Xenharmonic Wiki

6edo: Difference between revisions

← Older edit
VisualWikitext
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m →Music: Markup
Yourmusic Productions (talk | contribs)
autopatrolled
891 edits
→Instruments: expansion.
 
(21 intermediate revisions by 10 users not shown)
Line 7: Line 7:
{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|Whole tone scale}}
{{Wikipedia|Whole tone scale}}
'''6 equal divisions of the octave''' ('''6edo''') is the [[tuning system]] derived by dividing the [[octave]] into 6 equal steps of 200 [[cent]]s each, or the sixth root of 2. It is also known as the '''whole tone scale'''.  
{{ED intro}} It is also known as the '''whole tone scale'''.  


== Theory ==
== Theory ==
{{Harmonics in equal|6|intervals=odd}}
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:
 
* [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
 
Whichever way, this is very different from standard functional harmony.
 
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.


As a subset of [[12edo]], 6edo can be notated on a five-line staff with standard notation. It is the first [[edo]] that is not a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak]], has 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.
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.


While 6edo does not well approximate the 3rd harmonic, it does contain a good approximation of the 9th harmonic. Therefore, 6edo can be treated as a 2.5.7.9 subgroup temperament.
=== Notation ===
As a subset of [[12edo]], 6edo can be notated on a five-line staff with standard notation.  


Related edos:
=== Odd harmonics ===
* Subsets: [[2edo]], [[3edo]]
{{Harmonics in equal|6|intervals=odd}}
 
=== Subsets and supersets ===
* Subsets: [[2edo]] and [[3edo]]
* Supersets: [[12edo]], [[18edo]], [[24edo]] …
* Supersets: [[12edo]], [[18edo]], [[24edo]] …
* Neighbours: [[5edo]], [[7edo]]


== Intervals ==
== Intervals ==
{| class="wikitable right-1 right-2"
{| class="wikitable center-all"
! Steps
|+ style="font-size: 105%;" | Intervals of 6edo
! Cents
|-
! colspan="3" | Interval
! rowspan="2" | [[Degree]]
! Approximate JI Ratios*
! 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
|-
| 2
| 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
| 0
| 0
| unison
| Do
| P1
| Da
| D
| [[1/1]]
|-
|-
| 1
| 1
| 200
| 200
| major 2nd
| Re
| M2
| Ra
| E
| [[8/7]], [[9/8]], [[10/9]]
|-
|-
| 2
| 2
| 400
| 400
| major 3rd
| Mi
| M3
| Ma (M3)<br />Fo (d4)
| F#
| [[5/4]], [[9/7]]
|-
|-
| 3
| 3
| 600
| 600
| aug 4th, dim 5th
| Fi (A4)<br />Se (d5)
| A4, d5
| Pa (A4)<br />Sha (d5)
| G#, Ab
| [[7/5]], [[10/7]]
|-
|-
| 4
| 4
| 800
| 800
| minor 6th
| Si (A5)<br />Le (m6)
| m6
| Su (A5)<br />Fla (m6)
| Bb
| [[8/5]], [[14/9]]
|-
|-
| 5
| 5
| 1000
| 1000
| minor 7th
| Li (A6)<br />Te (m7)
| m7
| Lu (A6)<br />Tha (m7)
| C
| [[7/4]], [[9/5]], [[16/9]]
|-
|-
| 6
| 6
| 1200
| 1200
| perfect 8ve
| Do
| P8
| Da
| D
| [[2/1]]
|}
|}
<nowiki>*</nowiki> based on treating 6edo as a 2.5.7.9 subgroup temperament; other approaches are possible.


== Commas ==
== Regular temperament properties ==
6edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 6 10 14 17 21 22 }}.  
=== 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"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[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 group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 103: Line 260:
| 70.67
| 70.67
| Yoyo
| Yoyo
| Classic chromatic semitone
| Dicot comma, classic chroma
|-
|-
| 5
| 5
Line 110: Line 267:
| 41.06
| 41.06
| Trigu
| Trigu
| Diesis, augmented comma
| Augmented comma, diesis
|-
|-
| 5
| 5
Line 117: Line 274:
| 29.61
| 29.61
| Laquinyo
| Laquinyo
| Small diesis, magic comma
| Magic comma, small diesis
|-
|-
| 5
| 5
Line 138: Line 295:
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 145: Line 302:
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
Line 152: Line 309:
| 6.08
| 6.08
| Zozoquingu
| Zozoquingu
| Hemimean
| Hemimean comma
|-
|-
| 7
| 7
Line 159: Line 316:
| 5.36
| 5.36
| Sarurutrigu
| Sarurutrigu
| Porwell
| Porwell comma
|-
|-
| 7
| 7
Line 196: Line 353:
| Tridecimal neutral second
| Tridecimal neutral second
|}
|}
<references/>
 
== 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 ==
; Chimeratio
; [[Bryan Deister]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/6-chimeratio-bowser-breaks "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://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])
* [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
; [[Milan Guštar]]
* [http://www.uvnitr.cz/flaoyg/forgotten_works/dvandva.html ''Dvandva''] (1987/2007)
* [http://www.uvnitr.cz/flaoyg/forgotten_works/dvandva.html ''Dvandva''] (1987/2007)


; [[Aaron Andrew Hunt]]
; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-6et "Prelude in 6ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022) ([https://soundcloud.com/uz1kt3k/prelude-in-6et SoundCloud])
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-6et "Invention in 6ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-6et SoundCloud])
** "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
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AleKBhXifzY "Exiting"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
* [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]]
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/6edo/the-good-boundless-03.mp3 ''The Good Boundless''] (2011) ([http://chrisvaisvil.com/the-good-boundless/ details])
* ''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]
 
== Notes ==
<references group="note" />


[[Category:6edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|#]] <!-- 1-digit number -->
[[Category:6-tone scales]]
[[Category:6-tone scales]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Subgroup]]
Retrieved from "https://en.xen.wiki/w/6edo"