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'''6-EDO''' divides the 1200-[[cent]] octave into 6 equal parts, making its smallest interval exactly 200¢, or the sixth root of 2. It's known as the "whole tone" scale. As a subset of 12-edo, it can be notated on a five-line staff with standard notation.
{{Infobox ET}}
{{Wikipedia|Whole tone scale}}
{{ED intro}} It is also known as the '''whole tone scale'''.  


Related EDOs:
== Theory ==
* Subset: [[2edo]], [[3edo]]
6edo is identical to the 12edo whole-tone scale, however, it does have xenharmonic theoretical appeal. The 6-form is a simple basis for harmony in the [[2.5.7 subgroup]], somewhat like the [[Heptatonic|7-form]] for 2.3.5 and the [[Pentatonic|5-form]] for 2.3.7. This means that 6edo itself can be seen as a particularly crude tuning of temperaments like [[didacus]], in the same way as 7edo for [[meantone]] or 5edo for [[superpyth]]. The root chord in this harmonic system can be seen as:
* Supersets: [[12edo]], [[18edo]], [[24edo]]...
* Neighbours: [[5edo]], [[7edo]]


== Commas ==
* [0 2 5] = 4:5:7, with the harmonic seventh as a bounding interval
* [0 3 4] = 5:7:8, with the minor sixth as a bounding interval
* [0 1 3] = 7:8:10, with the large septimal tritone as a bounding interval


6 EDO tempers out the following commas. This assumes val {{val| 6 10 14 17 21 22 }}.  
Whichever way, this is very different from standard functional harmony.  


{| class="wikitable center-1 center-2 right-4 center-5"
If the prime 3 is added, it leads to absurd interpretations such as [[father]], as it is almost 100 cents sharp. However, in a composite subgroup, the 9th harmonic can be introduced, being directly approximated by 1 step of 6edo. In 2.9.5.7, the most salient fact about 6edo is that both [[64/63]] and [[81/80]] are tempered out, implying the restriction of both 2.3.7 [[superpyth]] and 2.3.5 [[meantone]] temperament (that is, [[Dominant (temperament)|dominant]] temperament) to 2.9.5.7.
 
6edo is the first edo to have lower [[Consistency limits of small EDOs|consistency]] than the one that precedes it, and the highest edo that has no single period mode of symmetry scales other than using the single step as a generator. This means it is relatively poor for its size at creating traditional tonal music, with 5edo and 7edo both having much better representations of the third harmonic, but has still seen more use than most edos other than 12, since it can be played on any 12-tone instrument.
 
=== Notation ===
As a subset of [[12edo]], 6edo can be notated on a five-line staff with standard notation.
 
=== Odd harmonics ===
{{Harmonics in equal|6|intervals=odd}}
 
=== Subsets and supersets ===
* Subsets: [[2edo]] and [[3edo]]
* Supersets: [[12edo]], [[18edo]], [[24edo]] …
 
== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 6edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals<ref group="note">{{sg|limit=subset of 12edo, itself treated as a 2.3.5.7.17.19 subgroup}} For example, for 6edo as a 2.5.7.9 subgroup temperament, ignore the "Other" column).</ref>
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
|
|
|
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)<br />[[8/7]] (−31.174)
| [[19/17]] (+7.442)<br />[[55/49]] (+0.020)<br />[[64/57]] (−0.532)<br />[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 2
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)<br />[[9/7]] (−35.084)
| [[34/27]] (+0.910)
| [[File:piano_1_3edo.mp3]]
|-
| 3
| 600
| [[Tritone]]
|
|
| [[7/5]] (+17.488)<br />[[10/7]] (−17.488)
| [[24/17]] (+3.000)<br />[[99/70]] (−0.088)<br />[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 4
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)<br />[[100/63]] (+0.108)
| [[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 5
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)<br />[[25/14]] (−3.802)
| [[30/17]] (+16.687)<br />[[57/32]] (+0.532)<br />[[98/55]] (−0.020)<br />[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 6
| 1200
| Octave
| [[2/1]] (just)
|
|
|
| [[File:piano_1_1edo.mp3]]
|}
 
== Notation ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 6edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[12edo]] [[subset notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 200
| '''Major second (M2)'''<br />Diminished third (d3)
| '''E'''<br />Fb
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
| 2
! [[Ratio]]
| 400
| Major third (M3)<br />Diminished fourth (d4)
| F#<br />Gb
|-
| 3
| 600
| Augmented fourth (A4)<br />Diminished fifth (d5)
| G#<br />Ab
|-
| 4
| 800
| Augmented fifth (A5)<br />Minor sixth (m6)
| A#<br />Bb
|-
| 5
| 1000
| Augmented sixth (A6)<br />'''Minor seventh (m7)'''
| B#<br />'''C'''
|-
| 6
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}
 
In 6edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
 
=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[12edo#Sagittal notation|12]], [[18edo#Sagittal notation|18]], [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]].
 
==== Evo flavor ====
<imagemap>
File:6-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[12-EDO#Sagittal_notation| 12-EDO notation]]
default [[File:6-EDO_Evo_Sagittal.svg]]
</imagemap>
 
Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
 
==== Revo flavor ====
<imagemap>
File:6-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 376 0 536 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 376 106 [[12-EDO#Sagittal_notation | 12-EDO notation]]
default [[File:6-EDO_Revo_Sagittal.svg]]
</imagemap>
 
== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 6edo
|-
! [[Degree]]
! [[Cents]]
! 12edo subset<br />standard [[solfege]]<br />(movable do)
! 12edo subset<br />[[uniform solfege]]<br />(2–3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 200
| Re
| Ra
|-
| 2
| 400
| Mi
| Ma (M3)<br />Fo (d4)
|-
| 3
| 600
| Fi (A4)<br />Se (d5)
| Pa (A4)<br />Sha (d5)
|-
| 4
| 800
| Si (A5)<br />Le (m6)
| Su (A5)<br />Fla (m6)
|-
| 5
| 1000
| Li (A6)<br />Te (m7)
| Lu (A6)<br />Tha (m7)
|-
| 6
| 1200
| Do
| Da
|}
 
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=6}}
 
=== Commas ===
6et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 6 10 14 17 21 22 }}.
 
