Mersenne's laws

A string half the length (1/2), square root the tension (√2), or inverse the mass per length (1/√2) is an octave higher (2/1).
If the tension on a string is ten lbs., it must be increased to 40 lbs. for a pitch an octave higher.^[1]

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^[1]

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. The equation was first proposed by French mathematician and music theorist Marin Mersenne in his 1637 work Traité de l'harmonie universelle.^[2] Mersenne's laws govern the construction and operation of string instruments, pianos, 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 unit length. They typically have lower tension. 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]

Equations

The fundamental 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 unit 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 unit 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:^[5]

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 unit 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]

References

1 2 3 4 Jeans, James Hopwood (1937/1968). Science & Music, p.62-4. Dover. ISBN 0-486-61964-8. Cited in "Mersenne's Laws", Wolfram.com
↑ Mersenne, Marin (1637). Traité de l'harmonie universelle, . via the Bavarian State Library. Cited in "Mersenne's Laws", Wolfram.com.
1 2 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", p.185-187.
↑ Steinhaus, Hugo (1999). Mathematical Snapshots, . Dover, ISBN 9780486409146. Cited in "Mersenne's Laws", Wolfram.com.

Acoustics

Engineering	Architectural acoustics Monochord Reverberation Soundproofing String vibration string resonance	Spectrogram

Psychoacoustics	Bark scale Combination tone Equal-loudness contour Fletcher–Munson curves Mel scale Missing fundamental

Frequency Pitch	Beat Formant Fundamental frequency Frequency spectrum harmonic spectrum Harmonic series inharmonicity Overtone Resonance Standing wave Node Subharmonic

Acousticians	John Backus Jens Blauert Ernst Chladni Hermann von Helmholtz Franz Melde Werner Meyer-Eppler Lord Rayleigh Joseph Sauveur D. Van Holliday Thomas Young

Related topics	Echo Infrasound Sound Ultrasound Mersenne's laws Musical acoustics piano violin

Marin Mersenne

Mersenne conjectures Mersenne's laws Mersenne prime Double Mersenne number Great Internet Mersenne Prime Search Mersenne Twister

Frequency and pitch

Notation	Concert pitch Enharmonic Helmholtz pitch notation Letter notation Piano key frequencies Pitch class Scientific pitch notation

Perception	Absolute pitch Ear training Pitch circularity Relative pitch Tonal memory Tone deafness Virtual pitch

See also	Electronic tuner Mersenne's laws Microtuner Musical tuning Pitch pipe Tuning fork

Musical strings, wires, and instruments (list/HS list)

Bow Bridge Third bridge Chordophone Course Drone Enharmonicity Fingerboard Fret Fundamental/Harmonics/Overtones Longitudinal wave Long-string instrument Melde's experiment Mersenne's laws Monochord Music On A Long Thin Wire Node Re-entry Scale length Soundboard Standing wave String vibration Transverse wave Tuning peg

Monochords and musical bows	Berimbau Bladder fiddle Boom-ba Đàn bầu Diddley bow Duxianqin Ektara Ground bow Ichigenkin Japanese fiddle Jaw harp Đàn môi Genggong Gogona Morsing Mukkuri Langeleik Lesiba Masenqo Onavillu Psalmodicon Tromba marina Tumbi Umuduri Washtub bass

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Mersenne's laws

Equations

See also

References