20edo

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20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 cents each. It contains smaller edos 2, 4, 5, and 10 and is part of the 5n Family of equal divisions of the octave. 20 edo fairly approximates the harmonics 7 (from 5edo), 11, 13 & 15 (from 10edo), 19 & 27 (from 4edo), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15.


Compositions

Intervals

Like 15edo, every note has many names. D is also C# and Eb. The major 3rd is also a perfect 4th and a dim 5th.

Degree Cents Approximate Ratios Ups and Downs Notation Nearest Harmonic
0 0 1/1 unison P1 D 1
1 60 29/28 upminor 2nd ^m2 D^ 33
2 120 14/13, 15/14 mid 2nd ~2 D^^, Evv 69
3 180 10/9 downmajor 2nd vM2 Ev 71
4 240 8/7, 15/13 major 2nd, minor 3rd M2, m3 E, F 37
5 300 13/11, 19/16 upminor 3rd ^m3 F^ 19
6 360 16/13, 5/4 mid 3rd ~3 F^^ 79
7 420 14/11, 51/40 downmajor 3rd vM3 F#v 41
8 480 25/19, 4/3 major 3rd, perfect fourth M3, P4 F#, G 21
9 540 15/11, 11/8 up-fourth ^4 G^ 11
10 600 7/5 double-up fourth, double-down fifth ^^4, vv5 G^^, Avv 91
11 660 22/15, 16/11 down-fifth v5 Av 47
12 720 38/25, 3/2 fifth P5, m6 A 97
13 780 11/7, 25/16 upminor 6th ^m6 A^ 25
14 840 13/8, 8/5 mid 6th ~6 A^^, Bvv 13
15 900 22/13, 32/19 downmajor 6th vM6 Bv 27
16 960 7/4, 26/15 major 6th, minor 7th M6, m7 B, C 7
17 1020 9/5 upminor 7th ^m7 C^ 115
18 1080 15/8, 13/7 mid 7th ~7 C^^, Dvv 15
19 1140 56/29 downmajor 7th vM7 Dv 31
20 1200 2/1 octave P8 D 2

Chord Names

20edo chord names using ups and downs:

  • 0-4-12 = D E A = Dsus2 = "D sus 2", or D F A = Dm = "D minor"
  • 0-5-12 = D F^ A = D.^m = "D upminor"
  • 0-6-12 = D F^^ A = D.~ = "D mid"
  • 0-7-12 = D F#v A = D.v = "D dot down" or "D downmajor"
  • 0-8-12 = D G A = Dsus4, or D F# A = D = "D" or "D major"
  • 0-4-12-16 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"
  • 0-5-12-16 = D F^ A C = Dm7(^3) = "D minor seven up-three", or D F^ A B = Dm6(^3) = "D minor six up-three"
  • 0-6-12-16 = D F^^ A C = D7(~3) = "D seven mid-three", or D F^ A B = D6(~3) = "D six mid-three"
  • 0-7-12-16 = D F#v A C = D7(v3) = "D seven down-three", or D F#v A B = D6(v3) = "D six down-three"
  • 0-8-12-16 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"
  • 0-7-12-19 = D F#v A C#v = D.vM7 = "D downmajor seven"
  • 0-5-12-17 = D F^ A C^ = D.^m7 = "D dot up minor-seven", or D F^ A B^ = D.^m6 = "D dot up minor-six"

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs. Because many intervals have several names, many chords do too.

Modes

20 tone equal modes:

3 1 3 1 3 1 3 1 3 1 Blackwood Major Decatonic (bi-equal decatonic, according to the MOS naming scheme)
1 3 1 3 1 3 1 3 1 3 Blackwood Minor Decatonic (also bi-equal decatonic)
2 3 2 2 2 3 2 2 2 Balzano Nine-tone (fair mavila, score9)
2 2 2 2 1 2 2 2 2 2 1 Balzano Eleven-tone, Agmon Diatonic DS4, score11
2 2 2 3 2 2 2 3 2 Balzano Nine-tone inverse (also fair mavila, score9)
1 2 2 2 2 2 1 2 2 2 2 Balzano Eleven-tone inverse (also score11)
2 3 2 3 2 3 2 3 Octatonic (diminished, according to the MOS naming scheme)
3 2 3 2 3 2 3 2 Diminished
4 3 1 4 3 4 1 Twenty-tone "Major"
4 1 3 4 1 4 3 Twenty-tone "Minor"
2 2 1 2 1 2 2 1 2 2 2 1 Twelve-tone Chromatic
2 2 2 2 1 2 2 2 2 1 2 Zweifel Major
2 1 2 2 2 2 2 1 2 2 2 Zweifel Natural Minor
3 3 3 3 3 3 2 Major quasi-equal Heptatonic (Grumpy heptatonic)
3 2 3 3 3 3 3 Minor quasi-equal Heptatonic (also Grumpy heptatonic)
3 2 2 2 2 3 2 2 2 Rothenberg Generalized Diatonic (also score9)
3 4 1 4 3 3 2 Stearns Major
7 2 7 2 2 score5 (classic pentatonic)
5 2 2 5 2 2 2 score7 (mavila, anti-diatonic)

Commas

20 EDO tempers out the following commas. (Note: This assumes the val < 20 32 46 56 69 74 |.)

Rational Monzo Size (Cents) Name 1 Name 2 Name 3
256/243 | 8 -5 > 90.22 Limma Pythagorean Minor 2nd
16875/16384 | -14 3 4 > 51.12 Negri Comma Double Augmentation Diesis
9931568/9752117 | -25 7 6 > 31.57 Ampersand's Comma
2048/2025 | 11 -4 -2 > 19.55 Diaschisma
525/512 | -9 1 2 1 > 43.41 Avicennma Avicenna's Enharmonic Diesis
49/48 | -4 -1 0 2 > 35.70 Slendro Diesis
50/49 | 1 0 2 -2 > 34.98 Tritonic Diesis Jubilisma
686/675 | 1 -3 -2 3 > 27.99 Senga
64/63 | 6 -2 0 -1 > 27.26 Septimal Comma Archytas' Comma Leipziger Komma
9859966/9733137 | -10 7 8 -7 > 22.41 Blackjackisma
1029/1024 | -10 1 0 3 > 8.43 Gamelisma
225/224 | -5 2 2 -1 > 7.71 Septimal Kleisma Marvel Comma
16875/16807 | 0 3 4 -5 > 6.99 Mirkwai
6772805/6751042 | 11 -10 -10 10 > 5.57 Linus
121/120 | -3 -1 -1 0 2 > 14.37 Biyatisma
91/90 | -1 -2 -1 1 0 1 > 19.13 Superleap
2401/2400 | -5 -1 2 4 > 0.72 Breedsma
676/675 | 2 -3 -2 0 0 2 > 2.56 Parizeksma

Books

External image: http://ronsword.com/images/20_tet_Coversm.jpg [dead link]

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"Icosaphonic Scales for Guitar" - Theory / Scale book with above modes and more by Ron Sword [dead link]