349edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 41: | Line 50: | ||
| 0.1756 | | 0.1756 | ||
| 5.11 | | 5.11 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 51: | Line 67: | ||
| 75/64 | | 75/64 | ||
| [[Orson]] | | [[Orson]] | ||
|} | |||
<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 13:11, 16 November 2024
| ← 348edo | 349edo | 350edo → |
Theory
349edo is only consistent to the 5-odd-limit. Omitting the harmonic 7, it is consistent to the 13-odd-limit with a flat tendency. In the 2.3.5.11.13 subgroup, the equal temperament tempers out 625/624, 17303/17280, 28561/28512, 41067/40960, 43940/43923, 85293/85184, 131625/131072 and 166375/165888.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.52 | -1.21 | +0.80 | -1.04 | -1.17 | -1.56 | +1.70 | +1.63 | +1.63 | +0.28 | +0.95 |
| Relative (%) | -15.2 | -35.3 | +23.3 | -30.4 | -34.2 | -45.3 | +49.5 | +47.5 | +47.3 | +8.1 | +27.7 | |
| Steps (reduced) |
553 (204) |
810 (112) |
980 (282) |
1106 (59) |
1207 (160) |
1291 (244) |
1364 (317) |
1427 (31) |
1483 (87) |
1533 (137) |
1579 (183) | |
Subsets and supersets
349edo is the 70th prime edo. 1047edo, which triples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-553 349⟩ | [⟨349 553]] | 0.1648 | 0.1648 | 4.79 |
| 2.3.5 | 2109375/2097152, [-31 43 -16⟩ | [⟨349 553 810]] | 0.2841 | 0.2158 | 6.28 |
| 2.3.5.11 | 166375/165888, 1366875/1362944, 1953125/1948617 | [⟨349 553 810 1207]] | 0.2980 | 0.1884 | 5.48 |
| 2.3.5.11.13 | 625/624, 17303/17280, 41067/40960, 216513/216320 | [⟨349 553 810 1207 1291]] | 0.3227 | 0.1756 | 5.11 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 79\349 | 271.63 | 75/64 | Orson |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct