The Mathematics of Musical Instruments

Abstract

1. INTRODUCTION The history of musical instruments goes back tens of thousands of years. Fragments of bone flutes and whistles have been found at Neanderthal sites. Recently, a 9,000-year-old flute found in China was shown to be the world’s oldest playable instrument (pictures and a recording of this flute are available at http://www.bnl.gov/bnlweb/flutes.html). These early instruments show that people have long been concerned with producing pitched sound—that is, sound containing predominantly a single frequency. Indeed, finger holes on the flutes indicate that these prehistoric musicians had some concept of a musical scale. The study of the mathematics of musical instruments dates back at least to the Pythagoreans, who discovered that certain combinations of pitches that they considered pleasing corresponded to simple ratios of frequencies such as 2:1 and 3:2. The problems of tuning, temperament, and acoustics have since occupied some the brightest minds in the natural sciences. Marin Marsenne’s treatise on tuning and acoustics Harmonie Universelle (1636) [19], H. v. Helmholtz’s On the Sensations of Tone (1870) [15], and Lord Rayleigh’s seminal The Theory of Sound (1877) [21 ]a re just three outstanding examples. Many pages have been written on this subject. We mean to present an overview and let the interested reader find more detailed discussions in the references, and on our web site www.sju.edu/∼rhall/newton. 2. THE WILLOW FLUTE In this section we consider the physical properties of a Norwegian folk flute called the seljefloyte, or willow flute. This instrument can be considered “primitive” in that it does not rely on finger holes to produce different pitches. Rather, by varying the strength with which he or she blows into the flute, the player selects from a series of pitches called harmonics, whose frequencies are integer multiples of the flute’s lowest tone, called the fundamental. The willow flute’s scale is approximately a major scale with a sharp fourth and flat sixth, and plus a flat seventh. The willow flute is a member of the recorder family, though it is held transversally. The flute is constructed from a hollow willow branch (or, more recently, a PVC pipe; see the quirky but informative web site http://www.geocities.com/SoHo/Museum/