559edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
559edo is a strong 5-limit system. | 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 === | ||
| Line 13: | Line 13: | ||
== 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 25: | Line 26: | ||
| {{monzo| 886 -559 }} | | {{monzo| 886 -559 }} | ||
| {{mapping| 559 886 }} | | {{mapping| 559 886 }} | ||
| | | −0.0040 | ||
| 0.0040 | | 0.0040 | ||
| 0.19 | | 0.19 | ||
| Line 32: | Line 33: | ||
| {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }} | | {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }} | ||
| {{mapping| 559 886 1298 }} | | {{mapping| 559 886 1298 }} | ||
| | | −0.0157 | ||
| 0.0168 | | 0.0168 | ||
| 0.78 | | 0.78 | ||
| Line 50: | Line 51: | ||
| 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 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 84: | Line 87: | ||
| [[Acrokleismic]] | | [[Acrokleismic]] | ||
|- | |- | ||
|13 | | 13 | ||
|232\559<br>(17\559) | | 232\559<br />(17\559) | ||
|498.03<br>(36.494) | | 498.03<br />(36.494) | ||
|4/3<br>(?) | | 4/3<br />(?) | ||
|[[Aluminium]] | | [[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>* | <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 | ||
Latest revision as of 13:31, 13 March 2026
| ← 558edo | 559edo | 560edo → |
559 equal divisions of the octave (abbreviated 559edo or 559ed2), also called 559-tone equal temperament (559tet) or 559 equal temperament (559et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 559 equal parts of about 2.15 ¢ each. Each step represents a frequency ratio of 21/559, or the 559th root of 2.
Theory
559edo is a very strong 5-limit system. It 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.
Prime harmonics
| 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) | |
Subsets and supersets
Since 559 factors into 13 × 43, 559edo contains 13edo and 43edo as subsets.
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 |
- 559et has a lower relative error than any previous equal temperaments in the 5-limit, past 441 and before 612.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 90\559 | 182.47 | 10/9 | Mitonic |
| 1 | 90\559 | 193.20 | 352/315 | Lunatic |
| 1 | 116\559 | 249.02 | [-27 11 3 1⟩ | Monzismic |
| 1 | 147\559 | 315.56 | 6/5 | Acrokleismic |
| 13 | 232\559 (17\559) |
498.03 (36.494) |
4/3 (?) |
Aluminium |
| 43 | 232\559 (2\559) |
498.03 (4.29) |
4/3 (385/384) |
Meridic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct