233edo: Difference between revisions
m Added "harmonics in equal" table |
No edit summary |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|233}} | |||
== Theory == | |||
233et has a generally flat tendency, in the sense that if the octave is pure, 3, 5, 7, 11, 13, and 17 are all flat. 233edo is accurate for the 5th harmonic (only 0.0476 cents flat), but less for the third harmonic (1.5258 cents flat). | |||
It tempers out 78732/78125 and |-53 32 1> in the 5-limit; 2401/2400, 65625/65536, and 177147/175616 in the 7-limit (supporting [[Breedsmic temperaments|tertiaseptal]] and [[Breedsmic temperaments|catafourth]]). Using the patent val, it tempers out 243/242, 441/440, 35937/35840, and 78408/78125 in the 11-limit; 351/350, 1001/1000, 1575/1573, 4225/4224, and 6656/6655 in the 13-limit. | It tempers out 78732/78125 and |-53 32 1> in the 5-limit; 2401/2400, 65625/65536, and 177147/175616 in the 7-limit (supporting [[Breedsmic temperaments|tertiaseptal]] and [[Breedsmic temperaments|catafourth]]). Using the patent val, it tempers out 243/242, 441/440, 35937/35840, and 78408/78125 in the 11-limit; 351/350, 1001/1000, 1575/1573, 4225/4224, and 6656/6655 in the 13-limit. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|233}} | |||
===Subsets and supersets === | |||
233edo is the 51st [[prime edo]]. | 233edo is the 51st [[prime edo]]. | ||
==Regular temperament properties== | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list|Comma List]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | |||
! colspan="2" |Tuning Error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3 | |||
|{{monzo|-369 233}} | |||
|{{val|233 369}} | |||
| 0.4813 | |||
| 0.4815 | |||
| 9.35 | |||
|- | |||
|2.3.5 | |||
|78732/78125, {{monzo|-53 32 1}} | |||
|{{val|233 369 541}} | |||
| 0.3277 | |||
| 0.4492 | |||
| 8.72 | |||
|- | |||
|2.3.5.7 | |||
|2401/2400, 65625/65536, 78732/78125 | |||
|{{val|233 369 541 654}} | |||
| 0.2979 | |||
| 0.3924 | |||
| 7.62 | |||
|- | |||
|2.3.5.7.11 | |||
|243/242, 441/440, 540/539, 2401/2400 | |||
|{{val|233 369 541 654 806}} | |||
| 0.2525 | |||
| 0.3625 | |||
| 7.04 | |||
|- | |||
|2.3.5.7.11.13 | |||
|243/242, 351/350, 441/440, 540/539, 1001/1000 | |||
|{{val|233 369 541 654 806 862}} | |||
| 0.2574 | |||
| 0.3311 | |||
| 6.43 | |||
|- | |||
|2.3.5.7.11.13.17 | |||
|351/350, 441/440, 540/539, 561/560, 936/935, 1156/1155 | |||
|{{val|233 369 541 654 806 862 952}} | |||
| 0.2888 | |||
| 0.3161 | |||
| 6.14 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
|1 | |||
|15\233 | |||
|77.25 | |||
|256/245 | |||
|[[Tertiaseptal]] | |||
|- | |||
|1 | |||
|22\233 | |||
|113.30 | |||
|16/15 | |||
|[[Misneb]] | |||
|- | |||
|1 | |||
|55\233 | |||
|283.26 | |||
|189/160 | |||
|[[Neominor]] | |||
|- | |||
|1 | |||
|77\233 | |||
|396.57 | |||
|98304/78125 | |||
|[[Squarschmidt]] | |||
|- | |||
|1 | |||
|86\233 | |||
|442.92 | |||
|9/7 | |||
|[[Sensi]] | |||
|- | |||
|1 | |||
|95\233 | |||
|489.27 | |||
|250/189 | |||
|[[Catafourth]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||