87edo: Difference between revisions

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 edo 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 ==
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 == Scales ==
-=== Mos scales ===
+=== Harmonic scales ===
-{{main|List of MOS scales in 87edo}}
+edo 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.
+==== (Mode 6) ====
+{| class="wikitable center-all"
+|-
+! Overtones
+| 12
+| 14
+| 16
+| 18
+| 20
+| 22
+| 24
+|-
+! JI Ratios
+| 1/1
+| 7/6
+| 4/3
+| 3/2
+| 5/3
+| 11/6
+| 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'''.
-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.
 ==== (Mode 8) ====
@@ Line 589: / Line 641: @@
 |}
-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 670: / Line 722: @@
 |}
-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'''.
 , 15, 16, 18, 20, and 22 are close matches.
 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 ===

87edo: Difference between revisions

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