497edo: Difference between revisions
Jump to navigation
Jump to search
Review |
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" Tags: Mobile edit Mobile web edit |
||
| (5 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
497 factors into 7 × | 497 factors into {{nowrap|7 × 71}}, with [[7edo]] and [[71edo]] 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 | ! 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]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| 788 -497 }} | | {{monzo| 788 -497 }} | ||
| {{mapping| 497 788 }} | | {{mapping| 497 788 }} | ||
| | | −0.2084 | ||
| 0.2084 | | 0.2084 | ||
| 8.63 | | 8.63 | ||
| Line 32: | Line 33: | ||
| {{monzo| 38 -2 -15 }}, {{monzo| 12 -31 16 }} | | {{monzo| 38 -2 -15 }}, {{monzo| 12 -31 16 }} | ||
| {{mapping| 497 788 1154 }} | | {{mapping| 497 788 1154 }} | ||
| | | −0.1396 | ||
| 0.1961 | | 0.1961 | ||
| 8.12 | | 8.12 | ||
| Line 39: | Line 40: | ||
=== 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 52: | Line 54: | ||
| [[Luna]] | | [[Luna]] | ||
|} | |} | ||
<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 13:32, 13 March 2026
| ← 496edo | 497edo | 498edo → |
497 equal divisions of the octave (abbreviated 497edo or 497ed2), also called 497-tone equal temperament (497tet) or 497 equal temperament (497et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 497 equal parts of about 2.41 ¢ each. Each step represents a frequency ratio of 21/497, or the 497th root of 2.
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 × 71, with 7edo and 71edo 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 distinct