220edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|220}}
{{EDO intro|220}}
== Theory ==
== Theory ==
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.  
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 ===
=== Odd harmonics ===
{{Harmonics in equal|220}}
{{Harmonics in equal|220}}
=== Subsets and supersets ===
=== Subsets and supersets ===
220 factors into 2<sup>2</sup> × 5 × 11, with subset edos {{EDOs| 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110}}
Since 220 factors into 2<sup>2</sup> × 5 × 11, 220edo has subset edos {{EDOs| 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 }}
==Regular temperament properties==
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|349 -220}}
| {{monzo| 349 -220 }}
|{{val|220 349}}
| {{mapping| 220 349 }}
| -0.5304
| -0.5304
| 0.5302
| 0.5302
| 9.72
| 9.72
|-
|-
|2.3.5
| 2.3.5
|{{monzo|20 -17 3}}, {{monzo|23 6 -14}}
| {{monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }}
|{{val|220 349 511}}
| {{mapping| 220 349 511 }}
| -0.4912
| -0.4912
| 0.4364
| 0.4364
| 8.00
| 8.00
|-
|-
|2.3.5.7
| 2.3.5.7
|6144/6125, 10976/10935, 390625/388962
| 6144/6125, 10976/10935, 390625/388962
|{{val|220 349 511 618}}
| {{mapping| 220 349 511 618 }}
| -0.5538
| -0.5538
| 0.3932
| 0.3932
| 7.21
| 7.21
|}
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|43\220
| 43\220
|234.55
| 234.55
|8/7
| 729/640
|[[Rodan]]
| [[Rodan]] (5-limit)
|-
|-
|1
| 1
|83\220
| 83\220
|452.73
| 452.73
|125/81
| 125/81
|[[Maja]]
| [[Maja]] (5-limit)
|-
|-
|2
| 2
|13\220
| 13\220
|70.91
| 70.91
|25/24
| 25/24
|[[Vishnu]]
| [[Vishnu]] (5-limit)
|-
|-
|11
| 11
|91\220<br>(9\220)
| 91\220<br>(9\220)
|496.36<br>(49.09)
| 496.36<br>(49.09)
|4/3<br>(36/35)
| 4/3<br>(36/35)
|[[Hendecatonic]]
| [[Hendecatonic]]
|}
|}
 
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 07:08, 2 April 2024

← 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

Template:EDO intro

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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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