In lieu of an abstract, here is a brief excerpt of the content:

19 � Tuning and Temperament herbert myers Imagine a world in which the units used for linear measurement were not quite commensurate —one in which, by some quirky royal decree, let us say, twelve official inches did not quite make an official foot, or three feet exactly a yard. Most citizens, presumably, would be aware of a problem only rarely, but anyone whose profession depended upon precise measurement would long since have become expert at making fine distinctions; we can be sure that architects and carpenters, for instance , would have come to distinguish unabashedly between “inches” and “twelfths of a foot.” The units of our musical world—those we call “intervals”—are, in fact, of a similarly incommensurate nature, although unlike the units of our metaphorical example, their size is not determined arbitrarily; their mathematical ratios reflect basic acoustical phenomena. And unlike the professionals in our metaphor, musicians —those who must deal constantly with the problem—are for the most part unaccustomed to discussing it intellectually, generally preferring an intuitive approach. In fact, so out of favor is a “scientific” approach to intonation that to mention it may arouse suspicion among other musicians as to one’s musical sensibilities. It was not always so; in centuries before the nineteenth, a firm grasp of the mathematical foundations of music was considered to be one of the highest attainments of a good musician. But perhaps more relevant to the present discussion is the fact that different musical priorities in earlier times led to solutions different from the usual modern ones. (In our linear analogy above, our experts made a provisional redefinition of the inch in terms of the official foot; they might just as well have found reason to redefine the foot in terms of the official inch instead.) In order to appreciate these earlier solutions—and certainly in order to put them into practice ourselves—we have to have an understanding of both the underlying theory and terminology. Neither the conceptual basis nor the attendant math is really all that complicated, although the full ramifications of some intonation schemes can appear rather threatening. Fortunately, all the hard work, both theoretical and practical, has been done—over and over, in fact—and we are in a position to reap the benefit. It is the purpose of this short chapter to introduce some of the basic concepts and terms, Tuning and Temperament 369 provide some historical context, and serve as a guide to the copious resources already available to the performer. Central to tuning theory is the idea of ratio or proportion. Before the nineteenth century, the intervallic ratios were understood in terms of string lengths on the monochord; more recently they have come to be understood in terms of vibration frequencies. Fortunately, the ratios themselves are the same, only inverted. Thus the octave, produced by a 2:1 ratio of string lengths, is also produced by a 1:2 ratio of frequencies ; the fifth can be thought of as either 3:2 or 2:3, and the fourth as 4:3 or 3:4. All that really matters is consistency: choose one form or the other and stay with it, at least in any one calculation. Remember, too, that in adding ratios, one multiplies; in subtracting, one divides. Thus, adding a fifth (3:2) to a fourth (4:3), we get a ratio of 12:6 (3 × 4: 2 × 3), which reduces to 2:1—the ratio of an octave. Subtracting a fourth from a fifth should give us a major second; as with fractions, dividing by a ratio is the same as multiplying by its reciprocal (i.e., inversion), so that 3:2 multiplied by 3:4 (the reciprocal of the fourth, 4:3) gives us 9:8 as the ratio of a major second. (There is, incidentally, some research to suggest that this subtraction of a fourth from a fifth is pretty much what our brains are doing naturally and subconsciously to determine the size of a second.) It was recognized from ancient times that a stack of six major seconds—a wholetone scale, if you will—exceeds an octave by a fractional amount, equaling about an eighth of a tone. (This discrepancy is called a “comma”—a “ditonic” or “Pythagorean ” comma, to be exact.) This small interval can be divided up and distributed equally along the chain of ascending fifths and descending fourths comprising the octave, all without most listeners being any the wiser; this, of course, is exactly what...

Share