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402edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|402}} == Theory == 402et is only consistent to the 5-limit, tempering out the semicomma. There are three possible mappings in the 7-limit: * {{..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|402}}
{{ED intro}}


== Theory ==
== Theory ==
402et is only consistent to the 5-limit, tempering out the [[semicomma]]. There are three possible mappings in the 7-limit:
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 '''934''' 1129}} (402c)
* {{val| 402 637 933 '''1128''' }} (402d)
* {{val|402 637 933 '''1128'''}} (402d)
* {{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 the [[semicomma]] in the 5-limit and [[4375/4374]], 7381125/7340032, and 3200000/3176523 in the 7-limit, [[support]]ing [[Abigail]].


Using the 402c val, it tempers out 3136/3125, 321489/320000 and 13060694016/12867859375 in the 7-limit. It supports [[bischismic]] and [[parahemwuer]].
Using the 402d val, it tempers out [[250047/250000]], and 1500625/1492992 and 2460375/2458624 in the 7-limit.  


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 the [[schisma]] in the 5-limit; and [[3136/3125]], 321489/320000, and 13060694016/12867859375 in the 7-limit, supporting [[bischismic]].


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
402 factors into 2 × 3 × 67, with subset edos {{EDOs|2, 3, 6, 67, 134, and 201}}. [[804edo]], which doubles it, gives a good correction to the harmonic 7.
Since 402 factors into {{nowrap|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 harmonics 5 and 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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-637 402}}
! rowspan="2" | [[Comma list]]
|{{mapping|402 637}}
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -637 402 }}
| {{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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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|91\402
| 91\402
|271.64
| 271.64
|75/64
| 75/64
|[[Orson]]
| [[Orson]] (402)
|-
| 1
| 115\402
| 343.28
| 8000/6561
| [[Raider]] (402)
|-
|-
|1
| 2
|115\402
| 70\402
|343.28
| 208.96
|8000/6561
| 44/39
|[[Raider]]
| [[Abigail]] (402)
|}
|-
| 2
| 167\402<br />(34\402)
| 498.51<br />(101.49)
| 4/3<br />(200/189)
| [[Bischismic]] (402c, 7-limit)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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