Diaharmonic

The Diaharmonic collection is the name proposed by Mason Green for a 9-note scale with step pattern L L S S L L L S S. It is a diatonic scale where each semitone is divided equally in half.

It can also be thought of as a cross between two of the three ancient Greek genera (diatonic and enharmonic) since the diatonic and enharmonic tetrachords "fuse together" nicely. The two whole tones of the diatonic tetrachord nest inside the major third of the enharmonic one, while the two quarter tones of the enharmonic tetrachord fit into the semitone of the diatonic one. This gives us a diaharmonic pentachord (portmanteau of diatonic and enharmonic), which can then be used to construct a 9-note scale.

19edo works extremely well as a tuning for the diaharmonic scale (in which it has step sizes 3 3 1 1 3 3 3 1 1). However, there are other options which include 24edo and 43edo.

The 19edo diaharmonic scale is interesting because it can combine the melodic advantages of neomedieval systems (like 17edo and Pythagorean) with the harmonic advantages of a meantone system. We can do this by shrinking the diatonic semitone in a melodic line while expanding the preceding whole tone. Essentially in major keys, we have a diatonic scale with two extra notes, one which can function either as an "ascending/augmented third" or a "descending/diminished" fourth, and the other which can function either as an "ascending/augmented seventh" or "descending/diminished" octave.