183edo: Difference between revisions

(18 intermediate revisions by 6 users not shown)

Line 3:

== Theory ==

183edo is notable as a higher limit system, ~~especially when 7 is left out of the picture. The equal temperament~~ [[~~tempering out~~|~~tempers out]] the [[schisma~~]] in the [[5-~~limit]]. In the [[7~~-limit]], ~~it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In~~ the [[11-~~limit]], it tempers out [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]], and [[8019/8000]]; in the [[13~~-limit]]~~, [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[1573/1568]], [[2080/2079]], [[4096/4095]], [[4225/4224]], and [[6656/6655]];~~ in ~~the~~ [[~~17-~~limit]] ~~[[442/441]]~~, [[~~561~~/~~560~~]]~~, [[715/714]], [[936/935]], [[1089/1088]], and [[1156/1155]]; and~~ in the [[19-limit~~]] [[456/455~~]]. It is ~~the~~ [[~~optimal patent val]] for 13- and 17~~-limit ~~[[mirkat~~]], the ~~72 & 111 temperament, and~~ an ~~excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]]. It allows [[~~essentially ~~tempered chord]] including~~ [[~~ratwolfsmic chords]], [[swetismic chords~~]]~~, [[squbemic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit~~.

183edo is notable as a higher-limit system, [[consistency|distinctly consistent]] in the [[17-odd-limit]], or the no-19 no-31 [[33-odd-limit]]. It has especially low errors in ''all'' [[prime limit]]s from 11 to 29, although its bad rendering of [[19/1|19]] makes it fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's [[29-limit]] system with the addition of an essentially perfectly accurate prime [[43/1|43]].

As a no-~~sevens~~ temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.

As an equal temperament, 183et [[tempering out|tempers out]] the [[schisma]] in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]], and [[8019/8000]]; in the [[13-limit]], [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[1573/1568]], [[2080/2079]], [[4096/4095]], [[4225/4224]], and [[6656/6655]]; in the [[17-limit]] [[442/441]], [[561/560]], [[715/714]], [[936/935]], [[1089/1088]], and [[1156/1155]]; and in the [[19-limit]] [[456/455]]. It is the [[optimal patent val]] for 13- and 17-limit [[mirkat]], the {{nowrap|72 & 111}} temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]]. It allows [[essentially tempered chord]] including [[ratwolfsmic chords]], [[swetismic chords]], [[squbemic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.

It is even stronger if 7 is left out of the picture. As a no-7 temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.

=== Prime harmonics ===

~~In the range of edos from 100 to 200, 183edo is notable as having especially low error~~ in ~~''all'' [[prime limit]]s from~~ 11 ~~to 29, compared using a variety~~ of ~~metrics (~~prime ~~error punishments), although it has a bad 19 which causes it to fail to be consistent~~ in ~~the [[19-odd-limit]]. It is however a strong no-19's system, being consistent in the no-19's no-35's 29-prime-limited 45-odd-limit add-43.~~ (The prime 43 is added in the set of odd harmonics due to its essentially perfect accuracy. The harmonic 35 is excluded due to the sharpness of 7 compounding and causing inconsistency in ''some'' cases such as for 39/35.) ~~It can also be considered to model the 2.17.29.43 subgroup with extreme accuracy.~~

{{Harmonics in equal|183|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 183edo (continued)}}

=== Subsets and supersets ===

Since 183 factors into primes as {{nowrap| 3 × 61 }}, 183edo contains [[3edo]] and [[61edo]] as its subsets.

== Approximation to JI ==

=== Interval mappings ===

== Regular temperament properties ==

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

|-

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

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

! rowspan="2" | [[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]] (¢)

Line 33:

Line 42:

| 32805/32768, {{val| 10 23 -20 }}

| {{mapping| 183 290 425 }}

| -0.0157

| −0.0157

| 0.182

| 2.78

Line 40:

Line 49:

| 6144/6125, 16875/16807, 19683/19600

| {{mapping| 183 290 425 514 }}

| -0.1601

| −0.1601

| 0.296

| 4.51

Line 47:

Line 56:

| 540/539, 1375/1372, 5632/5625, 8019/8000

| {{mapping| 183 290 425 514 633 }}

| -0.0993

| −0.0993

| 0.291

| 4.44

Line 54:

Line 63:

| 351/350, 540/539, 676/675, 1375/1372, 4096/4095

| {{mapping| 183 290 425 514 633 677 }}

| -0.0295

| −0.0295

| 0.308

| 4.70

Line 61:

Line 70:

| 351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095

| {{mapping| 183 290 425 514 633 677 748 }}

| -0.0240

| −0.0240

| 0.286

| 4.36

|}

* 183et has lower absolute errors in the 13-, 17-, 19-, and 23-limit than any previous equal temperaments, after [[130edo|130]], [[171edo|171]], [[161edo|161]], and [[159edo|159]], respectively. In the 13-, 19-, and 23-limit it is superseded by [[190edo|190g]]. In the 17-limit, where it is the strongest, by [[217edo|217]].

