233edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|233}} | {{EDO intro|233}} | ||
== Theory == | == Theory == | ||
233et has a generally flat tendency, in the sense that if the octave is pure, 3 | 233et has a generally flat tendency, in the sense that if the [[octave]] is pure, [[prime harmonic]]s 3 through 17 are all flat. 233edo is accurate for the [[5/1|5th harmonic]] (only 0.0476 cents flat), but less for the [[3/1|third harmonic]] (1.5258 cents flat). | ||
The equal temperament [[tempering out|tempers out]] [[78732/78125]] and {{monzo| -53 32 1 }} in the 5-limit; [[2401/2400]], [[65625/65536]], and 177147/175616 in the 7-limit (supporting [[tertiaseptal]] and [[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 === | === Odd harmonics === | ||
{{Harmonics in equal|233}} | {{Harmonics in equal|233}} | ||
===Subsets and supersets === | |||
=== Subsets and supersets === | |||
233edo is the 51st [[prime edo]]. | 233edo is the 51st [[prime edo]]. | ||
==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|-369 233}} | | {{monzo| -369 233 }} | ||
|{{ | | {{mapping| 233 369 }} | ||
| 0.4813 | | 0.4813 | ||
| 0.4815 | | 0.4815 | ||
| 9.35 | | 9.35 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|78732/78125, {{monzo|-53 32 1}} | | 78732/78125, {{monzo| -53 32 1 }} | ||
|{{ | | {{mapping| 233 369 541 }} | ||
| 0.3277 | | 0.3277 | ||
| 0.4492 | | 0.4492 | ||
| 8.72 | | 8.72 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|2401/2400, 65625/65536, 78732/78125 | | 2401/2400, 65625/65536, 78732/78125 | ||
|{{ | | {{mapping| 233 369 541 654 }} | ||
| 0.2979 | | 0.2979 | ||
| 0.3924 | | 0.3924 | ||
| 7.62 | | 7.62 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|243/242, 441/440, 540/539, 2401/2400 | | 243/242, 441/440, 540/539, 2401/2400 | ||
|{{ | | {{mapping| 233 369 541 654 806 }} | ||
| 0.2525 | | 0.2525 | ||
| 0.3625 | | 0.3625 | ||
| 7.04 | | 7.04 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|243/242, 351/350, 441/440, 540/539, 1001/1000 | | 243/242, 351/350, 441/440, 540/539, 1001/1000 | ||
|{{ | | {{mapping| 233 369 541 654 806 862 }} | ||
| 0.2574 | | 0.2574 | ||
| 0.3311 | | 0.3311 | ||
| 6.43 | | 6.43 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|351/350, 441/440, 540/539, 561/560, 936/935, 1156/1155 | | 351/350, 441/440, 540/539, 561/560, 936/935, 1156/1155 | ||
|{{ | | {{mapping| 233 369 541 654 806 862 952 }} | ||
| 0.2888 | | 0.2888 | ||
| 0.3161 | | 0.3161 | ||
| 6.14 | | 6.14 | ||
|} | |} | ||
=== 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 | |+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 | ||
|- | |- | ||
|1 | | 1 | ||
|15\233 | | 15\233 | ||
|77.25 | | 77.25 | ||
|256/245 | | 256/245 | ||
|[[Tertiaseptal]] | | [[Tertiaseptal]] | ||
|- | |- | ||
|1 | | 1 | ||
|22\233 | | 22\233 | ||
|113.30 | | 113.30 | ||
|16/15 | | 16/15 | ||
|[[Misneb]] | | [[Misneb]] | ||
|- | |- | ||
|1 | | 1 | ||
|55\233 | | 55\233 | ||
|283.26 | | 283.26 | ||
|189/160 | | 189/160 | ||
|[[Neominor]] | | [[Neominor]] | ||
|- | |- | ||
|1 | | 1 | ||
|77\233 | | 77\233 | ||
|396.57 | | 396.57 | ||
|98304/78125 | | 98304/78125 | ||
|[[Squarschmidt]] | | [[Squarschmidt]] | ||
|- | |- | ||
|1 | | 1 | ||
|86\233 | | 86\233 | ||
|442.92 | | 442.92 | ||
|9/7 | | 9/7 | ||
|[[Sensi]] | | [[Sensi]] | ||
|- | |- | ||
|1 | | 1 | ||
|95\233 | | 95\233 | ||
|489.27 | | 489.27 | ||
|250/189 | | 250/189 | ||
|[[Catafourth]] | | [[Catafourth]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
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