By Andrew Barker
This ebook examines, for the 1st time, the clinical tactics devised by way of Ptolemy (second century advert) for investigating the constructions underlying musical melody, a venture that he conceived as heavily on the topic of astronomy. Ptolemy's account of his tools is strangely particular, and he pursues them faithfully. by means of offering an research of Ptolemy's subtle theoretical gear, his concepts for integrating thought with remark, and his meticulous directions for the layout and behavior of experimental checks, the ebook deals historians of technological know-how a brand new starting-point for wider experiences of historical clinical approach.
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Extra info for Scientific Method in Ptolemy's Harmonics
In . 15 what's to be divided is the fourth, and the ratio 4:3. Ptolemy’s approach will contain major levels. The ﬁrst divides the ratio 4:3 into which are, in a slightly prolonged experience, ‘nearly equal’. the second one takes this kind of sub-ratios, divides it into 3 which are pretty much equivalent, after which places of them jointly back to go away a couple of signiﬁcantly unequal sub-ratios. the finished tetrachord hence comprises one of many ratios produced by way of the ﬁrst part of department and shaped by means of the second one. the method has created a department of the fourth into 3 via starting from a department into , and carrying on with through a approach that divides a sub-ratio into after a step regarding its department into 3. those quite a few ‘divisions into or 3’ are those to which our current sentence refers. The correct ‘near-equality’ holds no longer among the smaller time period of a ratio and the diﬀerence among its phrases, as at sixteen. 17–21, yet among the ratios into which a bigger ratio is split by means of smaller ones, as at 15. 29–16. 6. yet we needs to discover extra completely the explanations in the back of Ptolemy’s insistence that the divisions has to be ‘into or three’. they don't seem to be simple. we all know on different grounds, after all, that the tetrachord has to be prepare out of 3 durations or ratios; that may be a basic datum of musical perform. yet Ptolemy’s comment implies greater than that, because the define above has indicated. within the following few sentences he oﬀers an complex and reasonably baﬄing rationalization. by way of those divisions the diﬀerences within the ﬁrst concords have been additionally chanced on to be bounded, they usually pass up purely to the quantity three there too, considering the fact that that completes the entire durations. For starting from the octave homophone and the duple ratio, during which the diﬀerence among the extremes is the same as the person who is surpassed, we took for its aid by way of equals the hemiolic ratio of the harmony of a ﬁfth, in 1 See GMW2 p. 306 n. 124. 134 Melodic durations which the diﬀerence among the extremes features a half that that is surpassed, and the epitritic ratio of the harmony of a fourth, during which the diﬀerence among the extremes includes a 3rd a part of that that's handed; and for its augmentation through equals we took the triple ratio of the harmony of an octave and a ﬁfth, within which the diﬀerence among the extremes makes of that that is passed, antithetically to the part, and the quadruple ratio of the double octave homophone, within which the diﬀerence among the extremes makes 3 of that that is passed, antithetically, once more, to the 3rd half. (33. 9–22) those reﬂections look comparable to a passage we thought of prior, the place Ptolemy was once introducing the suggestion of tetrachordal department. The fourth, he acknowledged, is to be divided into 3 ratios, ‘so that the ﬁrst homophone, that is one, can be prepare from the ﬁrst concords, and the ﬁrst harmony from 3 melodics’ (28. 19–20). There too, and within the rapid sequel, we stumbled on a few complicated manoeuvres with the numbers 2 and three.