Here Socrates is insisting that the theory of music should be studied as a branch of mathematics, not by observation of the sounds and concords actually heard, about which musicians spend toil in vain. 'Yes,' says Glaucon, 'they talk of the close-fitting of intervals, and put their ears down to listen for the smallest possible interval, which is then to be the measure.' The smallest interval was of course the Enharmonic diesis or quarter of a tone, and this accordingly was the measure or unit into which the scale was divided. A group of notes separated by a diesis was called 'close' ([Greek: pyknon], or a [Greek: pyknôma]), and the filling up of the scale in that way was therefore a [Greek: katapyknôsis tou diagrammatos]--a filling up with 'close-set' notes, by the division of every tone into four equal parts. An example of a diagram of this kind has perhaps survived in a comparatively late writer, viz. Aristides Quintilianus, who gives a scale of two octaves, one divided into twenty-four dieses, the next into twelve semitones (_De Mus._ p. 15 Meib.). The characters used are not otherwise known, being quite different from the ordinary notation: but the nature of the diagram is plain from the accompanying words: [Greek: hautê estin hê para tois archaiois kata dieseis harmonia, heôs [=kd] dieseôn to proteron diagousa dia pasôn, to deuteron dia tôn hêmitoniôn auxêsasa]: 'this is the [Greek: harmonia] (division of the scale) according to dieses in use among the ancients, carried in the case of the first octave as far as twenty-four dieses, and dividing the second into semitones[1].' The phrase [Greek: hê kata dieseis harmonia], used for the division of an octave scale into quarter-tones, serves to explain the statement of Aristoxenus (in the third of the passages above quoted) that the writers who treated of octave Systems called them 'harmonies' ([Greek: ha ekaloun harmonias]). That statement has usually been taken to refer to the ancient Modes called [Greek: harmoniai] by Plato and Aristotle, and has been used accordingly as proof that the scales of these Modes were based upon the different species ([Greek: eidê]) of the Octave. But the form of the reference--'which _they called_ [Greek: harmoniai]'--implies some forgotten or at least unfamiliar use of the word by the older technical writers. It is very much more probable that the [Greek: harmoniai] in question are divisions of the octave scale, as shown in theoretical diagrams, and had no necessary connexion with the Modes. Apparently some at least of these diagrams were not musical scales, but tables of all the notes in the compass of an octave; and the Enharmonic diesis was used, not merely on account of the importance of that genus, but because it was the smallest interval, and therefore the natural unit of measurement[2].