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So working with true 3: 2 fifths, 4: 3 fourths, and 9: 8 tones does not produce a system that will stand up to much to-ing and fro-ing, or a division of the octave into twelve equal semitones, five in the fourth and seven in the fifth. That is achieved by fudging, otherwise known as Equal Temperament. The semitones are made equal, and the fourths and fifths allowed to become slightly impure. This compromise between the laws of nature and the needs of art has developed since the sixteenth century, being essential for music in which modulation between different keys is a regular resource. |
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The Greeks understood that the size of intervals is determined by numerical ratios; some of them, remarkably, came close to a correct physical explanation.14 A number of authors describe or prescribe tunings for musical scales in terms of ratios. They all take it as axiomatic that the octave is 2: 1, the fifth 3: 2, and the fourth 4: 3. There is no question of adjusting these intervals to conform with other requirements. |
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When it comes to the subdivisions of the fourth, they offer a variety of different formulae. From this and other evidence we know that Greek music used many intervals that cannot be adequately expressed by talking of semitones, tones, major thirds, etc. Nor is it helpful to leave them as ratios. If two notes are stated to be related in the ratio 32: 27, not many people will get any impression of what sort of interval is meant. |
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We shall therefore make use of the cent system invented by the tone-deaf philologist, mathematician, and musicologist A.J. Ellis (1814-90). On this system the octave is divided into 1,200 cents, 100 for each semitone of the equal-tempered scale. Any interval, however unconventional, can be accurately expressed and comprehended by the use of this form of measurement. For example, the 32: 27 interval converts into 294 cents, from which we can see at once that we are talking of an interval of the order of 1 1/2 tones.15 |
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The cent system has the further advantage that the sum of two or more intervals can be determined simply by adding the cent numbers instead of multiplying ratios or fractions. For example, the true |
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14 See ps.-Arist. Pr. 19.39 and De audibilibus 803b34-804d8, and especially Euc. Sect. Can. pp. 148-9 Jan with Barker's notes in GMW ii. 35 n. 29, 95, 107, 192f.; Burkert, LS 379-83. |
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15 The actual conversion from ratios to cents involves either a calculation using logarithms or consultation of a cent table. I have used the one published by E. M. von Hornbostel, Zeitschrift für Physik 6 (1921), 29ff., reproduced m Kunst, fig. 61. For further explanation see Kunst, 4-9; Sachs, WM 23-8. |
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