433edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
433edo 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 === | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{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 26: | 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 33: | 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 40: | 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 47: | 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 54: | 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 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 67: | Line 83: | ||
| 75/64 | | 75/64 | ||
| [[Orson]] | | [[Orson]] | ||
|} | |||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
== Music == | == Music == | ||