559edo: Difference between revisions
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+infobox; +RTT table and rank-2 temperaments |
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{{EDO intro|559}} 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]]. | {{Infobox ET | ||
| Prime factorization = 13 × 43 | |||
| Step size = 2.14669¢ | |||
| Fifth = 327\559 (701.97¢) | |||
| Semitones = 53:42 (113.77¢ : 90.16¢) | |||
| Consistency = 11 | |||
}} | |||
{{EDO intro|559}} | |||
== 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]]. | |||
{{Harmonics in equal|559|columns=11}} | {{Harmonics in equal|559|columns=11}} | ||
== 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 }} | |||
| [{{val| 559 886 }}] | |||
| -0.0040 | |||
| 0.0040 | |||
| 0.19 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }} | |||
| [{{val| 559 886 1298 }}] | |||
| -0.0157 | |||
| 0.0168 | |||
| 0.78 | |||
|- | |||
| 2.3.5.7 | |||
| 4375/4374, 2100875/2097152, {{monzo| -4 -2 -9 10 }} | |||
| [{{val| 559 886 1298 1569 }}] | |||
| +0.0478 | |||
| 0.1109 | |||
| 5.16 | |||
|- | |||
| 2.3.5.7.11 | |||
| 4375/4374, 12005/11979, 41503/41472, 172032/171875 | |||
| [{{val| 559 886 1298 1569 1934 }}] | |||
| 0.0161 | |||
| 0.1175 | |||
| 5.48 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! 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| -26 18 -1 }} | |||
| [[Monzismic]] | |||
|- | |||
| 1 | |||
| 147\559 | |||
| 315.56 | |||
| 6/5 | |||
| [[Acrokleismic]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 21:10, 29 August 2022
| ← 558edo | 559edo | 560edo → |
Theory
559edo tempers out the luna comma, [38 -2 -15⟩ and the minortone comma, [-16 35 -17⟩ in the 5-limit, as well as the monzisma, [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 supports include mitonic, lunatic, acrokleismic, monzism, and meridic.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.013 | +0.091 | -0.668 | +0.382 | +0.975 | +0.232 | +0.877 | +0.706 | +0.834 | -0.850 |
| Relative (%) | +0.0 | +0.6 | +4.2 | -31.1 | +17.8 | +45.4 | +10.8 | +40.9 | +32.9 | +38.9 | -39.6 | |
| Steps (reduced) |
559 (0) |
886 (327) |
1298 (180) |
1569 (451) |
1934 (257) |
2069 (392) |
2285 (49) |
2375 (139) |
2529 (293) |
2716 (480) |
2769 (533) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [886 -559⟩ | [⟨559 886]] | -0.0040 | 0.0040 | 0.19 |
| 2.3.5 | [38 -2 -15⟩, [-16 35 -17⟩ | [⟨559 886 1298]] | -0.0157 | 0.0168 | 0.78 |
| 2.3.5.7 | 4375/4374, 2100875/2097152, [-4 -2 -9 10⟩ | [⟨559 886 1298 1569]] | +0.0478 | 0.1109 | 5.16 |
| 2.3.5.7.11 | 4375/4374, 12005/11979, 41503/41472, 172032/171875 | [⟨559 886 1298 1569 1934]] | 0.0161 | 0.1175 | 5.48 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 90\559 | 182.47 | 10/9 | Mitonic |
| 1 | 90\559 | 193.20 | 352/315 | Lunatic |
| 1 | 116\559 | 249.02 | [-26 18 -1⟩ | Monzismic |
| 1 | 147\559 | 315.56 | 6/5 | Acrokleismic |