20edo: Difference between revisions

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autopatrolled, emailconfirmed
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m A bit more theory.
Yourmusic Productions (talk | contribs)
autopatrolled, emailconfirmed
1,965 edits
Ups and downs notation causes so many messy enharmonics here, adding less ambiguous alternative notation systems.
Line 22: Line 22:
! Approximate Ratios
! Approximate Ratios
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! [[Armodue]]/Balzano Notation
! Heptatonic Major 2nd Generated
! Nearest Harmonic
! Nearest Harmonic
|-
|-
Line 29: Line 31:
| unison
| unison
| P1
| P1
| D
| 1
| D
| D
| 1
| 1
Line 38: Line 42:
| ^1, ^m2
| ^1, ^m2
| ^D, ^Eb
| ^D, ^Eb
| 1#/2b
| D#
| 33
| 33
|-
|-
Line 46: Line 52:
| ^^1, ~2
| ^^1, ~2
| ^^D, vvE
| ^^D, vvE
| 2
| Eb
| 69
| 69
|-
|-
Line 54: Line 62:
| vM2
| vM2
| vE
| vE
| 2#/3b
| E
| 71
| 71
|-
|-
Line 62: Line 72:
| M2, m3
| M2, m3
| E, F
| E, F
| 3
| E#
| 37
| 37
|-
|-
Line 70: Line 82:
| ^m3
| ^m3
| ^F
| ^F
| 3#/4b
| Fb
| 19
| 19
|-
|-
Line 78: Line 92:
| ~3
| ~3
| ^^F, vvF#
| ^^F, vvF#
| 4
| F
| 79
| 79
|-
|-
Line 86: Line 102:
| vM3
| vM3
| vF#
| vF#
| 4#
| F#
| 41
| 41
|-
|-
Line 94: Line 112:
| M3, P4
| M3, P4
| F#, G
| F#, G
| 5b
| Gb
| 21
| 21
|-
|-
Line 102: Line 122:
| ^4
| ^4
| ^G
| ^G
| 5
| G
| 11
| 11
|-
|-
Line 110: Line 132:
| ~4, ~5
| ~4, ~5
| ^^G, vvA
| ^^G, vvA
| 5#/6b
| G#/Ab
| 91
| 91
|-
|-
Line 118: Line 142:
| v5
| v5
| vA
| vA
| 6
| A
| 47
| 47
|-
|-
Line 126: Line 152:
| P5, m6
| P5, m6
| A
| A
| 6#/7b
| A#
| 97
| 97
|-
|-
Line 134: Line 162:
| ^5, ^m6
| ^5, ^m6
| ^A, ^Bb
| ^A, ^Bb
| 7
| Bb
| 25
| 25
|-
|-
Line 142: Line 172:
| ~6
| ~6
| ^^A, vvB
| ^^A, vvB
| 7#/8b
| B
| 13
| 13
|-
|-
Line 150: Line 182:
| vM6
| vM6
| vB
| vB
| 8
| B#
| 27
| 27
|-
|-
Line 158: Line 192:
| M6, m7
| M6, m7
| B, C
| B, C
| 8#/9b
| Cb
| 7
| 7
|-
|-
Line 166: Line 202:
| ^m7
| ^m7
| ^C
| ^C
| 9
| C
| 115
| 115
|-
|-
Line 174: Line 212:
| ~7
| ~7
| ^^C, vvD
| ^^C, vvD
| 9#
| C#
| 15
| 15
|-
|-
Line 182: Line 222:
| vM7
| vM7
| vD
| vD
| 1b
| Db
| 31
| 31
|-
|-
Line 189: Line 231:
| octave
| octave
| P8
| P8
| D
| 1
| D
| D
| 2
| 2

Revision as of 12:53, 13 October 2020


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. 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 14/13, 15/14 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 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

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
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 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]

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

Footnotes:

  1. ↑ Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach
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