View source for Mersenne's laws

{{Short description|Laws describing the frequency of oscillation of a stretched string}}
[[File:Mersenne's laws.png|300px|thumb|right|A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1).<br/> If the tension on a string is ten lbs., it must be increased to 40 lbs. for a pitch an octave higher.<ref name="Jeans"/>]]
[[File:Monochord Jeans.png|thumb|300px|A string, tied at ''A'', is kept in tension by ''W'', a suspended weight, and two bridges, ''B'' and the movable bridge ''C'', while ''D'' is a freely moving wheel; all allowing one to demonstrate Mersenne's laws regarding tension and length<ref name="Jeans">[[James Hopwood Jeans|Jeans, James Hopwood]] (1937/1968). ''Science & Music'', pp.62-4. Dover. {{ISBN|0-486-61964-8}}. Cited in "[http://scienceworld.wolfram.com/physics/MersennesLaws.html Mersenne's Laws]", ''[[Wolfram Research|Wolfram.com]]''</ref>]]

'''Mersenne's laws''' are [[scientific law|laws]] describing the [[frequency]] of [[oscillation]] of a stretched [[Strings (music)|string]] or [[monochord]],<ref name="Jeans"/> 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]]''.<ref name="mersenne">Mersenne, Marin (1636). [https://gallica.bnf.fr/ark:/12148/bpt6k5471093v ''Harmonie universelle'']{{page needed|date=October 2013}}. Cited in "[http://scienceworld.wolfram.com/physics/MersennesLaws.html Mersenne's Laws]", ''[[Wolfram Research|Wolfram.com]]''.</ref> Mersenne's laws govern the construction and operation of [[string instruments]], such as [[piano]]s and [[harp]]s, 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 (physics)|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.{{notetag|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.}} Higher-pitched strings typically are thinner, have higher tension, and may be shorter. "This result does not differ substantially from [[Galileo Galilei|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<!--how does one count something that is too fast for one to see?-->).<ref name="Cohen">Cohen, H.F. (2013). ''Quantifying Music: The Science of Music at the First Stage of Scientific Revolution 1580–1650'', p.101. Springer. {{ISBN|9789401576864}}.</ref> "Mersenne investigated and refined these relationships by experiment but did not himself originate them".<ref name="Gozza">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.</ref> 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 beat]]s and [[metronome]]s.<ref>Beyer, Robert Thomas (1999). ''Sounds of Our Times: Two Hundred Years of Acoustics''. Springer. p.10. {{ISBN|978-0-387-98435-3}}.</ref>

==Equations==
The [[fundamental frequency|natural frequency]] is:

*a) Inversely [[Proportionality (mathematics)|proportion]]al to the [[length]] of the string (the law of Pythagoras<ref name="Jeans"/>),
*b) Proportional to the [[square root]] of the stretching force, and
*c) Inversely proportional to the square root of the [[mass]] per length.

:<math> f_0 \propto \tfrac{1}{L}.</math> (equation 26)

:<math> f_0 \propto \sqrt{F}.</math> (equation 27)

:<math>  f_0 \propto \frac{1}{\sqrt{\mu}}.</math> (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).
{| {{Table}}
! Harmonics !! Length, !! Tension, !! or Mass
|-
! 1
| 1
| 1
| 1
|-
! 2
| 1/2 = 0.5
| 2² = 4
| 1/2² = 0.25
|-
! 3
| 1/3 = 0.{{repitan|33}}
| 3² = 9
| 1/3² = 0.{{repitan|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:<ref name="Steinhaus">Steinhaus, Hugo (1999). ''Mathematical Snapshots''{{page needed|date=October 2013}}. Dover, {{ISBN|9780486409146}}. Cited in "[http://scienceworld.wolfram.com/physics/MersennesLaws.html Mersenne's Laws]", ''[[Wolfram Research|Wolfram.com]]''.</ref>
:<math> f_0 = \frac{\nu}{\lambda} = \frac{1}{2L}\sqrt{\frac{F}{\mu}}.</math>

The [[formula]] for the [[fundamental frequency]] is:
:<math> f_0=\frac{1}{2L}\sqrt{\frac{F}{\mu}}, </math>

where ''f'' is the frequency, ''L'' is the length, ''F'' is the force and ''&mu;'' 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#Physics of sound production|wind instrument pitch]] being dependent on longitudinal waves rather than "percussion".<ref name="Cohen"/>

==See also==
*[[Cycloid]]
*[[Long-string instrument]]
*[[Overtone]]
*[[Standing wave]]

==Notes==
{{notefoot}}

==References==
{{reflist}}

==External links==
*{{Commons category-inline}}

{{Acoustics}}
{{Mersenne}}
{{Pitch (music)}}
{{Strings (music)}}

[[Category:Musical tuning]]
[[Category:Empirical laws]]
[[Category:Eponyms]]