The E Flat Pythagorean Dorian diminished 4 Scale. In a world of 12-tone equal tempered scales the major third and diminished fourth are just different spellings of the same interval so no "Mixolydian diminished 4" scale would exist. However, in Pythagorean space the lack of enharmonic equivalence opens up the possibility of scales such as this one.
This is only the second scale encountered so far that contains two adjacent scale tones that physically descend (from the 81/64 major third to the 8192/6561 diminished fourth) when laid out in the ascending form as pictured above. And like the Pythagorean Lydian diminished 5 scale the interval between those two non-enharmonic tones is the Pythagorean Comma - 531441/524288 - an interval 23.46 cents wide. However, unlike the Pythagorean Lydian diminished 5 scale this comma is not located between two intervals that are inversions of one another. One of the interesting things about the comma found in the Pythagorean Mixolydian diminished 4 scale in particular is the striking sonic similarity between the Pythagorean diminished 4 (384.56 cents) and the 5-limit "classical" 5/4 major third (386.31 cents). In all likelihood the human auditory system would hear the Pythagorean diminished 4 as the more palatable version of the major third (much smoother than the 81/64 Pythagorean major third at 407.82 cents). Having both intervals in the same scale opens up an opportunity to explore two different shades of this 'consonance.'
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