183edo: Difference between revisions
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== Theory == | == Theory == | ||
183edo is notable as a higher limit system, especially when 7 is left out of the picture. | 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. | 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 === | === Prime harmonics === | ||
In the range of | 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}} | {{Harmonics in equal|183}} | ||
| Line 14: | Line 14: | ||
== 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]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 25: | Line 25: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -290 183 }} | | {{monzo| -290 183 }} | ||
| | | {{mapping| 183 290 }} | ||
| +0.0996 | | +0.0996 | ||
| 0.100 | | 0.100 | ||
| Line 32: | Line 32: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, {{val| 10 23 -20 }} | | 32805/32768, {{val| 10 23 -20 }} | ||
| | | {{mapping| 183 290 425 }} | ||
| -0.0157 | | -0.0157 | ||
| 0.182 | | 0.182 | ||
| Line 39: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 6144/6125, 16875/16807, 19683/19600 | | 6144/6125, 16875/16807, 19683/19600 | ||
| | | {{mapping| 183 290 425 514 }} | ||
| -0.1601 | | -0.1601 | ||
| 0.296 | | 0.296 | ||
| Line 46: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 1375/1372, 5632/5625, 8019/8000 | | 540/539, 1375/1372, 5632/5625, 8019/8000 | ||
| | | {{mapping| 183 290 425 514 633 }} | ||
| -0.0993 | | -0.0993 | ||
| 0.291 | | 0.291 | ||
| Line 53: | Line 53: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 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 }} | ||
| -0.0295 | | -0.0295 | ||
| 0.308 | | 0.308 | ||
| Line 60: | Line 60: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 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 }} | ||
| -0.0240 | | -0.0240 | ||
| 0.286 | | 0.286 | ||
| Line 70: | Line 70: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| Line 153: | Line 153: | ||
| [[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 | |||
== Music == | == Music == | ||
Revision as of 16:20, 22 April 2024
| ← 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, especially when 7 is left out of the picture. 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.
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 limits 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.
| 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) | |
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 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperament |
|---|---|---|---|---|
| 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 it is distinct