362edo: Difference between revisions
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-enfactored 362c val |
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== Theory == | == Theory == | ||
362edo is [[enfactoring|enfactored]] in the [[3-limit]] and is only [[consistent]] to the [[5-odd-limit]], with two mappings possible for the [[7-limit]]: | |||
* {{val|362 574 841 1016}} (patent val | * {{val| 362 574 841 1016 }} (patent val), | ||
* {{val| 362 574 841 '''1017''' }} (362d). | |||
* {{val|362 574 841 '''1017'''}} (362d). | |||
Using the patent val, it tempers out [[393216/390625]] and {{monzo|25 -48 22}} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]]. | Using the patent val, it tempers out [[393216/390625]] and {{monzo| 25 -48 22 }} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]]. | ||
Using the 362d val, it tempers out 393216/390625 and {{monzo| 25 -48 22 }} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit. | |||
Using the 362d val, it tempers out 393216/390625 and {{monzo|25 -48 22}} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 18: | Line 15: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
362 factors into 2 × 181, | 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 == | == 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]] (%) | ||
|- | |- | ||
| 2.3.5 | |||
| 393216/390625, {{monzo| 25 -48 22 }} | |||
| {{mapping| 362 574 841 }} | |||
|2.3.5 | |||
|393216/390625, {{monzo|25 -48 22}} | |||
|{{mapping|362 574 841}} | |||
| -0.3896 | | -0.3896 | ||
| 0.2822 | | 0.2822 | ||
| Line 55: | Line 45: | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|117\362 | | 117\362 | ||
|387.85 | | 387.85 | ||
|5/4 | | 5/4 | ||
|[[Würschmidt]] | | [[Würschmidt]] | ||
|} | |} | ||
<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 18:18, 5 February 2024
| ← 361edo | 362edo | 363edo → |
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 it is distinct