# 20edo

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 ← 19edo 20edo 21edo →
Prime factorization 22 × 5
Step size 60¢
Fifth 12\20 (720¢) (→3\5)
Semitones (A1:m2) 4:0 (240¢ : 0¢)
Consistency limit 3
Distinct consistency limit 3

20 equal divisions of the octave (abbreviated 20edo or 20ed2), also called 20-tone equal temperament (20tet) or 20 equal temperament (20et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 20 equal parts of exactly 60 ¢ each. Each step represents a frequency ratio of 21/20, or the 20th root of 2.

## Theory

20edo contains smaller EDOs2, 4, 5, and 10 and is part of the 5n family of equal divisions of the octave. It 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. As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0-7-9 steps) and its utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces Balzano nonatonic and undecatonic scales, which is probably the clearest arrangement for the black/white keys on a 20 tone keyboard. Treating the generator as 11\20 creates the same scale, but the primary triads are now 13:16:19 (0-6-11 steps) and its inversion instead. The 11\20 generator is a near-optimal tuning for both Mavericks temperament (which has a ~19/13 generator) and Score temperament (which has a ~16/11 generator).

Alternately, 20edo can be used as a tuning of the blackwood temperament, combining minor and major thirds to generate a highly symmetrical decatonic scale where every note is root to a major or minor triad and 7-limit tetrad that are heavily tempered, but in a useful way, as you can easily modulate to anywhere in the small cycle of 5ths, and build extended chords that use every note in the scale without clashing. Either of these works better than trying to force 20 into a diatonic framework.

20edo also possesses a 6L 1s scale generated using the narrow major second of 3\20 that is probably best interpreted as the sharp extreme of tetracot temperament and a 3L 5s generated by 7/20 that functions as the flat end of squares.

### Odd harmomics

Approximation of odd harmonics in 20edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +18.0 -26.3 -8.8 -23.9 -11.3 -0.5 -8.3 +15.0 +2.5 +9.2 -28.3
Relative (%) +30.1 -43.9 -14.7 -39.9 -18.9 -0.9 -13.8 +25.1 +4.1 +15.4 -47.1
Steps
(reduced)
32
(12)
46
(6)
56
(16)
63
(3)
69
(9)
74
(14)
78
(18)
82
(2)
85
(5)
88
(8)
90
(10)

## 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 Balzano Notation Archeotonic notation Nearest Harmonic
0 0 1/1 unison P1 D 1 D 1
1 60 29/28 up unison, upminor 2nd ^1, ^m2 ^D, ^Eb 1#/2b D# 33
2 120 15/14, 14/13 dup unison, mid 2nd ^^1, ~2 ^^D, vvE 2 Eb 69
3 180 10/9 downmajor 2nd vM2 vE 2#/3b E 71
4 240 8/7, 15/13 major 2nd, minor 3rd M2, m3 E, F 3 E# 37
5 300 13/11, 19/16 upminor 3rd ^m3 ^F 3#/4b Fb 19
6 360 16/13, 5/4 mid 3rd ~3 ^^F, vvF# 4 F 79
7 420 14/11, 51/40 downmajor 3rd vM3 vF# 4# F# 41
8 480 25/19, 4/3 major 3rd, perfect fourth M3, P4 F#, G 5b Gb 21
9 540 15/11, 11/8 up-fourth ^4 ^G 5 G 11
10 600 7/5 mid fourth, mid fifth ~4, ~5 ^^G, vvA 5#/6b G#/Ab 91
11 660 22/15, 16/11 down-fifth v5 vA 6 A 47
12 720 38/25, 3/2 fifth P5, m6 A 6#/7b A# 97
13 780 11/7, 25/16 upfifth, upminor 6th ^5, ^m6 ^A, ^Bb 7 Bb 25
14 840 13/8, 8/5 mid 6th ~6 ^^A, vvB 7#/8b B 13
15 900 22/13, 32/19 downmajor 6th vM6 vB 8 B# 27
16 960 7/4, 26/15 major 6th, minor 7th M6, m7 B, C 8#/9b Cb 7
17 1020 9/5 upminor 7th ^m7 ^C 9 C 115
18 1080 28/15, 15/8, 13/7 mid 7th ~7 ^^C, vvD 9# C# 15
19 1140 56/29 downmajor 7th vM7 vD 1b Db 31
20 1200 2/1 octave P8 D 1 D 2

