349edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|349}} == Theory == 349edo is only consistent to the 5-limit. Omitting the harmonic 7, it is consistent to the 13-limit; tempering out 625/624,..." |
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== Theory == | == Theory == | ||
349edo is only consistent to the 5-limit. Omitting the harmonic 7, it is consistent to the 13-limit | 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 === | ||
{{Harmonics in equal|349}} | {{Harmonics in equal|349}} | ||
===Subsets and supersets=== | === Subsets and supersets === | ||
349edo is the 70th [[prime edo]]. [[1047edo]], which triples it, gives a good correction to the harmonic 7. | 349edo is the 70th [[prime edo]]. [[1047edo]], which triples it, gives a good correction to the harmonic 7. | ||
== 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|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-553 349}} | | {{monzo| -553 349 }} | ||
|{{mapping|349 553}} | | {{mapping| 349 553 }} | ||
| 0.1648 | | 0.1648 | ||
| 0.1648 | | 0.1648 | ||
| 4.79 | | 4.79 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|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 | ||
|- | |- | ||
|2.3.5.11 | | 2.3.5.11 | ||
|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 | ||
|- | |- | ||
|2.3.5.11.13 | | 2.3.5.11.13 | ||
|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 55: | Line 55: | ||
|+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>Ratio* | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|79\349 | | 79\349 | ||
|271.63 | | 271.63 | ||
|75/64 | | 75/64 | ||
|[[Orson]] | | [[Orson]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||
Revision as of 18:50, 11 January 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