161edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 7 × 23~~

~~| Step size = 7.45324¢~~

~~| Fifth = 94\161 (700.62¢)~~

~~| Semitones = 14:13 (104.35¢ : 96.89)~~

~~| Consistency = 7~~

}}

The '''161 equal divisions of the octave''' ('''161edo'''), or the '''161(-tone) equal temperament''' ('''161tet''', '''161et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 161 [[equal]] parts of about 7.45 [[cent]]s each.

== Theory ==

161edo 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 [[~~1188~~/~~1183~~]], [[~~351~~/~~350~~]], [[~~847~~/~~845~~]], [[~~1575~~/~~1573~~]], [[~~1001~~/~~1000~~]] 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-~~limits~~.

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 ===

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

=== Subsets and supersets ===

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

== Intervals ==

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

Line 28:

Line 31:

| 2.3

| {{monzo| -255 161 }}

| [{{~~val~~| 161 255 }}]

| {{mapping| 161 255 }}

| +0.421

| 0.421

Line 35:

Line 38:

| 2.3.5

| 393216/390625, {{monzo| -17 21 -7 }}

| [{{~~val~~| 161 255 374 }}]

| {{mapping| 161 255 374 }}

| +0.099

| 0.570

Line 42:

Line 45:

| 2.3.5.7

| 2401/2400, 3136/3125, 177147/175000

| [{{~~val~~| 161 255 374 452 }}]

| {{mapping| 161 255 374 452 }}

| +0.064

| 0.498

Line 49:

Line 52:

| 2.3.5.7.11

| 243/242, 441/440, 3136/3125, 35937/35840

| [{{~~val~~| 161 255 374 452 557 }}]

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

| +0.037

| 0.448

Line 56:

Line 59:

| 2.3.5.7.11.13

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

| [{{~~val~~| 161 255 374 452 557 596 }}]

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

| -0.046

| −0.046

| 0.449

| 6.03

Line 63:

Line 66:

| 2.3.5.7.11.13.17

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

| [{{~~val~~| 161 255 374 452 557 596 658 }}]

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

| -0.018

| −0.018

| 0.422

| 5.66

Line 70:

Line 73:

| 2.3.5.7.11.13.17.19

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

| [{{~~val~~| 161 255 374 452 557 596 658 684 }}]

| {{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]].

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

Line 90:

Line 95:

| 16/15

| [[Vavoom]]

|-

| 1

| 16\161

| 119.25

| 15/14

| [[Septidiasemi]]

|-

| 1

Line 128:

Line 139:

|-

| 7

| 67\161 (2\161)

| 67\161 (2\161)

| 499.38 (14.91)

| 499.38 (14.91)

| 4/3 (81/80)

| 4/3 (81/80)

| [[Absurdity]]

|}

<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:Equal divisions of the octave]]~~

[[Category:Mintone]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 7 × 23
+{{ED intro}}
-| Step size = 7.45324¢
-| Fifth = 94\161 (700.62¢)
-| Semitones = 14:13 (104.35¢ : 96.89)
-| Consistency = 7
-}}
-The '''161 equal divisions of the octave''' ('''161edo'''), or the '''161(-tone) equal temperament''' ('''161tet''', '''161et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 161 [[equal]] parts of about 7.45 [[cent]]s each.
 == Theory ==
-edo 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 [[1188/1183]], [[351/350]], [[847/845]], [[1575/1573]], [[1001/1000]] 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-limits.
+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 ===
-edo is notable as being low in [[29-limit]] relative error in the 100 to 200 range.
 {{Harmonics in equal|161}}
+=== Subsets and supersets ===
+Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.
+== Intervals ==
+{{Interval table}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 28: / Line 31: @@
 | 2.3
 | {{monzo| -255 161 }}
-| [{{val| 161 255 }}]
+| {{mapping| 161 255 }}
 | +0.421
 | 0.421
@@ Line 35: / Line 38: @@
 | 2.3.5
 | 393216/390625, {{monzo| -17 21 -7 }}
-| [{{val| 161 255 374 }}]
+| {{mapping| 161 255 374 }}
 | +0.099
 | 0.570
@@ Line 42: / Line 45: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 177147/175000
-| [{{val| 161 255 374 452 }}]
+| {{mapping| 161 255 374 452 }}
 | +0.064
 | 0.498
@@ Line 49: / Line 52: @@
 | 2.3.5.7.11
 | 243/242, 441/440, 3136/3125, 35937/35840
-| [{{val| 161 255 374 452 557 }}]
+| {{mapping| 161 255 374 452 557 }}
 | +0.037
 | 0.448
@@ Line 56: / Line 59: @@
 | 2.3.5.7.11.13
 | 243/242, 351/350, 441/440, 847/845, 3136/3125
-| [{{val| 161 255 374 452 557 596 }}]
+| {{mapping| 161 255 374 452 557 596 }}
-| -0.046
+| −0.046
 | 0.449
 | 6.03
@@ Line 63: / Line 66: @@
 | 2.3.5.7.11.13.17
 | 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
-| [{{val| 161 255 374 452 557 596 658 }}]
+| {{mapping| 161 255 374 452 557 596 658 }}
-| -0.018
+| −0.018
 | 0.422
 | 5.66
@@ Line 70: / Line 73: @@
 | 2.3.5.7.11.13.17.19
 | 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
-| [{{val| 161 255 374 452 557 596 658 684 }}]
+| {{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]].
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 90: / Line 95: @@
 | 16/15
 | [[Vavoom]]
+|-
+| 1
+| 16\161
+| 119.25
+| 15/14
+| [[Septidiasemi]]
 |-
 | 1
@@ Line 128: / Line 139: @@
 |-
 | 7
-| 67\161<br>(2\161)
+| 67\161<br />(2\161)
-| 499.38<br>(14.91)
+| 499.38<br />(14.91)
-| 4/3<br>(81/80)
+| 4/3<br />(81/80)
 | [[Absurdity]]
 |}
+<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:Equal divisions of the octave]]
 [[Category:Mintone]]