### Selected 19-limit just intervals

Direct approximation (even if inconsistent)
Interval, complement Error (abs, ¢)
16/13 13/8 0.5276
15/14 28/15 0.5571
10/9 9/5 2.4037
19/16 32/19 2.4869
14/11 11/7 2.4920
19/13 26/19 3.0146
15/11 22/15 3.0492
15/13 26/15 7.7410
16/15 15/8 8.2687
14/13 13/7 8.2982
8/7 7/4 8.8259
13/11 22/13 10.7902
11/8 16/11 11.3179
11/9 18/11 12.5920
11/10 20/11 14.9957
17/16 32/17 15.0445
9/7 14/9 15.0840
6/5 5/3 15.6412
7/5 10/7 17.4878
3/2 4/3 18.0449
13/12 24/13 18.5726
9/8 16/9 23.9100
5/4 8/5 26.3137
7/6 12/7 26.8709
12/11 11/6 29.36294

## Chord names

20edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

• 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 vF# A = Dv = "D 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 = D^m,7 = "D upminor add-seven", or D ^F A B = D^m,6 = "D upminor add-six"
• 0-6-12-16 = D ^^F A C = D~,7 = "D mid add-seven", or D ^^F A B = D~,6 = "D mid add-six"
• 0-7-12-16 = D vF# A C = Dv,7 = "D down add-seven", or D vF# A B = Dv,6 = "D down add-six"
• 0-8-12-16 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"
• 0-7-12-19 = D vF# A vC# = DvM7 = "D downmajor seven"
• 0-5-12-17 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"

For a more complete list, see Ups and Downs Notation - Chords and Chord Progressions. 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 1 1 2 1 1 2 1 1 2 1 1 Blackwood Major Pentadecatonic (also tri-equal pentadecatonic) 1 1 2 1 1 2 1 1 2 1 1 2 Blackwood Diminished Pentadecatonic (also tri-equal pentadecatonic) 1 2 1 1 2 1 1 2 1 1 2 1 Blackwood Minor Pentadecatonic (also tri-equal pentadecatonic) 2 3 2 2 2 3 2 2 2 Balzano Nine-tone (fair mavila, score9) [1] 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 2 2 1 2 2 1 2 2 1 2 2 1 Dodecatonic 2 1 2 2 1 2 2 1 2 2 1 2 Diminished 1 2 2 1 2 2 1 2 2 1 2 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 (archaeotonic or Grumpy heptatonic) 3 2 3 3 3 3 3 Minor quasi-equal Heptatonic (also archaeotonic) 2 2 1 2 1 2 1 2 1 2 1 2 1 Major quasi-equal Triskaidecatonic (Grumpy triskaidecatonic) 2 1 2 1 2 1 2 1 2 1 2 1 2 Minor quasi-equal Triskaidecatonic A 1 2 1 2 1 2 1 2 1 2 1 2 2 Minor quasi-equal Triskaidecatonic B 2 1 2 1 2 1 2 1 2 1 2 2 1 Minor quasi-equal Triskaidecatonic C 3 2 2 2 2 3 2 2 2 Rothenberg Generalized Diatonic (also balzano or score9) 3 4 1 4 3 3 2 Stearns Major 7 2 7 2 2 score5 pentic, classic pentatonic 5 2 2 5 2 2 2 score7 (mavila, anti-diatonic)

