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}}
}}
{{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 harmonics ===
=== Odd harmonics ===
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== 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
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| 2
| 2
| 120
| 120
| [[15/14]], 14/13
|[[15/14]], 14/13
| dup unison, mid 2nd
| dup unison, mid 2nd
| ^^1, ~2
| ^^1, ~2
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| 6
| 6
| 360
| 360
| [[16/13]], 5/4
|[[16/13]], 5/4
| mid 3rd
| mid 3rd
| ~3
| ~3
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| 7
| 7
| 420
| 420
| [[14/11]], [[51/40]]
|[[14/11]], [[51/40]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
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| 9
| 9
| 540
| 540
| [[15/11]], 11/8
|[[15/11]], 11/8
| up-fourth
| up-fourth
| ^4
| ^4
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| 11
| 11
| 660
| 660
| [[22/15]], 16/11
|[[22/15]], 16/11
| down-fifth
| down-fifth
| v5
| v5
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| 14
| 14
| 840
| 840
| [[13/8]], 8/5
|[[13/8]], 8/5
| mid 6th
| mid 6th
| ~6
| ~6
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| 18
| 18
| 1080
| 1080
| [[28/15]], 15/8, 13/7
|[[28/15]], 15/8, 13/7
| mid 7th
| mid 7th
| ~7
| ~7
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| 29.36294
| 29.36294
|}
|}
== Notation ==
=== Ups and downs notation ===
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. 20edo can be notated with [[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}}
[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}
=== 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 ====
<imagemap>
File:20-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 351 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 351 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:20-EDO_Sagittal.svg]]
</imagemap>
==== Evo-SZ flavor ====
<imagemap>
File:20-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 327 0 487 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 327                                                                                                                                                                                                                                                                                                                                                                                                                                                          106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:20-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>


== Chord names ==
== Chord names ==
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* 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 [[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.


== Modes ==
== Modes ==
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|-
|-
| 3 1 3 1 3 1 3 1 3 1
| 3 1 3 1 3 1 3 1 3 1
| Blackwood Major Decatonic [[5L_5s|(bi-equal decatonic]], according to the [[MOSNamingScheme|MOS naming scheme]])
| Blackwood Major Decatonic [[5L_5s|(pentawood]], according to the [[MOSNamingScheme|MOS naming scheme]])
|-
|-
| 1 3 1 3 1 3 1 3 1 3
| 1 3 1 3 1 3 1 3 1 3
| Blackwood Minor Decatonic (also bi-equal decatonic)
| Blackwood Minor Decatonic (also pentawood)
|-
|-
| 2 1 1 2 1 1 2 1 1 2 1 1
| 2 1 1 2 1 1 2 1 1 2 1 1
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|-
|-
|2 3 2 2 2 3 2 2 2
|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>
| 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
|2 2 2 2 1 2 2 2 2 2 1
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|-
|-
|2 2 2 3 2 2 2 3 2
|2 2 2 3 2 2 2 3 2
|Balzano Nine-tone inverse (also fair mavila, score9)
|Balzano Nine-tone inverse (also balzano, score9)
|-
|-
|1 2 2 2 2 2 1 2 2 2 2
|1 2 2 2 2 2 1 2 2 2 2
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|-
|-
| 2 3 2 3 2 3 2 3
| 2 3 2 3 2 3 2 3
| Octatonic ([[4L_4s|diminished]], according to the [[MOSNamingScheme|MOS naming scheme]])
| Octatonic ([[4L_4s|tetrawood]], according to the [[MOSNamingScheme|MOS naming scheme]])
|-
|-
|3 2 3 2 3 2 3 2
|3 2 3 2 3 2 3 2
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|-
|-
|3 3 3 3 3 3 2
|3 3 3 3 3 3 2
|Major quasi-equal Heptatonic ([[6L_1s|archaeotonic or Grumpy heptatonic]])
|Major quasi-equal Heptatonic ([[6L_1s|archaeotonic]])
|-
|-
|3 2 3 3 3 3 3
|3 2 3 3 3 3 3
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== 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)
|-
|-
|3
| 3
|[[256/243]]
| [[256/243]]
|{{monzo| 8 -5 }}
| {{monzo| 8 -5 }}
|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
| Diaschisma
|Diaschisma
|-
|-
|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
|[[50/49]]
| [[50/49]]
|{{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
|34.98
| 34.98
|Biruyo
| Biruyo
|Tritonic Diesis, Jubilisma
| Jubilisma, tritonic diesis
|-
|-
|7
| 7
|[[686/675]]
| [[686/675]]
|{{monzo| 1 -3 -2 3 }}
| {{monzo| 1 -3 -2 3 }}
|27.99
| 27.99
|Trizo-agugu
| Trizo-agugu
|Senga
| Senga
|-
|-
| 7
| 7
|[[64/63]]
| [[64/63]]
|{{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
|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
|[[Blackjackisma]]
| [[Blackjackisma]]
|-
|-
|7
| 7
|[[1029/1024]]
| [[1029/1024]]
|{{monzo| -10 1 0 3 }}
| {{monzo| -10 1 0 3 }}
|8.43
| 8.43
|Latrizo
| Latrizo
|Gamelisma
| Gamelisma
|-
|-
|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
|[[16875/16807]]
| [[16875/16807]]
|{{monzo| 0 3 4 -5 }}
| {{monzo| 0 3 4 -5 }}
| 6.99
| 6.99
|Quinru-aquadyo
| Quinru-aquadyo
|Mirkwai
| Mirkwai comma
|-
|-
|7
| 7
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
|{{monzo| 11 -10 -10 10 }}
| {{monzo| 11 -10 -10 10 }}
|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
|Breedsma
| Breedsma
|-
|-
|11
| 11
|[[121/120]]
| [[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
|Biyatisma
| Biyatisma
|-
|-
|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
|[[676/675]]
| [[676/675]]
|{{monzo| 2 -3 -2 0 0 2 }}
| {{monzo| 2 -3 -2 0 0 2 }}
|2.56
| 2.56
|Bithogu
| Bithogu
|Parizeksma
| Island comma, parizeksma
|}
|}
<references />
<references />
Line 590: Line 627:
== 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 599: Line 639:
* [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/bqYA9flwL_g ''20edo prelude''] (2025)
* [https://www.youtube.com/watch?v=BFUTeFs7Ycg ''20edo groove''] (2025)


; [[E8 Heterotic]]
; [[E8 Heterotic]]
Retrieved from "https://en.xen.wiki/w/20edo"