407edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|407}} == Theory == 407et tempers out 32805/32768 in the 5-limit; 4096000/4084101, 134217728/133984375, 26873856/26796875, 78125000/781218..." |
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== Theory == | == Theory == | ||
407edo is a strong 5-limit system and 2.3.5.11.13.19.23 [[subgroup]] system. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; using the [[patent val]], [[16875/16807]], [[4096000/4084101]], and 26873856/26796875 in the 7-limit. It [[support]]s and provides the [[optimal patent val]] for the [[subsemifourth]] temperament in the 7- and 11-limit. [[Essentially tempered chord]]s available in 407et include [[pinkanberry chords]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos. | 407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos. [[814edo]], which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament. | ||
== 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|-645 407}} | | {{monzo| -645 407 }} | ||
|{{mapping|407 645}} | | {{mapping| 407 645 }} | ||
| 0.0742 | | +0.0742 | ||
| 0.0742 | | 0.0742 | ||
| 2.52 | | 2.52 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|32805/32768, {{monzo|30 47 -45}} | | 32805/32768, {{monzo| 30 47 -45 }} | ||
|{{mapping|407 645 945}} | | {{mapping| 407 645 945 }} | ||
| 0.0599 | | +0.0599 | ||
| 0.0638 | | 0.0638 | ||
| 2.16 | | 2.16 | ||
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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 | ||
|63\407 | | 63\407 | ||
|185.75 | | 185.75 | ||
|{{monzo|24 4 -13}} | | {{monzo| 24 4 -13 }} | ||
|[[Pirate]] | | [[Pirate]] | ||
|- | |||
| 1 | |||
| 83\407 | |||
| 244.72 | |||
| 15/13 | |||
| [[Subsemifourth]] (407f) | |||
|- | |- | ||
|1 | | 1 | ||
|169\407 | | 169\407 | ||
|498.28 | | 498.28 | ||
|4/3 | | 4/3 | ||
|[[Helmholtz]] | | [[Helmholtz]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Subsemifourth]] | |||