|
|
|
|
|
|
|
He obviously followed a quite different procedure from Archytas. It seems that his object was to find ratios that would correspond as closely as possible to the standard definitions of Aristoxenus. The 19:15 interval in the enharmonic is almost exactly a ditone, and the 6 · 5 interval in the chromatic is only 9 cents over 1 1/2 tones (Aristoxenus' tonic chromatic). Eratosthenes divides the remainder in each case, the pyknon, into two parts as nearly equal as possible. |
|
|
|
|
|
|
|
|
It would be interesting to know the relationship between Eratosthenes and Ptolemais, a female musicologist of uncertain date who also came from Cyrene. She wrote a work entitled Pythagorean Elements of Music, in the course of which she contrasted the Pythagoreans' mathematical approach with the Aristoxenians' empiricism, and argued that something of both was required. Reason and perception should go hand in hand.49 |
|
|
|
|
|
|
|
|
A theorist of the time of Nero called Didymus took a similar line in a work On the Difference between the Aristoxenians and the Pythagoreans. It was probably from him that Porphyry took the quotations from Ptolemais. Didymus offered another set of formulae for the intervals of the tetrachord: |
|
|
|
|
|
|
|
|
I have commented on these values in Chapter 6.50 |
|
|
|
|
|
|
|
|
It remains to give Ptolemy's ratios, from which the table on p. 170 is derived. As was noted there, Ptolemy differs systematically from his predecessors in making the middle interval in the enharmonic and chromatic tetrachords about twice as large as the one below it. |
|
|
|
 |
|
|
|
|
49 Excerpts are preserved by Porph. in Ptol. Harm. pp. 22-6; Thesleff, Pythagorean Texts, 242f. (abridged); Barker, GMW ii. 239-42. |
|
|
|
|
|
|
|
|
50 See p. 169 for both Eratosthenes and Didymus. |
|
|
|
|
|