Many theoreticians have disputed the existence of the Enharmonic genre in Byzantine music, at least with the definition of Chrysanthos (Mega Theoretikon, 1832). See, for instance, Panayiotopoulos' theory book. Simon Karas (Theoretikon, 1982) gives his own definition on what is Enharmonic genre and distinguishes it from "Hard Diatonic" in pieces that before were said to be all "Enharmonic" (e.g. Third mode). For now:

"Enharmonic Genre. The Enharmonic genre in holy melodies with respect to psaltiki lacks its own klimaka (scale) and its own fthorai (modulation signs). There is no troparion or sticheron that is chanted wholly in Enharmonic melody. There are fthorai that it may use, however, it cannot apply them because they cancel each other, and each one of them acts on a note with its own scale ("ekasth energei epi tinos fthongou idias klimakos"). It also lacks special martyriai. In contrast, the fthorai of the Diatonic and Chromatic genre, each of them dominates a whole scale. As for the Enharmonic genre, there are only some theseis or isolated lines between the other two genres and even those are rare.

The music teachers appear undecided as to the definition of the Enharmonic genre. For instance Theodoros Fokaeus composed an Axion Estin in Plagal Fourth Triphonos, similar to "Idou o Nymfios" and calls it "Third Mode Enharmonios" [example score]. See also in Pandekte, page 455. Chourmouzios names his Doxology from low ZW flat, Enharmonic. Georgios Biolakis and other music teachers want the sticheraric Plagal First mode to be Enharmonic or Minore of European music and each of them label enharmonic whatever melody as they like.

[He then presents the enharmonic scales from different contexts]

Our enharmonic genre. Our enharmonic genre is defined by its own scale, as in the other two genres, and has its own fthoras, from the so-called Enharmonic genre, acting on the range of a full scale. We preferred the fthora [spathe] which represents a sharp (diesis) and a flat (yphesis) and ditonon (two tones) from two elason tones. This fthora dominates a scale without need for any other fthora when descending or ascending, like this:

PA 12 BOY 6 GA 18 DI 6 KE(spathe) 6 ZW 18 NH 6 PA

This scale consists of two adjacent tetrachors and one appended tone (proslamvanomenos).

First tetrachord BOY-KE
Second tetrachord KE-PA
Appended tone PA-BOY
The low PA note is appended so as to perfect the octave interval with respect to high PA.

[Examples and ratios of the above scale for implementing it on the monochord instrument]

Note: The exercise of the tonal intervals of the Enharmonic genre on the monochordon can make a musicophile able to execute flawlesly this most noble genre."

Comments

See also the discussion on Third mode: Diatonic or Enharmonic? [html], Third Mode: a case study [html] and also the discussion on Enharmonic passages in Plagal First mode [html].