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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 |
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 24: | Line 24: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 349 -220 }} | ||
| {{ | | {{Mapping| 220 349 }} | ||
| −0.5304 | | −0.5304 | ||
| 0.5302 | | 0.5302 | ||
| Line 31: | Line 31: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }} | ||
| {{ | | {{Mapping| 220 349 511 }} | ||
| −0.4912 | | −0.4912 | ||
| 0.4364 | | 0.4364 | ||
| Line 39: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 6144/6125, 10976/10935, 390625/388962 | | 6144/6125, 10976/10935, 390625/388962 | ||
| {{ | | {{Mapping| 220 349 511 618 }} | ||
| −0.5538 | | −0.5538 | ||
| 0.3932 | | 0.3932 | ||
| Line 49: | Line 49: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 74: | Line 74: | ||
|- | |- | ||
| 11 | | 11 | ||
| 91\220<br | | 91\220<br>(9\220) | ||
| 496.36<br | | 496.36<br>(49.09) | ||
| 4/3<br | | 4/3<br>(36/35) | ||
| [[Hendecatonic]] | | [[Hendecatonic (temperament)|Hendecatonic]] | ||
|- | |- | ||
| 20 | | 20 | ||
| 91\220<br | | 91\220<br>(3\220) | ||
| 496.36<br | | 496.36<br>(16.36) | ||
| 4/3<br | | 4/3<br>(126/125) | ||
| [[Degrees]] | | [[Degrees]] | ||
|} | |} | ||
<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 | <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 12:19, 5 April 2026
| ← 219edo | 220edo | 221edo → |
220 equal divisions of the octave (abbreviated 220edo or 220ed2), also called 220-tone equal temperament (220tet) or 220 equal temperament (220et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 220 equal parts of about 5.45 ¢ each. Each step represents a frequency ratio of 21/220, or the 220th root of 2.
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 distinct