
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 vario ...
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 ...
of
oscillation
Oscillation is the repetitive or Periodic function, periodic variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. Familiar examples o ...
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 ...
or
monochord,
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
Tuni ...
and
musical instrument construction
A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. A person who pl ...
.
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 musica ...
s 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
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 element ...
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'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 beat
In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, ''perceived'' as a periodic variation in volume whose rate is the difference of the two frequencies.
With tuning instruments that can produce ...
s 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 p ...
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
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 element ...
per length.
:
(equation 26)
:
(equation 27)
:
(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''.
:
The
formula 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:
:
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".
See also
*
Cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another ...
*
Long-string instrument
The long-string instrument is a musical instrument in which the string is of such a length that the fundamental transverse wave is below what a person can hear as a tone (±20 Hz). If the tension and the length result in sounds with such ...
*
Overtone
An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
*
Standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
Notes
References
External links
*
Musical tuning
Empirical laws
Eponyms
{{Music-theory-stub