^{page 195 note 1} Gevaert, (Histoire et Théorie I., pp. 304–327)Google Scholar was the first to give these formulae the importance that is their due. This article owes much also to the work of Tannery, P.(Mémoires Scientifiques III, 97–115Google Scholar, reprinted from Revue des Et. Gr. XV, 336–352).

^{page 195 note 2} 22, 33 οὐχ before ὁμολογεῖται is rightly bracketed by Macran.

^{page 196 note 1} P. 27, 9: the text is corrupt, but Macran's χρωματικῆς τῆς βαρυτάτης for χρωματικῆς παρυπάτης is almost certainly right. Marquard imported the more general phrase used in the corresponding passage in Book II. There is no reason why Aristoxenus should not have been more precise here than there and in III, 73 (τῆς παρυπάτης ἐπὶ βαρὐ κινηΘείσης).

^{page 197 note 1} The same possibility of equal or unequal division of a pycnon is expressed in III, 73, 20. In I, 29, 16 he says that the small intervals are equal ὡς ἐπὶ τò πολὑ.

^{page 197 note 3} The latter is close to the enharmonic of Archytas, as we shall see, and the former not far from the chromatic of Didymus, both of which break this rule laid down by Aristoxenus and repeated by Ptolemy, (Harm. II, 14)Google Scholar. Was this chromatic in actual use at an early date? And is there a polemical purpose behind the selection of these two instances?

^{page 197 note 3} In order to give whole numbers I have allowed inconsistencies involving one cent in the figures for one and a half tones and between the three-quarter tone in the soft diatonic and the three-quarter tone pycnon of the ἡμιóλιου χρῶμα.

^{page 198 note 1} It may be, as Tannery suggests, that confronted with the incommensurability of the tone and varying musical practice they relegated the movable notes to the realm of ἄπειρου and refused to speculate upon them, till Archytas tackled the problem on more realistic lines. Aristoxenus himself, who in Books I and II selected certain magnitudes as γνώριμα from the infinite possibilities, in Book III, 69, 6 affirms that in these matters of pitch and magnitude scientific treatment is impossible, since the possibilities are unlimited; the functions and mutual arrangements of notes in the scale only are πεπερασμένα.

^{page 198 note 2} The ratios of Eratosthenes' enharmonic are dictated by his choice of the minor third

for the upper interval of his chromatic and by the assumption, made also by Didymus and Boethius, that the pycnon of the enharmonic should be equal to the lowest interval of the chromatic, Dividing his chromatic pycnon

into

, he then takes

as his enharmonic pycnon, leaving

for the highest interval.

^{page 200 note 1} The tempered thirds we tolerate are actually nearer to Plato than to Archytas.

^{page 200 note 2} Note that it is the practical musicians, not any theoretical κανονικοί, to whom it is ascribed. They tended, he says, to substitute it for the intonation

. And, as it was used in combination with a tetrachord of

, we can see clearly one reason why they did so:

in the E mode gives two false fourths and two false fifths; the substitution of

in the upper tetrachord gives all true fourths and only one false fifth [F-C]. It is also interesting to note that these intonations of the practical musicians of Ptolemy's time show a complete avoidance of both major and minor thirds, except in the tetrachord for which the ditonal type was substituted!

^{page 200 note 3} Tannery suggests that this is the legacy of some earlier theorist who wished to deny that Aristoxenus' quarter-tone was the smallest melodious interval. It seems to me more likely that he adopts here for the sake of uniformity the principle of division by tripling the terms

which gave him most satisfactory results for his chromatics. The enharmonic was extinct in his day.

^{page 202 note 1} The nearest approach to these equal semitones in mathematical dress is to be found in Aristides Quintilianus, p. 117, Meibom, who divides the tone as follows:

.

^{page 203 note 1} Similarly, the minor tone can be divided with approximate equality, as by Eratosthenes; but it is the division of Didymus that gives the more convincing intervals.

^{page 204 note 1} If Aristoxenus wished to represent this type of diatonic and was taking

as three-quarters of a tone, he was faced by a dilemma. Either he must grotesquely exaggerate the difference of the tones, as above, or, representing the major tone, as usual, by his own whole tone, make the lowest interval also three-quarters of a tone, which is absurd.

^{page 204 note 2} The interval of

will not really square with the account in Plutarch, which demands the possibility of confusion between the upper interval of the lower tetrachord (F-A) and the combination of disjunctive tone and σπονδειασμóς (A-C).

as σπονδειασμóς makes these intervals practically identical. On this point, and on the Spondeion in general, I would refer the reader to my article on ‘The Spondeion Scale’ in C.Q., Vol. XXII, 1928Google Scholar.

^{page 205 note 1} If this equation is correct, we have the interesting fact that the only type of chromatic that Ptolemy found in practical use in his time was actually regarded by Aristoxenus as diatonic!

^{page 205 note 2} κάὶ τἡν έκλυσιν καὶ τἠν ἐκβολἠν πολὑ μείζω πεποιηκέναι ϕασὶν αὐτòν. πολὑ μείζω certainly makes bad sense, and Reinach brackets it as concealing a marginal Πολὑμνηστον or Πολὑμνισγον and translates πεποιηκέναι by ‘il créa.’ But εὑρίσκω (or some compound) is usual in pseudo-Plutarch of musical ‘inventions,’ and I prefer Westphal's hypothesis of a lacuna after ἐκβολὴν; these accusatives then are constructed with ΠολυμνἡστႨ…ἀνατιΘέασι in the preceding phrase. It is conceivable that, just as Terpander is associated with the employment in melody of Dorian νἡτη, so Polymnestus popularized the addition of D below an enharmonic E octave. See below.

^{page 206 note 1} It is found also in the second Delphic Hymn, but there the pycnon is more probably chromatic.

^{page 206 note 2} This interval is the difference between the septimal tone

and the major tone

. It plays an important part in the theories of DrPerrett, W. (Some Questions of Musical Theory, 1926 and 1928)Google Scholar. Though I find difficulty in accepting them in detail, I believe with him that the Greeks used intervals strange to us with precision. They can scarcely, however, have used so small an interval as 27 cents. It is possible that the aulos-player, who could control the intonation at will, varied the pitch of the enharmonic parhypate according as it was employed in relation to hypate or to hyperhypate. If then in the course of the same piece he made the interval between hyperhypate and parhypate that of

and also employed the lichanos of Aristoxenus, a leimma above hypate, he would in effect be employing two notes distant only by

. Procedure of this sort might also have led Aristoxenus to evaluate

as frac;1 tones (see p. 207), which an interval of

between hyperhypate and parhypate would not do, as that he elsewhere equates with

tones.

^{page 208 note 1} Aristoxenus' remarks upon the lowering of lichanos here and his polemic against its raising in the enharmonic combine to show the primary importance of this string in determining genus and nuance. As for the two lowest intervals, those of Ptolemy's tetrachord might be expressed as

and

of a tone, and would thus both be irrational. It would be possible to fill up the tetrachord also with

, that is to say

, intervals more familiar to Aristoxenus. But this has no independent support.