6edo: Difference between revisions

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== Theory ==

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 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:

~~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~~.

* [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.

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 ===

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 == Theory ==
-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 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:
-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.
+* [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.
+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.
+=== Notation ===
+As a subset of [[12edo]], 6edo can be notated on a five-line staff with standard notation.
 === Odd harmonics ===