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Mathematical analysis of music started when Pythagoras made his observations about consonant intervals and ratios of string lengths.

ADDED:

In the paper Mathematical Music Theory -- Status Quo 2000, G. Mazzola, ETH Zürich, Departement GESS, and Universität Zürich, Institut für Informatik, available here, we have the following statement:

... These models use different types of mathematical approaches, such as—for instance—enumeration combinatorics, group and module theory, algebraic geometry and topology, vector fields and numerical solutions of differential equations, Grothendieck topologies, topos theory, and statistics. The results lead to good simulations of classical results of music and performance theory. There is a number of classifiaction theorems of determined categories of musical structures.

In conclusion, the author says:

If we review the overall power of mathematics in the description, analysis and performance of music, it turns out that it has a unique unifying character: Seemingly disparate subjects become related and comparable only through the universal language and methods of modern mathematics. Moreover, the operationalization of the abstract theories on the technical level of computers and software is an immediate and very important empirical and theoretical consequence of mathematization. For the first time, models and experimental setups can be applied in a scientific, i.e., precise and objective framework. Finally, the embedding of the historically grown existing theories in the mathematical concept framework preconizes a natural extension of facticity to fictitious variants, thereby opening the way to the comprehension of the crucial question of musicology: Why do we have this music and no other? Of course, there will be other musics. But mathematical methods and associated technologial tools will undoubtedly play a dominant role it their discovery and exploration, be it on the level of instrumental realization, be it on the very concept space which transcends pure intuition and catalyzes fantasy to an unprecedented degree.

The MODIFIED question is: from the modern mathematical point of view, is it possible to define (aspects of) music?

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That is not a mathematical question. It belongs in psychology, maybe. –  Gerald Edgar Jul 5 '12 at 17:52
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The question is pretty vague and nonspecific. This defect could be corrected in various ways, many of them quite interesting, but most of them not a good fit for MO. Voting to close. –  Kevin Walker Jul 5 '12 at 18:11
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Here's why I don't vote to close: a pure mathematician myself, I consider all applications of mathematics as good for business. Mathematics flourishes to the extent that it becomes relevant to the largest numbers of people thinking about the greatest diversity of things. Mathematics has powerful enemies who sometimes control funding, and they will eventually seize on seeming evidence of insularity. This question seems soft, but not so the 1335-page tome I recommended as framing a direct answer to it. Interesting mathematics can start from soft questions. –  David Feldman Jul 5 '12 at 18:49
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There are certainly attempts at connecting mathematics and music, but the question whether they actually get anywhere near making a connection is subjective and argumentative. –  user2035 Jul 5 '12 at 19:22
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I started a meta discussion about this question: tea.mathoverflow.net/discussion/1402/mathematics-and-music –  Artie Prendergast-Smith Jul 6 '12 at 1:06
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closed as not a real question by Will Sawin, Todd Trimble, Douglas Zare, Michael Greinecker, Tom Leinster Oct 8 '12 at 22:38

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

10 Answers

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I have to disagree strongly with Gerald Edgar.

The controversial book The Topos of Music by Guerino Mazzola could constitute a very serious attempt to answer your question. This is not a book that would be accessible to a typical musician, or even a typical expert in music theory - it is definitely a mathematics book about music, taken seriously by some eminent mathematicians as claimed here

http://en.wikipedia.org/wiki/Guerino_Mazzola

and probably in need a refinement.

The question "what is music?" might entertain a musician, a philosopher, an anthropologist, a sociologist or, as Edgar suggests, a psychologist, and they would have something (different) to say. The more precise question here might be "what mathematical structure provides an adequate model for anything someone might create and call a musical composition?" The problems here art that musical compositions have multiple realizations; realizations of a score are constrained by various axioms; and composer specify music compositions by choice with greater or lessor degrees of determinacy. Mazzola seeks a unified model that can represent the whole range of what modern composers (and not just of Western art music) actually do.

