327edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|327}} == Theory == 327et tempers out 283115520/282475249, 156250000/155649627, 16875/16807, 19683/19600, 95703125/95551488, 250047/250000..." |
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== Theory == | == Theory == | ||
327edo is [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation to the full 13-limit in its [[patent val]]. The equal temperament [[tempering out|tempers out]] the [[semicomma]] in the 5-limit; [[16875/16807]], [[19683/19600]], [[250047/250000]], and [[2100875/2097152]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[8019/8000]], 35937/35840, 46656/46585, 102487/102400, 137781/137500, and 160083/160000 in the 11-limit; and [[625/624]], [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4225/4224]], and [[10648/10647]] in the 13-limit. It supports [[mirkat]], [[pnict]], and the subgroup temperament [[petrtri]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
327 factors into 3 × 109, | Since 327 factors into 3 × 109, 327edo has [[3edo]] and [[109edo]] as its subsets. | ||
== 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|-518 327}} | | {{monzo| -518 327 }} | ||
|{{mapping|327 518}} | | {{mapping| 327 518 }} | ||
| 0.3273 | | 0.3273 | ||
| 0.3274 | | 0.3274 | ||
| 8.92 | | 8.92 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|2109375/2097152, {{monzo|-20 39 -18}} | | 2109375/2097152, {{monzo| -20 39 -18 }} | ||
|{{mapping|327 518 759}} | | {{mapping| 327 518 759 }} | ||
| 0.3608 | | 0.3608 | ||
| 0.2715 | | 0.2715 | ||
| 7.40 | | 7.40 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|19683/19600, | | 16875/16807, 19683/19600, 2100875/2097152 | ||
|{{mapping|327 518 759 918}} | | {{mapping| 327 518 759 918 }} | ||
| 0.2722 | | 0.2722 | ||
| 0.2807 | | 0.2807 | ||
| 7.65 | | 7.65 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|540/539, | | 540/539, 1375/1372, 8019/8000, 2100875/2097152 | ||
|{{mapping|327 518 759 918 1131}} | | {{mapping| 327 518 759 918 1131 }} | ||
| 0.2674 | | 0.2674 | ||
| 0.2512 | | 0.2512 | ||
| 6.85 | | 6.85 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|540/539, 625/624, | | 540/539, 625/624, 1575/1573, 2200/2197, 8019/8000 | ||
|{{mapping|327 518 759 918 1131 1210}} | | {{mapping| 327 518 759 918 1131 1210 }} | ||
| 0.2301 | | 0.2301 | ||
| 0.2441 | | 0.2441 | ||
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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 | ||
|74\327 | | 74\327 | ||
|271.56 | | 271.56 | ||
|75/64 | | 75/64 | ||
|[[Orson]] | | [[Orson]] | ||
|- | |- | ||
|3 | | 3 | ||
|44\327 | | 44\327 | ||
|161.47 | | 161.47 | ||
|192/175 | | 192/175 | ||
|[[Pnict]] | | [[Pnict]] | ||
|- | |- | ||
|3 | | 3 | ||
|50\327 | | 50\327 | ||
|183.49 | | 183.49 | ||
|10/9 | | 10/9 | ||
|[[Mirkat]] | | [[Mirkat]] | ||
|} | |} | ||
<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 | ||