93edo: Difference between revisions

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The 93 equal division divides the octave into 93 equal parts of 12.903 cents each. 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&50 temperament.

Since 93edo has good approximations of 13th, 17th and 19th ~~harmonics~~ unlike 31edo (as 838.~~710¢~~, 103.~~226¢~~, and 296.~~774¢~~ respectively), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.

== 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 & 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 ===

=== 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 ==

== Scales ==

* 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)

* Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)

* Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)

== Instruments ==

A [[Lumatone mapping for 93edo]] is available.

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025)

== See also ==

* [[93edo and stretched hemififths]]

@@ Line 1: / Line 1: @@
-The 93 equal division divides the octave into 93 equal parts of 12.903 cents each. 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.
+{{Infobox ET}}
+{{ED intro}}
-Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710¢, 103.226¢, and 296.774¢ respectively), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
+== 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}}
+=== 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 ==
+{{Interval table}}
+== Scales ==
+* 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)
+* Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)
+* Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)
+== Instruments ==
+A [[Lumatone mapping for 93edo]] is available.
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025)
+== See also ==
+* [[93edo and stretched hemififths]]