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the Greek specialist writers distinguish several different tunings within each of the named genera. |
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Apart from that, there is a problem about what exactly is meant by a 'tone'. The Greek writers define it as the interval by which a fifth is greater than a fourth. Strictly speaking, that is the interval given by the ratio 9 : 8, or 204 cents. But Aristoxenus regards it as being at the same time a unit of which a fourth (properly 498 cents) contains exactly two and a half. In effect he is operating with a tempered tone of 200 cents and a tempered fourth of 500 cents. He does not understand that that is what he is doing; he is simply working by ear. He dismisses as contrary to the evidence of the senses the mathematicians' proofs that a fourth less two tones leaves a remainder of something smaller than half a tone (4/3 divided twice by 9/8 = 256/243; in cents, 498 minus twice 204 = 90). Sometimes he speaks of intervals such as a third of a tone or three eighths of a tone. We must take these with a little pinch of salt, not as mathematically precise measurements but as approximations gauged by the ear. Writers of the Pythagorean tradition, on the other hand, insist on defining intervals as mathematical ratios, and are obliged to impose mathematical interpretations on whatever intervals were established in musical practice. They are, moreover, concerned to achieve a certain degree of mathematical elegance, and prepared to revise their formulae to this end. These authors, therefore, give us absolutely precise specifications for the genera and their varieties, but again we must take them with salt. The precision is specious. All the same, these data are valuable as providing some indication of the intervals in actual use. |
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The earliest of them, Philolaus (latter part of the fifth century BC), is in fact the least sophisticated mathematically, and his analyses, which in part have to be inferred from reports of the terms he used for various subdivisions of intervals,28 probably do reflect contemporary tuning practice in quite a direct way. In the diatonic tetra-chord he found two 9 : 8 tones and a remainder which he called by the name of diesis. In the chromatic the lowest interval was again the diesis, while the two lowest together added up to exactly a tone. In the enharmonic the two small intervals were obtained by bisecting one or other of those in the chromatic; Philolaus may have offered both alternatives. All these tunings could be arrived at in practice, and very likely were, by constructing steps of a tone from strings already tuned, by going up a fourth and down a fifth, or up a fifth and |
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28 See Ch. 8. |
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