161edo: Difference between revisions

Line 3:

== 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.

161et [[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 55:

| 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

Line 62:

| 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

Line 69:

| 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 ===

Line 139:

| [[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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Mintone]]

@@ Line 3: / Line 3: @@
 == 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.
+et [[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 55: / Line 55: @@
 | 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 62: @@
 | 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 69: @@
 | 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 139: @@
 | [[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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Mintone]]