559edo: Difference between revisions
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{{ | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | |||
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 === | |||
{{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 == | |||
{| 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 | |||
| {{monzo| 886 -559 }} | |||
| {{mapping| 559 886 }} | |||
| −0.0040 | |||
| 0.0040 | |||
| 0.19 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }} | |||
| {{mapping| 559 886 1298 }} | |||
| −0.0157 | |||
| 0.0168 | |||
| 0.78 | |||
|- | |||
| 2.3.5.7 | |||
| 4375/4374, 2100875/2097152, {{monzo| -4 -2 -9 10 }} | |||
| {{mapping| 559 886 1298 1569 }} | |||
| +0.0478 | |||
| 0.1109 | |||
| 5.16 | |||
|- | |||
| 2.3.5.7.11 | |||
| 4375/4374, 12005/11979, 41503/41472, 172032/171875 | |||
| {{mapping| 559 886 1298 1569 1934 }} | |||
| 0.0161 | |||
| 0.1175 | |||
| 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 === | |||
{| 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 | |||
| 90\559 | |||
| 182.47 | |||
| 10/9 | |||
| [[Mitonic]] | |||
|- | |||
| 1 | |||
| 90\559 | |||
| 193.20 | |||
| 352/315 | |||
| [[Lunatic]] | |||
|- | |||
| 1 | |||
| 116\559 | |||
| 249.02 | |||
| {{monzo| -27 11 3 1 }} | |||
| [[Monzismic]] | |||
|- | |||
| 1 | |||
| 147\559 | |||
| 315.56 | |||
| 6/5 | |||
| [[Acrokleismic]] | |||
|- | |||
| 13 | |||
| 232\559<br />(17\559) | |||
| 498.03<br />(36.494) | |||
| 4/3<br />(?) | |||
| [[Aluminium]] | |||
|- | |||
| 43 | |||
| 232\559<br />(2\559) | |||
| 498.03<br />(4.29) | |||
| 4/3<br />(385/384) | |||
| [[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 | |||