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{{Infobox ET}}
{{Infobox ET}}
'''221edo''' is the [[EDO|equal division of the octave]] into 221 parts of 5.4299 [[cent]]s each.
{{EDO intro|221}}
 
== Theory ==
It tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 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 [[Gamelismic clan|hemiseven temperament]].  
221et tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 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 [[Gamelismic clan|hemiseven temperament]].  


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.  
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.  


Using the 221ef val, 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 the 17-limit hemiseven and the 11-limit [[Semicomma family|triwell]].
Using the 221ef val, 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 the 17-limit hemiseven and the 11-limit [[Semicomma family|triwell]].
=== Odd harmonics ===
{{Harmonics in equal|221}}
=== Subsets and supersets ===
221 factors into 13 × 17, with its subset edos [[13edo]] and [[17edo]].
==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|-350 221}}
|{{val|221 350}}
| 0.4740
| 0.4742
| 8.73
|-
|2.3.5
|{{monzo|-21 3 7}}, {{monzo|-11 26 -13}}
|{{val|221 350 513}}
| 0.4299
| 0.3921
| 7.22
|-
|2.3.5.7
|1029/1024, 19683/19600, 235298/234375
|{{val|221 350 513 620}}
| 0.5282
| 0.3799
| 7.00
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
|1
|50\221
|271.49
|75/64
|[[Orson]]
|-
|1
|84\221
|456.11
|125/96
|[[Qak]]
|-
|1
|89\221
|483.26
|320/243
|[[Hemiseven]]
|-
|1
|93\221
|504.98
|104976/78125
|[[Countermeantone]]
|-
|1
|103\221
|559.28
|864/625
|[[Tritriple]]
|}


{{Harmonics in equal|221}}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 19:30, 28 October 2023

← 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

Template:EDO intro

Theory

221et tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 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 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.

Using the 221ef val, 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 the 17-limit hemiseven and the 11-limit triwell.

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

221 factors into 13 × 17, with its subset edos 13edo and 17edo.

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

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 50\221 271.49 75/64 Orson
1 84\221 456.11 125/96 Qak
1 89\221 483.26 320/243 Hemiseven
1 93\221 504.98 104976/78125 Countermeantone
1 103\221 559.28 864/625 Tritriple
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