## Regular temperament properties

### Uniform maps

13-limit uniform maps between 19.5 and 20.5
Min. size Max. size Wart notation Map
19.5000 19.5119 20bccddeeeeffff 20 31 45 55 67 72]
19.5119 19.5923 20bccddeeffff 20 31 45 55 68 72]
19.5923 19.5958 20bccddeeff 20 31 45 55 68 73]
19.5958 19.7695 20bddeeff 20 31 46 55 68 73]
19.7695 19.8009 20beeff 20 31 46 56 68 73]
19.8009 19.8625 20bff 20 31 46 56 69 73]
19.8625 19.8743 20b 20 31 46 56 69 74]
19.8743 20.0265 20 20 32 46 56 69 74]
20.0265 20.0900 20c 20 32 47 56 69 74]
20.0900 20.1257 20ce 20 32 47 56 70 74]
20.1257 20.1327 20cde 20 32 47 57 70 74]
20.1327 20.3791 20cdef 20 32 47 57 70 75]
20.3791 20.4030 20cdeeef 20 32 47 57 71 75]
20.4030 20.4571 20cdeeefff 20 32 47 57 71 76]
20.4571 20.4819 20cccdeeefff 20 32 48 57 71 76]
20.4819 20.5000 20cccdddeeefff 20 32 48 58 71 76]

### Commas

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

Prime
Limit
Ratio[1] Monzo Cents Color name Name(s)
3 256/243 [8 -5 90.22 Sawa Limma, Pythagorean Minor 2nd
5 16875/16384 [-14 3 4 51.12 Laquadyo Negri Comma, Double Augmentation Diesis
5 (16 digits) [-25 7 6 31.57 Lala-tribiyo Ampersand, Ampersand's Comma
5 2048/2025

[11 -4 -2

19.55 Sagugu Diaschisma
7 525/512 [-9 1 2 1 43.41 Lazoyoyo Avicennma, Avicenna's Enharmonic Diesis
7 49/48 [-4 -1 0 2 35.70 Zozo Slendro Diesis
7 50/49 [1 0 2 -2 34.98 Biruyo Tritonic Diesis, Jubilisma
7 686/675 [1 -3 -2 3 27.99 Trizo-agugu Senga
7 64/63 [6 -2 0 -1 27.26 Ru Septimal Comma, Archytas' Comma, Leipziger Komma
7 (18 digits) [-10 7 8 -7 22.41 Lasepru-aquadbiyo Blackjackisma
7 1029/1024 [-10 1 0 3 8.43 Latrizo Gamelisma
7 225/224 [-5 2 2 -1 7.71 Ruyoyo Septimal Kleisma, Marvel Comma
7 16875/16807 [0 3 4 -5 6.99 Quinru-aquadyo Mirkwai
7 (24 digits) [11 -10 -10 10 5.57 Saquinbizogu Linus
7 2401/2400 [-5 -1 -2 4 0.72 Bizozogu Breedsma
11 121/120 [-3 -1 -1 0 2 14.37 Lologu Biyatisma
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap
13 676/675 [2 -3 -2 0 0 2 2.56 Bithogu Parizeksma
1. Ratios longer than 10 digits are presented by placeholders with informative hints

## Instruments

• Like other members of the 5EDO family, 20-EDO lends itself well to guitar (and other fretted string instruments), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-4-4-2-0-0 (low to high), an "A" chord would be 0-0-4-4-2-0, and a "D" chord would be 2-0-0-4-4-2.
• Lumatone mapping for 20edo

## Books

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

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## Music

See also: Category:20edo tracks
Beheld
Pyotr Chernobrivets
E8 Heterotic
Francium
Andrew Heathwaite
Aaron Andrew Hunt
Mandrake
Claudi Meneghin
Herman Miller
NullPointerException Music
Sevish
Tancla
Chris Vaisvil
Chris Vaisvil and Bethan Mathis
Stephen Weigel