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.

Overview

The equation was first proposed by French mathematician and music theorist Marin Mersenne in his 1636 work 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 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]

Equations

The natural 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 length.

f_{0}\propto {\tfrac {1}{L}}.

(equation 26)

f_{0}\propto {\sqrt {F}}.

(equation 27)

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).

Harmonics	Length,	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]

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

The formula for the fundamental frequency is:

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 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]

Notes

^ 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.

References

^ ^a ^b ^c ^d Jeans, James Hopwood (1937/1968). Science & Music, pp.62-4. Dover. ISBN 0-486-61964-8. Cited in "Mersenne's Laws", Wolfram.com
^ Mersenne, Marin (1636). Harmonie universelle^{[page needed]}. Cited in "Mersenne's Laws", Wolfram.com.
^ ^a ^b Cohen, H.F. (2013). Quantifying Music: The Science of Music at the First Stage of Scientific Revolution 1580–1650, p.101. Springer. ISBN 9789401576864.
^ Gozza, Paolo; ed. (2013). Number to Sound: The Musical Way to the Scientific Revolution, p.279. Springer. ISBN 9789401595780. Gozza is referring to statements by Sigalia Dostrovsky's "Early Vibration Theory", pp.185-187.
^ Beyer, Robert Thomas (1999). Sounds of Our Times: Two Hundred Years of Acoustics. Springer. p.10. ISBN 978-0-387-98435-3.
^ Steinhaus, Hugo (1999). Mathematical Snapshots^{[page needed]}. Dover, ISBN 9780486409146. Cited in "Mersenne's Laws", Wolfram.com.

External links

Media related to Mersenne's laws at Wikimedia Commons

[3] 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.

[Jeans-1] Jeans, James Hopwood (1937/1968). Science & Music, pp.62-4. Dover. ISBN 0-486-61964-8. Cited in "Mersenne's Laws", Wolfram.com

[mersenne-2] Mersenne, Marin (1636). Harmonie universelle^{[page needed]}. Cited in "Mersenne's Laws", Wolfram.com.

[Cohen-4] Cohen, H.F. (2013). Quantifying Music: The Science of Music at the First Stage of Scientific Revolution 1580–1650, p.101. Springer. ISBN 9789401576864.

[Gozza-5] Gozza, Paolo; ed. (2013). Number to Sound: The Musical Way to the Scientific Revolution, p.279. Springer. ISBN 9789401595780. Gozza is referring to statements by Sigalia Dostrovsky's "Early Vibration Theory", pp.185-187.

[6] Beyer, Robert Thomas (1999). Sounds of Our Times: Two Hundred Years of Acoustics. Springer. p.10. ISBN 978-0-387-98435-3.

[Steinhaus-7] Steinhaus, Hugo (1999). Mathematical Snapshots^{[page needed]}. Dover, ISBN 9780486409146. Cited in "Mersenne's Laws", Wolfram.com.

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