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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 ==
{| class="wikitable center-all right-2 left-3 left-4"
{| class="wikitable center-all right-2 left-3 left-4"
|-
|-
! rowspan="2" | &#35;
! 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 36: Line 38:
|-
|-
| 1
| 1
| 13.793
| 13.8
| [[91/90]], [[100/99]], [[126/125]]
| [[91/90]], [[100/99]], [[126/125]]
|
|
Line 43: Line 45:
|-
|-
| 2
| 2
| 27.586
| 27.6
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
|
|
Line 50: Line 52:
|-
|-
| 3
| 3
| 41.379
| 41.4
| [[40/39]], [[45/44]], [[50/49]]
| [[40/39]], [[45/44]], [[50/49]]
| [[39/38]]
| [[39/38]]
Line 57: Line 59:
|-
|-
| 4
| 4
| 55.172
| 55.2
| ''[[28/27]]'', [[33/32]], [[36/35]]
| ''[[28/27]]'', [[33/32]], [[36/35]]
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
Line 64: Line 66:
|-
|-
| 5
| 5
| 68.966
| 69.0
| [[25/24]], [[26/25]], [[27/26]]
| [[25/24]], [[26/25]], [[27/26]]
| [[24/23]]
| [[24/23]]
Line 71: 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 78: Line 80:
|-
|-
| 7
| 7
| 96.552
| 96.6
| [[35/33]]
| [[35/33]]
| [[18/17]], [[19/18]]
| [[18/17]], [[19/18]]
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|-
|-
| 8
| 8
| 110.345
| 110.3
| [[16/15]]
| [[16/15]]
| [[17/16]], [[31/29]], [[33/31]]
| [[17/16]], [[31/29]], [[33/31]]
Line 92: Line 94:
|-
|-
| 9
| 9
| 124.138
| 124.1
| [[14/13]], [[15/14]]
| [[14/13]], [[15/14]]
| [[29/27]]
| [[29/27]]
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|-
|-
| 10
| 10
| 137.931
| 137.9
| [[13/12]], [[27/25]]
| [[13/12]], [[27/25]]
| [[25/23]]
| [[25/23]]
Line 106: Line 108:
|-
|-
| 11
| 11
| 151.724
| 151.7
| [[12/11]], [[35/32]]
| [[12/11]], [[35/32]]
|
|
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|-
|-
| 12
| 12
| 165.517
| 165.5
| [[11/10]]
| [[11/10]]
| [[32/29]], [[34/31]]
| [[32/29]], [[34/31]]
Line 120: Line 122:
|-
|-
| 13
| 13
| 179.310
| 179.3
| [[10/9]]
| [[10/9]]
|
|
Line 127: Line 129:
|-
|-
| 14
| 14
| 193.103
| 193.1
| [[28/25]]
| [[28/25]]
| [[19/17]], [[29/26]]
| [[19/17]], [[29/26]]
Line 134: Line 136:
|-
|-
| 15
| 15
| 206.897
| 206.9
| [[9/8]]
| [[9/8]]
| [[26/23]]
| [[26/23]]
Line 141: Line 143:
|-
|-
| 16
| 16
| 220.690
| 220.7
| [[25/22]]
| [[25/22]]
| [[17/15]], [[33/29]]
| [[17/15]], [[33/29]]
Line 148: Line 150:
|-
|-
| 17
| 17
| 234.483
| 234.5
| [[8/7]]
| [[8/7]]
| [[31/27]]
| [[31/27]]
Line 155: Line 157:
|-
|-
| 18
| 18
| 248.276
| 248.3
| [[15/13]]
| [[15/13]]
| [[22/19]], [[23/20]], [[38/33]]
| [[22/19]], [[23/20]], [[38/33]]
Line 162: Line 164:
|-
|-
| 19
| 19
| 262.089
| 262.1
| [[7/6]]
| [[7/6]]
| [[29/25]], [[36/31]]
| [[29/25]], [[36/31]]
Line 169: Line 171:
|-
|-
| 20
| 20
| 275.862
| 275.9
| [[75/64]]
| [[75/64]]
| [[20/17]], [[27/23]], [[34/29]]
| [[20/17]], [[27/23]], [[34/29]]
Line 176: Line 178:
|-
|-
| 21
| 21
| 289.655
| 289.7
| [[13/11]], [[32/27]], [[33/28]]
| [[13/11]], [[32/27]], [[33/28]]
|
|
Line 183: Line 185:
|-
|-
| 22
| 22
| 303.448
| 303.4
| [[25/21]]
| [[25/21]]
| [[19/16]], [[31/26]]
| [[19/16]], [[31/26]]
Line 190: Line 192:
|-
|-
| 23
| 23
| 317.241
| 317.2
| [[6/5]]
| [[6/5]]
|
|
Line 197: Line 199:
|-
|-
| 24
| 24
| 331.034
| 331.0
| [[40/33]]
| [[40/33]]
| [[23/19]], [[29/24]]
| [[23/19]], [[29/24]]
Line 204: Line 206:
|-
|-
| 25
| 25
| 344.828
| 344.8
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
Line 211: Line 213:
|-
|-
| 26
| 26
| 358.621
| 358.6
| [[16/13]], [[27/22]]
| [[16/13]], [[27/22]]
| [[38/31]]
