About maximally even sets in 22edo

Lately,I have delved a little into the details of maximally even sets ( http://xenharmonic.wikispaces.com/maximal+evenness ). Namely, I was interested in 22edo and its 7-limit tetrads (major with intervals 7 6 5 4 and minor with intervals 6 7 4 5), and the question whether there would be a relation like in 12edo, where the major and minor triads are second-order maximally even (which means they are not maximally even directly in 12edo, but maximally even inside the diatonic scale, which in turn is maximally even in 12edo).

Some general first results confirm that maximally even sets tend to be musically relevant scale collections. E.g., Porcupine[7] as well as the symmetric decatonic scale of Pajara are maximally even in 22edo. (I listed more results on the xenwiki page linked above.) The major and the minor tetrad, however, are not maximally even, nor first- nor second-order (the latter tested with 7-, 8-, 9-, 10-, 11- and 12-element scales).

Now the idea arose whether these tetrads might be third-order maximally even - i.e. maximally even in some "albitonic" scale, which in turn is maximally even in some "chromatic scale", which in turn is maximally even in 22edo. I investigated this with 7 as number of elements for the "albitonic" scale, 10 and 12 as number of elements for the "chromatic" scale - and indeed there exist second-order maximally even scales with one or two of the 7-limit tetrads maximally even (making the tetrads third-order maximally even) in both cases. With other choices for the "albitonic" and/or "chromatic" scale, there may be even more.

Taken without any context, these scales are not very "regular" - they all have 3 or even 4 different sizes of interval steps. Notable is that each scale contains only one or two tetrads of the given form. This seems to suggest to use these scales not as tonalities of their own but let the tonality be defined be the containing "chromatic" scale and use the heptatonic scales for melodic lines in the context of a given chord.

Details:

Maximally even 7-12 set: the familiar diatonic scale
{0, 2, 4, 5, 7, 9, 11}

Maximally even 12-22 set:
{0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20}

There are, different up to translations, 6 second-order maximally even 7-12-22 scales. They divide into 3 groups of 2 elements that are inversions of each other. 2 contain a major tetrad, 2 contain a minor tetrad and 2 none at all.

{0, 4, 6, 9, 13, 17, 19} - minor tetrad
{0, 4, 6, 9, 13, 17, 18} - minor tetrad
{0, 4, 7, 9, 13, 17, 18} - major tetrad
{0, 4, 7, 9, 13, 16, 18} - major tetrad
{0, 4, 7, 9, 13, 16, 20} -
{0, 4, 6, 9, 13, 15, 19}

Maximally even 7-10 set:
{0, 1, 2, 4, 5, 6, 7, 8}

Maximally even 10-22 set: symmetric tetrachordal scale of Pajara
{0, 2, 4, 7, 9, 11, 13, 15, 18, 20}

There are, different up to translations, 5 second-order maximally even 7-10-22 scales. They divide into 3 groups as well, 2 with 2 elements that are inversions of each other and one with 1 element that is inversion-symmetric. Each scale contains two tetrads of major (7 6 5 4), minor (6 7 4 5), augmented (7 7 4 4) or diminished (6 5 6 5) type.

{0, 4, 7, 11, 13, 18, 20} - major and augmented tetrad
{0, 5, 7, 11, 13, 18, 20} - major and minor tetrad
{0, 2, 6, 9, 13, 15, 17} - minor and augmented tetrad
[0, 4, 6, 11, 13, 17, 19} - minor, major and diminished tetrad
{0, 4, 6, 11, 13, 17, 20} - minor, major and diminished tetrad

I think I like the 7-10-22 scales more than the 7-12-22 ones. They offer more choice in the tetrads, appear generally more "xen" to me , and the containing tonalities are already known and established decatonic scales - with, not to neglect, 2 notes less than the 12-22 tonalities. Anyway, even the decatonics contain many notes - nearly too many for my taste. I always had a little problem with Pajara for this reason - maybe the heptatonic second-order ME scales are a solution for this dilemma!

How this actually works in music is to be tested yet.

Hm ... if I can trust my intuition, then 2nd order ME scales are - unless they're MOS's -
related to rank-3 temperaments, which have two generators and a period.
If you want to have scales with many occurances of a 4:5:6:7 chord (or its inversion),
then it's probably a good idea to look for a low-complexity 7-limit rank-3 temperament
that is supported by the target-EDO, and then find 2nd order ME scales that are related.

For example, in 31EDO, gamelan with generators 8/7 and 5/4 (and an octave period) looks
promising. Three 8/7s upwards give 3/2, a 5/4 upwards gives 5/4 (who would have thought ^^),
and an 8/7 downwards gives 7/4. Now the only thing to do is to find a 2nd order ME scale
that consists of two or three chains of 8/7s separated by 5/4s.

I suppose for 22EDO, the 7-limit rank-3 temperament that tempers out 64:63 might be a good
idea. It has a 3/2 and a 5/4 generator (and again an octave period), and two 4/3s give 7/4,
so the complexity for a 4:5:6:7 chord is even smaller than for gamelan.
In this case, finding a scale that consists of two or three chains of 3/2s separated by 5/4s
should do the job.