By carving out of the Greater Perfect System (which we may call simply the Complete System) overlapping octave sections, each beginning on a different note, the Greek theorists found these to correspond in their intervals to each of the seven different modes, as follows: Thus all scales came to be thought of theoretically as transpositions of the corresponding octave sections in the Complete System (Foundation Scale). Indeed, the entire _system_ was considered as transposed and the individual tones retained their names regardless of pitch, i. e., in the Dorian mode the _mese_ would always be the fourth note from the bottom, in the Phrygian the fifth, etc. [Illustration: Music] As an example, let us transpose the Foundation Scale one tone above its natural pitch: [Illustration: Music] The middle octave will now be seen to be Phrygian (corresponding to No. 3 above) instead of Dorian as before. Now in their system of transposition scales (in reality transposed Complete Systems) the Greeks gave to every scale the name corresponding to the mode of its middle octave. Before the time of Aristoxenus only seven of these transposition scales, or keys ([Greek: tonoi]) were in use. That theoretician eventually rounded out the scheme to eighteen (of which six appear in modern notation as duplicates or octave transpositions). He did this systematically by taking the interval of the perfect fifth as a basis and building on each semi-tone degree a group of three scales (natural, hypo, and hyper). As there were not enough of the original modes to supply names for all of the new scales, it was, of course, necessary to invent arbitrary names for the superfluous ones. By this achievement it was possible to transpose a melody into any one of the eighteen (or really twelve) keys without changing its modal character. We may therefore assume with some justification that Aristoxenus’ system in a way did for the Greeks what our own equal temperament has done for modern music. We end our brief sketch of Greek theory at this point, which may be assumed as the highest development of the system. Later systems were either based on Aristoxenus or were of reactionary nature. We must, however, for a moment retrace our steps to explain briefly the achievements of an earlier theoretician, the great philosopher Pythagoras, in the field of musical acoustics. IV Like many of the ancient philosophers, Pythagoras (_ca._ 600 B. C.) is known only by his disciples and by their quotations from or commentaries on his teaching. Of these the most important are Archytas (400-365 B. C.) and the great mathematician Euclid (_ca._ 300 B. C), though there is some reason to suppose the part of Euclid’s work dealing with music to have been written by Cleonides (_ca._ 200 B. C.) and by the later Pythagorean Nichomachus (_ca._ 150 A. D.).