349edo: Difference between revisions
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== Theory == | == Theory == | ||
349edo is only [[consistent]] to the [[5-odd-limit]]. Omitting the [[harmonic]] [[7/1|7]], it is consistent to the [[13-odd-limit]] with a flat tendency. In the 2.3.5.11.13 [[subgroup]], the equal temperament [[tempering out|tempers out]] [[625/624]], 17303/17280, 28561/28512, 41067/40960, 43940/43923, 85293/85184, 131625/131072 and 166375/165888. | 349edo is only [[consistent]] to the [[5-odd-limit]]. Omitting the [[harmonic]] [[7/1|7]], it is consistent to the [[13-odd-limit]] with a flat tendency. In the 2.3.5.11.13 [[subgroup]], the equal temperament [[tempering out|tempers out]] [[625/624]], 17303/17280, 28561/28512, 41067/40960, 43940/43923, 85293/85184, 131625/131072, and 166375/165888. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 26: | Line 26: | ||
| {{monzo| -553 349 }} | | {{monzo| -553 349 }} | ||
| {{mapping| 349 553 }} | | {{mapping| 349 553 }} | ||
| 0.1648 | | +0.1648 | ||
| 0.1648 | | 0.1648 | ||
| 4.79 | | 4.79 | ||
| Line 33: | Line 33: | ||
| 2109375/2097152, {{monzo| -31 43 -16 }} | | 2109375/2097152, {{monzo| -31 43 -16 }} | ||
| {{mapping| 349 553 810 }} | | {{mapping| 349 553 810 }} | ||
| 0.2841 | | +0.2841 | ||
| 0.2158 | | 0.2158 | ||
| 6.28 | | 6.28 | ||
| Line 40: | Line 40: | ||
| 166375/165888, 1366875/1362944, 1953125/1948617 | | 166375/165888, 1366875/1362944, 1953125/1948617 | ||
| {{mapping| 349 553 810 1207 }} | | {{mapping| 349 553 810 1207 }} | ||
| 0.2980 | | +0.2980 | ||
| 0.1884 | | 0.1884 | ||
| 5.48 | | 5.48 | ||
| Line 47: | Line 47: | ||
| 625/624, 17303/17280, 41067/40960, 216513/216320 | | 625/624, 17303/17280, 41067/40960, 216513/216320 | ||
| {{mapping| 349 553 810 1207 1291 }} | | {{mapping| 349 553 810 1207 1291 }} | ||
| 0.3227 | | +0.3227 | ||
| 0.1756 | | 0.1756 | ||
| 5.11 | | 5.11 | ||
| Line 68: | Line 68: | ||
| [[Orson]] | | [[Orson]] | ||
|} | |} | ||
<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 | ||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
* "chill futuristic lounge" from ''End Of Sartorius Membranes'' (2024) | * "chill futuristic lounge" from ''End Of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/3p2RNlmQieZA2fh3ZwQzVw Spotify] | [https://francium223.bandcamp.com/track/chill-futuristic-lounge Bandcamp] | [https://www.youtube.com/watch?v=B_P7zp5Zmio YouTube] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
Revision as of 16:59, 15 January 2025
| ← 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 distinct