Tetrachord System: The early form of the system now known as Tonic Sol-fa. (Stainer, John; Barrett, W.A.; A Dictionary of Musical Terms; Novello, Ewer and Co., London, pre-1900)
Interesting comments about the terms 'diatonic' 'chromatic' and 'enharmonic:
First of all, the basis for Greek scale construction was the tetrachord (= '4 strings'). Their theory (at least Aristoxenus and after) was based on the lyre (a sort of small harp), and not on any wind instruments.
So the tetrachord designates 4 notes, of which two are fixed and two are moveable.
The fixed notes are those bounding the tetrachord, which are always assumed to be the interval of the Pythagorean 'perfect 4th', with the ratio 3:4. It's the position of the two moveable notes that was argued about so much, and which makes this stuff so interesting to tuning theorists.
(BTW, John Chalmers's book Divisions of the Tetrachord is entirely about specifically this.)
Those various divisions are what determine the different genera (plural of genus - the actual Greek word is genos, but commentators writing in English generally use the Latin form). There were 3 basic genera: Diatonic (= 'thru tones'), Chromatic (= 'colored' or 'thru the shades'), and Enharmonic (= 'properly attuned').
Apparently the Enharmonic derived from the ancient scales which were called harmonia, thus its name. That was the one with 'quarter-tones'. The chromatic had a pattern that more-or-less involved a succession of 2 semitones, and the Diatonic is the one we're most familiar with, using mainly 'whole tones' with a few semitones. from http://www.ixpres.com/interval/monzo/aristoxenus/tutorial.htm