According to Ptolemy (i. 13) the Pythagorean philosopher Archytas was the author of a new division of the tetrachord for each of the three genera. In it the natural Major Third (5: 4) was given for the large interval of the Enharmonic, in place of the Pythagorean ditone (81: 64); and the Diatonic was the same as the Middle Soft Diatonic of Ptolemy. But, as Westphal long ago pointed out (_Harmonik und Melopöie_, p. 230, ed. 1863), this scheme is probably the work of the later Pythagorean school. It seems to be unknown to Plato and Aristoxenus,--the latter wrote a life of Archytas--and also to Euclid, as we have seen. The next scheme of musical ratios is that of Eratosthenes, who makes no use of the natural Major Third.] In Plato's time, as we can see from a well-known passage of the _Republic_ (quoted on p. 53), the Enharmonic and Chromatic scales were the object of much zealous study and experiment on the part of musicians of different schools,--some seeking to measure and compare the intervals directly by the ear, others to find numbers in the consonances which they heard, and both, from the Platonic point of view, 'setting ears above intelligence,' and therefore labouring in vain[1]. The multiplicity of intervals, then, which surprises us in the doctrine of the _genera_ and 'colours' was not an accident or excrescence. And although some of the finer varieties, such as the Enharmonic, belong only to the early or classical period, there is enough to show that it continued to be characteristic of the Greek musical system, at least until the revival of Hellenism in the age of the Antonines. The grounds of this peculiarity may be sought partly in the Greek temperament. We can hardly deny the Greeks the credit of a fineness of sensibility upon which civilisation, to say the least, has made no advance. We may note further how entirely it is in accordance with the analogies of Greek art to find a series of artistic types created by subtle variations within certain well-defined limits. For the present purpose, however, it will be enough to consider how the phenomenon is connected with other known characteristics of Greek music,--its limited compass and probably imperfect tonality, the thin and passionless quality of its chief instrument, on the other hand the keen sense of differences of pitch, the finely constructed rhythm, and finally the natural adaptation, on which we have already dwelt, between the musical form and the language. The last is perhaps the feature of greatest significance, especially in a comparison of the ancient and modern types of the art. The beauty and even the persuasive effect of a voice depend, as we are more or less aware, in the first place upon the pitch or key in which it is set, and in the second place upon subtle variations of pitch, which give emphasis, or light and shade. Answering to the first of these elements ancient music, if the main contention of this essay is right, has its system of Modes or keys.