93edo: Difference between revisions

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== Theory ==

93 = 3 × 31, ~~and~~ 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]] (43 & 50) temperament. It is the 13th no-3s [[zeta peak edo]]. The bd val is close to the 9-odd limit minimax tuning for [[superpyth]].

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. It is the 13th no-3s [[zeta peak edo]]. The 93bd val is close to the 9-odd limit minimax tuning for [[superpyth]] and approximates {{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{{~~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.

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 ===

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 {{Infobox ET}}
-{{EDO intro|93}}
+{{ED intro}}
 == Theory ==
-= 3 × 31, and 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]] (43 &amp; 50) temperament. It is the 13th no-3s [[zeta peak edo]]. The bd val is close to the 9-odd limit minimax tuning for [[superpyth]].
+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. It is the 13th no-3s [[zeta peak edo]]. The 93bd val is close to the 9-odd limit minimax tuning for [[superpyth]] and approximates {{frac|2|7}}-[[64/63|septimal&nbsp;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{{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.
+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 ===