220edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.3
| {{monzo| 349 -220 }}
| {{monzo| 349 -220 }}
| {{mapping| 220 349 }}
| {{mapping| 220 349 }}
| -0.5304
| &minus;0.5304
| 0.5302
| 0.5302
| 9.72
| 9.72
Line 32: Line 24:
| {{monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }}
| {{monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }}
| {{mapping| 220 349 511 }}
| {{mapping| 220 349 511 }}
| -0.4912
| &minus;0.4912
| 0.4364
| 0.4364
| 8.00
| 8.00
Line 39: Line 31:
| 6144/6125, 10976/10935, 390625/388962
| 6144/6125, 10976/10935, 390625/388962
| {{mapping| 220 349 511 618 }}
| {{mapping| 220 349 511 618 }}
| -0.5538
| &minus;0.5538
| 0.3932
| 0.3932
| 7.21
| 7.21
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 72: Line 58:
|-
|-
| 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]]
|-
|-
| 20
| 20
| 91\220<br>(3\220)
| 91\220<br />(3\220)
| 496.36<br>(16.36)
| 496.36<br />(16.36)
| 4/3<br>(126/125)
| 4/3<br />(126/125)
| [[Degrees]]
| [[Degrees]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 02:08, 16 November 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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