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It is well-known that there is a relationship between music and mathematics, and there are many references that explore this topic (for example, Benson's book).

However, I would like to ask if there is any current research going on between the relationship between mathematics and music which produced results significative to the development of mathematics (other than to the development of music theory). In case, where can I find a comprehensive survey paper of such results?

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    $\begingroup$ I doubt a comprehensive survey exists. Even to find one good example of what you seek may be a stretch... $\endgroup$ – Joseph O'Rourke Jan 24 '15 at 14:23
  • $\begingroup$ Dear Prof. @JosephO'Rourke, still I really hope that some examples of 'useful' results exist. $\endgroup$ – Dal Jan 24 '15 at 14:42
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    $\begingroup$ You may be interested in the book "The topos of music". It seems that there are very few people in this world who can even assess whether any of it makes sense, but it seems like it may. At some point in my life I would like to tackle it, maybe when I am an old man... $\endgroup$ – Steven Gubkin Jan 24 '15 at 16:35
  • $\begingroup$ @StevenGubkin I have heard of it, but I really couldn't understand what it is about. Could you tell me using simple (not-too-technical) terms, please? $\endgroup$ – Dal Jan 24 '15 at 16:38
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    $\begingroup$ There is also A Geometry of Music by Dimitri Tymoczko, but I don't think it illustrates what you seek. $\endgroup$ – Joseph O'Rourke Jan 24 '15 at 17:16
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One narrow example is the Hexachordal Theorem, which in one version says that a chord composed of any six notes on a twelve-tone scale has the same "interval content" as the chord composed of the complementary six notes. Some believed this underpinned Schoenberg's use of hexachords. The short abstract below provides a number of references to proofs of the theorem, starting from Milton Babbitt in the 1950s, through to proofs by crystallographers interested in distinct point patterns that lead to the same X-ray diffraction patterns. This could be considered an example where the mathematics that developed out of a musical notion was perhaps more interesting than its musical origin.

Toussaint, Godfried. Abstract: "Interlocking rhythms, duration interval content, cyclotomic sets, and the hexachordal theorem." Fourth International Workshop on Computational Music Theory, Universidad Politecnica de Madrid, Escuela Universitaria de Informatica. 2006. (PDF download)

And this illustrates the meaning of "interval content":


HexachordalFig2


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  • $\begingroup$ Very nice. I'll have a look at this. Thank you very much. $\endgroup$ – Dal Jan 24 '15 at 16:40
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if you allow computer science to be a branch of mathematics, the computational modeling of music similarity could fit your description; this workshop described the challenge of the field as follows:

The dramatic increase in the digitization of music calls for the development of computational methods in Music Information Research, such as content-based querying and retrieval, automatic music classification, music recommendation, and digital rights management. A fundamental topic involved in these different aspects of processing music information is the computational modeling of music similarity. Similarity in music is a highly context dependent notion and poses serious challenges for computational modeling, so much so that state-of-the-art retrieval methods have recently hit a glass ceiling.

For an overview of the literature, this Ph.D. thesis might be a good starting point.

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  • $\begingroup$ It is not quite what I have in mind. Still, it is interesting. Thank you. $\endgroup$ – Dal Jan 24 '15 at 16:28
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I suggest you look into the work of my colleague Jack Douthette.

http://scholar.google.com/citations?user=q99kfOQAAAAJ&hl=en

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