When chanting using the enharmonic scale, like this hymn in the third tone, I always thought that you chant in the diatonic scale until you come across an enharmonic ftora like #4 in the picture. At that point, only then would you chant using the enharmonic scale...Am I mistaken?
Thanks so much!
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Generally, to go to the enharmonic scale for the Third mode, you need Atzem phthora on both Ga and Zo.
Otherwise, it is the diatonic scale below Ga, and a diminished Zo above the Ga.
The enharmonic scale has the enharmonic diesis, by its definition, which some have said, but that is the Ancient Greek Tonic Diatonic semitone;
And actual Enharmonic diesis is 36/35, anything above that is not truly enharmonic.
What you actually need to have is a bigger Ga-Di interval, by having a lower Ga and Ni.
You can lower it so you have a septimal tone 8/7 between Ga and Di, and then Di-Ke will be the result of (4/3)/[(36/35)*(8/7)], which is 245/216.
This would be the enharmonic tetrachord in 72-EDO:
The only real time that you find a true enharmonic scale is Chroa Kliton (and sometimes, you do find the ditone 5/4 too, that is between Pa and Vu, when the Vu is raised to the Ga, while Chroa Kliton is put on Di); Echos Tritos, on the other hand, is for all intents and purposes, a diatonic mode.