Mersenne's laws are

laws Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...

describing the

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

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 pendul ...

of a stretched string or

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 h ...

, useful in

musical tuning In music, there are two common meanings for tuning: * Tuning practice, the act of tuning an instrument or voice. * Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases. Tuning practice Tun ...

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 String instruments, stringed instruments, or chordophones are musical instruments that produce sound from vibrating strings when a performer plays or sounds the strings in some manner. Musicians play some string instruments by plucking the st ...

, such as

piano The piano is a stringed keyboard instrument in which the strings are struck by wooden hammers that are coated with a softer material (modern hammers are covered with dense wool felt; some early pianos used leather). It is played using a keyboa ...

s and

harp The harp is a stringed musical instrument that has a number of 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 orc ...

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 property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...

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) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...

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

metronome A metronome, from ancient Greek μέτρον (''métron'', "measure") and νομός (nomós, "custom", "melody") is a device that produces an audible click or other sound at a regular interval that can be set by the user, typically in beats pe ...

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

Equations

The

natural frequency Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...

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 distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...

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 ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...

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'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...

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

* Musical tuning Empirical laws Eponyms {{Music-theory-stub

Overview

Equations

See also

Notes

References

External links