Mersenne's laws are

laws Law is a set of rules that are created and are law enforcement, enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a Socia ...

describing the

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

of a stretched

string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...

monochord A monochord, also known as sonometer (see below), is an ancient musical and scientific laboratory instrument, involving one (mono-) string ( chord). The term ''monochord'' is sometimes used as the class-name for any musical stringed instrument ...

, useful in

musical tuning In music, there are two common meanings for tuning: * #Tuning practice, Tuning practice, the act of tuning an instrument or voice. * #Tuning systems, Tuning systems, the various systems of Pitch (music), pitches used to tune an instrument, and ...

and musical instrument construction.

Overview

The equation was first proposed by French mathematician and music theorist

Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...

in his 1636 work '' Harmonie universelle''.Mersenne, Marin (1636)
''Harmonie universelle''
Cited in

, '' Wolfram.com''. Mersenne's laws govern the construction and operation of

string instruments In musical instrument classification, string instruments, or chordophones, are musical instruments that produce sound from vibrating strings when a performer strums, plucks, strikes or sounds the strings in varying manners. Musicians play some ...

, such as

piano A piano is a keyboard instrument that produces sound when its keys are depressed, activating an Action (music), action mechanism where hammers strike String (music), strings. Modern pianos have a row of 88 black and white keys, tuned to a c ...

s and

harp The harp is a stringed musical instrument that has individual strings running at an angle to its soundboard; the strings are plucked with the fingers. Harps can be made and played in various ways, standing or sitting, and in orchestras or ...

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 Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

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. Higher-pitched strings typically are thinner, have higher tension, and may be shorter. "This result does not differ substantially from

Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...

's, yet it is rightly known as Mersenne's law," because Mersenne physically proved their truth through experiments (while Galileo considered their proof impossible).Cohen, H.F. (2013). ''Quantifying Music: The Science of Music at the First Stage of Scientific Revolution 1580–1650'', p.101. Springer. . "Mersenne investigated and refined these relationships by experiment but did not himself originate them".Gozza, Paolo; ed. (2013). ''Number to Sound: The Musical Way to the Scientific Revolution'', p.279. Springer. . Gozza is referring to statements by Sigalia Dostrovsky's "Early Vibration Theory", pp.185-187. Though his theories are correct, his measurements are not very exact, and his calculations were greatly improved by

Joseph Sauveur Joseph Sauveur (; 24 March 1653 – 9 July 1716) was a French mathematician and physicist. He was a professor of mathematics and in 1696 became a member of the French Academy of Sciences. Life Joseph Sauveur was born in La Flèche, the son of a ...

(1653–1716) through the use of acoustic beats and

metronome A metronome () is a device that produces an audible click or other sound at a uniform interval that can be set by the user, typically in beats per minute (BPM). Metronomes may also include synchronized visual motion, such as a swinging pendulum ...

s.Beyer, Robert Thomas (1999). ''Sounds of Our Times: Two Hundred Years of Acoustics''. Springer. p.10. .

Equations

The

natural frequency Natural frequency, measured in terms of '' eigenfrequency'', is the rate at which an oscillatory system tends to oscillate in the absence of disturbance. A foundational example pertains to simple harmonic oscillators, such as an idealized spring ...

is: *a) Inversely proportional to the

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

of the string (the law of Pythagoras), *b) Proportional to the

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

of the stretching force, and *c) Inversely proportional to the square root of the

per length. :

f_0 \propto \tfrac.

(equation 26) :

f_0 \propto \sqrt.

(equation 27) :

f_0 \propto \frac.

(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:Steinhaus, Hugo (1999). ''Mathematical Snapshots''. Dover, . Cited in
Mersenne's Laws
, '' Wolfram.com''. :

f_0 = \frac = \frac\sqrt.

The

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

for the

fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...

is: :

f_0=\frac\sqrt,

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".

Notes

References

External links

* {{Strings (music) Musical tuning Empirical laws Eponymous rules

Overview

Equations

See also

Notes

References

External links