20edo
Theory
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.
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 | Nearest Harmonic | ||
|---|---|---|---|---|---|---|
| 0 | 0 | 1/1 | unison | P1 | D | 1 |
| 1 | 60 | 29/28 | up unison, upminor 2nd | ^1, ^m2 | ^D, ^Eb | 33 |
| 2 | 120 | 14/13, 15/14 | double-up unison, mid 2nd | ^^1, ~2 | ^^D, vvE | 69 |
| 3 | 180 | 10/9 | downmajor 2nd | vM2 | vE | 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, vvF# | 79 |
| 7 | 420 | 14/11, 51/40 | downmajor 3rd | vM3 | vF# | 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 | mid fourth, mid fifth | ~4, ~5 | ^^G, vvA | 91 |
| 11 | 660 | 22/15, 16/11 | down-fifth | v5 | vA | 47 |
| 12 | 720 | 38/25, 3/2 | fifth | P5, m6 | A | 97 |
| 13 | 780 | 11/7, 25/16 | upfifth, upminor 6th | ^5, ^m6 | ^A, ^Bb | 25 |
| 14 | 840 | 13/8, 8/5 | mid 6th | ~6 | ^^A, vvB | 13 |
| 15 | 900 | 22/13, 32/19 | downmajor 6th | vM6 | vB | 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, vvD | 15 |
| 19 | 1140 | 56/29 | downmajor 7th | vM7 | vD | 31 |
| 20 | 1200 | 2/1 | octave | P8 | D | 2 |
Selected just intervals
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | ||
|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.00 | +18.0449 | -26.3137 | -8.8259 | -11.3179 | -0.5276 | +15.0445 | +2.4869 |
| relative (%) | 0.0 | +30.074 | -43.856 | -14.709 | -18.863 | -0.879 | +25.074 | +4.144 | |
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 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 |
| 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 |.)
| Ratio | Monzo | Cents | Color Name | Name 1 | Name 2 | Name 3 |
|---|---|---|---|---|---|---|
| 256/243 | [8 -5⟩ | 90.22 | Sawa | Limma | Pythagorean Minor 2nd | |
| 16875/16384 | [-14 3 4⟩ | 51.12 | Laquadyo | Negri Comma | Double Augmentation Diesis | |
| 34171875/33554432 | [-25 7 6⟩ | 31.57 | Lala-tribiyo | Ampersand's Comma | ||
| 2048/2025 | [11 -4 -2⟩ | 19.55 | Sagugu | Diaschisma | ||
| 525/512 | [-9 1 2 1⟩ | 43.41 | Lazoyoyo | Avicennma | Avicenna's Enharmonic Diesis | |
| 49/48 | [-4 -1 0 2⟩ | 35.70 | Zozo | Slendro Diesis | ||
| 50/49 | [1 0 2 -2⟩ | 34.98 | Biruyo | Tritonic Diesis | Jubilisma | |
| 686/675 | [1 -3 -2 3⟩ | 27.99 | Trizo-agugu | Senga | ||
| 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal Comma | Archytas' Comma | Leipziger Komma |
| 854296875/843308032 | [-10 7 8 -7⟩ | 22.41 | Lasepru-aquadbiyo | Blackjackisma | ||
| 1029/1024 | [-10 1 0 3⟩ | 8.43 | Latrizo | Gamelisma | ||
| 225/224 | [-5 2 2 -1⟩ | 7.71 | Ruyoyo | Septimal Kleisma | Marvel Comma | |
| 16875/16807 | [0 3 4 -5⟩ | 6.99 | Quinru-aquadyo | Mirkwai | ||
| 6772805/6751042 | [11 -10 -10 10⟩ | 5.57 | Saquinbizogu | Linus | ||
| 121/120 | [-3 -1 -1 0 2⟩ | 14.37 | Lologu | Biyatisma | ||
| 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozogu | Superleap | ||
| 2401/2400 | [-5 -1 -2 4⟩ | 0.72 | Bizozogu | Breedsma | ||
| 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]
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"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
Footnotes: