161edo: Difference between revisions

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== Theory ==

161et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.

161edo has a [[perfect fifth]] slightly sharp of that of [[12edo]], such that it maps the [[Pythagorean comma]] to one step. It approximates many of the low primes fairly well; however, it is only consistent to the [[7-odd-limit]], due to [[10/9]] being mapped too sharply from prime [[5/1|5]] being sharp, while [[3/1|3]] is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the [[23-odd-limit]] being 9/5, [[13/9]], [[23/13]], and their [[octave complement]]s, and additional inconsistencies in the [[25-odd-limit]] include [[25/18]], [[25/23]], and their octave complements. Prime [[29/1|29]] is also accurate, though harmonic [[27/1|27]] is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes [[28/27]] to be mapped wider than [[27/26]], meaning 161edo is at most [[diamond monotone]] in the 25-odd-limit.

As an equal temperament, 161et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.

=== Prime harmonics ===

~~In the range of edos from 100 to 200, 161edo is notable as being low in [[29-limit]] relative error.~~

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=== Subsets and supersets ===

Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.

== Intervals ==

== Regular temperament properties ==

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| [[Absurdity]]

|}

<nowiki />* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

[[Category:Mintone]]

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 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
-et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.
+edo has a [[perfect fifth]] slightly sharp of that of [[12edo]], such that it maps the [[Pythagorean comma]] to one step. It approximates many of the low primes fairly well; however, it is only consistent to the [[7-odd-limit]], due to [[10/9]] being mapped too sharply from prime [[5/1|5]] being sharp, while [[3/1|3]] is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the [[23-odd-limit]] being 9/5, [[13/9]], [[23/13]], and their [[octave complement]]s, and additional inconsistencies in the [[25-odd-limit]] include [[25/18]], [[25/23]], and their octave complements. Prime [[29/1|29]] is also accurate, though harmonic [[27/1|27]] is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes [[28/27]] to be mapped wider than [[27/26]], meaning 161edo is at most [[diamond monotone]] in the 25-odd-limit.
+As an equal temperament, 161et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.
 === Prime harmonics ===
-In the range of edos from 100 to 200, 161edo is notable as being low in [[29-limit]] relative error.
 {{Harmonics in equal|161}}
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 === Subsets and supersets ===
 Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.
+== Intervals ==
+{{Interval table}}
 == Regular temperament properties ==
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 | [[Absurdity]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 [[Category:Mintone]]