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


[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic.
[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic. Multiples like [[696edo]] and [[783edo]] are also of theoretical interest.
 
One step of 87edo is ''extremely'' close to [[126/125]], with the associated comma, 87-126/125-comma ({{monzo|86 174 -261 87}}) being tempered in equal temperaments well over 100,000 and with less than 0.001{{c}} error, with the [[optimal patent val]] for the 7-limit temperament being an enormous [[425082edo]]. But more notably, 87edo itself does the mapping consistently, making it a true circle of 126/125's.
 
See also [[87th-octave temperaments]].


== Intervals ==
== Intervals ==
Line 524: Line 528:


== Scales ==
== Scales ==
=== Mos scales ===
=== Harmonic scales ===
{{main|List of MOS scales in 87edo}}
87edo very accurately approximates [[Overtone scale|modes]] up to 8 of the [[harmonic series]], and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently.
* [[Kleismic]][7]: 5 18 5 18 5 18 18
 
* Kleismic[15]: 5 8 5 5 5 8 5 5 5 8 5 5 5 8 5
==== (Mode 6) ====
* Kleismic[19]: 5 5 3 5 5 5 5 3 5 5 5 3 5 5 5 5 3 5 5
{| class="wikitable center-all"
* Kleismic[34]: 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3
|-
* [[Misty]][6]: 22 7 22 7 22 7
! Overtones
* Misty[12]: 7 8 7 7 7 8 7 7 7 8 7 7
| 12
* Misty[15]: 7 7 1 7 7 7 7 1 7 7 7 7 1 7 7
| 14
* Misty[27]: 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1
| 16
* Misty[39]: 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1
| 18
* [[Luna]][6]: 14 14 14 17 14 14
| 20
* Luna[13]: 3 11 3 11 3 11 3 11 3 11 3 11 3
| 22
* Luna[19]: 3 8 3 3 8 3 3 8 3 3 3 8 3 3 8 3 3 8 3
| 24
* Luna[25]: 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3
|-
* Luna[31]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
! JI Ratios
* Luna[56]: 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2
| 1/1
* [[Rodan]][5]: 17 17 17 17 19
| 7/6
* Rodan[11]: 2 15 2 15 2 15 2 15 2 15 2
| 4/3
* Rodan[16]: 2 13 2 2 13 2 2 2 13 2 2 13 2 2 13 2
| 3/2
* Rodan[21]: 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2
| 5/3
* Rodan[36]: 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2
| 11/6
* Rodan[41]: 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2
| 2/1
|-
! … in cents
| 0.0
| 266.9
| 498.0
| 702.0
| 884.4
| 1049.4
| 1200.0
|-
! Degrees in 87edo
| 0
| 19
| 36
| 51
| 64
| 76
| 87
|-
! … in cents
| 0.0
| 262.1
| 496.6
| 703.4
| 882.8
| 1048.3
| 1200.0
|}


=== Harmonic scales ===
The scale in adjacent steps is '''19 17 15 13 12 11'''.  
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) ====
Line 696: Line 727:


14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
=== Mos scales ===
{{main|List of MOS scales in 87edo}}
* [[Kleismic]][7]: 5 18 5 18 5 18 18
* Kleismic[15]: 5 8 5 5 5 8 5 5 5 8 5 5 5 8 5
* Kleismic[19]: 5 5 3 5 5 5 5 3 5 5 5 3 5 5 5 5 3 5 5
* Kleismic[34]: 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3
* [[Misty]][6]: 22 7 22 7 22 7
* Misty[12]: 7 8 7 7 7 8 7 7 7 8 7 7
* Misty[15]: 7 7 1 7 7 7 7 1 7 7 7 7 1 7 7
* Misty[27]: 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1
* Misty[39]: 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1
* [[Luna]][6]: 17 14 14 14 14 14
* Luna[13]: 3 11 3 11 3 11 3 11 3 11 3 11 3
* Luna[19]: 3 8 3 3 8 3 3 8 3 3 3 8 3 3 8 3 3 8 3
* Luna[25]: 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3
* Luna[31]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
* Luna[56]: 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2
* [[Rodan]][5]: 17 17 17 17 19
* Rodan[11]: 2 15 2 15 2 15 2 15 2 15 2
* Rodan[16]: 2 13 2 2 13 2 2 2 13 2 2 13 2 2 13 2
* Rodan[21]: 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2
* Rodan[36]: 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2
* Rodan[41]: 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2


=== Other scales ===
=== Other scales ===
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