362edo: 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]] (%) | ||
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| 393216/390625, {{monzo| 25 -48 22 }} | | 393216/390625, {{monzo| 25 -48 22 }} | ||
| {{mapping| 362 574 841 }} | | {{mapping| 362 574 841 }} | ||
| | | −0.3896 | ||
| 0.2822 | | 0.2822 | ||
| 8.51 | | 8.51 | ||
| Line 53: | Line 53: | ||
| [[Würschmidt]] | | [[Würschmidt]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
Latest revision as of 12:18, 21 February 2025
| ← 361edo | 362edo | 363edo → |
362 equal divisions of the octave (abbreviated 362edo or 362ed2), also called 362-tone equal temperament (362tet) or 362 equal temperament (362et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 362 equal parts of about 3.31 ¢ each. Each step represents a frequency ratio of 21/362, or the 362nd root of 2.
Theory
362edo is enfactored in the 3-limit and is only consistent to the 5-odd-limit, with two mappings possible for the 7-limit:
- ⟨362 574 841 1016] (patent val),
- ⟨362 574 841 1017] (362d).
Using the patent val, it tempers out 393216/390625 and [25 -48 22⟩ in the 5-limit; 4375/4374, 458752/455625 and 11529602/11390625 in the 7-limit, supporting barbados.
Using the 362d val, it tempers out 393216/390625 and [25 -48 22⟩ in the 5-limit; 5120/5103, 118098/117649 and 1959552/1953125 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.81 | +1.53 | -0.87 | +1.61 | -1.04 | +1.46 | -0.98 | +1.12 | +0.83 | -0.06 | +1.56 |
| Relative (%) | +24.4 | +46.2 | -26.2 | +48.7 | -31.4 | +44.1 | -29.4 | +33.8 | +25.0 | -1.9 | +47.1 | |
| Steps (reduced) |
574 (212) |
841 (117) |
1016 (292) |
1148 (62) |
1252 (166) |
1340 (254) |
1414 (328) |
1480 (32) |
1538 (90) |
1590 (142) |
1638 (190) | |
Subsets and supersets
Since 362 factors into 2 × 181, 372edo has 2edo and 181edo as its subsets. 1448edo, which quadruples it, is a strong full 13-limit system.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 393216/390625, [25 -48 22⟩ | [⟨362 574 841]] | −0.3896 | 0.2822 | 8.51 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 117\362 | 387.85 | 5/4 | Würschmidt |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct