183edo
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. The equal temperament 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.
| 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.
Regular temperament properties
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf