183edo: Difference between revisions
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== Theory == | == Theory == | ||
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 [[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. | 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 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. | 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 === | ||
{{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 === | === Subsets and supersets === | ||
Since 183 factors into 3 × 61, 183edo contains [[3edo]] and [[61edo]] as its subsets. | Since 183 factors into primes as{{nowrap| 3 × 61 }}, 183edo contains [[3edo]] and [[61edo]] as its subsets. | ||
== Approximation to JI == | == Approximation to JI == | ||
Revision as of 16:23, 18 January 2026
| ← 182edo | 183edo | 184edo → |
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
| 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) | |
| 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 as3 × 61, 183edo contains 3edo and 61edo as its subsets.
Approximation to JI
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 1210zpi | 182.999728 | 6.557387 | 11.020824 | 11.020734 | 1.64341 | 19.731996 | 1200.001781 | 0.001781 | 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
| 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