220edo: Difference between revisions

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'''220edo''' is the [[EDO|equal division of the octave]] into 220 parts of 5.4545 [[cent]]s each. Using the patent val, it tempers out 131072000/129140163 (rodan comma) and 6115295232/6103515625 (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. It is consistent to the 7-odd-limit.
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[[Category:edo]]
== Theory ==
Using the patent val, it tempers out {{monzo| 20 -17 3 }} ([[rodan comma]]) and {{monzo| 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 ===
{{Harmonics in equal|220}}
 
=== Subsets and supersets ===
Since 220 factors into {{factorisation|220}}, 220edo has subset edos {{EDOs| 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 }}.
 
== 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| 349 -220 }}
| {{Mapping| 220 349 }}
| −0.5304
| 0.5302
| 9.72
|-
| 2.3.5
| {{Monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }}
| {{Mapping| 220 349 511 }}
| −0.4912
| 0.4364
| 8.00
|-
| 2.3.5.7
| 6144/6125, 10976/10935, 390625/388962
| {{Mapping| 220 349 511 618 }}
| −0.5538
| 0.3932
| 7.21
|}
 
=== 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
| 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<br>(9\220)
| 496.36<br>(49.09)
| 4/3<br>(36/35)
| [[Hendecatonic (temperament)|Hendecatonic]]
|-
| 20
| 91\220<br>(3\220)
| 496.36<br>(16.36)
| 4/3<br>(126/125)
| [[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

Latest revision as of 12:19, 5 April 2026

← 219edo 220edo 221edo →
Prime factorization 22 × 5 × 11
Step size 5.45455 ¢ 
Fifth 129\220 (703.636 ¢)
Semitones (A1:m2) 23:15 (125.5 ¢ : 81.82 ¢)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of odd harmonics in 220edo
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

Table of rank-2 temperaments by generator
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

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