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]]. Since {{nowrap|87 {{=}} 3 × 29}}, 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]], and 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.

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]].

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=== 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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=== Subsets and supersets ===

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

[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic.

== Intervals ==

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| …

|}

== Notation ==

=== Ups and downs notation ===

87edo can be written using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down):

Mapping an arrow to 2\87 rather than 1\87 is an alternative approach which takes advantage of 87edo being a tuning of akea temperament. This way, one arrow is equivalent to 81/80~64/63, and two arrows are equivalent to 33/32~1053/1024.

== Approximation to JI ==

=== 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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| 2.3.5

| 15625/15552, 67108864/66430125

| {{~~mapping~~| 87 138 202 }}

| {{Mapping| 87 138 202 }}

| −0.299

| 0.455

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| 2.3.5.7

| 245/243, 1029/1024, 3136/3125

| {{~~mapping~~| 87 138 202 244 }}

| {{Mapping| 87 138 202 244 }}

| +0.070

| 0.752

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| 2.3.5.7.11

| 245/243, 385/384, 441/440, 3136/3125

| {{~~mapping~~| 87 138 202 244 301 }}

| {{Mapping| 87 138 202 244 301 }}

| +0.033

| 0.676

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| 2.3.5.7.11.13

| 196/195, 245/243, 352/351, 364/363, 625/624

| {{~~mapping~~| 87 138 202 244 301 322 }}

| {{Mapping| 87 138 202 244 301 322 }}

| −0.011

| 0.625

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| 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 }}

| {{Mapping| 87 138 202 244 301 322 356 }}

| −0.198

| 0.738

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| 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 }}

| {{Mapping| 87 138 202 244 301 322 356 370 }}

| −0.348

| 0.796

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| [[Mystery]]

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

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 ==

; [[ALLY195]]

* [https://www.bilibili.com/video/BV16h411g7QM/ ''Root note and subharmonic series cadence - 103EDO, 87EDO, 94EDO''] (2023)

* [https://www.bilibili.com/video/BV1N84y1T792/ ''A comparison between 87edo and 12edo''] (2023)

; [[Bryan Deister]]

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

* [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025)

* [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover 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

~~[[Category:Zeta|##]] ~~

[[Category:Listen]]

[[Category:Clyde]]

@@ Line 3: / Line 3: @@
 == 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]]. Since {{nowrap|87 {{=}} 3 × 29}}, 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]], and 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.
 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]].
@@ Line 12: / Line 12: @@
 === 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)}}
@@ Line 19: / Line 18: @@
 === Subsets and supersets ===
 edo contains [[3edo]] and [[29edo]] as subset edos.
+[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic.
 == Intervals ==
@@ Line 346: / Line 347: @@
 | …
 |}
+== Notation ==
+=== Ups and downs notation ===
+edo can be written using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down):
+{{Ups and downs sharpness}}
+Mapping an arrow to 2\87 rather than 1\87 is an alternative approach which takes advantage of 87edo being a tuning of akea temperament. This way, one arrow is equivalent to 81/80~64/63, and two arrows are equivalent to 33/32~1053/1024.
 == Approximation to JI ==
 === 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 396: / Line 372: @@
 | 2.3.5
 | 15625/15552, 67108864/66430125
-| {{mapping| 87 138 202 }}
+| {{Mapping| 87 138 202 }}
 | −0.299
 | 0.455
@@ Line 403: / Line 379: @@
 | 2.3.5.7
 | 245/243, 1029/1024, 3136/3125
-| {{mapping| 87 138 202 244 }}
+| {{Mapping| 87 138 202 244 }}
 | +0.070
 | 0.752
@@ Line 410: / Line 386: @@
 | 2.3.5.7.11
 | 245/243, 385/384, 441/440, 3136/3125
-| {{mapping| 87 138 202 244 301 }}
+| {{Mapping| 87 138 202 244 301 }}
 | +0.033
 | 0.676
@@ Line 417: / Line 393: @@
 | 2.3.5.7.11.13
 | 196/195, 245/243, 352/351, 364/363, 625/624
-| {{mapping| 87 138 202 244 301 322 }}
+| {{Mapping| 87 138 202 244 301 322 }}
 | −0.011
 | 0.625
@@ Line 424: / Line 400: @@
 | 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 }}
+| {{Mapping| 87 138 202 244 301 322 356 }}
 | −0.198
 | 0.738
@@ Line 431: / Line 407: @@
 | 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 }}
+| {{Mapping| 87 138 202 244 301 322 356 370 }}
 | −0.348
 | 0.796
@@ Line 540: / Line 516: @@
 | [[Mystery]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 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 ==
@@ Line 551: / Line 527: @@
 {{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.
@@ Line 613: / Line 589: @@
 |}
-* 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) ====
@@ Line 694: / Line 670: @@
 |}
-* 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 ==
+; [[ALLY195]]
+* [https://www.bilibili.com/video/BV16h411g7QM/ ''Root note and subharmonic series cadence - 103EDO, 87EDO, 94EDO''] (2023)
+* [https://www.bilibili.com/video/BV1N84y1T792/ ''A comparison between 87edo and 12edo''] (2023)
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025)
+* [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025)
+* [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover 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
-[[Category:Zeta|##]] <!-- 2-digit number -->
 [[Category:Listen]]
 [[Category:Clyde]]