367edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|367}} == Theory == 367et is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit: * {{val|367 582 852 1030}} (paten..." |
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== Theory == | == Theory == | ||
367et is only consistent to the [[5-odd-limit]], with three mappings possible for the 7-limit: | 367et is only [[consistent]] to the [[5-odd-limit]], with three mappings possible for the 7-limit: | ||
* {{val|367 582 852 1030}} (patent val) | * {{val| 367 582 852 1030 }} (patent val) | ||
* {{val|367 582 852 '''1031'''}} (367d val) | * {{val| 367 582 852 '''1031''' }} (367d val) | ||
* {{val|367 582 '''853''' '''1031'''}} (367cd val) | * {{val| 367 582 '''853''' '''1031''' }} (367cd val) | ||
Using the patent val, it tempers out [[15625/15552]] and {{monzo|102 -57 -5}} in the 5-limit; [[5120/5103]] | Using the patent val, it tempers out [[15625/15552]] and {{monzo| 102 -57 -5 }} in the 5-limit; [[5120/5103]] and 40353607/39858075 in the 7-limit. | ||
Using the 367d val, it tempers out 15625/15552 and {{monzo|102 -57 -5}} in the 5-limit; | Using the 367d val, it tempers out 15625/15552 and {{monzo| 102 -57 -5 }} in the 5-limit; 2460375/2458624 and [[2097152/2083725]] in the 7-limit. | ||
Using the 367cd val, it tempers out 268435456/263671875 and {{monzo|33 -34 9}} in the 5-limit; 5120/5103, 7558272/7503125 and 235298/234375 in the 7-limit. | Using the 367cd val, it tempers out 268435456/263671875 and {{monzo| 33 -34 9 }} in the 5-limit; 5120/5103, 7558272/7503125 and 235298/234375 in the 7-limit. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 18: | Line 18: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
367edo is the 73rd [[prime | 367edo is the 73rd [[prime edo]]. [[1101edo]], 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|582 -367}} | | {{monzo| 582 -367 }} | ||
|{{mapping|367 582}} | | {{mapping| 367 582 }} | ||
| -0.3288 | | -0.3288 | ||
| 0.3287 | | 0.3287 | ||
| 10.05 | | 10.05 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|15625/15552, {{monzo|102 -57 -5}} | | 15625/15552, {{monzo| 102 -57 -5 }} | ||
|{{mapping|367 582 852}} | | {{mapping| 367 582 852 }} | ||
| -0.1500 | | -0.1500 | ||
| 0.3688 | | 0.3688 | ||
| Line 55: | Line 55: | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|28\367 | | 28\367 | ||
|91.55 | | 91.55 | ||
|{{monzo|46 -7 -15}} | | {{monzo| 46 -7 -15 }} | ||
|[[Gross]] | | [[Gross]] | ||
|- | |- | ||
|1 | | 1 | ||
|97\367 | | 97\367 | ||
|317.17 | | 317.17 | ||
|6/5 | | 6/5 | ||
|[[Hanson]] | | [[Hanson]] | ||
|} | |} | ||
<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 14:09, 12 January 2024
| ← 366edo | 367edo | 368edo → |
Theory
367et is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit:
- ⟨367 582 852 1030] (patent val)
- ⟨367 582 852 1031] (367d val)
- ⟨367 582 853 1031] (367cd val)
Using the patent val, it tempers out 15625/15552 and [102 -57 -5⟩ in the 5-limit; 5120/5103 and 40353607/39858075 in the 7-limit.
Using the 367d val, it tempers out 15625/15552 and [102 -57 -5⟩ in the 5-limit; 2460375/2458624 and 2097152/2083725 in the 7-limit.
Using the 367cd val, it tempers out 268435456/263671875 and [33 -34 9⟩ in the 5-limit; 5120/5103, 7558272/7503125 and 235298/234375 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.04 | -0.48 | -0.98 | -1.19 | +1.27 | -0.20 | +0.56 | -0.32 | +0.03 | +0.06 | -0.48 |
| Relative (%) | +31.9 | -14.8 | -29.9 | -36.2 | +38.9 | -6.1 | +17.1 | -9.9 | +1.1 | +2.0 | -14.7 | |
| Steps (reduced) |
582 (215) |
852 (118) |
1030 (296) |
1163 (62) |
1270 (169) |
1358 (257) |
1434 (333) |
1500 (32) |
1559 (91) |
1612 (144) |
1660 (192) | |
Subsets and supersets
367edo is the 73rd prime edo. 1101edo, 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 | [582 -367⟩ | [⟨367 582]] | -0.3288 | 0.3287 | 10.05 |
| 2.3.5 | 15625/15552, [102 -57 -5⟩ | [⟨367 582 852]] | -0.1500 | 0.3688 | 11.28 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 28\367 | 91.55 | [46 -7 -15⟩ | Gross |
| 1 | 97\367 | 317.17 | 6/5 | Hanson |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct