Mea Culpa (Dorian/Phrygian Mode)

Paul E is correct. I added 22 as an afterthought just as I was posting and
it was a mistake, though my observation that a recognizable
Pythagorean-type Greek Phrygian/Ecclesiastical Dorian exists in 22 and 17
is correct.

It is difficult to come up with a satisfactory tuning for the Greek
Phrygian/Eccl. Dorian mode in 22. The best I have found is either the
all-pythagorean 4 1 4 4 4 1 4 (as Paul mentioned) or the mixed 4 2 3 4 4 2
3 which has an inharmonic pythagorean triad on its subdominant and the
usual 22-tet minor triads on the tonic and dominant. The trichordal variety
with a major triad on the subdominant has an equally-divided upper
tetrachord: 4 2 3 4 3 3 3. The major (Lydian) mode itself is 4 3 2 4 3 4 2.
Twenty-two, at least has a well-defined Pythagorean diatonic 4 4 1 4 4 4 1.

Similar problems arise in 15 tet as well where the "nearly just" major mode
is 3 2 1 3 2 3 1 and the P/D mode is 3 1 2 3 2 2 2 or 3 1 2 3 3 1 2 with
the chord on the subdominant as 6 3 rather than 4 5 or 4 5. One might also
try 3 1 2 3 2 1 3, but then the chord on the dominant will be 3 6 (0 3 9 on
degree 9).

--John

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