183edo: Difference between revisions

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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 183 factors into 3 × 61, 183edo contains [[3edo]] and [[61edo]] as its subsets.
Since 183 factors into 3 × 61, 183edo contains [[3edo]] and [[61edo]] as its subsets.
== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[1210zpi]]
| 182.999728466810
| 6.55738677895164
| 11.020824
| 1.643410
| 19.731996
| 183edo
| 1200.00178054815
| 18
| 18
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 21: Line 53:
! rowspan="2" | [[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
|-
|-
Line 75: Line 107:
|+ style="font-size: 105%;" | 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*
! Temperaments
! Temperaments
|-
|-
Line 142: Line 174:
|-
|-
| 3
| 3
| 38\183<br />(23\183)
| 38\183<br>(23\183)
| 249.18<br />(150.82)
| 249.18<br>(150.82)
| 15/13<br />(12/11)
| 15/13<br>(12/11)
| [[Hemiterm]]
| [[Hemiterm]]
|-
|-
| 3
| 3
| 76\183<br />(15\183)
| 76\183<br>(15\183)
| 498.36<br />(98.36)
| 498.36<br>(98.36)
| 4/3<br />(200/189)
| 4/3<br>(200/189)
| [[Term]] / terminator
| [[Term]] / terminator
|-
|-
| 61
| 61
| 38\183<br />(2\183)
| 38\183<br>(2\183)
| 249.18<br />(13.11)
| 249.18<br>(13.11)
| 13750/11907<br />(?)
| 13750/11907<br>(?)
| [[Promethium]]
| [[Promethium]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[1210zpi]]
| 182.999728466810
| 6.55738677895164
| 11.020824
| 1.643410
| 19.731996
| 183edo
| 1200.00178054815
| 18
| 18
|}


== Music ==
== Music ==

Revision as of 10:22, 19 January 2025

← 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 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

In the range of edos from 100 to 200, 183edo is notable as having especially low error in all prime limits from 11 to 29, compared using a variety of prime error punishments, although it has a bad 19 and fails to be consistent in the 19-odd-limit. It is however a strong no-19's 29-limit system with an essentially perfectly accurate prime 43. It can also be considered to model the 2.17.29.43 subgroup with extreme accuracy.


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)

Subsets and supersets

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

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
1210zpi 182.999728466810 6.55738677895164 11.020824 1.643410 19.731996 183edo 1200.00178054815 18 18

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 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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