World Library  
Flag as Inappropriate
Email this Article

Mersenne's laws

Article Id: WHEBN0010897780
Reproduction Date:

Title: Mersenne's laws  
Author: World Heritage Encyclopedia
Language: English
Subject: Acoustics, Mersenne, Musical tuning, Great Internet Mersenne Prime Search, Mel scale
Publisher: World Heritage Encyclopedia

Mersenne's laws


Mersenne's laws are laws describing the frequency of oscillation of a stretched string.[1] 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.


The fundamental frequency is:

  • a) Inversely proportional to the length of the string,
  • 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_1 \propto \tfrac{1}{L}. (equation 26)
f_1 \propto \sqrt{F}. (equation 27)
f_1 \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 one would need either to halve its length, or to increase the tension by a factor of 4, or to decrease its mass per unit length by a factor of 4.

These laws are derived from Mersenne's equation 22:[3]

f_1 = \frac{\nu}{\lambda} = \frac{1}{2L}\sqrt{\frac{F}{\mu}}.

The formula for the fundamental frequency is:


where f is the frequency, L is the length, F is the force and μ is the mass per unit length.

See also


  1. ^ Jeans, James H. (1968). Science and Music, . Dover, ISBN 978-0486619644. Cited in "Mersenne's Laws",
  2. ^ Mersenne, Marin (1637). Traité de l'harmonie universelle, . via the Bavarian State Library. Cited in "Mersenne's Laws",
  3. ^ Steinhaus, Hugo (1999). Mathematical Snapshots, . Dover, ISBN 9780486409146. Cited in "Mersenne's Laws",

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.