Here are some collections of information about the interplay between music and mathematics. I have collected a bibliography of such items as they float past me in cyberspace, and I went hunting for references in the reviewing journal Mathematical Reviews for mentions of music. In addition I am interested in a few other specific topics.

Special note to students: I am delighted that so many people seem to find this page when they are considering illustrating the connections between music and mathematics. Evidently this relationship has been explored in quite a few Science Fair projects; it is the basis for a number of classroom modules prepared by student teachers; it is the theme for essays in many general-studies courses in music or mathematics departments. At least, so I must surmise from the flow of mail!

Yes, I am pleased that so many people find this topic interesting and I hope to hear about novel angles from which you will study it. But please, before you write to me, make sure you have something specific to tell me or ask me about, and make sure the question you ask isn't already answered here. As much as I like to yak about these ideas, I grow rather testy when I get mail which simply says "Saw your website; can you suggest a science fair topic?", or "Can you find any other way to explain to me why there are 12 notes in a scale? I need an explanation by tomorrow". I'm not hiding anything from you: everything I know is already here -- I can't "send you something else"; and I've already made it as math-free as I can without losing sight of the coolness of the math involved -- I can try to clarify a passage or two but won't "make it all really simple".

Let's do it this way instead: once you have explored some new twist to these ideas, send me a summary or a URL and I'll put it here for others to share and enjoy -- OK? Sample exchanges

Another science fair project: abstract and full paper [LONG -- ZIP archive]

A similar report has been updated by its author and is no longer here at this site but cn be found here

With that in mind: read and enjoy what's here, and write if there's any
*appropriate* comment or question.

A frequently-asked question is, why are there 12 tones in an octave? The answer has to do in part with the nature of sound. (It also has to do with the human perception of sound, a delightful analysis of which I saw somewhere in a discussion of compressing audio data onto audio CDs, but I can't remember where I saw this -- ring any bells with you?) This in turn raises some questions about the theory of chords and so on. (This reference includes a discussion of tuba-playing which, as it happens, is of personal interest to me. There are also further information and some corrections elsewhere.). This information is also useful in piano-tuning. (I do it but I haven't checked these references. Here is more information.)

There are other places to look on the web for readings and samplings of tunings, more from a musician's perspective than a mathematician's. See e.g. John Starrett's Microtonal Music Page, and some comments about tuning and musical "color". More microtonal tuning links are in a letter I received.

Just for fun I tried an alternative tuning of the scale.

- 41tone.pas - Turbo Pascal 3 source to make an AT keyboard into a music keyboard illustrating the division of an octave into 41 tones
- 41tone.com - PC executable of the above
- Mark McConnell has picked up this theme and really run with it.

Odds and ends about tunings:

- A list of the frequencies on an evenly-tuned piano (tuned to an A440)
- Maybe the 12-tone scale (and rhythm patterns) are cultural constructs?
- References to tuning in mathematics; favoring other chords besides fifths
- Review of Barbour article using other continued-fraction techniques
- Tuning problems lead to the Riemann zeta function
- Nineteen-tone tunings also have supporters
- Pointers to recordings in alternative tunings.
- Once more slowly: why are logarithms in here?
- ...and what do you do with the logarithms?
- Any other suggestions for a High School project?
- ...and another attempt to be very basic about "why 12 tones".
- Setting theory aside, how did the twelve tones arise historically?
- discussion about acoustics and tunings through history

Historical Tunings on the Modern Concert Grand

What many suspect seems to be true -- that there is a connection between music and the learning of math. Here is a link to commentary about this connection, from MENC (the (U.S.) National Association for Music Education). There are other studies that have been done, e.g. University of Sarasota Study (Jeffrey Lynn Kluball); East Texas State University Study (Daryl Erick Trent); here is a AMC web page. Even brain researchers are getting into the act.

One of the most publicized links between music and academic subjects (including mathematics) is the "Mozart effect": the claim that exposure to certain types of music -- especially exposure to _early classical_ music, very _early in life_ -- can lead to improved performance on test scores, including tests of spatial visualization, abstract reasoning, and so on.

I cannot comment authoritatively about the validity of this claim. On the one hand, I myself had considerable exposure to music of this type, and indeed do well with mathematical and other abstract tasks; this association seems fairly comon among people of my acquaintance. On the other hand, I am extremely skeptical of proposed "scientific truths" which are discussed in the absence of thorough experimentation, analysis, and corroboration; I am particularly dubious of truths expounded by those who stand to profit from their popularization.

For further reading, you might follow some Internet links regarding the Mozart Effect. Capitalized, it refers to a book by Don Campbell, complete now with its own "Resource Center" and web site. For a contrasting point of view, you might consult, say, Steven Halpern's site or the Skeptic's Dictionary. I was asked to compare the conflicting claims at these sites; here is my response. Perhaps a better response ...

There are a couple of other connections between math and music. Check out this tidbit about Bartok's music and this intermarriage between math and music. (I'm still waiting for someone to set me straight on this one.)

A new twist: mathematical proofs set to music! (No, it's not "Deep Truth", but it is kind of fun!)

Some other additions to the odds-and-ends file:

- Math and Chopin's music; reciprocally, we have
- Music and Birkhoff's math
- What are sharp and flat?
- Appearances of the golden mean in music
- Change-ringing and other campanological aspects of Group Theory
- Why is the harp shaped the way it is? See also other web links about mathematics and the harp.
- Beat frequencies and some hidden trigonometry
- Defining musical styles mathematically
- Psychology of music and other interplays between math and music.
- "I'm great at music but lousy at math; can you help?" -- a somewhat exaggerated attempt to draw parallels between mastering math and mastering music.
- So is it true that music has anything to do with sums of series? (yes)
- A combinatorial partition problem due to Babbitt.
- A student essay on the relationship between math and music.
- Entropy in music (and a link to EigenRadio)

Some good reflections and book reviews on the math-and-music theme.

There are other places on the web to look for math'n'music information; most of the links I could mention are already e.g. here. Here's a discussion of the acoustics of the piano. (On the other hand, I can't help but be flummoxed by the International Society for Mathematical and Computational Aesthetics !)

Just for good measure, I'll throw in a few links to sites which offer something musical (mathematical or not). I should probably start with a link to Music Resources on the Internet and specifically a Welcome to the World of Classical Music I saw a link to some PC software -- at that site you'll find many other competing products; I have no recommendation.

Finally let me suggest this page of Pathways between Mathematics and the Arts for some explorations of the links between mathematics and the visual arts. (There are some more themes there relating mathematics and music as well.)

Other pages at my own web site you may find of interest:

- My home page is just a short introduction to the pages I maintain
- The parent page to this one is another short way-station, this time to a few pages concerning "applications" of mathematics (well, let's call them "investigations only a mathematician would love").
- I keep a large set of pages which provide a thematic introduction to the subject areas of modern mathematics, complete with answers to thousands of more or less common questions.

This page is

http://www.math.niu.edu/~rusin/uses-math/music/index.htmlLast modified 2006/07/29 by Dave Rusin, rusin@math.niu.edu