183edo: Difference between revisions

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== Theory ==

183edo is notable as a higher limit system, especially when 7 is left out of the picture. It 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, 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.

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.

=== 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.

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.

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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| -290 183 }}

| [{{~~val~~| 183 290 }}]

| {{mapping| 183 290 }}

| +0.0996

| 0.100

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| 2.3.5

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

| [{{~~val~~| 183 290 425 }}]

| {{mapping| 183 290 425 }}

| -0.0157

| 0.182

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| 2.3.5.7

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

| [{{~~val~~| 183 290 425 514 }}]

| {{mapping| 183 290 425 514 }}

| -0.1601

| 0.296

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| 2.3.5.7.11

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

| [{{~~val~~| 183 290 425 514 633 }}]

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

| -0.0993

| 0.291

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| 2.3.5.7.11.13

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

| [{{~~val~~| 183 290 425 514 633 677 }}]

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

| -0.0295

| 0.308

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| 2.3.5.7.11.13.17

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

| [{{~~val~~| 183 290 425 514 633 677 748 }}]

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

| -0.0240

| 0.286

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperament

|-

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| [[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

== Music ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is notable as a higher limit system, especially when 7 is left out of the picture. It 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, 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.
 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.
 === 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.
+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}}
@@ Line 14: / Line 14: @@
 == 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 25: / Line 25: @@
 | 2.3
 | {{monzo| -290 183 }}
-| [{{val| 183 290 }}]
+| {{mapping| 183 290 }}
 | +0.0996
 | 0.100
@@ Line 32: / Line 32: @@
 | 2.3.5
 | 32805/32768, {{val| 10 23 -20 }}
-| [{{val| 183 290 425 }}]
+| {{mapping| 183 290 425 }}
 | -0.0157
 | 0.182
@@ Line 39: / Line 39: @@
 | 2.3.5.7
 | 6144/6125, 16875/16807, 19683/19600
-| [{{val| 183 290 425 514 }}]
+| {{mapping| 183 290 425 514 }}
 | -0.1601
 | 0.296
@@ Line 46: / Line 46: @@
 | 2.3.5.7.11
 | 540/539, 1375/1372, 5632/5625, 8019/8000
-| [{{val| 183 290 425 514 633 }}]
+| {{mapping| 183 290 425 514 633 }}
 | -0.0993
 | 0.291
@@ Line 53: / Line 53: @@
 | 2.3.5.7.11.13
 | 351/350, 540/539, 676/675, 1375/1372, 4096/4095
-| [{{val| 183 290 425 514 633 677 }}]
+| {{mapping| 183 290 425 514 633 677 }}
 | -0.0295
 | 0.308
@@ Line 60: / Line 60: @@
 | 2.3.5.7.11.13.17
 | 351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095
-| [{{val| 183 290 425 514 633 677 748 }}]
+| {{mapping| 183 290 425 514 633 677 748 }}
 | -0.0240
 | 0.286
@@ Line 70: / Line 70: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperament
 |-
@@ Line 153: / Line 153: @@
 | [[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
 == Music ==