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== Theory ==
== 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 ===
=== 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
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| 32805/32768, {{val| 10 23 -20 }}
| 32805/32768, {{val| 10 23 -20 }}
| {{mapping| 183 290 425 }}
| {{mapping| 183 290 425 }}
| -0.0157
| −0.0157
| 0.182
| 0.182
| 2.78
| 2.78
Line 40: Line 49:
| 6144/6125, 16875/16807, 19683/19600
| 6144/6125, 16875/16807, 19683/19600
| {{mapping| 183 290 425 514 }}
| {{mapping| 183 290 425 514 }}
| -0.1601
| −0.1601
| 0.296
| 0.296
| 4.51
| 4.51
Line 47: Line 56:
| 540/539, 1375/1372, 5632/5625, 8019/8000
| 540/539, 1375/1372, 5632/5625, 8019/8000
| {{mapping| 183 290 425 514 633 }}
| {{mapping| 183 290 425 514 633 }}
| -0.0993
| −0.0993
| 0.291
| 0.291
| 4.44
| 4.44
Line 54: Line 63:
| 351/350, 540/539, 676/675, 1375/1372, 4096/4095
| 351/350, 540/539, 676/675, 1375/1372, 4096/4095
| {{mapping| 183 290 425 514 633 677 }}
| {{mapping| 183 290 425 514 633 677 }}
| -0.0295
| −0.0295
| 0.308
| 0.308
| 4.70
| 4.70
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| 351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095
| 351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095
| {{mapping| 183 290 425 514 633 677 748 }}
| {{mapping| 183 290 425 514 633 677 748 }}
| -0.0240
| −0.0240
| 0.286
| 0.286
| 4.36
| 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 ===
=== 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
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperament
! Temperaments
|-
|-
| 1
| 1
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| 27/26
| 27/26
| [[Luminal]]
| [[Luminal]]
|-
| 1
| 16\183
| 104.92
| 17/16
| [[Septendesemi]]
|-
|-
| 1
| 1
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| 498.36
| 498.36
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 1
| 1
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| [[Promethium]]
| [[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 ==
== Music ==

Latest revision as of 13:32, 13 March 2026

← 182edo 183edo 184edo →
Prime factorization 3 × 61
Step size 6.55738 ¢ 
Fifth 107\183 (701.639 ¢)
Semitones (A1:m2) 17:14 (111.5 ¢ : 91.8 ¢)
Consistency limit 17
Distinct consistency limit 17

The 183 equal divisions of the octave (183edo), or the 183(-tone) equal temperament (183tet, 183et) when viewed from a regular temperament perspective, divides the octave into 183 equal parts of about 6.56 cents each, a size close to 243/242, the rastma.

Theory

183edo is notable as a higher-limit system, distinctly consistent in the 17-odd-limit, or the no-19 no-31 33-odd-limit. It has especially low errors in all prime limits from 11 to 29, although its bad rendering of 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.

As an equal temperament, 183et 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 temperaments 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

Approximation of prime harmonics in 183edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.32 +0.57 +1.67 -0.50 -1.18 -0.04 -2.43 +1.23 -0.07 +2.51
Relative (%) +0.0 -4.8 +8.7 +25.4 -7.6 -18.0 -0.6 -37.1 +18.8 -1.1 +38.2
Steps
(reduced) 		183
(0) 		290
(107) 		425
(59) 		514
(148) 		633
(84) 		677
(128) 		748
(16) 		777
(45) 		828
(96) 		889
(157) 		907
(175)
Approximation of prime harmonics in 183edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -2.16 -2.83 -0.04 -3.21 -1.37 +3.12 -2.13 -0.62 -2.65 +1.72 +2.68
Relative (%) -33.0 -43.2 -0.6 -49.0 -20.9 +47.6 -32.5 -9.4 -40.4 +26.2 +40.8
Steps
(reduced) 		953
(38) 		980
(65) 		993
(78) 		1016
(101) 		1048
(133) 		1077
(162) 		1085
(170) 		1110
(12) 		1125
(27) 		1133
(35) 		1154
(56)

Subsets and supersets

Since 183 factors into primes as 3 × 61, 183edo contains 3edo and 61edo as its subsets.

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 183edo. Prime harmonics are in bold.

As 183edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 183edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/9, 18/11 0.133 2.0
11/6, 12/11 0.183 2.8
15/8, 16/15 0.256 3.9
3/2, 4/3 0.316 4.8
11/8, 16/11 0.498 7.6
13/9, 18/13 0.552 8.4
5/4, 8/5 0.572 8.7
9/8, 16/9 0.631 9.6
13/11, 22/13 0.685 10.4
15/11, 22/15 0.754 11.5
13/12, 24/13 0.868 13.2
5/3, 6/5 0.887 13.5
11/10, 20/11 1.070 16.3
7/5, 10/7 1.094 16.7
13/8, 16/13 1.183 18.0
9/5, 10/9 1.203 18.3
15/14, 28/15 1.410 21.5
15/13, 26/15 1.439 21.9
7/4, 8/7 1.666 25.4
13/10, 20/13 1.755 26.8
7/6, 12/7 1.982 30.2
11/7, 14/11 2.164 33.0
9/7, 14/9 2.297 35.0
13/7, 14/13 2.849 43.5

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-290 183⟩ [⟨183 290]] +0.0996 0.100 1.52
2.3.5 32805/32768, ⟨10 23 -20] [⟨183 290 425]] −0.0157 0.182 2.78
2.3.5.7 6144/6125, 16875/16807, 19683/19600 [⟨183 290 425 514]] −0.1601 0.296 4.51
2.3.5.7.11 540/539, 1375/1372, 5632/5625, 8019/8000 [⟨183 290 425 514 633]] −0.0993 0.291 4.44
2.3.5.7.11.13 351/350, 540/539, 676/675, 1375/1372, 4096/4095 [⟨183 290 425 514 633 677]] −0.0295 0.308 4.70
2.3.5.7.11.13.17 351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095 [⟨183 290 425 514 633 677 748]] −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 130, 171, 161, and 159, respectively. In the 13-, 19-, and 23-limit it is superseded by 190g. In the 17-limit, where it is the strongest, by 217.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 10\183 65.57 27/26 Luminal
1 16\183 104.92 17/16 Septendesemi
1 17\183 111.48 16/15 Stockhausenic
1 38\183 249.18 15/13 Hemischis
1 58\183 380.33 56/45 Quanharuk
1 59\183 386.89 5/4 Grendel
1 76\183 498.36 4/3 Helmholtz
1 77\183 504.92 104976/78125 Countermeantone
3 21\183 137.70 13/12 Avicenna
3 24\183 157.38 35/32 Nessafof
3 28\183 183.61 10/9 Mirkat
3 38\183
(23\183) 		249.18
(150.82) 		15/13
(12/11) 		Hemiterm
3 76\183
(15\183) 		498.36
(98.36) 		4/3
(200/189) 		Term / terminator
61 38\183
(2\183) 		249.18
(13.11) 		13750/11907
(?) 		Promethium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

birdshite stalactite
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