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{{interwiki
{{Interwiki
| en = 20edo
| de =
| de =
| en = 20edo
| es = 20 EDO
| es = 20 EDO
| ja = 20平均律
| ja = 20平均律
}}
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|20}}
{{ED intro}}
 
== Theory ==
== Theory ==
20edo contains smaller {{EDOs|legend=1| 2, 4, 5, and 10 }} and is part of the 5''n'' 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 [[No-threes_subgroup_temperaments#Mavericks|Mavericks]] temperament (which has a ~[[19/13]] generator) and [[Chromatic pairs#Score|Score]] temperament (which has a ~[[16/11]] generator).
20edo contains smaller edos {{EDOs| 2, 4, 5, and 10 }} and is part of the 5''n'' 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, 20edo does a reasonably convincing approximation of harmonics 4:7:11:13:15.  
 
20edo is around the point where 5edo's 3rd harmonic starts to become notably inaccurate relative to the size of the edo (that is, it is over 25 relative cents off). It thus inherits 5edo's crude [[archy]] temperament, with its particularly accurate approximation of 7/4 at 960 cents. 
 
As 7, 11 and 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 [[no-threes subgroup temperaments #Mavericks|mavericks]] temperament (which has a ~[[19/13]] generator) and [[no-threes subgroup temperaments #Score|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.
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_family#Tetracot|tetracot]] temperament and a [[3L 5s]] generated by 7/20 that functions as the flat end of [[squares]].
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]].
 
Thanks to its sevenths, 20edo is an ideal tuning for its size for [[metallic harmony]].


=== Odd harmomics ===
=== Odd harmonics ===
{{Harmonics in equal|20}}
{{Harmonics in equal|20}}


== Intervals ==
== 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.
{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
|-
|-
! [[Degree]]
! [[Degree]]
! [[Cents]]
! [[Cents]]
! Approximate Ratios
! Approximate ratios
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" |[[Ups and downs notation]]
! [[2L 7s|Balzano]] Notation
! [[2L 7s|Balzano]]<br>Notation
! [[6L 1s|Archeotonic]] notation
! [[6L 1s|Archeotonic]]<br>notation
! Nearest Harmonic
! Nearest<br>harmonic
|-
|-
| 0
| 0
Line 52: Line 59:
| 2
| 2
| 120
| 120
| [[15/14]], 14/13
|[[15/14]], 14/13
| dup unison, mid 2nd
| dup unison, mid 2nd
| ^^1, ~2
| ^^1, ~2
Line 92: Line 99:
| 6
| 6
| 360
| 360
| [[16/13]], 5/4
|[[16/13]], 5/4
| mid 3rd
| mid 3rd
| ~3
| ~3
Line 102: Line 109:
| 7
| 7
| 420
| 420
| [[14/11]], [[51/40]]
|[[14/11]], [[51/40]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 122: Line 129:
| 9
| 9
| 540
| 540
| [[15/11]], 11/8
|[[15/11]], 11/8
| up-fourth
| up-fourth
| ^4
| ^4
Line 142: Line 149:
| 11
| 11
| 660
| 660
| [[22/15]], 16/11
|[[22/15]], 16/11
| down-fifth
| down-fifth
| v5
| v5
Line 172: Line 179:
| 14
| 14
| 840
| 840
| [[13/8]], 8/5
|[[13/8]], 8/5
| mid 6th
| mid 6th
| ~6
| ~6
Line 212: Line 219:
| 18
| 18
| 1080
| 1080
| [[28/15]], 15/8, 13/7
|[[28/15]], 15/8, 13/7
| mid 7th
| mid 7th
| ~7
| ~7
Line 241: Line 248:
|}
|}


=== Selected 19-limit just intervals ===
=== Chord names in color notation ===
{{Q-odd-limit intervals|20}}
 
== 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).
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"
* 0-4-12 = D E A = Dsus2 = "D sus 2", or D F A = Dm = "D minor"
Line 259: Line 263:
* 0-5-12-17 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"
* 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|Ups and Downs Notation - Chords and Chord Progressions]]. Because many intervals have several names, many chords do too.
For a more complete list, see [[Kite's ups and downs notation #Chords and chord Progressions]]. Because many intervals have several names, many chords do too.
 
