161edo: Difference between revisions

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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~~.

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 ===

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== Regular temperament properties ==

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

! rowspan="2" | Subgroup

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

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

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

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

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

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

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| -255 161 }}

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

| {{mapping| 161 255 }}

| +0.421

| 0.421

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| 2.3.5

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

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

| {{mapping| 161 255 374 }}

| +0.099

| 0.570

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| 2.3.5.7

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

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

| {{mapping| 161 255 374 452 }}

| +0.064

| 0.498

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| 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

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| 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

| 0.449

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| 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

| 0.422

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| 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

| 0.397

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

=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

! Periods per ~~octave~~

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

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| [[Absurdity]]

|}

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

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:Mintone]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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.
+{{EDO intro}}
 == 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.
+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.
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 22: / Line 22: @@
 | 2.3
 | {{monzo| -255 161 }}
-| [{{val| 161 255 }}]
+| {{mapping| 161 255 }}
 | +0.421
 | 0.421
@@ Line 29: / Line 29: @@
 | 2.3.5
 | 393216/390625, {{monzo| -17 21 -7 }}
-| [{{val| 161 255 374 }}]
+| {{mapping| 161 255 374 }}
 | +0.099
 | 0.570
@@ Line 36: / Line 36: @@
 | 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 43: / Line 43: @@
 | 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 50: / Line 50: @@
 | 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
 | 0.449
@@ Line 57: / Line 57: @@
 | 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
 | 0.422
@@ Line 64: / Line 64: @@
 | 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
 | 0.397
 | 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]].
-et has lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent. The same subgroup is only better tuned by [[183edo|183et]].
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
 |+Table of rank-2 temperaments by generator
-! Periods<br>per octave
+! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 135: / Line 134: @@
 | [[Absurdity]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Mintone]]