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69 5 Purely in Tune Exploring Just Intonation Systems tempered tuning systems were designed to solve a problem inherent in acoustics: no mathematical process can generate a scale that includes both pure (3:2) fifths and pure (2:1) octaves; that is, no power of 2:1 will ever equal a power of 3:2. The ancient Greeks (by legend, Pythagoras ) discovered that pure musical intervals are related by proportions measurable by the comparative lengths of vibrating strings: the octave 2:1, the fifth 3:2, the fourth 4:3, the major third 5:4, the minor third 6:5, and so on (demonstrated on the enclosed CD, track 12.) Mathematicians also recognized the impossibility of tuning a fixed-pitch instrument so that all of these intervals adhere to pure proportions. For example, we can tune a series of pure 3:2 fifths (C–G–D–A–E– B–F#–C#–G#–D#–A#–E#–B#) by multiplying the vibration frequency of each successive note by 3 ⁄2, but the twelfth fifth, B#, which is (3 ⁄2)12 , will be higher than the seventh octave, C (27 ), by about an eighth of a tone (the precise difference is 129.75:128). This small interval is called the “Pythagorean comma.” There is no inherent reason why any two numbers in the series of fifths and octaves should coincide, but the fact that the twelfth fifth is close enough to the seventh octave to sound like an out-of-tune version of the same note has created headaches for musicians for centuries. The close relationship of these two l o u h a r r i s o n | Purely in Tune 70 notes gave rise to the Western division of the octave into twelve parts, but also cemented a scale system that included the intractable Pythagorean comma. There are two main solutions to the non-intersection of the twelfth fifth and the seventh octave: either the size of some or all of the fifths can be reduced in order to close the circle, or the natural spiral can be extended as needed. The ancient Chinese, restricted by their belief that all intervals should be generated by 3:2 fifths, developed practical solutions using eighteen, and later sixty, pitches. By the fifth century a.d. Ch’ien Lo-Chih theorized a cycle of 359 fifths, at which point variance from the octave series is imperceptible to the human ear. European theorists in the medieval period and the Renaissance used research string instruments with movable bridges to demonstrate the effect of dividing a string in various numbers of equal parts. Harrison championed the ancient monochord in his own research and recommended it for general music teaching. Harrison and Colvig built several monochords for the study of tuning systems and recommended that such instruments be used in elementary schools to teach “numeracy” through sound. Building on the work of Marin Mersenne (1588–1648), John Wallis (1616– 1703), Joseph Sauveur (1653–1716), and others, eighteenth-century scientists such as Brook Taylor (1685–1731) and Daniel Bernoulli (1700–82) began to develop mathematical models describing the vibrations of bowed strings, showing in the process that multiple modes of vibration occur simultaneously. Therefore the higher pitches (“overtones” or “harmonics”) that characterize the subdivisions of the string sound simultaneously with the string’s fundamental. The discovery of overtones did not solve the acoustical problem of the nonequivalence of the twelfth fifth and the seventh octave, however. In fact, it exacerbated the issue by demonstrating that pure intervals are components of musical tones; therefore if two fundamental pitches are not tuned in pure intonation, all of their overtones will also be dissonant, creating a relatively harsh timbre. Theorists and practicing musicians over the centuries devised a variety of possible temperaments (systems using alterations of pure intervals) to solve this problem. For the music of the middle ages, a Pythagorean temperament, in which all fifths but one are pure, was often workable. The single impure fifth (called the “wolf”) was tuned so as to close the circle and thus form a pure octave with the starting pitch. In the process, this fifth became so small as to be virtually useless , at least as a consonant interval. But if the wolf occurred between notes not used in a composition, its effect was imperceptible. Harrison used Pythagorean temperament on occasion and kept his portable organ tuned to that system. This organ (and its characteristic tuning) plays a crucial role in La Koro...

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