6edo: Difference between revisions

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{{ED intro}} It is also known as the '''whole tone scale'''.

6edo is ~~also known as~~ the ~~'''~~whole tone scale~~'''~~.

== Theory ==

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.

~~== Theory ==~~

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 [[zeta peak edo|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 ~~2.9.5.7~~-~~[[subgroup]] temperament~~.

=== Notation ===

As a subset of [[12edo]], 6edo can be notated on a five-line staff with standard notation.

=== Odd harmonics ===

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== Intervals ==

{| class="wikitable center-all"

|+ Intervals of 6edo

|+ style="font-size: 105%;" | Intervals of 6edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! rowspan="2" | [[Interval region]]

! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])

! 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

|-

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| 200

| Major second

| [[9/8]] (-3.910)

| [[9/8]] (−3.910)

| [[10/9]] (+17.596)

| [[28/25]] (+3.802) [[8/7]] (~~-31~~.174)

| [[28/25]] (+3.802) [[8/7]] (−31.174)

| [[19/17]] (+7.442) [[55/49]] (+0.020) [[64/57]] (-0.532) [[17/15]] (~~-16~~.687)

| [[19/17]] (+7.442) [[55/49]] (+0.020) [[64/57]] (−0.532) [[17/15]] (−16.687)

| [[File:piano_1_6edo.mp3]]

|-

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| 400

| Major third

| [[81/64]] (-7.820)

| [[81/64]] (−7.820)

| [[5/4]] (+13.686)

| [[63/50]] (-0.108) [[9/7]] (~~-35~~.084)

| [[63/50]] (−0.108) [[9/7]] (−35.084)

| [[34/27]] (+0.910)

| [[File:piano_1_3edo.mp3]]

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|

| [[7/5]] (+17.488) [[10/7]] (~~-17~~.488)

| [[7/5]] (+17.488) [[10/7]] (−17.488)

| [[24/17]] (+3.000) [[99/70]] (-0.088) [[17/12]] (-3.000)

| [[24/17]] (+3.000) [[99/70]] (−0.088) [[17/12]] (−3.000)

| [[File:piano_1_2edo.mp3]]

|-

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| Minor sixth

| [[128/81]] (+7.820)

| [[8/5]] (~~-13~~.686)

| [[8/5]] (−13.686)

| [[14/9]] (+35.084) [[100/63]] (+0.108)

| [[14/9]] (+35.084) [[100/63]] (+0.108)

| [[27/17]] (-0.910)

| [[27/17]] (−0.910)

| [[File:piano_2_3edo.mp3]]

|-

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| Minor seventh

| [[16/9]] (+3.910)

| [[9/5]] (~~-17~~.596)

| [[9/5]] (−17.596)

| [[7/4]] (+31.174) [[25/14]] (-3.802)

| [[7/4]] (+31.174) [[25/14]] (−3.802)

| [[30/17]] (+16.687) [[57/32]] (+0.532) [[98/55]] (-0.020) [[34/19]] (-7.442)

| [[30/17]] (+16.687) [[57/32]] (+0.532) [[98/55]] (−0.020) [[34/19]] (−7.442)

| [[File:piano_5_6edo.mp3]]

|-

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| [[File:piano_1_1edo.mp3]]

|}

<nowiki>*</nowiki> based on treating 6edo as a subset of 12edo, itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible (e.g. for 6edo as a 2.5.7.9 subgroup temperament, ignore the "Other" column).

== Notation ==

{| class="wikitable center-all"

|+ Notation of 6edo

|+ style="font-size: 105%;" | Notation of 6edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

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| 1

| 200

| '''Major second (M2)''' Diminished third (d3)

| '''Major second (M2)''' Diminished third (d3)

| '''E''' Fb

| '''E''' Fb

|-

| 2

| 400

| Major third (M3) Diminished fourth (d4)

| Major third (M3) Diminished fourth (d4)

| F# Gb

| F# Gb

|-

| 3

| 600

| Augmented fourth (A4) Diminished fifth (d5)

| Augmented fourth (A4) Diminished fifth (d5)

| G# Ab

| G# Ab

|-

| 4

| 800

| Augmented fifth (A5) Minor sixth (m6)

| Augmented fifth (A5) Minor sixth (m6)

| A# Bb

| A# Bb

|-

| 5

| 1000

| Augmented sixth (A6) '''Minor seventh (m7)'''

| Augmented sixth (A6) '''Minor seventh (m7)'''

| B# '''C'''

| B# '''C'''

|-

| 6

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* 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===

=== Sagittal notation ===

~~====Evo flavor====~~

==== Evo flavor ====

File:6-EDO_Evo_Sagittal.svg

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Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

