Game of avoiding the tritone: From modes via first twelvetone to major and minor

There are two problems in just intonation, as in "octaves and other pure intervals". One I leave aside, it being that to get all major and all but one minor thirds pure, one fifth in the scale needs to be reduced, thus approaching tritone. Which is exactly why pythagorean tuning was preferred, even though the thirds were impure (64:81 rather than 64:80=4:5) The other is so much less necessary, but was felt more keenly: how to avoid the tritone.

The tritone being between final and fourth in lydian, to get a pure (minor) fourth down to final, or up from it, you lower the B to a B flat. The alternative would have been to avoid approaching them: which is so much easier in modes where the tones concerned involve neither final nor dominant. The first accidental was then a B flat. And a lydian (with its original final, not transposed) with a flattened B, is the scale of F major. The major mode transposed from modified "lydian" to pure naturals, called ionic by Glareanus, is of course between lydian and mixolydian:

lydian:
F-CGDAEB
major/ionic:
F-C-GDAEB
mixolydian:
FC-G-DAEB

He also instituted the eolic mode, between doric and phrygian:

doric:
FCG-D-AEB
eolic:
FCGD-A-EB
phrygian:
FCGDA-E-B

Eolic looks like minor, right? Well, when melodic minor descends, it is simply eolic. Melodic ascending, as well as harmonic minors are not pure eolic. We are not quite there yet.