Early Musical Acoustics

All things which can be known have number; for it is not possible that without number anything can be either conceived or known.

-- Philolaus (ca. 400 BC)

Vibrating strings were studied by the Pythagoreans (6th-5th century
BC). Pythagorus noticed that harmonics were produced by dividing the
string length by whole numbers, and he was interested in understanding
consonant pitch intervals in terms of simple ratios of string lengths.
``Harmony theory'' from the
Pythagoreans was taught throughout the Middle Ages as one of the seven
liberal arts: the *quadrivium*, consisting of arithmetic,
geometry, astronomy, and music (harmony theory); and
the *trivium*, consisting of grammar, logic, and
rhetoric [414]. The correspondence between musical pitch
and frequency of physical vibration was not discovered until the
seventeenth century [113].

It took until Galileo (1564-1642) to be free of the formulation of
Aristotle (384-322 BC) that all motion required an ongoing applied
force, thereby opening the way for modern differential equations of
motion. The ideas of Galileo were formalized and extended by Newton
(1642-1727), whose famous second law of motion ``
'' lies at the
foundation of essentially all classical mechanics and acoustics.
Newton's *Principia* (1686) describes sound as traveling
pressure pulses, and single-frequency sound waves were analyzed.

The first to publish a one-dimensional wave equation for the vibrating
string was the applied mathematician Jean Le Rond d'Alembert
(1717-1783) [100,103].^{A.1}The 1D wave equation can be written as

where

is time in seconds, and denotes position along the string. (See discussion in §C.1.) It can be derived directly from Newton's second law applied to a differential string element by equating the string restoring force about a small element (first described by Brook Taylor in 1713) to Newton's mass times acceleration of that element, which is how d'Alembert in fact proceeded in his derivation [103]. In addition to introducing the 1D wave equation as an equation among differentials, d'Alembert derived its solution in terms of traveling waves:

where denotes the wave propagation speed. Earlier, in 1717, Brook Taylor (1685-1731)

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University