559edo: Difference between revisions
5-limit notability |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{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 24: | ||
| {{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 49: | Line 41: | ||
| 0.1175 | | 0.1175 | ||
| 5.48 | | 5.48 | ||
{{comma basis end}} | |||
* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo|441]] and before [[612edo|612]]. | * 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 === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 85: | Line 71: | ||
| [[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]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
Revision as of 03:09, 16 November 2024
| ← 558edo | 559edo | 560edo → |
Theory
559edo is a very strong 5-limit system. The equal temperament 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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 559et has a lower relative error than any previous equal temperaments in the 5-limit, past 441 and before 612.
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf