87edo: Difference between revisions

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== 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]]. 87 = 3 × 29, ~~and~~ 87edo shares the same perfect fifth with [[29edo]].

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 0.00061 ~~cents~~ sharper than the 13-limit [[CWE tuning|CWE generator]]. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.01479 ~~cents~~ sharp of the 13-limit CWE 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 ===

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|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}

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=== Interval mappings ===

~~=== 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 ==

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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 ==

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=== 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.

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|}

* 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) ====

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|}

* 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 ===

* [[Sequar5m]]

== Instruments ==

* [[Lumatone mapping for 87edo]]

* [[Skip fretting system 87 2 17]]

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025)

; [[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

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 == Theory ==
-edo 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]]. 87 = 3 × 29, and 87edo shares the same perfect fifth with [[29edo]].
+edo 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]].
-edo 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.
+edo 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]].
-edo is a particularly good tuning for [[rodan]], the {{nowrap|41 & 46}} temperament. The 8/7 generator of 17\87 is a remarkable 0.00061 cents sharper than the 13-limit [[CWE tuning|CWE generator]]. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.01479 cents sharp of the 13-limit CWE generator.
+edo 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 ===
-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|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
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 === Interval mappings ===
 {{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 ==
@@ Line 544: / Line 512: @@
 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 ==
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 {{main|List of MOS scales in 87edo}}
-=== Harmonic scale ===
+=== Harmonic scales ===
 edo 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.
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 |}
-* 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) ====
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 |}
-* 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.
+, 15, 16, 18, 20, and 22 are close matches.
+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 ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025)
 ; [[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