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Various mathematicians, most famously G.Birkhoff, have attempted mathematical theories of aesthetics. They are mostly failures...mathematics won't tell you what is good and what is bad art because the question isn't well-defined -- the best one could hope for is that an individual might devise a mathematical model of his or her own biases. The Topos of Music is not a book about aesthetics. Rather it developments value-neutral mathematics that might have applications to machine environments for music composition and performance, and also to creating databases for musicological research. –  David Feldman Jul 5 '12 at 18:28
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@David Feldman: "They are mostly failures...mathematics won't tell you what is good and what is bad art because the question isn't well-defined". Probably nitpicking, but saying the question is "ill-defined" is itself an answer of sorts to the question of what is good or bad art. A more correct answer is simply to say that the question, like all general philosophical questions, is not amenable to a mathematical treatment because what counts as "good" or "bad" art is not quantifiable or measurable -- words taken in the broadest possible sense. –  G. Rodrigues Jul 5 '12 at 18:48
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@G. Rodrigues: One could reasonably hope to do this (and mathematics would matter): from an empirical body of value judgements by a population of listeners, deduce structural correlates. Beyond mere statistical methods, mathematics would contribute potentially relevant abstract models, sine qua non even for formulating statistical questions. Mathematical methods also might predict or postdict those structural correlates from known or hypothetical neural dynamics. So one might even succeed in quantifying some received idea of "good" and "bad" associated with certain stakeholders. –  David Feldman Jul 5 '12 at 19:45
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@David Feldman: "from an empirical body of value judgements by a population of listeners, deduce structural correlates" That could constitute an interesting exercise in a theory of taste but is hardly relevant for the consideration of the total, aesthetic musical order. More general, from what little you wrote, methinks we would violently disagree on some general philosophical points: on the nature of the Good or about the reduction of the mind to the brain. But this is off-topic, so I will just shut-up. –  G. Rodrigues Jul 6 '12 at 15:20
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Are there (online) samples of music generated with the category theoretic techniques from the book? –  o a Jul 11 '12 at 17:42
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I guess that this is a much an additional question as a response, but here is a question that I have frequently wondered about:

Is it conceivable that one could develop mathematical tools for analyzing the style and the content of the music of a great composer and use that analysis to create an algorithm that, together with a "seed theme" as input, would create a composition approximating the level of quality of that composer's actual work?

It is hard for me to believe that any purely mathematical emulation could reach the sublime beauty of some of Mozart's later works. But even a good approximation would be worth a lot of effort, and there are many examples of later composers writing music "in the style of X", that are pretty good, and that suggests there is some hope.

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en.wikipedia.org/wiki/David_Cope His (machine's) work has occasionally fooled experts. –  David Feldman Jul 5 '12 at 19:52
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@David Feldman: THANKS! Just the sort of thing I was looking for. Interesting that he is now an emeritus professor at UCSC. I was there for a couple of years (1975-77) and wonder if we overlapped. –  Dick Palais Jul 5 '12 at 20:34
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@Dick Palais Cope started at UCSC in 1977, so you just missed him. But I'll bet he'd be happy to hear that your interested in all this. –  David Feldman Jul 5 '12 at 20:57
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You may be interested in Dimitri Tymoczko's 2011 book A Geometry of Music (author link; Oxford link): he "describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock." As an example his work, he's made a video of Chopin's E minor prelude "as it travels through a slice of the four-dimensional space containing seventh chords."
          Trichord Prism

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The following paper will be a good starting point if by modern mathematical point of view you take in consideration topology and group theory, where it discusses how Beethoven's Ninth Symphony makes a torus and "chord progression" is a path on it: Music and Mathematics by Thomas M. Fiore.

Edit: Since the term modern is so broad with no specific field specified, you may also be interested in use of stochastic approaches in algorithmic composition as used by Xenakis:

Xenakis used the computer's high-speed computations to calculate various probability theories to aid in compositions like Atrées (1962) and Morsima-Amorsima (1962). The program would "deduce" a score from a "list of note densities and probabilistic weights supplied by the programmer, leaving specific decisions to a random number generator" (Alpern, 1995). "Stochastic" is a term from mathematics which designates such a process, "in which a sequence of values is drawn from a corresponding sequence of jointly distributed random variables" (Webster's dictionary). As in the previous example of the Illiac Suite, these scores were performed by live performers on traditional instruments.

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There was time when the paradigm of self-similarity was applied to pretty much everything, including music. This article is one of the attempts made in this direction (and mentions some other ones):

http://www-acad.sheridanc.on.ca/~degazio/pdf/Musical%20Aspects.pdf

It includes e.g. a reference to "the study of music as a scaling (1/f) noise".

One can find a lot of "fractal music" on the web, but the connection with mathematics is usually very remote.

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Mathematics is a toolbox (the unreasonable efficiency of Mathematics in natural Sciences of Eugene Wigner). It is not a philosophy. It does not explain everything. When it tries to explain the sensible world, it becomes Physics. But it cannot explain the human world. For instance, Sciences don't prove or disprove the existence of God. Likewise, I am an amateur musician, yet I don't look in Mathematics for the justification of the emotions I feel when playing or listening Mozart.

