20edo

← 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

Theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19
Error	absolute (¢)	0.0	+18.0	-26.3	-8.8	-11.3	-0.5	+15.0	+2.5
	relative (%)	0	+30	-44	-15	-19	-0.8	+25	+4
nearest edomapping		20	12	6	16	9	14	2	5

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. 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 it's 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 it's inversion instead.

Alternately, 20edo can be used as a 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.

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			Armodue/Balzano Notation	Heptatonic Major 2nd Generated	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	double-up 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 just intervals

Direct mapping (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)
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 (Grumpy heptatonic)
3 2 3 3 3 3 3	Minor quasi-equal Heptatonic (also Grumpy heptatonic)
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 (also Grumpy triskaidecatonic)
1 2 1 2 1 2 1 2 1 2 1 2 2	Minor quasi-equal Triskaidecatonic B (also Grumpy triskaidecatonic)
2 1 2 1 2 1 2 1 2 1 2 2 1	Minor quasi-equal Triskaidecatonic C (also Grumpy triskaidecatonic)
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].)

Prime Limit	Ratio	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

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.

Books

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

WARNING: MediaWiki doesn't have very good support for external images.

Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

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

Music

• 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