| [[38/31]]
Line 218: Line 220:
|-
|-
| 27
| 27
| 372.414
| 372.4
| [[26/21]]
| [[26/21]]
| [[31/25]], [[36/29]]
| [[31/25]], [[36/29]]
Line 225: Line 227:
|-
|-
| 28
| 28
| 386.207
| 386.2
| [[5/4]]
| [[5/4]]
|
|
Line 232: Line 234:
|-
|-
| 29
| 29
| 400.000
| 400.0
| [[44/35]]
| [[44/35]]
| [[24/19]], [[29/23]], [[34/27]]
| [[24/19]], [[29/23]], [[34/27]]
Line 239: Line 241:
|-
|-
| 30
| 30
| 413.793
| 413.8
| [[14/11]], [[33/26]], [[81/64]]
| [[14/11]], [[33/26]], [[81/64]]
| [[19/15]]
| [[19/15]]
Line 246: Line 248:
|-
|-
| 31
| 31
| 427.586
| 427.6
| [[32/25]]
| [[32/25]]
| [[23/18]]
| [[23/18]]
Line 253: 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 260: Line 262:
|-
|-
| 33
| 33
| 455.172
| 455.2
| [[13/10]]
| [[13/10]]
| [[30/23]]
| [[30/23]]
Line 267: 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 274: Line 276:
|-
|-
| 35
| 35
| 482.759
| 482.8
| [[33/25]]
| [[33/25]]
|
|
Line 281: Line 283:
|-
|-
| 36
| 36
| 496.552
| 496.6
| [[4/3]]
| [[4/3]]
|
|
Line 288: Line 290:
|-
|-
| 37
| 37
| 510.345
| 510.3
| [[35/26]]
| [[35/26]]
| [[31/23]]
| [[31/23]]
Line 295: Line 297:
|-
|-
| 38
| 38
| 524.138
| 524.1
| [[27/20]]
| [[27/20]]
| [[23/17]]
| [[23/17]]
Line 302: Line 304:
|-
|-
| 39
| 39
| 537.931
| 537.9
| [[15/11]]
| [[15/11]]
| [[26/19]], [[34/25]]
| [[26/19]], [[34/25]]
Line 309: Line 311:
|-
|-
| 40
| 40
| 551.724
| 551.7
| [[11/8]], [[48/35]]
| [[11/8]], [[48/35]]
|
|
Line 316: Line 318:
|-
|-
| 41
| 41
| 565.517
| 565.5
| [[18/13]]
| [[18/13]]
| [[32/23]]
| [[32/23]]
Line 323: Line 325:
|-
|-
| 42
| 42
| 579.310
| 579.3
| [[7/5]]
| [[7/5]]
| [[46/33]]
| [[46/33]]
Line 330: Line 332:
|-
|-
| 43
| 43
| 593.103
| 593.1
| [[45/32]]
| [[45/32]]
| [[24/17]], [[31/22]], [[38/27]]
| [[24/17]], [[31/22]], [[38/27]]
Line 354: Line 356:
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 410: Line 412:
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
Line 477: Line 479:
|-
|-
| 3
| 3
| 18\87<br />(11\87)
| 18\87<br>(11\87)
| 248.276<br />(151.724)
| 248.276<br>(151.724)
| 15/13<br />(12/11)
| 15/13<br>(12/11)
| [[Hemimist]]
| [[Hemimist]]
|-
|-
| 3
| 3
| 23\87<br />(6\87)
| 23\87<br>(6\87)
| 317.241<br />(82.759)
| 317.241<br>(82.759)
| 6/5<br />(21/20)
| 6/5<br>(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|-
|-
| 3
| 3
| 28\87<br />(1\87)
| 28\87<br>(1\87)
| 386.207<br />(13.793)
| 386.207<br>(13.793)
| 5/4<br />(126/125)
| 5/4<br>(126/125)
| [[Mutt]]
| [[Mutt]]
|-
|-
| 3
| 3
| 36\87<br />(7\87)
| 36\87<br>(7\87)
| 496.552<br />(96.552)
| 496.552<br>(96.552)
| 4/3<br />(18/17~19/18)
| 4/3<br>(18/17~19/18)
| [[Misty]]
| [[Misty]]
|-
|-
| 29
| 29
| 28\87<br />(1\87)
| 28\87<br>(1\87)
| 386.207<br />(13.793)
| 386.207<br>(13.793)
| 5/4<br />(121/120)
| 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
<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:


* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 &amp; 270]]) {{multival| 24 -9 -66 12 27 … }}
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])  
* [[Breed|87 &amp; 494]] {{multival| 51 -1 -133 11 32 … }}
* [[Breed|87 & 494]]  
 
== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! 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
|}


== Scales ==
== Scales ==
=== MOS scales ===
=== Mos scales ===
{{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 611: 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 692: 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
Retrieved from "https://en.xen.wiki/w/87edo"