Well, Mathematics are very useful (our modern world is full of uses of Mathematics), but only if you apply them to relevant topics. It is OK to employ Mathematics to design a CAT scan. It is not appropriate when designing softwares for financial trade.

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I'm confused. What exactly is inappropriate about applying mathematics to algorithmic trading? –  Vidit Nanda Jul 5 '12 at 23:49
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@Vel. Juge the tree from its fruits. I could give other examples. Each time a human activity is completely automatized, Human beings loose their abilities and their feelings. Here, the result is a financial crash. There, the crash of an airplane (AF447 Rio-Paris for instance). –  Denis Serre Jul 6 '12 at 6:41
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@Denis Serre: I do not believe your argument would be maintainable under scrutinity. Yet, since I am convinced that MO is not the place for philosophical and/or political debates, I will leave it at that. –  quid Jul 6 '12 at 11:16
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I would no more expect a mathematical answer to the question "What is music" than I would to "What is fire" or "What is an electron".

Which is to say, I would expect those questions to be primarily addressed by other branches of inquiry, but perhaps with a healthy serving of mathematics in the answer.

There certainly are a lot of applications of physics, biology, and psychology to Music, and underlying those there is certainly a lot of mathematics. One of my favorite old books on this topic is by Juan Roederer, Introduction to the physics and psychophysics of music.

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I agree, philosophically speaking, mathematics doesn't answer "what is an electron." But it would not surprise me to hear a physicist say "well, actually, an electron is a representation of this-or-that group" or "...an equivalence class of sections of this-or-that sort of vector bundle." For that matter, if you know Feynmann's "I don't even believe in the inside of a brick," electrons are perhaps only theoretical entities (so fundamentally mathematical) that physicists invented to explain more concrete things. –  David Feldman Jul 5 '12 at 20:19
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If we had to grant someone a license to formulate a mathematical definition of an electron, it would have to be a particle physicist, who at least has the necessary intuitive physical understanding. Similarly, if we had to grant someone a license to formulate a mathematical definition of music, it probably should be an M.D. Psychologist/Biophysicist who plays in a subprofessional string quartet. But a mathematician..... –  Lee Mosher Jul 5 '12 at 22:20
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I am enjoying this debate, but wonder if MO is the right place for it? –  David Feldman Jul 6 '12 at 1:14
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I think by far the most useful applications of mathematics to music come with a limitation on the scope of what is being attempted, and are found in the theory of tuning and scales. A great deal of information can be found here: http://xenharmonic.wikispaces.com/Mathematical+Theory A striking aspect of this sort of thing is that even though some of the things on this wiki are highly mathematical, people are making use of it to compose actual music.

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If you define

$$Discretise_n\(x\) = \{ \frac{m}{n} | \min_{m \in \mathbb{Z}}|x - \frac{m}{n}| \}$$

then you can calculate

$$Dissonance \(x\) = \sum_{j=1}^{\infty} \( x - Discretise_j \(x\) \)^2$$

as a good metric of the amount of dissonance two notes with a ratio of x in frequency will cause. If you look at the graph of this function, you will see it has fractal qualities and several local maxima. The maxima are good starting points for generating strongly dissonant noises, if you are into the noise/japanoise/extreme noise genres (ala Merzbow, Massona, Government Alpha, etc.). I have used this to good effect in my own projects.

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Well, I guess that answer didn't go over very well. It is intended to be serious, though, but I guess if you are not into that scene it may seem facetious. To add more math in the hopes that it is actually seen as a legitimate response to the question, note: the Dissonance function is continuous, has local minima at the standard consonant intervals of Pythogrean tunings, has applications in continued fractions (such as bounding the error of certain truncation forms), and generally has a lot more structure than might appear from above. –  ex0du5 Oct 9 '12 at 16:52
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Mathematics itself is music. See Mathematics, the Music of Reason. The point I am making here is that to "define (aspects of) music" is not a mathematician's goal. That is perhaps the task of someone who is more interested in music and has little interest in mathematics, or perhaps someone who needs to find credibility in what he does by resorting to popular culture, in order to show others that mathematics is useful and can even define aspects of music. What is special about music in this question other than it is a popular form of art that can be enjoyed more easily by more people? Asking "if mathematics can define (aspects of) music" is similar to asking if mathematics can define (aspects) of cinema, which may be a good question but not for MO.

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