple badness|relative]] (%)
| 3.42
| 3.30
| 5.45
| 4.90
| 4.53
| 5.35
| 5.77
| 5.77
| 5.92
| 5.74
| 5.60
|}
|}


== 13-limit detempering of 87et ==
=== 13-limit detempering ===
 
{{Main|87edo/13-limit detempering}}
:''See also: [[Detempering]]''
 
{{main|87edo/13-limit detempering}}
 
== Rank two temperaments ==


{| class="wikitable center-all right-3 left-5"
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br> per octave
! Periods<br>per 8ve
! Generator
! Generator*
! Cents
! Cents*
! Associated<br>ratio
! Associated<br>ratio*
! Temperament
! Temperament
|-
| 1
| 2\87
| 27.586
| 64/63
| [[Arch]]
|-
|-
| 1
| 1
| 4\87
| 4\87
| 55.172
| 55.172
| [[33/32]]
| 33/32
| [[Sensa]]
| [[Escapade]] / [[escaped]] / [[alphaquarter]]
|-
|-
| 1
| 1
| 10\87
| 10\87
| 137.931
| 137.931
| [[13/12]]
| 13/12
| [[Quartemka]]
| [[Quartemka]]
|-
|-
Line 466: Line 439:
| 14\87
| 14\87
| 193.103
| 193.103
| [[28/25]]
| 28/25
| [[Luna]] / [[Hemithirds]]
| [[Luna]] / [[didacus]] / [[hemithirds]]
|-
|-
| 1
| 1
| 17\87
| 17\87
| 234.483
| 234.483
| [[8/7]]
| 8/7
| [[Rodan]]
| [[Slendric]] / [[rodan]]
|-
|-
| 1
| 1
| 23\87
| 23\87
| 317.241
| 317.241
| [[6/5]]
| 6/5
| [[Hanson]] / [[Countercata]] / [[Metakleismic]]
| [[Hanson]] / [[countercata]] / [[metakleismic]]
|-
|-
| 1
| 1
| 26\87
| 26\87
| 358.621
| 358.621
| [[16/13]]
| 16/13
| [[Restles]]
| [[Restles]]
|-
|-
Line 490: Line 463:
| 32\87
| 32\87
| 441.379
| 441.379
| [[9/7]]
| 9/7
| [[Clyde]]
| [[Clyde]]
|-
|-
Line 496: Line 469:
| 38\87
| 38\87
| 524.138
| 524.138
| [[65/48]]
| 65/48
| [[Widefourth]]
| [[Widefourth]]
|-
|-
Line 502: Line 475:
| 40\87
| 40\87
| 551.724
| 551.724
| [[11/8]]
| 11/8
| [[Emkay]]
| [[Emka]] / [[emkay]]
|-
| 3
| 18\87<br>(11\87)
| 248.276<br>(151.724)
| 15/13<br>(12/11)
| [[Hemimist]]
|-
|-
| 3
| 3
| 23\87
| 23\87<br>(6\87)
| 317.241
| 317.241<br>(82.759)
| [[6/5]]
| 6/5<br>(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|-
| 3
| 28\87<br>(1\87)
| 386.207<br>(13.793)
| 5/4<br>(126/125)
| [[Mutt]]
|-
| 3
| 36\87<br>(7\87)
| 496.552<br>(96.552)
| 4/3<br>(18/17~19/18)
| [[Misty]]
|-
|-
| 29
| 29
| 28\87
| 28\87<br>(1\87)
| 386.207
| 386.207<br>(13.793)
| [[5/4]]
| 5/4<br>(121/120)
| [[Mystery]]
| [[Mystery]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


87 can serve as a MOS in these:
87 can serve as a mos in these:


* [[M&N temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])
* [[M&N temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||
* [[Breed|87 & 494]]  


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


=== Harmonic Scale ===
=== Harmonic scales ===
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).  
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 ====
==== (Mode 8) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
| Overtones
|-
! Overtones
| 8
| 8
| 9
| 9
Line 541: Line 536:
| 16
| 16
|-
|-
| JI Ratios
! JI Ratios
| 1/1
| 1/1
| 9/8
| 9/8
Line 552: Line 547:
| 2/1
| 2/1
|-
|-
| … in cents
! … in cents
| 0.0
| 0.0
| 203.9
| 203.9
Line 563: Line 558:
| 1200.0
| 1200.0
|-
|-
| Degrees in 87edo
! Degrees in 87edo
| 0
| 0
| 15
| 15
Line 574: Line 569:
| 87
| 87
|-
|-
| … in cents
! … in cents
| 0.0
| 0.0
| 206.9
| 206.9
Line 586: 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 16 ====
==== (Mode 12) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
| Odd overtones
|-
! Overtones
| 12
| 13
| 14
| 15
| 16
| 17
| 17
| 18
| 19
| 19
| 20
| 21
| 21
| 22
| 23
| 23
| 25
| 24
| 27
| 29
| 31
|-
|-
| JI Ratios
! JI Ratios
| 17/16
| 1/1
| 19/16
| 13/12
| 21/16
| 7/6
| 23/16
| 5/4
| 25/16
| 4/3
| 27/16
| 17/12
| 29/16
| 3/2
| 31/16
| 19/12
| 5/3
| 7/4
| 11/6
| 23/12
| 2/1
|-
|-
| … in cents
! … in cents
| 105.0
| 0.0
| 297.5
| 138.6
| 470.8
| 266.9
| 628.3
| 386.3
| 772.6
| 498.0
| 905.9
| 603.0
| 1029.6
| 702.0
| 1145.0
| 795.6
| 884.4
| 968.8
| 1049.4
| 1126.3
| 1200.0
|-
|-
| Degrees in 87edo
! Degrees in 87edo
| 8
| 0
| 22
| 10
| 34
| 19
| 46
| 28
| 56
| 36
| 66
| 44
| 75
| 51
| 83
| 58
| 64
| 70
| 76
| 82
| 87
|-
|-
| … in cents
! … in cents
| 110.3
| 0.0
| 303.4
| 137.9
| 469.0
| 262.1
| 634.5
| 386.2
| 772.4
| 496.6
| 910.3
| 606.9
| 1034.5
| 703.4
| 1144.8
| 800.0
| 882.8
| 965.5
| 1048.3
| 1131.0
| 1200.0
|}
|}


* The scale in adjacent steps is 8, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 4, 5, 4, 4, 4.
The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.
* 25 and 31 are close matches.  
 
* 21 is a little bit flat, but still decent.
13, 15, 16, 18, 20, and 22 are close matches.  
* The others (17, 19, 23, 27 and 29) are extremely sharp, but the intervals between them are close.  
 
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
 
=== Other scales ===
* [[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)


* [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[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


[[Category:theory]]
[[Category:Zeta|##]] <!-- 2-digit number -->
[[Category:edo]]
[[Category:Listen]]
[[Category:87edo]]
[[Category:Clyde]]
[[Category:listen]]
[[Category:Countercata]]
[[Category:clyde]]
[[Category:Hemithirds]]
[[Category:countercata]]
[[Category:Mystery]]
[[Category:hemithirds]]
[[Category:Rodan]]
[[Category:mystery]]
[[Category:Tritikleismic]]
[[Category:rodan]]
[[Category:tritikleismic]]

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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