2053edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|2053}} == Theory == 2053edo is consistent to the 5-odd-limit, its harmonic 7 is exactly between its steps. The equal temperamen..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro | {{EDO intro}} | ||
== Theory == | == Theory == | ||
2053edo is [[consistent]] to the [[5-odd-limit]], its [[harmonic]] [[7/1|7]] is exactly between its steps. | 2053edo is [[consistent]] to the [[5-odd-limit]], its [[harmonic]] [[7/1|7]] is exactly between its steps. It is strong in the 2.3.5.13.19.23.31 [[subgroup]], [[tempering out]] 359424/359375, 22816/22815, 96876/96875, 497705/497664, 4784000/4782969, and 369664/369603. Using the 2.3.5.13.19.37 subgroup, it tempers out [[7696/7695]]. | ||
=== 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" |[[Comma list | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
|- | ! colspan="2" | Tuning error | ||
![[TE error|Absolute]] (¢) | |- | ||
![[TE simple badness|Relative]] (%) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo|3254 -2053}} | | {{monzo|3254 -2053}} | ||
| {{mapping|2053 3254}} | | {{mapping|2053 3254}} | ||
| | | −0.0133 | ||
| 0.0133 | | 0.0133 | ||
| 2.28 | | 2.28 | ||
| Line 32: | Line 33: | ||
| {{monzo|37 25 -33}}, {{monzo|93 -66 5}} | | {{monzo|37 25 -33}}, {{monzo|93 -66 5}} | ||
| {{mapping|2053 3254 4767}} | | {{mapping|2053 3254 4767}} | ||
| | | −0.0157 | ||
| 0.0114 | | 0.0114 | ||
| 1.95 | | 1.95 | ||
| 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: | ||
| [[Whoosh]] | | [[Whoosh]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | |||
Revision as of 14:59, 16 January 2025
| ← 2052edo | 2053edo | 2054edo → |
Theory
2053edo is consistent to the 5-odd-limit, its harmonic 7 is exactly between its steps. It is strong in the 2.3.5.13.19.23.31 subgroup, tempering out 359424/359375, 22816/22815, 96876/96875, 497705/497664, 4784000/4782969, and 369664/369603. Using the 2.3.5.13.19.37 subgroup, it tempers out 7696/7695.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.042 | +0.048 | -0.292 | -0.125 | -0.002 | +0.256 | +0.003 | +0.074 | -0.254 | +0.020 |
| Relative (%) | +0.0 | +7.2 | +8.2 | -50.0 | -21.3 | -0.3 | +43.9 | +0.5 | +12.7 | -43.5 | +3.5 | |
| Steps (reduced) |
2053 (0) |
3254 (1201) |
4767 (661) |
5763 (1657) |
7102 (943) |
7597 (1438) |
8392 (180) |
8721 (509) |
9287 (1075) |
9973 (1761) |
10171 (1959) | |
Subsets and supersets
2053edo is the 310th prime edo. 4106edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [3254 -2053⟩ | [⟨2053 3254]] | −0.0133 | 0.0133 | 2.28 |
| 2.3.5 | [37 25 -33⟩, [93 -66 5⟩ | [⟨2053 3254 4767]] | −0.0157 | 0.0114 | 1.95 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 959\2053 | 560.546 | 864/625 | Whoosh |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct