93edo: Difference between revisions
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== Theory == | == 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 | 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. | 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. | ||
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=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|93}} | {{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 == | == Intervals == | ||
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* Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 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) | * 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 == | == See also == | ||
* [[93edo and stretched hemififths]] | * [[93edo and stretched hemififths]] | ||