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime<br />limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 27: Line 250:
| 3
| 3
| [[32/27]]
| [[32/27]]
| {{Monzo| 5 -3 }}
| {{monzo| 5 -3 }}
| 294.13
| 294.13
| Wa
| Wa
Line 34: Line 257:
| 5
| 5
| [[25/24]]
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| {{monzo| -3 -1 2 }}
| 70.67
| 70.67
| Yoyo
| Yoyo
| Classic chromatic semitone
| Dicot comma, classic chroma
|-
|-
| 5
| 5
| [[128/125]]
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| {{monzo| 7 0 -3 }}
| 41.06
| 41.06
| Trigu
| Trigu
| Diesis, augmented comma
| Augmented comma, diesis
|-
|-
| 5
| 5
| [[3125/3072]]
| [[3125/3072]]
| {{Monzo| -10 -1 5 }}
| {{monzo| -10 -1 5 }}
| 29.61
| 29.61
| Laquinyo
| Laquinyo
| Small diesis, magic comma
| Magic comma, small diesis
|-
|-
| 5
| 5
| [[393216/390625]]
| [[393216/390625|(12 digits)]]
| {{Monzo| 17 1 -8 }}
| {{monzo| 17 1 -8 }}
| 11.45
| 11.45
| Saquadbigu
| Saquadbigu
| Würschmidt comma
| [[Würschmidt comma]]
|-
|-
| 5
| 5
| [[Mutt comma|…]]
| [[476837158203125/474989023199232|(30 digits)]]
| {{Monzo| -44 -3 21 }}
| {{monzo| -44 -3 21 }}
| 6.72
| 6.72
| Trila-septriyo
| Trila-septriyo
| Mutt comma
| [[Mutt comma]]
|-
|-
| 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
| [[3136/3125]]
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquingu
| Hemimean
| Hemimean comma
|-
|-
| 7
| 7
| [[6144/6125]]
| [[6144/6125]]
| {{Monzo| 11 1 -3 -2 }}
| {{monzo| 11 1 -3 -2 }}
| 5.36
| 5.36
| arurutrigu
| Sarurutrigu
| Porwell
| Porwell comma
|-
|-
| 7
| 7
| [[2401/2400]]
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| {{monzo| -5 -1 -2 4 }}
| 0.72
| 0.72
| Bizozogu
| Bizozogu
Line 104: Line 327:
| 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 111: Line 334:
| 11
| 11
| [[176/175]]
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
Line 118: Line 341:
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|-
| 13
| [[13/12]]
| {{monzo| -2 -1 0 0 0 1 }}
| 138.57
| tho 2nd
| Tridecimal neutral second
|}
|}
== Instruments ==
Any instruments that can play the full gamut of 12edo (ie, not diatonic harmonicas, dulcimers or harps) can obviously also play 6edo as well, although it is significantly more ergonomic on some instruments than others.
If you want a more specialist design making 6edo music easy to play without having to worry about hitting out of key 12edo notes, a [[Lumatone mapping for 6edo]] is available, or you could remove all the unwanted bars on a xylophone or marimba.


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=knvdNFjKj-o ''6edo improvisation''] (2024)
; [[Chimeratio]]
* "Bowser breaks into Arnold Schoenberg's house and steals six of the twelve Tone Crystals (every other one), activating The 666666-Year-Curse Mechanism", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/6-chimeratio-bowser-breaks SoundCloud] | [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/6-bowser-breaks-into-arnold-schoenbergs-house-and-steals-six-of-the-twelve-tone-crystals-every-other-one-activating-the-666666-year-curse-mechanism Bandcamp]
; [[Milan Guštar]]
* [http://www.uvnitr.cz/flaoyg/forgotten_works/dvandva.html ''Dvandva''] (1987/2007)
; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** "Prelude in 6ET" – [https://aaronandrewhunt.bandcamp.com/track/prelude-in-6et Bandcamp] | [https://soundcloud.com/uz1kt3k/prelude-in-6et SoundCloud]{{dead link}}
** "Invention in 6ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-6et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-6et SoundCloud]{{dead link}}
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AleKBhXifzY "Exiting"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=6V97NrhUaps ''Heximal''] (2024)
; [[Chris Vaisvil]]
* ''The Good Boundless'' (2011) – [https://www.chrisvaisvil.com/the-good-boundless/ blog] | [https://web.archive.org/web/20230530111053/http://micro.soonlabel.com/6edo/the-good-boundless-03.mp3 play]


* [http://www.uvnitr.cz/flaoyg/forgotten_works/dvandva.html Dvandva] by Milan Guštar.
== Notes ==
* [http://micro.soonlabel.com/6edo/the-good-boundless-03.mp3 The Good Boundless] by Chris Vaisvil
<references group="note" />
* [https://soundcloud.com/uz1kt3k/prelude-in-6et Prelude In 6ET &#124; SoundCloud] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/invention-in-6et Invention In 6ET &#124; SoundCloud] by Aaron Andrew Hunt


[[Category:6-tone]]
[[Category:6-tone scales]]
[[Category:6edo]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Scale]]
[[Category:Subgroup]]
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