432edo: Difference between revisions
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m changed EDO intro to ED intro |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
432edo has a reasonable approximation to the 7-limit, where the equal temperament [[tempering out|tempers out]] [[5120/5103], [[703125/702464]], [[40353607/40310784]], 102760448/102515625, and 283115520/282475249. It [[support]]s and provides a good tuning for the 5-limit [[maja]] temperament. | 432edo has a reasonable approximation to the 7-limit, where the equal temperament [[tempering out|tempers out]] [[5120/5103]], [[703125/702464]], [[40353607/40310784]], 102760448/102515625, and 283115520/282475249. It [[support]]s and provides a good tuning for the 5-limit [[maja]] temperament. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 13: | Line 13: | ||
== 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| 685 -432 }} | | {{monzo| 685 -432 }} | ||
| {{mapping| 432 685 }} | | {{mapping| 432 685 }} | ||
| | | −0.2596 | ||
| 0.2595 | | 0.2595 | ||
| 9.34 | | 9.34 | ||
| Line 32: | Line 33: | ||
| {{monzo| 41 -20 -4 }}, {{monzo| -3 -23 17 }} | | {{monzo| 41 -20 -4 }}, {{monzo| -3 -23 17 }} | ||
| {{mapping| 432 685 1003 }} | | {{mapping| 432 685 1003 }} | ||
| | | −0.1440 | ||
| 0.2676 | | 0.2676 | ||
| 9.63 | | 9.63 | ||
| Line 39: | Line 40: | ||
| 5120/5103, 703125/702464, 6565234375/6530347008 | | 5120/5103, 703125/702464, 6565234375/6530347008 | ||
| {{mapping| 432 685 1003 1213 }} | | {{mapping| 432 685 1003 1213 }} | ||
| | | −0.1631 | ||
| 0.2341 | | 0.2341 | ||
| 8.43 | | 8.43 | ||
| 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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 60: | Line 62: | ||
|- | |- | ||
| 4 | | 4 | ||
| 179\432<br>(37\432) | | 179\432<br />(37\432) | ||
| 497.22<br>(102.78) | | 497.22<br />(102.78) | ||
| 4/3 | | 4/3 | ||
| [[Undim]] | | [[Undim]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||