The want of harmony is to be connected not only with the defective tonality which was probably characteristic of Greek music,--we have seen (p. 42) that there is some evidence of tonality,--but still more with the non-harmonic quality of many of the intervals of which their scales were composed. We have repeatedly dwelt upon the variety and strangeness (to our apprehension) of these intervals. Modern writers are usually disposed to underrate their importance, or even to explain them away. The Enharmonic, they point out, was produced by the interpolation of a note which may have been only a passing note or _appoggiatura_. The Chromatic also, it is said, was regarded as too difficult for ordinary performers, and most of its varieties went out of use at a comparatively early period. Yet the accounts which we find in writers so remote in time and so opposed in their theoretical views as Aristoxenus and Ptolemy, bear the strongest testimony to the reality and persistence of [Footnote 1: Plato, _Legg_. p. 812 d [Greek: panta oun ta toiauta mê prospherein tois mellousin en trisin etesi to tês mousikês chrêsimon eklêpsesthai dia tachous.]] these non-diatonic scales. And we have the decisive fact that of the six scales of the cithara given by Ptolemy (see p. 85) not one is diatonic in the modern sense of the word. It may be alleged on the other side that the ideal scale in the _Timaeus_ of Plato is purely diatonic, and exhibits the strictest Pythagorean division. But that scale is primarily a framework of mathematical ratios, and could not take notice of intervals which had not yet been identified with ratios. It is not certain when the discovery of Pythagoras was extended to the non-diatonic scales. Even in the _Sectio Canonis_ of Euclid there is no trace of knowledge that any intervals except those of the Pythagorean diatonic scale had a numerical or (as we should say) physical basis[1]. [Footnote 1: In Euclid's _Sectio Canonis_ the Pythagorean division is assumed, and there is no hint of any other ratio than those which Pythagoras discovered. Prop. xvii shows how to find the Enharmonic Lichanos and Paranêtê by means of the Fourth and Fifth. Prop. xviii proves against Aristoxenus (of course without naming him), that a [Greek: pyknon] cannot be divided into two equal intervals; but there is no attempt to explain the nature of the Enharmonic diesis. It is worth notice that in these propositions the Lichanos and Paranêtê of the Enharmonic scale are called [Greek: lichanos] and [Greek: paranêtê] simply, as though the Enharmonic were the only genus--a usage which agrees with that of the Aristotelian _Problems_ (supra, p. 33).