Mersenne's laws

Mersenne's laws are laws describing the frequency of oscillation of a stretched string or monochord, useful in musical tuning and musical instrument construction. The equation was first proposed by French mathematician and music theorist Marin Mersenne in his 1637 work Traité de l'harmonie universelle. Mersenne's laws govern the construction and operation of string instruments, such as pianos and harps, which must accommodate the total tension force required to keep the strings at the proper pitch. Lower strings are thicker, thus having a greater mass per unit length. They typically have lower tension. Higher-pitched strings typically are thinner, have higher tension, and may be shorter. "This result does not differ substantially from Galileo's, yet it is rightly known as Mersenne's law," because Mersenne physically proved their truth through experiments (while Galileo considered their proof impossible). "Mersenne investigated and refined these relationships by experiment but did not himself originate them". Though his theories are correct, his measurements are not very exact, and his calculations where greatly improved by Joseph Sauveur (1653–1716) through the use of acoustic beats and metronomes.

The fundamental frequency is:

Thus, for example, all other properties of the string being equal, to make the note one octave higher (2/1) one would need either to decrease its length by half (1/2), to increase the tension to the square (4), or to decrease its mass per unit length by the inverse square (1/4).

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