Pythagorean Paper Folding: A Study in Tuning and Temperament

IN TUNING the tones of a scale the ear wrestles with some fascinating prob lems concerning vibration ratios. This paper describes a convenient method for displaying to the eye the mathematical relationships involved.1 The proportions embodied in musical intervals and the essential features of three tuning systems?Pythagorean, Just (or Pure), and Equal Temperament?can be demonstrated by a succession of paper folding operations. Musicians may be de lighted?and mathematicians amused?to discover that even such fabulous numbers as the "Pythagorean comma" of ratio 524288/531441 or "the twelth root of 2" 0V2)? generally simplified to 1.059463+ ?will exhibit their meaning with as much accuracy as anyone deserves?that is, with as much as he can hear?after a rela tively few folds. The procedure outlined below approaches the elegance of geom etry by relying upon proportional division rather than on mathematical computation. The folding procedure is a variant of the old monochord demonstration in which frequency is inversely proportional to string length; a strip of paper?calculating machine tape proves convenient?is sub stituted for the vibrating string. If a