220edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 34: | Line 43: | ||
| 0.3932 | | 0.3932 | ||
| 7.21 | | 7.21 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 68: | Line 84: | ||
| 4/3<br />(126/125) | | 4/3<br />(126/125) | ||
| [[Degrees]] | | [[Degrees]] | ||
|} | |||
<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 13:10, 16 November 2024
| ← 219edo | 220edo | 221edo → |
Theory
Using the patent val, it tempers out [20 -17 3⟩ (rodan comma) and [23 6 -14⟩ (vishnuzma) in the 5-limit; 6144/6125, 10976/10935, and 390625/388962 in the 7-limit; 1331/1323, 1375/1372, 2200/2187, and 16384/16335 in the 11-limit; 325/324, 352/351, 1001/1000, 1573/1568, and 2704/2695 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.68 | +0.96 | +2.08 | -2.09 | -0.41 | -0.53 | +2.64 | -1.32 | +2.49 | -1.69 | -1.00 |
| Relative (%) | +30.8 | +17.6 | +38.2 | -38.4 | -7.5 | -9.7 | +48.4 | -24.2 | +45.6 | -31.0 | -18.4 | |
| Steps (reduced) |
349 (129) |
511 (71) |
618 (178) |
697 (37) |
761 (101) |
814 (154) |
860 (200) |
899 (19) |
935 (55) |
966 (86) |
995 (115) | |
Subsets and supersets
Since 220 factors into 22 × 5 × 11, 220edo has subset edos 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [349 -220⟩ | [⟨220 349]] | −0.5304 | 0.5302 | 9.72 |
| 2.3.5 | [20 -17 3⟩, [23 6 -14⟩ | [⟨220 349 511]] | −0.4912 | 0.4364 | 8.00 |
| 2.3.5.7 | 6144/6125, 10976/10935, 390625/388962 | [⟨220 349 511 618]] | −0.5538 | 0.3932 | 7.21 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 43\220 | 234.55 | 729/640 | Rodan (5-limit) |
| 1 | 83\220 | 452.73 | 125/81 | Maja (5-limit) |
| 2 | 13\220 | 70.91 | 25/24 | Vishnu (5-limit) |
| 11 | 91\220 (9\220) |
496.36 (49.09) |
4/3 (36/35) |
Hendecatonic |
| 20 | 91\220 (3\220) |
496.36 (16.36) |
4/3 (126/125) |
Degrees |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct