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'''221edo''' is the [[EDO|equal division of the octave]] into 221 parts of 5.4299 [[cent]]s each. 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|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 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]].
{{Infobox ET}}
{{ED intro}}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Theory ==
221edo has a flat tendency, with [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] all tuned flat. The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| -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 ===
{{Harmonics in equal|221}}
 
=== Subsets and supersets ===
Since 221 factors into 13 × 17, 221edo has [[13edo]] and [[17edo]] as its subsets.
 
== 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
| {{monzo| -350 221 }}
| {{mapping| 221 350 }}
| +0.4740
| 0.4742
| 8.73
|-
| 2.3.5
| {{monzo| -21 3 7 }}, {{monzo| -11 26 -13 }}
| {{mapping| 221 350 513 }}
| +0.4299
| 0.3921
| 7.22
|-
| 2.3.5.7
| 1029/1024, 19683/19600, 235298/234375
| {{mapping| 221 350 513 620 }}
| +0.5282
| 0.3799
| 7.00
|-
| 2.3.5.7.11
| 385/384, 441/440, 19683/19600, 235298/234375
| {{mapping| 221 350 513 620 764 }} (221e)
| +0.5904
| 0.3618
| 6.66
|}
 
=== 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
| 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)
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 19:31, 20 February 2025

← 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 distinct

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