337edo: Difference between revisions
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== Theory == | == Theory == | ||
337edo is [[consistent]] to the [[9-odd-limit]], but the error of [[harmonic]] [[5/1|5]] is quite large. If the harmonic is used at all, it tends very flat. The equal temperament [[tempering out|tempers out]] [[16875/16807]], 420175/419904, and 5250987/5242880 in the 7-limit. It [[support]]s | 337edo is [[consistent]] to the [[9-odd-limit]], but the error of [[harmonic]] [[5/1|5]] is quite large. If the harmonic is used at all, it tends very flat. The equal temperament [[tempering out|tempers out]] [[16875/16807]], [[420175/419904]], and 5250987/5242880 in the 7-limit. It [[support]]s [[tokko]] and [[sqrtphi]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 43: | Line 43: | ||
| 8.10 | | 8.10 | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
| Line 63: | Line 64: | ||
| 6/5 | | 6/5 | ||
| [[Hanson]] | | [[Hanson]] | ||
|- | |||
| 1 | |||
| 117\337 | |||
| 416.62 | |||
| 14/11 | |||
| [[Sqrtphi]] (337, 11-limit) | |||
|} | |} | ||
<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 | ||
| Line 68: | Line 75: | ||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
* "The End Of A Good Day" from ''Mysteries'' (2023) [https://open.spotify.com/track/1yaSXn4u2TVWClwXBtMVs4 Spotify] | [https://francium223.bandcamp.com/track/the-end-of-a-good-day Bandcamp] | [https://www.youtube.com/watch?v=2ixL8YQmJig YouTube] | * "The End Of A Good Day" from ''Mysteries'' (2023) – [https://open.spotify.com/track/1yaSXn4u2TVWClwXBtMVs4 Spotify] | [https://francium223.bandcamp.com/track/the-end-of-a-good-day Bandcamp] | [https://www.youtube.com/watch?v=2ixL8YQmJig YouTube] | ||
Revision as of 07:22, 22 January 2024
| ← 336edo | 337edo | 338edo → |
Theory
337edo is consistent to the 9-odd-limit, but the error of harmonic 5 is quite large. If the harmonic is used at all, it tends very flat. The equal temperament tempers out 16875/16807, 420175/419904, and 5250987/5242880 in the 7-limit. It supports tokko and sqrtphi.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.47 | -1.74 | -0.28 | -0.94 | +0.61 | -0.17 | +1.35 | -1.69 | +1.60 | -0.75 | -1.57 |
| Relative (%) | -13.2 | -49.0 | -7.9 | -26.5 | +17.2 | -4.8 | +37.8 | -47.5 | +44.8 | -21.1 | -44.0 | |
| Steps (reduced) |
534 (197) |
782 (108) |
946 (272) |
1068 (57) |
1166 (155) |
1247 (236) |
1317 (306) |
1377 (29) |
1432 (84) |
1480 (132) |
1524 (176) | |
Subsets and supersets
337edo is the 68th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-534 337⟩ | [⟨337 534]] | 0.1487 | 0.1487 | 4.18 |
| 2.3.5 | 15625/15552, [-88 57 -1⟩ | [⟨337 534 782]] | 0.3495 | 0.3089 | 8.67 |
| 2.3.5.7 | 15625/15552, 16875/16807, 7381125/7340032 | [⟨337 534 782 946]] | 0.2870 | 0.2886 | 8.10 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 67\337 | 238.58 | 147/128 | Tokko |
| 1 | 89\337 | 316.91 | 6/5 | Hanson |
| 1 | 117\337 | 416.62 | 14/11 | Sqrtphi (337, 11-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct