Note 21 — § 22. The Scales as treated by Aristoxenus.
Reading edition
This point is one which Aristoxenus is fond of insisting upon: cp. p. 10, 16 ou pros tên katapyknôsin blepontas hôsper hoi harmonikoi: p. 38, 3 hoti de estin hê katapyknôsis ekmelês kai panta tropon achrêstos phaneron: p. 53, 3 kata tên tou melous physin zêtêteon to hexês kai ouch hôs hoi eis tên katapyknôsin blepontes eiôthasin apodidonai to hexês.
The statement that the ancient diagrams gave a series of twenty-eight successive dieses or quarter-tones has not been explained. The number of quarter-tones in an octave is only twenty-four. Possibly it is a mere error of transcription (κη for κδ). If not, we may perhaps connect it with the seven intervals of the ordinary octave scale, and the simple method by which the enharmonic intervals were expressed in the instrumental notation. It has been explained that raising a note a quarter of a tone was shown by turning it through a quarter of a circle. Thus, our c being denoted by , c* was , and c♯ was . Now the ancient diagrams, which divided every tone into four parts, must have had a character for c♯*, or the note three-quarters of a tone above c. Naturally this would be the remaining position of , namely . Again, we have seen that when the interval between two notes on the diatonic scale is only a semitone, the result of the notation is to produce a certain number of duplicates, so to speak. Thus: stands for b, and therefore for c: but c is a note of the original scale, and as such is written . It may be that the diagrams to which Aristoxenus refers made use of these duplicates: that is to say, they may have made use of all four positions of a character (such as ) whether the interval to be filled was a tone or a semitone. If so, the seven intervals would give twenty-eight characters (besides the upper octave-note), and apparently therefore twenty-eight dieses. Some traces of this use of characters in four positions have been noticed by Bellermann (Tonleitern, p. 65).]
