402edo: Difference between revisions
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== Theory == | == Theory == | ||
402edo is only [[consistent]] to the [[5-odd-limit]]. There are three possible mappings in the 7-limit: | |||
* {{val|402 637 933 1129}} (patent val) | * {{val| 402 637 933 1129 }} ([[patent val]]) | ||
* {{val|402 637 ''' | * {{val| 402 637 933 '''1128''' }} (402d) | ||
* {{val|402 637 | * {{val| 402 637 '''934''' 1129 }} (402c) | ||
Using the patent val, it tempers out 4375/4374, 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports [[ragismic]] and [[abigail]]. | Using the patent val, it tempers out [[4375/4374]], 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports [[ragismic]] and [[abigail]]. | ||
Using the | Using the 402d val, it tempers out [[250047/250000]], 1500625/1492992 and 2460375/2458624 in the 7-limit. It supports the [[landscape]] temperament. | ||
Using the | Using the 402c val, it tempers out [[3136/3125]], 321489/320000 and 13060694016/12867859375 in the 7-limit. It supports [[bischismic]] and [[parahemwuer]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
402 factors into 2 × 3 × 67, | Since 402 factors into 2 × 3 × 67, 402edo has subset edos {{EDOs| 2, 3, 6, 67, 134, and 201 }}. [[804edo]], which doubles 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|-637 402}} | | {{monzo| -637 402 }} | ||
|{{mapping|402 637}} | | {{mapping| 402 637 }} | ||
| 0.1459 | | 0.1459 | ||
| 0.1459 | | 0.1459 | ||
| 4.89 | | 4.89 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|2109375/2097152, {{monzo|25 -48 22}} | | 2109375/2097152, {{monzo| 25 -48 22 }} | ||
|{{mapping|402 637 933}} | | {{mapping| 402 637 933 }} | ||
| 0.2752 | | 0.2752 | ||
| 0.2182 | | 0.2182 | ||
| Line 50: | Line 50: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio* | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|91\402 | | 91\402 | ||
|271.64 | | 271.64 | ||
|75/64 | | 75/64 | ||
|[[Orson]] | | [[Orson]] | ||
|- | |- | ||
|1 | | 1 | ||
|115\402 | | 115\402 | ||
|343.28 | | 343.28 | ||
|8000/6561 | | 8000/6561 | ||
|[[Raider]] | | [[Raider]] | ||
|} | |} | ||
<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 12:31, 15 April 2024
| ← 401edo | 402edo | 403edo → |
Theory
402edo is only consistent to the 5-odd-limit. There are three possible mappings in the 7-limit:
- ⟨402 637 933 1129] (patent val)
- ⟨402 637 933 1128] (402d)
- ⟨402 637 934 1129] (402c)
Using the patent val, it tempers out 4375/4374, 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports ragismic and abigail.
Using the 402d val, it tempers out 250047/250000, 1500625/1492992 and 2460375/2458624 in the 7-limit. It supports the landscape temperament.
Using the 402c val, it tempers out 3136/3125, 321489/320000 and 13060694016/12867859375 in the 7-limit. It supports bischismic and parahemwuer.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.46 | -1.24 | +1.32 | -0.92 | +0.92 | +1.26 | +1.28 | -0.48 | +0.99 | +0.86 | -1.41 |
| Relative (%) | -15.5 | -41.5 | +44.3 | -31.0 | +30.8 | +42.3 | +43.0 | -16.0 | +33.3 | +28.8 | -47.2 | |
| Steps (reduced) |
637 (235) |
933 (129) |
1129 (325) |
1274 (68) |
1391 (185) |
1488 (282) |
1571 (365) |
1643 (35) |
1708 (100) |
1766 (158) |
1818 (210) | |
Subsets and supersets
Since 402 factors into 2 × 3 × 67, 402edo has subset edos 2, 3, 6, 67, 134, and 201. 804edo, which doubles 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 | [-637 402⟩ | [⟨402 637]] | 0.1459 | 0.1459 | 4.89 |
| 2.3.5 | 2109375/2097152, [25 -48 22⟩ | [⟨402 637 933]] | 0.2752 | 0.2182 | 7.31 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 91\402 | 271.64 | 75/64 | Orson |
| 1 | 115\402 | 343.28 | 8000/6561 | Raider |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct