433edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
443edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. To start with, the [[patent val]] {{val| 433 686 1005 '''1216''' }} as well as the 433d [[val]] {{val| 433 686 1005 '''1215''' }} are worth considering. | 443edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. To start with, the [[patent val]] {{val| 433 686 1005 '''1216''' }} as well as the 433d [[val]] {{val| 433 686 1005 '''1215''' }} are worth considering. | ||
Using the patent val, | Using the patent val, it [[tempering out|tempers out]] [[19683/19600]] and 4096000/4084101 in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 161280/161051, and 180224/180075 in the 11-limit. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 11: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
433edo is the 84th [[prime edo]]. | 433edo is the 84th [[prime edo]]. It might be interesting due to being the smallest subset edo of the nanotemperament [[2901533edo]], an extremely high-precision/complexity [[microtemperament]]. | ||
== 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]] (¢) | ||
| Line 27: | Line 28: | ||
| {{monzo| -686 433 }} | | {{monzo| -686 433 }} | ||
| {{mapping| 433 686 }} | | {{mapping| 433 686 }} | ||
| 0.2525 | | +0.2525 | ||
| 0.2525 | | 0.2525 | ||
| 9.11 | | 9.11 | ||
| Line 34: | Line 35: | ||
| 2109375/2097152, {{monzo| -29 52 -23 }} | | 2109375/2097152, {{monzo| -29 52 -23 }} | ||
| {{mapping| 433 686 1005 }} | | {{mapping| 433 686 1005 }} | ||
| 0.3254 | | +0.3254 | ||
| 0.2306 | | 0.2306 | ||
| 8.32 | | 8.32 | ||
| Line 41: | Line 42: | ||
| 19683/19600, 4096000/4084101, 2109375/2097152 | | 19683/19600, 4096000/4084101, 2109375/2097152 | ||
| {{mapping| 433 686 1005 1216 }} | | {{mapping| 433 686 1005 1216 }} | ||
| 0.1414 | | +0.1414 | ||
| 0.3759 | | 0.3759 | ||
| 13.56 | | 13.56 | ||
| Line 48: | Line 49: | ||
| 3025/3024, 6250/6237, 30375/30184, 180224/180075 | | 3025/3024, 6250/6237, 30375/30184, 180224/180075 | ||
| {{mapping| 433 686 1005 1216 1498 }} | | {{mapping| 433 686 1005 1216 1498 }} | ||
| 0.1026 | | +0.1026 | ||
| 0.3451 | | 0.3451 | ||
| 12.45 | | 12.45 | ||
| Line 55: | Line 56: | ||
| 2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405 | | 2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405 | ||
| {{mapping| 433 686 1005 1216 1498 1602 }} | | {{mapping| 433 686 1005 1216 1498 1602 }} | ||
| 0.1217 | | +0.1217 | ||
| 0.3179 | | 0.3179 | ||
| 11.47 | | 11.47 | ||
| Line 62: | Line 63: | ||
| 2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803 | | 2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803 | ||
| {{mapping| 433 686 1005 1216 1498 1602 1770 }} | | {{mapping| 433 686 1005 1216 1498 1602 1770 }} | ||
| 0.0919 | | +0.0919 | ||
| 0.3033 | | 0.3033 | ||
| 10.94 | | 10.94 | ||
| Line 69: | Line 70: | ||
=== 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 82: | Line 84: | ||
| [[Orson]] | | [[Orson]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||