====Revo flavor====

==== Revo flavor ====

File:6-EDO_Revo_Sagittal.svg

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== Solfege ==

{| class="wikitable center-all"

|+ Solfege of 6edo

|+ style="font-size: 105%;" | Solfege of 6edo

|-

! [[Degree]]

! [[Cents]]

! 12edo subset standard [[solfege]] (movable do)

! 12edo subset standard [[solfege]] (movable do)

! 12edo subset [[uniform solfege]] (~~2-3~~ vowels)

! 12edo subset [[uniform solfege]] (2–3 vowels)

|-

| 0

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| 400

| Mi

| Ma (M3) Fo (d4)

| Ma (M3) Fo (d4)

|-

| 3

| 600

| Fi (A4) Se (d5)

| Fi (A4) Se (d5)

| Pa (A4) Sha (d5)

| Pa (A4) Sha (d5)

|-

| 4

| 800

| Si (A5) Le (m6)

| Si (A5) Le (m6)

| Su (A5) Fla (m6)

| Su (A5) Fla (m6)

|-

| 5

| 1000

| Li (A6) Te (m7)

| Li (A6) Te (m7)

| Lu (A6) Tha (m7)

| Lu (A6) Tha (m7)

|-

| 6

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== Regular temperament properties ==

=== Uniform maps ===

=== Commas ===

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{| class="commatable wikitable center-1 center-2 right-4 center-5"

|-

! [[Harmonic limit|Prime limit]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref group="note">~~Ratios longer than 10 digits are presented by placeholders with informative hints~~</ref>

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

! [[Cent]]s

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| Tridecimal neutral second

|}

~~<references group~~=~~"note"/>~~

== 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 ==

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; [[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]

== Notes ==

[[Category:6-tone scales]]

[[Category:Listen]]

