412edo: Difference between revisions
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== 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 2<sup>2</sup> × 103, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to | 412 factors into 2<sup>2</sup> × 103, 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" | ||
! 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|-653 412}} | | {{monzo|-653 412}} | ||
|{{mapping|412 653}} | | {{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 | | -0.1501 | ||
| 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 | | -0.2085 | ||
| 0.2143 | | 0.2143 | ||
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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 | ||
|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]] | | [[Septisuperfourth]] (7-limit) | ||
|} | |} | ||
<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 | ||