|
|
|
|
|
|
|
Except for the ditonic diatonic, which Ptolemy includes as a concession to common musical practice, he contrives throughout to make each ratio superparticular. He regards this as an essential requisite of melodic intervals for reasons explained in an earlier chapter of his work.51 The smooth progression in the right-hand column (the interval between Lichanos and Mese) from 5:4 to 10:9 will be noted. |
|
|
|
|
|
|
|
|
A few words should be added about devices for measuring, testing, and demonstrating intervals with various ratios. The zithers used by some early lecturers on harmonics could have served this purpose only if the strings were equally tuned and the soundboard under them so calibrated that a string could be stopped at precisely measurable fractions of its length. More probably these instruments were used with open strings, each differently tuned. The string-torturers described by Plato (above, p. 225) are not measuring intervals but trying to settle on an interval to measure by. If they had been measuring by ratios they would not have been looking for a minimal unit. |
|
|
|
|
|
|
|
|
However, the harmonicist who apparently gave his name to one of these zithers, Simos, is also associated with the invention of the monochord, the instrument commonly used for ratio measurement. It had a single string stretched over a graduated rule (kanon), with a movable bridge by which the vibrating length of the string could be shortened or divided in measured proportions. Pythagoreans later claimed that Simos had appropriated knowledge of the kanon from an exposition of Pythagoras' wisdom set up on an inscription at Samos by a son of Pythagoras, and promulgated it as his own. The story implies that Simos was widely credited with the invention.52 Perhaps it was on the monochord that Archytas tried out the ratios of 5:4, 6:5, etc., that he incorporated in his system. It was certainly in use by the time of the Sectio Canonis. The last two propositions of the work give directions for dividing the rule so as to construct a diatonic two-octave system. |
|
|
|
|
|
|
|
|
For a long time measurements were taken only from one end of the string. Didymus in the first century AD realized that one could use the portions on both sides of the bridge and take the ratio of one part |
|
|
|
 |
|
|
|
|
51Harm. 1.7p. 15. 18ff., cf. 1.15p. 33.5ff. |
|
|
|
|
|
|
|
|
52 Duris, FGrH 76 F 23; cf. U. yon Wilamowitz-Moellendorff, Platon ii (Berlin, 1920), 93f.; Burkert, LS 455 n. 40. For Pythagoras as inventor of the kanon cf. Nicom. p. 248.16, Gaud. p. 341.13, Aristid. Quint. p. 97.3, Diog. Laert. 8.12, Anecdota p. 9 Studemund; other references in Burkert, LS 375 n. 22. |
|
|
|
|
|