The structure of the Scale so far as was necessary for the development of the Greek modes was comprised in The Disjunct or Greater System Complete; yet at various times the extent of the diatonic scale by degrees was increased, tetrachord was added to tetrachord until in the days of Plato its compass was stated to have been made to comprehend four octaves, a fifth, and a tone.
Archytas and Aristoxenus were both of Tarentum, a noted Greek colony in Southern Italy, founded by Sparta about 705 B.C. Archytas was a contemporary of Plato (b 429 d 347). The period was one of artistic luxury, the Parthenon had been completed, and Greece had her golden age of Art, Science, and Philosophy. Here Praxiteles, the great sculptor, second only to Phidias, comes upon the scene, and we may with confidence accept his design of Apollo’s Lyre as a true representation of the instrument as it existed in his day, and, it may be assumed as used in Apollo’s Temple, and by the master-musicians. The date of this sculptor has not been ascertained precisely, Prof. E. A. Gardner gives in a guarded way 400 B.C.
Aristoxenus was a musician, the son of a musician, he came at a time when great mathematicians were engaged in battle over fine distinctions in Pythagorean systems, to them of superlative interest and importance. Aristoxenus opposed the Pythagoreans and held that “it was absurd to aim at an artificial accuracy in gratifying the ear beyond its own power of distinction,” a decision very natural, coming from a musician. He was a great writer and theorist, wrote it was said more than four hundred treatises, all of which have been lost except three on “Harmonic Elements,” and this is the oldest musical work at present known.
In those years from Archytas to Aristoxenus the evolution of Greek music had passed from the poet-musicians, the real masters of the lyre, into the hands of philosophers and disputants, men learned in all the subtleties of Pythagorean lore, who busied themselves with recondite demonstrations of the proportions of numbers, and applied them to the theoretical division of the octave, to an extent which transcended altogether the range of the practical art of the cithara players, nevertheless the labour was not wholly lost, since it went to the strengthening of the foundations of the science of music.
A new era had arrived, Greece lost her position and became a dependency in the Macedonian empire. The centre of Greek life and thought had been transferred to Alexandria, and here at the great library which had been founded B.C. 332 by Alexander the Great, Eratosthenes was librarian, and his name figures largely in the mathematics of Music. His lifetime extended from 276 to 196 B.C.
Two other Alexandrians complete the record so far as the present simple treatment of the development of the scale is concerned. They lived within the Christian era.
Didymus, A.D. 60, introduced the minor tone into the scale, and consequently the practical major third. He demonstrated the lesser or minor tone to be necessary to the right division of fourths and fifths.
Claudius Ptolemy, A.D. 130, accepted the scheme, but altered the arrangement of the tones.
Didymus and Claudius Ptolemy, the two latest philosophers who sought to perfect the diatonic scale, achieved highly important results by simple means; whereas the octochord as left by Pythagoras, comprised but two kinds of divisions, the tone and the hemitone (not exactly half a tone, it was the overplus after the measurement of the two whole tones in the tetrachord)—and these, taking C as the starting point for our convenience, may be represented thus:—
| C | ....... | D | ....... | E | ...... | F | ....... | G | ....... | A | ....... | B | ...... | C |
| major tone | major tone | hemi tone | major tone | major tone | major tone | hemi tone |