Working with Cents: A Survey
Fred Lieberman · Ethnomusicology · 1971
The cent, defined by Alexander Ellis as 0.01 of an equal-tempered semitone, has proven to be a useful tool for comparing musical intervals. Information on the mathematics and history of logarithmic representations of frequency ratios, and on applications of the cents system, is readily available and need not be reviewed here (Apel 1969; Husmann 1952). The purpose of the present article is to compare the most common methods for the calculation of cents, and determine their relative accuracy and convenience. The necessity for this became apparent when, in the course of preparing a new computer-generated cents table (Lieberman and Larrabee 1970), I noticed errors of significant magnitude in influential and widely disseminated ethnomusicological sources. For most musical purposes it is sufficient to state figures to the nearest whole cent, rounding off (not simply dropping) fractions. It should be noted that in calculations involving a sequence of operations (for instance in calculating a scale or cyclic tuning system) what begin as small errors can easily cumulate into much larger ones. Fritz A. Kuttner found that inconsistencies on the order of 3 cents . .. are much too great to furnish useful foundations for the interpretation of tonal systems. (1953:3) Therefore it is not only sufficient but also necessary to state cents accurately to the nearest whole cent, and only those methods which produce this degree of accuracy should be employed-thus insuring that published results can be confidently used by future researchers. The following survey evaluates accuracy and convenience for various cents calculation methods; the reader will then be able to choose the one most suited to his particular need. As an example for comparison a single arbitrarily chosen frequency ratio will be converted into cents. The ratio x:y will be assumed to be 756:546; the correct figure for this ratio, to the nearest 0.01 cent, is 563.38 C.