Thus it appears that it was Didymus, and not Ptolemy, who proposed the tuning of the tetrachord which is now accepted as correct. It is very evident from the entire course of the discussion as conducted by Ptolemy that his calculations were purely abstract. He is to be reckoned among the Pythagoreans, who held that in time and number all things consist. It was not until some centuries later that the happy thought of Didymus came to recognition as the true statement of the mathematical relation of the first four tones of the scale, and then only through the ears of a race of musicians following the great thesis of Aristoxenos, that in music it is always the ear which must be the arbiter, and not abstract reasoning or calculation. The ratios of the major and minor third also occur among the calculations of Didymus; but here, again, they count for nothing in the history of art, because these intervals derive their value and expressive quality from their harmonic relation, while Didymus and all the Greeks employed them as melodic skips only, and reckoned them in with a multitude of other skips and progressions, without distinguishing them in any way. The one characteristic instrument of Greek music from the earliest to the latest days was the lyre. In the oldest times, those of Homer and Hesiod, it was called phorminx, which is believed to have been the form so often represented on Greek vases of a turtle shell with side pieces like horns, an instrument having but little effective resonance. The later form was the so-called cithara, the most common shape of which is that made familiar to all by the pedal piece of the square pianoforte. This instrument rarely had more than six strings, and as it had no finger board it could have had no more notes than strings. Chappell, the English historian, attempts to demonstrate that certain ones of these instruments had a bridge dividing the string into two parts, thus largely increasing the compass, but the evidence supporting this hypothesis is not satisfactory. Plato speaks of instruments of many strings imported from Asia, which seem to have been the fashion or fad in his day. He disapproved of them very heartily, but the terms in which he speaks of them show that he cannot have been very familiar with their appearance, for it is impossible to make out what he is driving at. [Illustration: Fig. 13. LYRE.] [Illustration: Fig. 14. CITHARA.] There is considerable doubt as to the extent to which the larger instruments of Asiatic origin penetrated the general musical practice of Greece. Athenaeus, in his "Banquets of the Learned" (B. xvi, C), quotes Anakreon as saying: "I hold my magadis, and sing, Striking loud the twentieth string, Leucaspis at the rapid hour Leads you to youth and beauty's bower." Most certainly the lyre of Terpander had no twenty strings.