== Notation ==
Like [[15edo]], every note has many names. D is also C# and Eb. The major third is also a perfect fourth and a diminished fifth.
 
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp4-szg}}
 
=== Kite's ups and downs notation ===
20edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}
 
=== Sagittal notation ===
This notation is a superset of the notations for edos [[10edo #Sagittal notation|10]] and [[5edo #Sagittal notation|5]].
 
==== Evo and Revo flavors ====
{{Sagittal chart|}}


== Modes ==
==== Evo-SZ flavor ====
20 tone equal modes:
{{Sagittal chart|Evo-SZ}}


{| class="wikitable"
== Approximation to JI ==
|-
=== Interval mappings ===
| 3 1 3 1 3 1 3 1 3 1
{{Q-odd-limit intervals|20}}
| Blackwood Major Decatonic [[5L_5s| (bi-equal decatonic]], according to the [[MOSNamingScheme| 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 ([[2L_7s| fair mavila]], [[score9]]) <ref>[https://www.jstor.org/stable/833490?seq=1 Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach]</ref>
|-
| 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 ([[4L_4s| diminished]], according to the [[MOSNamingScheme| 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 ([[6L_1s| 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 [[2L 7s| balzano]] or [[score9]])
|-
| 3 4 1 4 3 3 2
| Stearns Major
|-
| 7 2 7 2 2
| [[score5]] [[2L_3s| pentic, classic pentatonic]]
|-
| 5 2 2 5 2 2 2
| [[score7]] ([[2L_5s| mavila, anti-diatonic]])
|}
<references />


== Regular temperament properties ==
== Regular temperament properties ==
=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|19.5|20.5}}
{{Uniform map|edo=20}}


=== Commas ===
=== Commas ===
20 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 20 32 46 56 69 74 }}.)
20et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 20 32 46 56 69 74 }}.


