# 221edo

 ← 220edo 221edo 222edo →
Prime factorization 13 × 17
Step size 5.42986¢
Fifth 129\221 (700.452¢)
Semitones (A1:m2) 19:18 (103.2¢ : 97.74¢)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of odd harmonics in 221edo
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

Table of rank-2 temperaments by generator
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