2022edo: Difference between revisions
Rework theory; +subsets and supersets |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 21: | Line 21: | ||
== 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]] | ||
| Line 33: | Line 34: | ||
| {{monzo| 3205 -2022 }} | | {{monzo| 3205 -2022 }} | ||
| {{mapping| 2022 3205 }} | | {{mapping| 2022 3205 }} | ||
| | | −0.038534 | ||
| 0.038533 | | 0.038533 | ||
| 6.493 | | 6.493 | ||
| Line 40: | Line 41: | ||
| {{monzo| 25 -48 22 }}, {{monzo| -90 -15 49 }} | | {{monzo| 25 -48 22 }}, {{monzo| -90 -15 49 }} | ||
| {{mapping| 2022 3205 4695 }} | | {{mapping| 2022 3205 4695 }} | ||
| | | −0.030920 | ||
| 0.033254 | | 0.033254 | ||
| 5.603 | | 5.603 | ||
| Line 47: | Line 48: | ||
| 2431/2430, 2755/2754, 3520/3519, 142025/141984, 2582624/2581875, 9096256/9092061, 11293425/11290976, 51054848/51046875 | | 2431/2430, 2755/2754, 3520/3519, 142025/141984, 2582624/2581875, 9096256/9092061, 11293425/11290976, 51054848/51046875 | ||
| {{mapping| 2022 3205 4695 6955 7482 8265 8589 9147 9283 }} | | {{mapping| 2022 3205 4695 6955 7482 8265 8589 9147 9283 }} | ||
| | | −0.010752 | ||
| 0.036910 | | 0.036910 | ||
| 6.219 | | 6.219 | ||
| Line 55: | Line 56: | ||
; [[Eliora]] | ; [[Eliora]] | ||
* [https://www.youtube.com/watch?v=IulKlMjxAM8 ''Noble Gas''] (2021) | * [https://www.youtube.com/watch?v=IulKlMjxAM8 ''Noble Gas''] (2021) | ||
[[Category:Listen]] | |||
Latest revision as of 22:56, 20 February 2025
| ← 2021edo | 2022edo | 2023edo → |
2022 equal divisions of the octave (abbreviated 2022edo or 2022ed2), also called 2022-tone equal temperament (2022tet) or 2022 equal temperament (2022et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2022 equal parts of about 0.593 ¢ each. Each step represents a frequency ratio of 21/2022, or the 2022nd root of 2.
Theory
2022edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. Nonetheless, it offers good appoximations of the 2.3.5.11.17.29.41.43.53.61 subgroup. When using smaller numbers, 2.3.5.11 is a good choice, and if rougher errors are allowed, no-sevens 29-limit is a satisfactory choice.
In the 5-limit, 2022edo supports the pirate temperament, 323 & 407, and tempers out the [-90 -15 49⟩ comma.
In the 2.3.5.11 subgroup, 2022edo supports the rank-3 temperament that eliminates the [25 -17 -23 16⟩ comma. If the 11-limit is taken as a whole, 2022edo tempers out 3025/3024 and 4375/4374 when it is 7/4 is put on the 1633rd step (2022d val), and 41503/41472 with 250047/250000 when using the 1632nd step of the patent val.
In the 2.5.11.17.29.41.43.53.61 subgroup, 2022edo tempers out 17630/17629, 18491/18490, 21200/21199, and 22528/22525.
If the 29-limit is taken as a whole even including the 7-limit inconsistency, 2022edo tempers out 2002/2001, 3451/3450, 5104/5103, and 16445/16443.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.122 | +0.036 | -0.280 | +0.017 | -0.172 | +0.089 | -0.184 | +0.212 | +0.096 | -0.228 |
| Relative (%) | +0.0 | +20.6 | +6.1 | -47.2 | +2.9 | -28.9 | +15.0 | -30.9 | +35.8 | +16.2 | -38.5 | |
| Steps (reduced) |
2022 (0) |
3205 (1183) |
4695 (651) |
5676 (1632) |
6995 (929) |
7482 (1416) |
8265 (177) |
8589 (501) |
9147 (1059) |
9823 (1735) |
10017 (1929) | |
Subsets and supersets
Since 2022 factors into 2 × 3 × 337, 2022edo has subset edos 2, 3, 6, 337, 674, and 1011.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [3205 -2022⟩ | [⟨2022 3205]] | −0.038534 | 0.038533 | 6.493 |
| 2.3.5 | [25 -48 22⟩, [-90 -15 49⟩ | [⟨2022 3205 4695]] | −0.030920 | 0.033254 | 5.603 |
| 2.3.5.11.13.17.19.23.29 | 2431/2430, 2755/2754, 3520/3519, 142025/141984, 2582624/2581875, 9096256/9092061, 11293425/11290976, 51054848/51046875 | [⟨2022 3205 4695 6955 7482 8265 8589 9147 9283]] | −0.010752 | 0.036910 | 6.219 |
Music
- Noble Gas (2021)