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
![[Harmonic limit| Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
! [[Monzo]]
![[Cent]]s
! [[Cent]]s
! [[Color name]]
! [[Color name]]
! Name(s)
! Name(s)
Line 376: Line 310:
| 90.22
| 90.22
| Sawa
| Sawa
| [[Limma]], Pythagorean Minor 2nd
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
| [[16875/16384]]
| [[16875/16384]]
| {{monzo|-14 3 4 }}
| {{monzo| -14 3 4 }}
| 51.12
| 51.12
| Laquadyo
| Laquadyo
| Negri Comma, Double Augmentation Diesis
| Negri comma
|-
|-
| 5
| 5
| [[34171875/33554432| (16 digits)]]
| [[34171875/33554432|(16 digits)]]
| {{monzo|-25 7 6 }}
| {{monzo| -25 7 6 }}
| 31.57
| 31.57
| Lala-tribiyo
| Lala-tribiyo
| [[Ampersand]], Ampersand's Comma
| Ampersand comma
|-
|-
| 5
| 5
| [[2048/2025]]
| [[2048/2025]]
|  
| {{monzo| 11 -4 -2 }}
{{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Sagugu
Line 402: Line 335:
| 7
| 7
| [[525/512]]
| [[525/512]]
| {{monzo|-9 1 2 1 }}
| {{monzo| -9 1 2 1 }}
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma, Avicenna's Enharmonic Diesis
| Avicennma, Avicenna's enharmonic diesis
|-
|-
| 7
| 7
| [[49/48]]
| [[49/48]]
| {{monzo|-4 -1 0 2 }}
| {{monzo| -4 -1 0 2 }}
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro Diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
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| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic Diesis, Jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
Line 433: Line 366:
| 27.26
| 27.26
| Ru
| Ru
| Septimal Comma, Archytas' Comma, Leipziger Komma
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|-
| 7
| 7
| <abbr title="854296875/843308032">(18 digits)</abbr>
| <abbr title="854296875/843308032">(18 digits)</abbr>
| {{monzo|-10 7 8 -7 }}
| {{monzo| -10 7 8 -7 }}
| 22.41
| 22.41
| Lasepru-aquadbiyo
| Lasepru-aquadbiyo
Line 444: Line 377:
| 7
| 7
| [[1029/1024]]
| [[1029/1024]]
| {{monzo|-10 1 0 3 }}
| {{monzo| -10 1 0 3 }}
| 8.43
| 8.43
| Latrizo
| Latrizo
Line 451: Line 384:
| 7
| 7
| [[225/224]]
| [[225/224]]
| {{monzo|-5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal Kleisma, Marvel Comma
| Marvel comma, septimal kleisma
|-
|-
| 7
| 7
Line 461: Line 394:
| 6.99
| 6.99
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 468: Line 401:
| 5.57
| 5.57
| Saquinbizogu
| Saquinbizogu
| [[Linus comma| Linus]]
| [[Linus comma]]
|-
|-
| 7
| 7
| [[2401/2400]]
| [[2401/2400]]
| {{monzo|-5 -1 -2 4 }}
| {{monzo| -5 -1 -2 4 }}
| 0.72
| 0.72
| Bizozogu
| Bizozogu
Line 479: Line 412:
| 11
| 11
| [[121/120]]
| [[121/120]]
| {{monzo|-3 -1 -1 0 2 }}
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
Line 486: Line 419:
| 13
| 13
| [[91/90]]
| [[91/90]]
| {{monzo|-1 -2 -1 1 0 1 }}
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 496: Line 429:
| 2.56
| 2.56
| Bithogu
| Bithogu
| Parizeksma
| Island comma, parizeksma
|}
|}
<references />
<references />
Line 504: Line 437:
* [[Lumatone mapping for 20edo]]
* [[Lumatone mapping for 20edo]]


== Books ==
== Modes ==
{{External Image| http://ronsword.com/images/20_tet_Coversm.jpg {{dead link}} }}
20 tone equal modes:
[http://www.ronsword.com/ "Icosaphonic Scales for Guitar" - Theory / Scale book with above modes and more] by [[Ron Sword]] {{dead link}}
 
{| class="wikitable"
|-
| 3 1 3 1 3 1 3 1 3 1
| Blackwood Major Decatonic [[5L_5s|(pentawood]], according to the [[MOSNamingScheme|MOS naming scheme]])
|-
| 1 3 1 3 1 3 1 3 1 3
| Blackwood Minor Decatonic (also pentawood)
|-
| 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 ([[2L_7s|balzano]], [[score9]]) <ref>[https://www.jstor.org/stable/833490?seq=1 Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach]</ref>
|-
|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 balzano, 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 ([[4L_4s|tetrawood]], according to the [[MOSNamingScheme|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 ([[6L_1s|archaeotonic]])
|-
|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 [[2L 7s|balzano]] or [[score9]])
|-
| 3 4 1 4 3 3 2
|Stearns Major
|-
|7 2 7 2 2
|[[score5]] [[2L_3s|pentic, classic pentatonic]]
|-
|5 2 2 5 2 2 2
|[[score7]] ([[2L_5s|mavila, anti-diatonic]])
|-
|4 1 3 2 2 3 1 4
|[[polymicrotonal|8-tone 4&5edo scale]]
|}
<references />


== Music ==
== Music ==
{{Catrel| 20edo tracks }}
{{Catrel| 20edo tracks }}
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=xRNZJhFcoL8 ''20EDO Tablet of Union | Enochian Vision Magick''] (2025)


