20edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Dave Keenan (talk | contribs)
autopatrolled, emailconfirmed
1,671 edits
→Notation: added Sagittal notation
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Text replacement - "Ups and Downs Notation" to "Ups and downs notation"
 
(18 intermediate revisions by 10 users not shown)
Line 6: Line 6:
}}
}}
{{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.  
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.  


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.  
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.


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).
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]].
Thanks to its sevenths, 20edo is an ideal tuning for its size for [[metallic harmony]].
Line 24: Line 27:


== Intervals ==
== Intervals ==
=== Selected 19-limit just intervals ===
{| class="wikitable center-all"
|+[[Direct approximation]] (even if [[inconsistent]])
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
|-
| [[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
|}
== 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.
{| 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 143: 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 183: Line 99:
| 6
| 6
| 360
| 360
| [[16/13]], 5/4
|[[16/13]], 5/4
| mid 3rd
| mid 3rd
| ~3
| ~3
Line 193: Line 109:
| 7
| 7
| 420
| 420
| [[14/11]], [[51/40]]
|[[14/11]], [[51/40]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 213: Line 129:
| 9
| 9
| 540
| 540
| [[15/11]], 11/8
|[[15/11]], 11/8
| up-fourth
| up-fourth
| ^4
| ^4
Line 233: Line 149:
| 11
| 11
| 660
| 660
| [[22/15]], 16/11
|[[22/15]], 16/11
| down-fifth
| down-fifth
| v5
| v5
Line 263: Line 179:
| 14
| 14
| 840
| 840
| [[13/8]], 8/5
|[[13/8]], 8/5
| mid 6th
| mid 6th
| ~6
| ~6
Line 303: 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 332: Line 248:
|}
|}


===Sagittal notation===
=== Selected 19-limit just intervals ===
This notation is a subset of the notations for EDOs [[36edo#Sagittal notation|36]] and [[72edo#Sagittal notation|72]] and a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
{| class="wikitable center-all"
====Evo flavor====
|+[[Direct approximation]] (even if [[inconsistent]])
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
|-
| [[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
|}
 
== 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>
<imagemap>
File:18-EDO_Evo_Sagittal.svg
File:20-EDO_Sagittal.svg
desc none
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 80 0 300 50 [[Sagittal_notation]]
rect 463 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 351 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 463 106 [[36-EDO#Sagittal_notation | 36-EDO notation]]
rect 20 80 351 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:18-EDO_Evo_Sagittal.svg]]
default [[File:20-EDO_Sagittal.svg]]
</imagemap>
</imagemap>


====Revo flavor====
==== Evo-SZ flavor ====
 
<imagemap>
<imagemap>
File:18-EDO_Revo_Sagittal.svg
File:20-EDO_Evo-SZ_Sagittal.svg
desc none
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 80 0 300 50 [[Sagittal_notation]]
rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 327 0 487 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 447 106 [[36-EDO#Sagittal_notation | 36-EDO notation]]
rect 20 80 327                                                                                                                                                                                                                                                                                                                                                                                                                                                          106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:18-EDO_Revo_Sagittal.svg]]
default [[File:20-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
</imagemap>


Line 371: Line 377:
* 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 ==
Line 379: Line 385:
|-
|-
| 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
Line 394: Line 400:
|-
|-
|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
Line 400: Line 406:
|-
|-
|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
Line 406: Line 412:
|-
|-
| 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
Line 436: Line 442:
|-
|-
|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
Line 469: Line 475:
== 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 622: 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 631: 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]]

Latest revision as of 01:07, 20 August 2025

← 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

Selected 19-limit just intervals

Direct approximation (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

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.

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

Alternative symbols for ups and downs notation uses sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:

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

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

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

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

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

See also: Category:20edo tracks
Ambient Esoterica
Beheld
Pyotr Chernobrivets
Bryan Deister
E8 Heterotic
Francium
Andrew Heathwaite
Aaron Andrew Hunt
Mandrake
Claudi Meneghin
Herman Miller
NullPointerException Music
Sevish
Tancla
Chris Vaisvil
Chris Vaisvil and Bethan Mathis
Stephen Weigel
Frostburn
Retrieved from "https://en.xen.wiki/index.php?title=20edo&oldid=207611"