497edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|497}} == Theory == 497et only is consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 48828125/48771072, 2100875/20..." |
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== Theory == | == Theory == | ||
497et only is consistent to the 5-limit. Using the patent val, | 497et only is [[consistent]] to the [[5-odd-limit]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[2100875/2097152]], 48828125/48771072, 67108864/66976875, and 200120949/200000000 in the 7-limit; [[5632/5625]], 42875/42768, 43923/43904, [[131072/130977]], 151263/151250, 160083/160000, and 391314/390625 in the 11-limit. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
497 factors into 7 × 41, with [[7edo]] and [[41edo]] as its subset edos. [[1491edo]], which triples it, gives a good correction to | 497 factors into 7 × 41, with [[7edo]] and [[41edo]] as its subset edos. [[1491edo]], which triples it, gives a good correction to harmonics 3 and 7. | ||
== 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|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|788 -497}} | | {{monzo| 788 -497 }} | ||
|{{mapping|497 788}} | | {{mapping| 497 788 }} | ||
| -0.2084 | | -0.2084 | ||
| 0.2084 | | 0.2084 | ||
| 8.63 | | 8.63 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|38 -2 -15}}, {{ | | {{monzo| 38 -2 -15 }}, {{monzo| 12 -31 16 }} | ||
|{{mapping|497 788 1154}} | | {{mapping| 497 788 1154 }} | ||
| -0.1396 | | -0.1396 | ||
| 0.1961 | | 0.1961 | ||
| Line 41: | Line 41: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio* | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|80\497 | | 80\497 | ||
|193.16 | | 193.16 | ||
|262144/234375 | | 262144/234375 | ||
|[[Luna]] | | [[Luna]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||
Revision as of 15:28, 13 December 2023
| ← 496edo | 497edo | 498edo → |
Theory
497et only is consistent to the 5-odd-limit. Using the patent val, the equal temperament tempers out 2100875/2097152, 48828125/48771072, 67108864/66976875, and 200120949/200000000 in the 7-limit; 5632/5625, 42875/42768, 43923/43904, 131072/130977, 151263/151250, 160083/160000, and 391314/390625 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.66 | +0.00 | -0.62 | -1.09 | -0.81 | -0.29 | +0.66 | -1.13 | -0.53 | +0.04 | -0.51 |
| Relative (%) | +27.4 | +0.2 | -25.5 | -45.3 | -33.8 | -11.9 | +27.5 | -46.9 | -22.0 | +1.8 | -21.0 | |
| Steps (reduced) |
788 (291) |
1154 (160) |
1395 (401) |
1575 (84) |
1719 (228) |
1839 (348) |
1942 (451) |
2031 (43) |
2111 (123) |
2183 (195) |
2248 (260) | |
Subsets and supersets
497 factors into 7 × 41, with 7edo and 41edo as its subset edos. 1491edo, which triples it, gives a good correction to harmonics 3 and 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [788 -497⟩ | [⟨497 788]] | -0.2084 | 0.2084 | 8.63 |
| 2.3.5 | [38 -2 -15⟩, [12 -31 16⟩ | [⟨497 788 1154]] | -0.1396 | 0.1961 | 8.12 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 80\497 | 193.16 | 262144/234375 | Luna |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct