|
|
|
|
|
|
|
of the tetrachord in each genus.45 As ratios of 3:2, 4:3, and 9:8 were already established as being of harmonic importance, Archytas thought it logical to look for 5:4, 6:5, 7:6, and 8:7. He probably constructed these intervals experimentally on some instrument and then decided where they belonged in the scale. Two of them, 6:5 and 7:6, he found only between non-adjacent notes. 6:5 suited the interval from the enharmonic Paranete diezeugmenon to Mese (or Lichanos to Hyperhypate); 7:6 suited that from Trite diezeugmenon (in any genus) to Mese, or Parhypate to Hyperhypate. More recherché figures had to be found for the various intervals of less than a tone, but he was able to deduce them from those which he had already put in place (Table 8.3). Nearly all these ratios are 'super-particular', that is, of the type n + 1:n. Ptolemy speaks as if Archytas sought this on principle. Archytas was certainly interested in superparticulars, and worked out a proof that they could not be divided into two equal parts; the 9:8 tone, for example, cannot be divided into precisely equal semitones.46 |
|
|
|
|
|
|
|
|
This and a number of other general propositions about super-particular and multiple ratios (n:1) are demonstrated in the Euclidean Sectio Canonis. The work also contains proofs of various basic facts about the relationships of the concordant intervals and the tone, for example that the octave is less than six tones, and the fourth less than two and a half. Archytas' elegantly proportioned scales are ignored: the diatonic tetrachord is assumed to contain two 9:8 tonal steps, as in Philolaus, and the enharmonic tetrachord a ditone step.47 |
|
|
|
|
|
|
|
|
This form of diatonic is also presupposed in Plato48 and in several later writers. Among them is Eratosthenes of Cyrene, the versatile scholar and scientist who presided over the Alexandrian Library for most of the second half of the third century BC. Ptolemy cites Eratosthenes' sets of ratios for all three genera. From Hypate to Parhypate to Lichanos to Mese they are: |
|
|
|
 |
|
|
|
|
45 Cf. above, p. 168. The following analysis owes much to Barker, GMW ii. 46-52. Plato, Resp. 531 a/c, apparently contrasts Philolaus' with Archytas' approach. |
|
|
|
|
|
|
|
|
46 Ptol. Harm. 1.13 p. 30.9; Boeth. Inst. Mus. 3.11 (Diels, Vorsokr. 47 A 19); cf. Barker, GMW ii. 43 n. 63, 46f., 195 n. 12. |
|
|
|
|
|
|
|
|
4747 Cf. p. 169. |
|
|
|
|
|
|
|
|
48Tim. 35b-36b; see Barker, GMW ii. 59f. |
|
|
|
|
|