# Mersenne's laws

Mersenne's laws are laws describing the frequency of oscillation of a stretched string or monochord,[1] 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.[2] 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. 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 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).[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 by Joseph Sauveur (1653–1716) through the use of acoustic beats and metronomes.[5]

## Notes

1. Mass is typically added by increasing cross-section area. This increases the string's force constant (k). Higher k doesn't affect pitch per se, but fretting a string stretches it in addition to shortening it, and the pitch increase due to stretching is larger for higher k values. Thus intonation requires more compensation for lower strings, and (markedly) for steel vs nylon. This effect still applies to strings where mass is increased with windings, albeit to a lesser extent, because the core that supports string tension generally needs to be larger to support larger masses of winding.

## Equations

The fundamental frequency is:

• a) Inversely proportional to the length of the string (the law of Pythagoras[1]),
• b) Proportional to the square root of the stretching force, and
• c) Inversely proportional to the square root of the mass per unit length.
${\displaystyle f_{0}\propto {\tfrac {1}{L}}.}$ (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 unit length by the inverse square (1/4).

HarmonicsLength,Tension,or Mass
1 1 1 1
2 1/2 = 0.5 2² = 4 1/2² = 0.25
3 1/3 = 0.33 3² = 9 1/3² = 0.11
4 1/4 = 0.25 4² = 16 1/4² = 0.0625
8 1/8 = 0.125 8² = 64 1/8² = 0.015625

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

${\displaystyle f_{0}={\frac {\nu }{\lambda }}={\frac {1}{2L}}{\sqrt {\frac {F}{\mu }}}.}$

The formula for the fundamental frequency is:

${\displaystyle f_{0}={\frac {1}{2L}}{\sqrt {\frac {F}{\mu }}},}$

where f is the frequency, L is the length, F is the force and μ is the mass per unit length.

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