; [[Beheld]]
; [[Beheld]]
Line 519: Line 548:
* [https://www.youtube.com/watch?v=duJHUo_ACgk ''Composition for cello and clavier in 20-tone equal temperament''] (2017)
* [https://www.youtube.com/watch?v=duJHUo_ACgk ''Composition for cello and clavier in 20-tone equal temperament''] (2017)
* [https://www.youtube.com/watch?v=u6k4lgCnHqs ''5 compositions for violin and clavier in 20-tone equal temperament''] (2017)
* [https://www.youtube.com/watch?v=u6k4lgCnHqs ''5 compositions for violin and clavier in 20-tone equal temperament''] (2017)
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/-GTh0EyxCd0 ''20edo''] (2023)
* [https://www.youtube.com/shorts/bqYA9flwL_g ''20edo prelude''] (2025)
* [https://www.youtube.com/watch?v=BFUTeFs7Ycg ''20edo groove''] (2025)
* ''Nocturne in 20edo'' (2025)
** [https://www.youtube.com/shorts/GxElRqwqz-o Lumatone view]
** [https://www.youtube.com/watch?v=PMgYSG4-XaY Visualizer view]
* [https://www.youtube.com/shorts/peypGQ_h0ys ''another 20edo groove''] (2026)


; [[E8 Heterotic]]
; [[E8 Heterotic]]
* [https://youtu.be/GRW1GFHFIys?si=l-I64VcbItcZkvZz ''T w e n t y / T w e n t y''] (2019) – synthwave, blackwood[10] in 20edo tuning
* [https://www.youtube.com/watch?v=GRW1GFHFIys ''T w e n t y / T w e n t y''] (2019) – synthwave, Blackwood[10] in 20edo tuning
* [https://youtu.be/zcvsvc6Hl5g?si=H7i3kia2XKxUYt_U "Elements - Wood"] from ''Elements'' (2019–2020) – tetracot in 20edo tuning
* [https://www.youtube.com/watch?v=zcvsvc6Hl5g "Wood"], from [https://www.youtube.com/playlist?list=PLbe8k6ZUb2FiSx8Jal1N0XjzWJM_cjgsp ''Elements''] (2019–2020) – tetracot in 20edo tuning
* [https://youtu.be/KnSSdiv_rS0?si=d1DI3fTnrEIJvsCQ ''Decade''] (2020) – synthwave
* [https://www.youtube.com/watch?v=KnSSdiv_rS0 ''Decade''] (2020) – synthwave


; [[Francium]]
; [[Francium]]
* [https://www.youtube.com/watch?v=rkgXkQ2D3mw ''Horrible Night Of Horror''] (2023)
* [https://www.youtube.com/watch?v=rkgXkQ2D3mw ''Horrible Night Of Horror''] (2023)
* [https://www.youtube.com/watch?v=26XqPsnx16w ''Madeira Preta''] (2023) – blackwood[10] in 20edo
* [https://www.youtube.com/watch?v=26XqPsnx16w ''Madeira Preta''] (2023) – Blackwood[10] in 20edo tuning
 
; [[Frostburn]]
* [https://www.youtube.com/watch?v=nlF6al8MEuk ''Galloping Nightmares''] (2024) – synth metal
 
; [[groundfault]]
* "The Lake Reflects a Black Sky" from ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/track/the-lake-reflects-a-black-sky-29-31-20edo Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo YouTube (0:00–2:38)] – in part, the rest being in 29edo and 31edo


; [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
Line 533: Line 577:


; [[Aaron Andrew Hunt]]
; [[Aaron Andrew Hunt]]
* [http://aaronandrewhunt.zentral.zone/mp3/20ETPreludeFugue.mp3 20ET — ''Prelude &amp; Fugue a3'']
* [http://aaronandrewhunt.zentral.zone/mp3/20ETPreludeFugue.mp3 ''Prelude & Fugue a3'']


