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.~~

=== Subsets and supersets ===

Since 161 factors into ~~{{factorization|161}}~~, 161edo contains [[7edo]] and [[23edo]] as its subsets.

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

== Intervals ==

== Regular temperament properties ==

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| 243/242, 351/350, 441/440, 847/845, 3136/3125

| {{mapping| 161 255 374 452 557 596 }}

| ~~−0~~.046

| −0.046

| 0.449

| 6.03

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| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088

| {{mapping| 161 255 374 452 557 596 658 }}

| ~~−0~~.018

| −0.018

| 0.422

| 5.66

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| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512

| {{mapping| 161 255 374 452 557 596 658 684 }}

| ~~−0~~.034

| −0.034

| 0.397

| 5.32

|}

* 161et has a lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo~~|183et~~]].

* 161et has a lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo]].

=== Rank-2 temperaments ===

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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 ~~it is~~ 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]]

@@ Line 1: / Line 1: @@
 {{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}}
 === Subsets and supersets ===
-Since 161 factors into {{factorization|161}}, 161edo contains [[7edo]] and [[23edo]] as its subsets.
+Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.
+== Intervals ==
+{{Interval table}}
 == Regular temperament properties ==
@@ Line 55: / Line 60: @@
 | 243/242, 351/350, 441/440, 847/845, 3136/3125
 | {{mapping| 161 255 374 452 557 596 }}
-| &minus;0.046
+| −0.046
 | 0.449
 | 6.03
@@ Line 62: / Line 67: @@
 | 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
 | {{mapping| 161 255 374 452 557 596 658 }}
-| &minus;0.018
+| −0.018
 | 0.422
 | 5.66
@@ Line 69: / Line 74: @@
 | 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
 | {{mapping| 161 255 374 452 557 596 658 684 }}
-| &minus;0.034
+| −0.034
 | 0.397
 | 5.32
 |}
-* 161et has a lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo|183et]].
+* 161et has a lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo]].
 === Rank-2 temperaments ===
@@ Line 139: / Line 144: @@
 | [[Absurdity]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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]]