Mersenne's laws arelaws describing thefrequency ofoscillation of a stretchedstring ormonochord,^[1] useful inmusical tuning andmusical instrument construction.

The equation was first proposed by French mathematician and music theoristMarin Mersenne in his 1636 workHarmonie universelle.^[2] Mersenne's laws govern the construction and operation ofstring instruments, such aspianos andharps, which must accommodate the total tension force required to keep the strings at the proper pitch. Lower strings are thicker, thus having a greatermass per length. They typically have lowertension. Guitars are a familiar exception to this: string tensions are similar, for playability, so lower string pitch is largely achieved with increased mass per length.^{[note 1]} Higher-pitched strings typically are thinner, have higher tension, and may be shorter. "This result does not differ substantially fromGalileo's, yet it is rightly known as Mersenne's law," because Mersenne physically proved their truth through experiments (while Galileo considered their proof impossible).^[3] "Mersenne investigated and refined these relationships by experiment but did not himself originate them".^[4] Though his theories are correct, his measurements are not very exact, and his calculations were greatly improved byJoseph Sauveur (1653–1716) through the use ofacoustic beats andmetronomes.^[5]

Thenatural frequency is:

• a) Inverselyproportional to thelength of the string (the law of Pythagoras^[1]),
• b) Proportional to thesquare root of the stretching force, and
• c) Inversely proportional to the square root of themass per length.

${\displaystyle f_0\propto \frac{1}{L}.}$ (equation 26)

${\displaystyle f_0\propto \sqrt{F}.}$ (equation 27)

${\displaystyle f_0\propto \frac{1}{\sqrt{\mu }}.}$ (equation 28)

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 length by the inverse square (1/4).

These laws are derived from Mersenne's equation 22:^[6]

${\displaystyle f_0=\frac{\nu }{\lambda }=\frac{1}{2L}\sqrt{\frac{F}{\mu }}.}$

Theformula for thefundamental frequency is:

${\displaystyle f_0=\frac{1}{2L}\sqrt{\frac{F}{\mu }},}$

wheref is the frequency,L is the length,F is the force andμ is the mass per length.

Similar laws were not developed for pipes and wind instruments at the same time since Mersenne's laws predate the conception ofwind instrument pitch being dependent on longitudinal waves rather than "percussion".^[3]

• Cycloid
• Long-string instrument
• Overtone
• Standing wave

• Media related toMersenne's laws at Wikimedia Commons

• Musical tuning
• Empirical laws
• Eponymous rules

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• Short description matches Wikidata
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