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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 5 edo), 11, 13 & 15 (from 10 edo), 19 & 27 (from 4 edo), 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 | Properties | Chord Names

Compositions


Properties

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 20_tet_Coversm.jpg
"Icosaphonic Scales for Guitar" - Theory / Scale book with above modes and more by Ron Sword