~~[[Category:Macrotonal]]~~

@@ Line 7: / Line 7: @@
 {{Infobox ET}}
 {{Wikipedia|Whole tone scale}}
-{{EDO intro|6}}
+{{ED intro}} It is also known as the '''whole tone scale'''.
-edo is also known as the '''whole tone scale'''.
+== Theory ==
+edo 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.
-== Theory ==
+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 [[zeta peak edo|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.
+edo 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 2.9.5.7-[[subgroup]] temperament.
+=== Notation ===
+As a subset of [[12edo]], 6edo can be notated on a five-line staff with standard notation.
 === Odd harmonics ===
@@ Line 26: / Line 34: @@
 == Intervals ==
 {| class="wikitable center-all"
-|+ Intervals of 6edo
+|+ style="font-size: 105%;" | Intervals of 6edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
 ! rowspan="2" | [[Interval region]]
-! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
+! 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
 |-
@@ Line 50: / Line 59: @@
 | 200
 | Major second
-| [[9/8]] (-3.910)
+| [[9/8]] (−3.910)
 | [[10/9]] (+17.596)
-| [[28/25]] (+3.802)<br>[[8/7]] (-31.174)
+| [[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)
+| [[19/17]] (+7.442)<br />[[55/49]] (+0.020)<br />[[64/57]] (−0.532)<br />[[17/15]] (−16.687)
 | [[File:piano_1_6edo.mp3]]
 |-
@@ Line 59: / Line 68: @@
 | 400
 | Major third
-| [[81/64]] (-7.820)
+| [[81/64]] (−7.820)
 | [[5/4]] (+13.686)
-| [[63/50]] (-0.108)<br>[[9/7]] (-35.084)
+| [[63/50]] (−0.108)<br />[[9/7]] (−35.084)
 | [[34/27]] (+0.910)
 | [[File:piano_1_3edo.mp3]]
@@ Line 70: / Line 79: @@
 |
 |
-| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)
+| [[7/5]] (+17.488)<br />[[10/7]] (−17.488)
-| [[24/17]] (+3.000)<br>[[99/70]] (-0.088)<br>[[17/12]] (-3.000)
+| [[24/17]] (+3.000)<br />[[99/70]] (−0.088)<br />[[17/12]] (−3.000)
 | [[File:piano_1_2edo.mp3]]
 |-
@@ Line 78: / Line 87: @@
 | Minor sixth
 | [[128/81]] (+7.820)
-| [[8/5]] (-13.686)
+| [[8/5]] (−13.686)
-| [[14/9]] (+35.084)<br>[[100/63]] (+0.108)
+| [[14/9]] (+35.084)<br />[[100/63]] (+0.108)
-| [[27/17]] (-0.910)
+| [[27/17]] (−0.910)
 | [[File:piano_2_3edo.mp3]]
 |-
@@ Line 87: / Line 96: @@
 | Minor seventh
 | [[16/9]] (+3.910)
-| [[9/5]] (-17.596)
+| [[9/5]] (−17.596)
-| [[7/4]] (+31.174)<br>[[25/14]] (-3.802)
+| [[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)
+| [[30/17]] (+16.687)<br />[[57/32]] (+0.532)<br />[[98/55]] (−0.020)<br />[[34/19]] (−7.442)
 | [[File:piano_5_6edo.mp3]]
 |-
@@ Line 101: / Line 110: @@
 | [[File:piano_1_1edo.mp3]]
 |}
-<nowiki>*</nowiki> based on treating 6edo as a subset of 12edo, itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible (e.g. for 6edo as a 2.5.7.9 subgroup temperament, ignore the "Other" column).
 == Notation ==
 {| class="wikitable center-all"
-|+ Notation of 6edo
+|+ style="font-size: 105%;" | Notation of 6edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
@@ Line 121: / Line 129: @@
 | 1
 | 200
-| '''Major second (M2)'''<br>Diminished third (d3)
+| '''Major second (M2)'''<br />Diminished third (d3)
-| '''E'''<br>Fb
+| '''E'''<br />Fb
 |-
 | 2
 | 400
-| Major third (M3)<br>Diminished fourth (d4)
+| Major third (M3)<br />Diminished fourth (d4)
-| F#<br>Gb
+| F#<br />Gb
 |-
 | 3
 | 600
-| Augmented fourth (A4)<br>Diminished fifth (d5)
+| Augmented fourth (A4)<br />Diminished fifth (d5)
-| G#<br>Ab
+| G#<br />Ab
 |-
 | 4
 | 800
-| Augmented fifth (A5)<br>Minor sixth (m6)
+| Augmented fifth (A5)<br />Minor sixth (m6)
-| A#<br>Bb
+| A#<br />Bb
 |-
 | 5
 | 1000
-| Augmented sixth (A6)<br>'''Minor seventh (m7)'''
+| Augmented sixth (A6)<br />'''Minor seventh (m7)'''
-| B#<br>'''C'''
+| B#<br />'''C'''
 |-
 | 6
@@ Line 154: / Line 162: @@
 * 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===
+=== 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====
+==== Evo flavor ====
 <imagemap>
 File:6-EDO_Evo_Sagittal.svg
@@ Line 169: / Line 177: @@
 Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:6-EDO_Revo_Sagittal.svg
@@ Line 182: / Line 189: @@
 == Solfege ==
 {| class="wikitable center-all"
-|+ Solfege of 6edo
+|+ style="font-size: 105%;" | Solfege of 6edo
+|-
 ! [[Degree]]
 ! [[Cents]]
-! 12edo subset<br>standard [[solfege]]<br>(movable do)
+! 12edo subset<br />standard [[solfege]]<br />(movable do)
-! 12edo subset<br>[[uniform solfege]]<br>(2-3 vowels)
+! 12edo subset<br />[[uniform solfege]]<br />(2–3 vowels)
 |-
 | 0
@@ Line 201: / Line 209: @@
 | 400
 | Mi
-| Ma (M3)<br>Fo (d4)
+| Ma (M3)<br />Fo (d4)
 |-
 | 3
 | 600
-| Fi (A4)<br>Se (d5)
+| Fi (A4)<br />Se (d5)
-| Pa (A4)<br>Sha (d5)
+| Pa (A4)<br />Sha (d5)
 |-
 | 4
 | 800
-| Si (A5)<br>Le (m6)
+| Si (A5)<br />Le (m6)
-| Su (A5)<br>Fla (m6)
+| Su (A5)<br />Fla (m6)
 |-
 | 5
 | 1000
-| Li (A6)<br>Te (m7)
+| Li (A6)<br />Te (m7)
-| Lu (A6)<br>Tha (m7)
+| Lu (A6)<br />Tha (m7)
 |-
 | 6
@@ Line 226: / Line 234: @@
 == Regular temperament properties ==
 === Uniform maps ===
-{{Uniform map|13|5.5|6.5}}
+{{Uniform map|edo=6}}
 === Commas ===
@@ Line 233: / Line 241: @@
 {| class="commatable wikitable center-1 center-2 right-4 center-5"
 |-
-! [[Harmonic limit|Prime<br>limit]]
+! [[Harmonic limit|Prime<br />limit]]
-! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
+! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]
 ! [[Cent]]s
@@ Line 345: / Line 353: @@
 | Tridecimal neutral second
 |}
-<references group="note"/>
+== 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 ==
@@ Line 370: / Line 382: @@
 ; [[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]
+== Notes ==
+<references group="note" />
 [[Category:6-tone scales]]
 [[Category:Listen]]
-[[Category:Macrotonal]]