22edo: Difference between revisions

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In the 7-limit 22edo tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (the [[jubilee comma]]), and [[64/63]], (the [[septimal comma]]), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22-et.

In the 11-limit, 22edo tempers out [[~~Quartismic temperaments~~|117440512/117406179]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small EDOs such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property- although it should be noted that the related [[159edo]] ''does''.

In the 11-limit, 22edo tempers out [[Quartisma|117440512/117406179]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small EDOs such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property- although it should be noted that the related [[159edo]] ''does''.

As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo|11edo]], in the same way that 12edo can play 6edo (the whole tone scale). 11-equal is interesting for sounding melodically very similar to 12-equal (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal]], 11 can be notated as every other note of 22.

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 In the 7-limit 22edo tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (the [[jubilee comma]]), and [[64/63]], (the [[septimal comma]]), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22-et.
-In the 11-limit, 22edo tempers out [[Quartismic temperaments|117440512/117406179]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third.  This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small EDOs such as [[17edo]], [[26edo]] and [[34edo]].  In fact, not even the famous [[53edo]] has this property- although it should be noted that the related [[159edo]] ''does''.
+In the 11-limit, 22edo tempers out [[Quartisma|117440512/117406179]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third.  This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small EDOs such as [[17edo]], [[26edo]] and [[34edo]].  In fact, not even the famous [[53edo]] has this property- although it should be noted that the related [[159edo]] ''does''.
 As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo|11edo]], in the same way that 12edo can play 6edo (the whole tone scale). 11-equal is interesting for sounding melodically very similar to 12-equal (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal]], 11 can be notated as every other note of 22.