639edo: Difference between revisions
Jump to navigation
Jump to search
ArrowHead294 (talk | contribs) m Partial undo |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| Line 26: | Line 26: | ||
| {{monzo| 1013 -639 }} | | {{monzo| 1013 -639 }} | ||
| {{mapping| 639 1013 }} | | {{mapping| 639 1013 }} | ||
| | | −0.1238 | ||
| 0.1238 | | 0.1238 | ||
| 6.59 | | 6.59 | ||
| Line 33: | Line 33: | ||
| {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }} | | {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }} | ||
| {{mapping| 639 1013 1484 }} | | {{mapping| 639 1013 1484 }} | ||
| | | −0.1601 | ||
| 0.1134 | | 0.1134 | ||
| 6.04 | | 6.04 | ||
| Line 40: | Line 40: | ||
| 2401/2400, 4375/4374, {{monzo| 58 -14 -13 -2 }} | | 2401/2400, 4375/4374, {{monzo| 58 -14 -13 -2 }} | ||
| {{mapping| 639 1013 1484 1794 }} | | {{mapping| 639 1013 1484 1794 }} | ||
| | | −0.1369 | ||
| 0.1062 | | 0.1062 | ||
| 5.65 | | 5.65 | ||
| Line 47: | Line 47: | ||
| 2401/2400, 4375/4374, 5632/5625, 161280/161051 | | 2401/2400, 4375/4374, 5632/5625, 161280/161051 | ||
| {{mapping| 639 1013 1484 1794 2211 }} | | {{mapping| 639 1013 1484 1794 2211 }} | ||
| | | −0.1554 | ||
| 0.1020 | | 0.1020 | ||
| 5.43 | | 5.43 | ||
| Line 54: | Line 54: | ||
| 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449 | | 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449 | ||
| {{mapping| 639 1013 1484 1794 2211 2365 }} | | {{mapping| 639 1013 1484 1794 2211 2365 }} | ||
| | | −0.1650 | ||
| 0.0955 | | 0.0955 | ||
| 5.08 | | 5.08 | ||
| Line 61: | Line 61: | ||
| 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625 | | 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625 | ||
| {{mapping| 639 1013 1484 1794 2211 2365 2612 }} | | {{mapping| 639 1013 1484 1794 2211 2365 2612 }} | ||
| | | −0.1487 | ||
| 0.0970 | | 0.0970 | ||
| 5.16 | | 5.16 | ||
| Line 68: | Line 68: | ||
| 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400 | | 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400 | ||
| {{mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h) | | {{mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h) | ||
| | | −0.1618 | ||
| 0.0971 | | 0.0971 | ||
| 5.17 | | 5.17 | ||
| Line 101: | Line 101: | ||
| [[Ennealimmal]] / ennealimmalis | | [[Ennealimmal]] / ennealimmalis | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Revision as of 19:22, 15 January 2025
| ← 638edo | 639edo | 640edo → |
Theory
639edo is distinctly consistent in the 17-odd-limit. It has a sharp tendency, with harmonics of 3 to 17 all tuned sharp. The 639h val gives a reasonable approximation of harmonic 19, where it tempers out [1 27 -18⟩ (ennealimma) and [55 -1 -23⟩ (counterwürschmidt comma) in the 5-limit; 2401/2400 and 4375/4374 in the 7-limit; 5632/5625 and 19712/19683 in the 11-limit; 2080/2079 and 4459/4455 in the 13-limit; 1156/1155, 2058/2057, and 2601/2600 in the 17-limit; 1216/1215, 1445/1444, 1540/1539, 2376/2375, and 2926/2925 in the 19-limit. It supports 11-limit ennealimmal and its 13-limit extension ennealimmalis.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.392 | +0.541 | +0.188 | +0.795 | +0.787 | +0.209 | -0.799 | +0.834 | -0.469 | +0.504 |
| Relative (%) | +0.0 | +20.9 | +28.8 | +10.0 | +42.3 | +41.9 | +11.1 | -42.6 | +44.4 | -25.0 | +26.9 | |
| Steps (reduced) |
639 (0) |
1013 (374) |
1484 (206) |
1794 (516) |
2211 (294) |
2365 (448) |
2612 (56) |
2714 (158) |
2891 (335) |
3104 (548) |
3166 (610) | |
Subsets and supersets
Since 639 = 32 × 71, it has subset edos 3, 9, 71, and 213.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1013 -639⟩ | [⟨639 1013]] | −0.1238 | 0.1238 | 6.59 |
| 2.3.5 | [1 -27 18⟩, [55 -1 -23⟩ | [⟨639 1013 1484]] | −0.1601 | 0.1134 | 6.04 |
| 2.3.5.7 | 2401/2400, 4375/4374, [58 -14 -13 -2⟩ | [⟨639 1013 1484 1794]] | −0.1369 | 0.1062 | 5.65 |
| 2.3.5.7.11 | 2401/2400, 4375/4374, 5632/5625, 161280/161051 | [⟨639 1013 1484 1794 2211]] | −0.1554 | 0.1020 | 5.43 |
| 2.3.5.7.11.13 | 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449 | [⟨639 1013 1484 1794 2211 2365]] | −0.1650 | 0.0955 | 5.08 |
| 2.3.5.7.11.13.17 | 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625 | [⟨639 1013 1484 1794 2211 2365 2612]] | −0.1487 | 0.0970 | 5.16 |
| 2.3.5.7.11.13.17.19 | 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400 | [⟨639 1013 1484 1794 2211 2365 2612, 2715]] (639h) | −0.1618 | 0.0971 | 5.17 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 53\639 | 99.53 | 18/17 | Quindro |
| 1 | 206\639 | 386.85 | 5/4 | Counterwürschmidt |
| 9 | 168\639 (26\639) |
315.49 (48.83) |
6/5 (36/35) |
Ennealimmal / ennealimmalis |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct