20edo

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.

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Compositions

• Underwater Spontaniety (lo-fi ambient) by Stephen Weigel
• God's Favorite Tuning (xen-pop) by Stephen Weigel
• You Lied, and I'm not Mentioning your Name (xen-pop) by Stephen Weigel
• Etude in 20-tone equal tuning play by Herman Miller
• 20ET — Prelude & Fugue a3 by Aaron Hunt
• We Call It Sharing Breath play by Andrew Heathwaite
• The Walrus and the Carpenter byChris Vaisvil and Bethan Mathis
• 20 edo Electric Guitar Improvisation byChris Vaisvil
• Göbekli Tepe (20 edo guitar solo) by Chris Vaisvil
• A Paradigm Shift (20 edo) by Chris Vaisvil
• Composition for unconventional tempered clavier by Pyotr Chernobrivets
• 5 compositions for violin and clavier in 20-EDO by Pyotr Chernobrivets
• Composition for cello and clavier in 20-tone equal temperament by Pyotr Chernobrivets
• 5 compositions for violin and clavier in 20-tone equal temperament by Pyotr Chernobrivets

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]}