=== Rank-2 temperaments ===

{| class="wikitable center-all ~~right-3~~ left-5"

{| 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 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! ~~Temperament~~

! Temperaments

|-

| 1

Line 80:

Line 91:

| 27/26

| [[Luminal]]

|-

| 1

| 16\183

| 104.92

| 17/16

| [[Septendesemi]]

|-

| 1

Line 109:

Line 126:

| 498.36

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|-

| 1

Line 153:

Line 170:

| [[Promethium]]

|}

<nowiki>*~~</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

== Music ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is notable as a higher limit system, especially when 7 is left out of the picture. The equal temperament [[tempering out|tempers out]] the [[schisma]] in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]], and [[8019/8000]]; in the [[13-limit]], [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[1573/1568]], [[2080/2079]], [[4096/4095]], [[4225/4224]], and [[6656/6655]]; in the [[17-limit]] [[442/441]], [[561/560]], [[715/714]], [[936/935]], [[1089/1088]], and [[1156/1155]]; and in the [[19-limit]] [[456/455]]. It is the [[optimal patent val]] for 13- and 17-limit [[mirkat]], the 72 &amp; 111 temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]]. It allows [[essentially tempered chord]] including [[ratwolfsmic chords]], [[swetismic chords]], [[squbemic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.
+edo is notable as a higher-limit system, [[consistency|distinctly consistent]] in the [[17-odd-limit]], or the no-19 no-31 [[33-odd-limit]]. It has especially low errors in ''all'' [[prime limit]]s from 11 to 29, although its bad rendering of [[19/1|19]] makes it fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's [[29-limit]] system with the addition of an essentially perfectly accurate prime [[43/1|43]].
-As a no-sevens temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.
+As an equal temperament, 183et [[tempering out|tempers out]] the [[schisma]] in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]], and [[8019/8000]]; in the [[13-limit]], [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[1573/1568]], [[2080/2079]], [[4096/4095]], [[4225/4224]], and [[6656/6655]]; in the [[17-limit]] [[442/441]], [[561/560]], [[715/714]], [[936/935]], [[1089/1088]], and [[1156/1155]]; and in the [[19-limit]] [[456/455]]. It is the [[optimal patent val]] for 13- and 17-limit [[mirkat]], the {{nowrap|72 &amp; 111}} temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]]. It allows [[essentially tempered chord]] including [[ratwolfsmic chords]], [[swetismic chords]], [[squbemic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.
+It is even stronger if 7 is left out of the picture. As a no-7 temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.
 === Prime harmonics ===
-In the range of edos from 100 to 200, 183edo is notable as having especially low error in ''all'' [[prime limit]]s from 11 to 29, compared using a variety of metrics (prime error punishments), although it has a bad 19 which causes it to fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's system, being consistent in the no-19's no-35's 29-prime-limited 45-odd-limit add-43. (The prime 43 is added in the set of odd harmonics due to its essentially perfect accuracy. The harmonic 35 is excluded due to the sharpness of 7 compounding and causing inconsistency in ''some'' cases such as for 39/35.) It can also be considered to model the 2.17.29.43 subgroup with extreme accuracy.
+{{Harmonics in equal|183|columns=11}}
+{{Harmonics in equal|183|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 183edo (continued)}}
-{{Harmonics in equal|183}}
+=== Subsets and supersets ===
+Since 183 factors into primes as {{nowrap| 3 × 61 }}, 183edo contains [[3edo]] and [[61edo]] as its subsets.
+== Approximation to JI ==
+=== Interval mappings ===
+{{Q-odd-limit intervals}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[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 33: / Line 42: @@
 | 32805/32768, {{val| 10 23 -20 }}
 | {{mapping| 183 290 425 }}
-| -0.0157
+| −0.0157
 | 0.182
 | 2.78
@@ Line 40: / Line 49: @@
 | 6144/6125, 16875/16807, 19683/19600
 | {{mapping| 183 290 425 514 }}
-| -0.1601
+| −0.1601
 | 0.296
 | 4.51
@@ Line 47: / Line 56: @@
 | 540/539, 1375/1372, 5632/5625, 8019/8000
 | {{mapping| 183 290 425 514 633 }}
-| -0.0993
+| −0.0993
 | 0.291
 | 4.44
@@ Line 54: / Line 63: @@
 | 351/350, 540/539, 676/675, 1375/1372, 4096/4095
 | {{mapping| 183 290 425 514 633 677 }}
-| -0.0295
+| −0.0295
 | 0.308
 | 4.70
@@ Line 61: / Line 70: @@
 | 351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095
 | {{mapping| 183 290 425 514 633 677 748 }}
-| -0.0240
+| −0.0240
 | 0.286
 | 4.36
 |}
+* 183et has lower absolute errors in the 13-, 17-, 19-, and 23-limit than any previous equal temperaments, after [[130edo|130]], [[171edo|171]], [[161edo|161]], and [[159edo|159]], respectively. In the 13-, 19-, and 23-limit it is superseded by [[190edo|190g]]. In the 17-limit, where it is the strongest, by [[217edo|217]].
 === Rank-2 temperaments ===
-{| class="wikitable center-all right-3 left-5"
+{| 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 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br>ratio*
-! Temperament
+! Temperaments
 |-
 | 1
@@ Line 80: / Line 91: @@
 | 27/26
 | [[Luminal]]
+|-
+| 1
+| 16\183
+| 104.92
+| 17/16
+| [[Septendesemi]]
 |-
 | 1
@@ Line 109: / Line 126: @@
 | 498.36
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |-
 | 1
@@ Line 153: / Line 170: @@
 | [[Promethium]]
 |}
-<nowiki>*</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
 == Music ==