87edo: Difference between revisions

{{Infobox ET}}

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 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.

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]])⁸⁷ only very slightly exceeds the octave.

{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}

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

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

{| class="wikitable center-all right-2 left-3 left-4"

! rowspan="2" | #

! colspan="2" | Approximated ratios

! colspan="2" rowspan="2" | [[Ups and downs notation]]

|-

! 13-limit

! 31-limit extension

| [[1/1]]

| 13.8

| [[91/90]], [[100/99]], [[126/125]]

| ^D

| 27.6

| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]

| ^^1

| 41.4

| [[40/39]], [[45/44]], [[50/49]]

| [[39/38]]

| 55.2

| ''[[28/27]]'', [[33/32]], [[36/35]]

| vvEb

| 69.0

| [[25/24]], [[26/25]], [[27/26]]

| [[24/23]]

| 82.8

| [[21/20]], [[22/21]]

| Eb

| 96.6

| [[35/33]]

| [[18/17]], [[19/18]]

| 110.3

| [[16/15]]

| [[17/16]], [[31/29]], [[33/31]]

| ^^m2

| 124.1

| [[14/13]], [[15/14]]

| [[29/27]]

| vv~2

| 137.9

| [[13/12]], [[27/25]]

| ^⁴Eb

| 151.7

| [[12/11]], [[35/32]]

|

| 165.5

| [[11/10]]

| [[32/29]], [[34/31]]

| 179.3

| [[10/9]]

|

| 193.1

| [[28/25]]

| [[19/17]], [[29/26]]

| 206.9

| [[9/8]]

| [[26/23]]

| 220.7

| [[25/22]]

| [[17/15]], [[33/29]]

| 234.5

| [[8/7]]

| [[31/27]]

| 248.3

| [[15/13]]

| [[22/19]], [[23/20]], [[38/33]]

| 262.1

| [[7/6]]

| [[29/25]], [[36/31]]

| 275.9

| [[75/64]]

| [[20/17]], [[27/23]], [[34/29]]

| 289.7

| [[13/11]], [[32/27]], [[33/28]]

| F

| 303.4

| [[25/21]]

| [[19/16]], [[31/26]]

| 317.2

| [[6/5]]

|

| 331.0

| [[40/33]]

| [[23/19]], [[29/24]]

| 344.8

| [[11/9]], [[39/32]]

|

| 358.6

| [[16/13]], [[27/22]]

| v⁴F#

| 372.4

| [[26/21]]

| [[31/25]], [[36/29]]

| 386.2

| [[5/4]]

|

| 400.0

| [[44/35]]

| [[24/19]], [[29/23]], [[34/27]]

| 413.8

| [[14/11]], [[33/26]], [[81/64]]

| F#

| 427.6

| [[32/25]]

| [[23/18]]

| 441.4

| [[9/7]], [[35/27]]

| ^^F#

| 455.2

| [[13/10]]

| [[30/23]]

| 469.0

| [[21/16]]

| [[17/13]], [[25/19]], [[38/29]]

| 482.8

| [[33/25]]

|

| 496.6

| [[4/3]]

|

| 510.3

| [[35/26]]

| [[31/23]]

| ^4

| 524.1

| [[27/20]]

| [[23/17]]

| ^^4

| 537.9

| [[15/11]]

| [[26/19]], [[34/25]]

| 551.7

| [[11/8]], [[48/35]]

| ^⁴4

| 565.5

| [[18/13]]

| [[32/23]]

| 579.3

| [[7/5]]

| [[46/33]]

| v³A4, d5

| 593.1

| [[45/32]]

| [[24/17]], [[31/22]], [[38/27]]

| …

| …

|}

 

=== 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.

 

 

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

| 2.3.5

| 15625/15552, 67108864/66430125

| {{Mapping| 87 138 202 }}

| −0.299

| 0.455

| 2.3.5.7

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

| {{Mapping| 87 138 202 244 }}

| +0.070

| 0.752

| 2.3.5.7.11

| +0.033

| 0.676

| 4.90

| 2.3.5.7.11.13

| −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

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

| −0.198

| 0.738

| 5.35

| 2.3.5.7.11.13.17.19

| 5.77

 

 

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

| 1

| 2\87

| 27.586

| 64/63

| [[Arch]]

| 1

| 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

| 248.276<br>(151.724)

| 15/13<br>(12/11)

| [[Hemimist]]

| 3

| 23\87<br>(6\87)

| 317.241<br>(82.759)

| 6/5<br>(21/20)

| [[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

| 28\87<br>(1\87)

| 386.207<br>(13.793)

| 5/4<br>(121/120)

<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]])

* [[Breed|87 & 494]]  

 

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

 

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) ====

| 11

| 12

| 13

| 15

| 16

| 79

| 87

| 386.2

| 551.7

| 703.5

| 1089.7

| 1200.0

 

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

 

| 12

| 13

| 14

| 16

| 17

| 4/3

| 17/12

| 3/2

| 19/12

| 5/3

| 2/1

| 0.0

| 138.6

| 266.9

| 386.3

| 603.0

| 702.0

| 795.6

| 884.4

| 968.8

| 1049.4

| 1126.3

| 1200.0

| 87

* ''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:Listen]]

[[Category:Clyde]]

[[Category:Countercata]]

[[Category:Hemithirds]]

[[Category:Mystery]]

[[Category:Rodan]]

[[Category:Tritikleismic]]