93edo: Difference between revisions

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@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|93}}
+{{ED intro}}
 == Theory ==
+Since {{nowrap|93 {{=}} 3 × 31}}, 93edo is a [[contorted]] [[31edo]] through the [[7-limit]]. In the 11-limit the [[patent val]] [[tempering out|tempers out]] [[4000/3993]] and in the 13-limit [[144/143]], [[1188/1183]], and [[364/363]]. It provides the [[optimal patent val]] for the 11-limit [[31st-octave_temperaments#Prajapati|prajapati]] and 13-limit [[31st-octave_temperaments#Kumhar|kumhar]] temperaments and the 11- and 13-limit [[Meantone family#Trimean|trimean]] ({{nowrap|43 &amp; 50}}) temperament, and is the 13th no-3s [[zeta peak edo]]. The 93bd val is close to the 9-odd limit minimax tuning for [[superpyth]] and approximates {{nowrap|{{frac|2|7}}-[[64/63|septimal comma]]}} superpyth very well.
+Since 93edo has good approximations of [[13/1|13th]], [[17/1|17th]] and [[19/1|19th]] [[harmonic]]s unlike 31edo (as 838.710{{c}}, 103.226{{c}}, and 296.774{{c}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
+=== Odd harmonics ===
 {{Harmonics in equal|93}}
-= 3 * 31, and 93 is a [[contorted]] 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&amp;50 temperament. It is the 13th no-3s zeta peak edo.
-Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710{{cent}}, 103.226{{cent}}, and 296.774{{cent}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
+=== No-3 approach ===
+If prime 3 is ignored, 93edo represents the no-3 35-odd-limit consistently. 93edo is distinctly consistent within the no-3 19-integer-limit.
 == Intervals ==
@@ Line 12: / Line 17: @@
 == Scales ==
-Meantone Chromatic
+* Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)
+* Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93)
-* 116.129
+* Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)
-* 193.548
+* Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)
-* 309.677
-* 387.097
-* 503.226
-* 580.645
-* 696.774
-* 812.903
-* 890.323
-* 1006.452
-* 1083.871
-* 1200.000
-Superpyth Chromatic
-* 51.613
-* 219.355
-* 270.968
-* 438.710
-* 490.323
-* 658.065
-* 709.677
-* 761.290
-* 929.032
-* 980.645
-* 1148.387
-* 1200.000
-Superpyth Shailaja
-* 270.968
-* 709.677
-* 761.290
-* 980.645
-* 1200.000
-Superpyth Subminor Hexatonic
-* 219.355
+== Instruments ==
-* 270.968
-* 490.323
-* 709.677
-* 980.645
-* 1200.000
+A [[Lumatone mapping for 93edo]] is available.
-Superpyth Subminor Pentatonic
+== Music ==
+; [[Bryan Deister]]
-* 270.968
+* [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025)
-* 490.323
-* 709.677
-* 980.645
-* 1200.000
-== See Also ==
+== See also ==
 * [[93edo and stretched hemififths]]
-[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->