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 | ||
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|+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 | ||