; [[Mandrake]]
; [[Mandrake]]
Line 542: Line 586:
* ''Canon on Twinkle Twinkle Little Star'' (2023–2024)
* ''Canon on Twinkle Twinkle Little Star'' (2023–2024)
** [https://www.youtube.com/watch?v=6n_a9SfAjsY for organ] · [https://www.youtube.com/watch?v=UuIsIeEc-7w for baroque oboe & viola]
** [https://www.youtube.com/watch?v=6n_a9SfAjsY for organ] · [https://www.youtube.com/watch?v=UuIsIeEc-7w for baroque oboe & viola]
* [https://www.youtube.com/watch?v=5cIYrrxFN9Q ''Chacony & Fugue, in 20edo for Baroque Consort''] (2026)


; [[Herman Miller]]
; [[Herman Miller]]
* ''Etude in 20-tone equal tuning'' – [https://web.archive.org/web/20070219044535/http://www.io.com/~hmiller/midi/20tet.mid midi file] [https://web.archive.org/web/20201127014350/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/20-et-Pianoteq-Schantz.mp3 play]
* ''Etude in 20-tone equal tuning'' – [https://web.archive.org/web/20070219044535/http://www.io.com/~hmiller/midi/20tet.mid midi file] [https://web.archive.org/web/20201127014350/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/20-et-Pianoteq-Schantz.mp3 play]
* [https://soundcloud.com/morphosyntax-1/spinning-the-web ''Spinning the Web''] (2017)
* [https://soundcloud.com/morphosyntax-1/ivari-na-taniravi ''Ivari na taniravi''] (2020)
* [https://soundcloud.com/morphosyntax-1/drifting ''Drifting''] (2020)


; [[NullPointerException Music]]
; [[NullPointerException Music]]
Line 570: Line 618:
* [https://soundcloud.com/overtoneshock/gods-favorite-tuning-20-edo ''God's Favorite Tuning''] (2017) – xen-pop
* [https://soundcloud.com/overtoneshock/gods-favorite-tuning-20-edo ''God's Favorite Tuning''] (2017) – xen-pop
* [https://soundcloud.com/overtoneshock/you-lied-and-im-not-mentioning-your-name-20-edo ''You Lied, and I'm not Mentioning your Name''] (2017) – xen-pop
* [https://soundcloud.com/overtoneshock/you-lied-and-im-not-mentioning-your-name-20-edo ''You Lied, and I'm not Mentioning your Name''] (2017) – xen-pop
== Books ==
{{External Image| http://ronsword.com/images/20_tet_Coversm.jpg {{dead link}} }}
[http://www.ronsword.com/ "Icosaphonic Scales for Guitar" - Theory / Scale book with above modes and more] by [[Ron Sword]] {{dead link}}


[[Category:Listen]]
[[Category:Listen]]
[[Category:Twentuning]]
[[Category:Twentuning]]
[[Category:Todo:add rank 2 temperaments table]]
[[Category:Todo:add rank 2 temperaments table]]

Latest revision as of 11:51, 12 May 2026

← 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 edos 2, 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, 20edo does a reasonably convincing approximation of harmonics 4:7:11:13:15.

20edo is around the point where 5edo's 3rd harmonic starts to become notably inaccurate relative to the size of the edo (that is, it is over 25 relative cents off). It thus inherits 5edo's crude archy temperament, with its particularly accurate approximation of 7/4 at 960 cents.

As 7, 11 and 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.

Thanks to its sevenths, 20edo is an ideal tuning for its size for metallic harmony.

Odd harmonics

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

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

Chord names in color notation

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 Kite's ups and downs notation #Chords and chord Progressions. Because many intervals have several names, many chords do too.

Notation

Like 15edo, every note has many names. D is also C# and Eb. The major third is also a perfect fourth and a diminished fifth.

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats with arrows:

Step offset 0 1 2 3 4 5 6 7 8 9
Sharp symbol
Flat symbol

Kite's ups and downs notation

20edo can also be notated with Kite's ups and downs, spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

Step offset 0 1 2 3 4 5 6 7 8 9
Sharp symbol               
Flat symbol
                

Sagittal notation

This notation is a superset of the notations for edos 10 and 5.

Evo and Revo flavors

20-EDO__Sagittal.svg

Evo-SZ flavor

20-EDO_Evo-SZ_Sagittal.svg

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 20edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 20edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.528 0.9
15/14, 28/15 0.557 0.9
9/5, 10/9 2.404 4.0
11/7, 14/11 2.492 4.2
15/11, 22/15 3.049 5.1
15/13, 26/15 7.741 12.9
15/8, 16/15 8.269 13.8
13/7, 14/13 8.298 13.8
7/4, 8/7 8.826 14.7
13/11, 22/13 10.790 18.0
11/8, 16/11 11.318 18.9
11/9, 18/11 12.592 21.0
11/10, 20/11 14.996 25.0
9/7, 14/9 15.084 25.1
5/3, 6/5 15.641 26.1
7/5, 10/7 17.488 29.1
3/2, 4/3 18.045 30.1
13/12, 24/13 18.573 31.0
13/9, 18/13 23.382 39.0
9/8, 16/9 23.910 39.9
13/10, 20/13 25.786 43.0
5/4, 8/5 26.314 43.9
7/6, 12/7 26.871 44.8
11/6, 12/11 29.363 48.9
15-odd-limit intervals in 20edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.528 0.9
15/14, 28/15 0.557 0.9
11/7, 14/11 2.492 4.2
15/11, 22/15 3.049 5.1
15/13, 26/15 7.741 12.9
15/8, 16/15 8.269 13.8
13/7, 14/13 8.298 13.8
7/4, 8/7 8.826 14.7
13/11, 22/13 10.790 18.0
11/8, 16/11 11.318 18.9
11/10, 20/11 14.996 25.0
7/5, 10/7 17.488 29.1
3/2, 4/3 18.045 30.1
13/12, 24/13 18.573 31.0
13/10, 20/13 25.786 43.0
5/4, 8/5 26.314 43.9
7/6, 12/7 26.871 44.8
11/6, 12/11 29.363 48.9
9/8, 16/9 36.090 60.1
13/9, 18/13 36.618 61.0
5/3, 6/5 44.359 73.9
9/7, 14/9 44.916 74.9
11/9, 18/11 47.408 79.0
9/5, 10/9 62.404 104.0

Regular temperament properties

Uniform maps

13-limit uniform maps between 19.8 and 20.2
Min. size Max. size Wart notation Map
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]

Commas

20et tempers out the following commas. 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 Blackwood comma, Pythagorean limma
5 16875/16384 [-14 3 4⟩ 51.12 Laquadyo Negri comma
5 (16 digits) [-25 7 6⟩ 31.57 Lala-tribiyo Ampersand 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 Semaphoresma, slendro diesis
7 50/49 [1 0 2 -2⟩ 34.98 Biruyo Jubilisma, tritonic diesis
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 Marvel comma, septimal kleisma
7 16875/16807 [0 3 4 -5⟩ 6.99 Quinru-aquadyo Mirkwai comma
7 (24 digits) [11 -10 -10 10⟩ 5.57 Saquinbizogu Linus comma
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 comma, biome comma
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Bithogu Island comma, 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

Modes

20 tone equal modes:

3 1 3 1 3 1 3 1 3 1 Blackwood Major Decatonic (pentawood, according to the MOS naming scheme)
1 3 1 3 1 3 1 3 1 3 Blackwood Minor Decatonic (also pentawood)
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 (balzano, 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 balzano, 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 (tetrawood, 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)
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)
4 1 3 2 2 3 1 4 8-tone 4&5edo scale
  1. ↑ Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach

Music

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

Books

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

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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]

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