412edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|412}} == Theory == 412et tempers out 2460375/2458624, 6144/6125, 102760448/102515625, 1640558367/1638400000 and 200120949/200000000 in the..." |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
412edo has a very accurate [[3/2|perfect fifth]], but it is not quite accurate beyond that. The equal temperament [[tempering out|tempers out]] {{monzo| 32 -7 -9 }} ([[escapade comma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the 5-limit; [[6144/6125]], 118098/117649, 2460375/2458624, 49009212/48828125, and notably the [[nanisma]] in the 7-limit. It supports [[nanic]] and [[counterschismic]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
412 factors into | 412 factors into {{factorisation|412}}, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|-653 412}} | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|412 653}} | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo|-653 412}} | |||
| {{mapping| 412 653 }} | |||
| +0.0042 | | +0.0042 | ||
| 0.0042 | | 0.0042 | ||
| 0.14 | | 0.14 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|32 -7 -9}}, {{monzo|-5 31 -19}} | | {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }} | ||
|{{mapping|412 653 957}} | | {{mapping| 412 653 957 }} | ||
| | | −0.1501 | ||
| 0.2182 | | 0.2182 | ||
| 7.49 | | 7.49 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|6144/6125, 2460375/2458624, | | 6144/6125, 2460375/2458624, 49009212/48828125 | ||
|{{mapping|412 653 957 1157}} | | {{mapping| 412 653 957 1157 }} | ||
| | | −0.2085 | ||
| 0.2143 | | 0.2143 | ||
| 7.36 | | 7.36 | ||
| Line 46: | Line 47: | ||
=== 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 | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|9\412 | | 9\412 | ||
|26.21 | | 26.21 | ||
|49/48 | | 49/48 | ||
|[[Sfourth]] | | [[Sfourth]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|19\412 | | 19\412 | ||
|55.34 | | 55.34 | ||
|16875/16384 | | 16875/16384 | ||
|[[Escapade]] | | [[Escapade]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|171\412 | | 171\412 | ||
|498.06 | | 498.06 | ||
|4/3 | | 4/3 | ||
|[[Counterschismic]] | | [[Counterschismic]]<br>[[Nanic]] | ||
|- | |- | ||
|2 | | 2 | ||
|19\412 | | 19\412 | ||
|55.34 | | 55.34 | ||
|16875/16384 | | 16875/16384 | ||
|[[ | | [[Septisuperfourth]] (7-limit) | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
<nowiki>* | |||