337edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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|- | |- | ||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br />8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 26: | Line 26: | ||
| {{monzo| -534 337 }} | | {{monzo| -534 337 }} | ||
| {{mapping| 337 534 }} | | {{mapping| 337 534 }} | ||
| 0.1487 | | +0.1487 | ||
| 0.1487 | | 0.1487 | ||
| 4.18 | | 4.18 | ||
| Line 33: | Line 33: | ||
| 15625/15552, {{monzo| -88 57 -1 }} | | 15625/15552, {{monzo| -88 57 -1 }} | ||
| {{mapping| 337 534 782 }} | | {{mapping| 337 534 782 }} | ||
| 0.3495 | | +0.3495 | ||
| 0.3089 | | 0.3089 | ||
| 8.67 | | 8.67 | ||
| Line 40: | Line 40: | ||
| 15625/15552, 16875/16807, 7381125/7340032 | | 15625/15552, 16875/16807, 7381125/7340032 | ||
| {{mapping| 337 534 782 946 }} | | {{mapping| 337 534 782 946 }} | ||
| 0.2870 | | +0.2870 | ||
| 0.2886 | | 0.2886 | ||
| 8.10 | | 8.10 | ||
| Line 52: | Line 52: | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br /> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Sqrtphi]] (337, 11-limit) | | [[Sqrtphi]] (337, 11-limit) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Music == | == Music == | ||
Latest revision as of 12:51, 21 February 2025
| ← 336edo | 337edo | 338edo → |
337 equal divisions of the octave (abbreviated 337edo or 337ed2), also called 337-tone equal temperament (337tet) or 337 equal temperament (337et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 337 equal parts of about 3.56 ¢ each. Each step represents a frequency ratio of 21/337, or the 337th root of 2.
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 distinct