221edo
| ← 220edo | 221edo | 222edo → |
221 equal divisions of the octave (abbreviated 221edo or 221ed2), also called 221-tone equal temperament (221tet) or 221 equal temperament (221et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 221 equal parts of about 5.43 ¢ each. Each step represents a frequency ratio of 21/221, or the 221st root of 2.
Theory
221edo has a flat tendency, with harmonics 3, 5, and 7 all tuned flat. The equal temperament tempers out 2109375/2097152 (semicomma) and [-11 26 -13⟩ in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the optimal patent val for the 7-limit hemiseven temperament.
Using the 221ef val, which does the best into the 17-limit, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting 17-limit hemiseven and 11-limit triwell.
Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.50 | -0.79 | -2.31 | +2.42 | +2.53 | +1.10 | -2.30 | -1.79 | +1.13 | +1.62 | +1.59 |
| Relative (%) | -27.7 | -14.6 | -42.5 | +44.7 | +46.6 | +20.3 | -42.3 | -32.9 | +20.8 | +29.8 | +29.3 | |
| Steps (reduced) |
350 (129) |
513 (71) |
620 (178) |
701 (38) |
765 (102) |
818 (155) |
863 (200) |
903 (19) |
939 (55) |
971 (87) |
1000 (116) | |
Subsets and supersets
Since 221 factors into 13 × 17, 221edo has 13edo and 17edo as its subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-350 221⟩ | [⟨221 350]] | +0.4740 | 0.4742 | 8.73 |
| 2.3.5 | [-21 3 7⟩, [-11 26 -13⟩ | [⟨221 350 513]] | +0.4299 | 0.3921 | 7.22 |
| 2.3.5.7 | 1029/1024, 19683/19600, 235298/234375 | [⟨221 350 513 620]] | +0.5282 | 0.3799 | 7.00 |
| 2.3.5.7.11 | 385/384, 441/440, 19683/19600, 235298/234375 | [⟨221 350 513 620 764]] (221e) | +0.5904 | 0.3618 | 6.66 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 50\221 | 271.49 | 75/64 | Orson |
| 1 | 57\221 | 309.50 | 448/375 | Triwell (221e) |
| 1 | 84\221 | 456.11 | 125/96 | Qak |
| 1 | 89\221 | 483.26 | 320/243 | Hemiseven (221ef) |
| 1 | 93\221 | 504.98 | 104976/78125 | Countermeantone |
| 1 | 103\221 | 559.28 | 864/625 | Tritriple (221e) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct