What is the relationship between classical music and mathematics?

Ed: The journal articles cited below are mostly pulled from the eDatabase, JSTOR. You can access this database for FREE, if you are a NLB Digital Library Member. Registration is FREE. For details, please visit http://eresources.nlb.gov.sg/.

To the answer proper then.

To say the least, this is a complex question on which debates and discussions continue to wage today. Based on what I’ve read about this topic, I think one can conclude that while music and mathematics share similarities and may be tightly-related, there is no proven, direct link between the two subjects.

To begin, let’s look at how H.L.F. Helmhotz describes music and mathematics as highly contrasting, yet inseparable “intellectual activities”:

Mathematics and Music, the most sharply contrasted fields of intellectual activity which one can discover, and yet bound together, supporting one another as if they would demonstrate the hidden bond which draws together all activities of our mind, and which also in the revelations of artistic genius leads us to surmise unconscious expressions of a mysteriously active intelligence (qtd. in Archibald 1).

Bonded they may be, but note how Helmhotz characterises this tie between the two subjects as a “hidden bond”. One might then say that mathematics and music are not exactly one and the same, and while somehow seemingly related, the link might perhaps be tenuous at best.

We only have to refer to Guy Warrack’s wry observation on this supposition to understand why the study of one does not necessarily mean you’ll excel in the other:

How often has it been said in conversation that ‘Music and Mathematics go together’? It is certainly no more absurd than most other such generalizations, but it is certainly no less so . . . Certainly many mathematicians are interested in music, and many musicians in mathematics, but that proves nothing. It would be easy to point to at least as many tone-deaf mathematicians and to as many musicians whose idea of mathematics is making an income-tax return ([and] probably getting it wrong) (Warrack 21).

What then, are we left with when we talk about Music and Mathematics? Professor Thomas Fiore gives a good suggestion:

Music and Mathematics are intricately related [but there] is not an equation that will model all works of music and we should not spend time looking for it. Nevertheless, there are certain mathematical structures inherent in all works of music, and these mathematical structures are not given by equations. The language of mathematics is a convenient tool for comprehending and communicating this underlying structure . . . [so as] to find a good way to hear a piece of music and to communicate that way of hearing (Fiore 5).

Needless to say, everyone will have his or her own ideas how best to listen to a piece of music, music theorists, mathematicians and musicians included.

Here then, is a list of selected journal articles that explore the relationship between music and mathematics, between art and science, between structure and flow within the enclosure of our minds:

[i]
Boettcher,Wendy S., et al. “Mathematics and Music: A Search for Insight into Higher Brain Function.” Leonardo Music Journal 4 (1994): 53-58. http:// www.jstor.org/stable/1513181. 16 Dec 2008.

Abstract:
A profound problem of historical origin and continuing interest is the relationships and similarities among higher brain functions relating to music, mathematics and chess. The authors present the results of detailed interviews with 14 professors of mathematics concerning their research and possible relationships between mathematics and music. The results provide interesting insights into higher brain function, suggesting a common theme of pattern development. In addition, the authors propose some new behavioral experiments.

[ii]
Hsu, Kenneth J., et al. “Fractal Geometry of Music.” Proceedings of the National Academy of Sciences of the United States of America 87.3 (Feb 1990): 938-941. http:// www.jstor.org/stable/2353742. 16 Dec 2008.

Abstract:
Music critics have compared Bach’s music to the precision of mathematics. What “mathematics” and what “precision” are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot’s fractal geometry and the precision is the deviation from a log-log linear plot.

[iii]
Vaughn, Kathryn. “Music and Mathematics: Modest Support for the Oft-Claimed Relationship.” Journal of Aesthetic Education 34.3/4. 2000: 149-166. http:// www.jstor.org/stable/3333641. 16 Dec 2008.

Librarian’s note:
This article investigates the relationship between music and mathematics based on the supposition that “if music is based on mathematical principles, and if an understanding of music requires some understanding of these principles, then it is possible that music education can lead to an improved understanding.” Its focus is on the relation between music and mathematics in education.

[iv]
Budden, F. J. “Modern Mathematics and Music.” The Mathematical Gazette 51.377 (Oct 1967): 204-215. http:// www.jstor.org/stable/3613237. 16 Dec 2008.

Librarian’s note:
Mr. Budden talks music with mathematical formula. A technical discussion of music and mathematics that should be rewarding for readers with an aptitude for both. Which basically means not me, heh heh.

[v]
Henle, Jim. “Classical Mathematics.” The American Mathematical Monthly 103.1 (Jan 1996): 18-29. http:// www.jstor.org/stable/2975210. 16 Dec 2008.

Librarian’s note:
An interesting treatise that argues that it is not music that is mathematical; instead it is mathematics that is musical.

[vi]
Siddharthan, Rahul. “Music, Mathematics and Bach.” Indian Institute of Science. http://www.iisc.ernet.in/. 16 Dec 2008.

Librarian’s note:
Mr. Siddharthan delves into the science of music with Bach as his focus.

List of Works Cited

• Archibald, R. C. . “Mathematicians and Music.” The American Mathematical Monthly 31.1 (Jan 1924): 1-25. http:// www.jstor.org/stable/2298868. 16 Dec 2008.

• Fiore, Thomas M. “Music and Mathematics.” Lectures on Mathematical Music Theory. http://www.math.uchicago.edu/~fiore/1/musictotal.pdf. 16 Dec 2008.

• Warrack, Guy. “Music and Mathematics.” Music & Letters 26.1 (Jan 1945): 21-27. http:// www.jstor.org/stable/727804. 16 Dec 2008.

Note:

It isn’t quite possible for me to go into the historical breadth and depth of analysis that has been devoted to this question, so for a good overview, do consider looking up the entry, “Numbers and music” in The New Grove Dictionary of Music and Musicians, Vol. 18. This is available at the library@esplanade.

Title: The New Grove Dictionary of Music and Musicians
Editor: Stanley Sadie
Call No.: RART q780.3 NEW v. 18
For Reference Only
Location: Arts Central

Another useful book for further reading would be Leon Harkleroad’s The Math Behind the Music:

Title: The Math Behind the Music
Author: Leon Harkleroad
Call No.: R 510 HAR
For Reference Only at Lee Kong Chian Reference Library