To put the essential principles of Aristoxenus in the simplest possible form it may be said that he established two principal rules: (1) that music accepts Sound as sounds heard by the ear, and that the science of music must be built upon the foundation of sounds that are heard; (2) that sound-functions exist, possessing properties of sonance not directly reducible to any simple or elemental numerical ratio. The work of Aristoxenus was a revolution in musical philosophy based upon the principle of music as an organic whole of sounds bearing a dynamic relation each to the other. Aristotle had not been able to break away from the old Pythagorean conception, but Aristoxenus brushed away the misty speculations of morality, the mathematical entanglements and the musty formalism that surround the music of his time and set himself to answer the one vital question: Why does Music employ certain sounds and reject certain others? The third stage of development of Greek music may be represented by Claudius Ptolemy, who lived in the second century of the _Christian_ era. He may, with considerable authority, be deemed the inventor of the first interpreter of the equal tempered scale. R. C. Phillips has thrown considerable light upon the disputed questions involved in this matter, and to his monograph on the 'Harmonic Tetrachords of Claudius Ptolemy’ (1904) we may refer the reader desirous of detailed information. Leaving the question of theory, we now proceed to pick up the thread of mythical story and trace what we can of the history of Greek composition. V