559edo: Difference between revisions
→Regular temperament properties: +meridic |
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{{Infobox ET | {{Infobox ET}} | ||
{{ED intro}} | |||
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{{ | |||
== Theory == | == Theory == | ||
559edo [[tempering out|tempers out]] the luna comma, {{monzo| 38 -2 -15 }} and the minortone comma, {{monzo| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]]. | 559edo is a very strong 5-limit system. It [[tempering out|tempers out]] the luna comma, {{monzo| 38 -2 -15 }} and the minortone comma, {{monzo| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|559|columns=11}} | {{Harmonics in equal|559|columns=11}} | ||
=== Subsets and supersets === | |||
Since 559 factors into {{factorization|559}}, 559edo contains [[13edo]] and [[43edo]] as subsets. | |||
== 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 25: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 886 -559 }} | | {{monzo| 886 -559 }} | ||
| | | {{mapping| 559 886 }} | ||
| | | −0.0040 | ||
| 0.0040 | | 0.0040 | ||
| 0.19 | | 0.19 | ||
| Line 34: | Line 32: | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }} | | {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }} | ||
| | | {{mapping| 559 886 1298 }} | ||
| | | −0.0157 | ||
| 0.0168 | | 0.0168 | ||
| 0.78 | | 0.78 | ||
| Line 41: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, 2100875/2097152, {{monzo| -4 -2 -9 10 }} | | 4375/4374, 2100875/2097152, {{monzo| -4 -2 -9 10 }} | ||
| | | {{mapping| 559 886 1298 1569 }} | ||
| +0.0478 | | +0.0478 | ||
| 0.1109 | | 0.1109 | ||
| Line 48: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 4375/4374, 12005/11979, 41503/41472, 172032/171875 | | 4375/4374, 12005/11979, 41503/41472, 172032/171875 | ||
| | | {{mapping| 559 886 1298 1569 1934 }} | ||
| 0.0161 | | 0.0161 | ||
| 0.1175 | | 0.1175 | ||
| 5.48 | | 5.48 | ||
|} | |} | ||
* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo|441]] and before [[612edo|612]]. | |||
=== 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 | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 86: | Line 86: | ||
| 6/5 | | 6/5 | ||
| [[Acrokleismic]] | | [[Acrokleismic]] | ||
|- | |||
| 13 | |||
| 232\559<br />(17\559) | |||
| 498.03<br />(36.494) | |||
| 4/3<br />(?) | |||
| [[Aluminium]] | |||
|- | |- | ||
| 43 | | 43 | ||
| 232\559<br>(2\559) | | 232\559<br />(2\559) | ||
| 498.03<br>(4.29) | | 498.03<br />(4.29) | ||
| 4/3<br>(385/384) | | 4/3<br />(385/384) | ||
| [[Meridic]] | | [[Meridic]] | ||
|} | |} | ||
<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 | |||
[[ | |||