Music: mathematical point of view (revised) - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-19T12:07:41Z http://mathoverflow.net/feeds/question/101420 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised Music: mathematical point of view (revised) PaPiro 2012-07-05T17:48:18Z 2012-10-08T22:21:38Z <p>Mathematical analysis of music started when Pythagoras made his observations about consonant intervals and ratios of string lengths.</p> <p><strong>ADDED:</strong></p> <p>In the paper <strong>Mathematical Music Theory -- Status Quo 2000</strong>, G. Mazzola, ETH Zürich, Departement GESS, and Universität Zürich, Institut für Informatik, available <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.2.8716&amp;rep=rep1&amp;type=pdf" rel="nofollow">here</a>, we have the following statement: </p> <p><em>... These models use different types of mathematical approaches, such asâ€”for instanceâ€”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.</em></p> <p>In conclusion, the author says:</p> <p><em>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: <strong>Seemingly disparate subjects become related and comparable only through the universal language and methods of modern mathematics</strong>. 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.</em></p> <blockquote> <blockquote> <p>The <strong>MODIFIED</strong> question is: from the <em>modern</em> mathematical point of view, is it possible to define (<em>aspects of</em>) <strong>music</strong>?</p> </blockquote> </blockquote> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101424#101424 Answer by Mahmud for Music: mathematical point of view (revised) Mahmud 2012-07-05T18:20:28Z 2012-07-05T18:46:26Z <p>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: <a href="http://www-personal.umd.umich.edu/~tmfiore/1/musictotal.pdf" rel="nofollow">Music and Mathematics</a> by Thomas M. Fiore.</p> <p>Edit: Since the term <em>modern</em> is so broad with no specific field specified, you may also be interested in use of <a href="https://ccrma.stanford.edu/~blackrse/algorithm.html#computer" rel="nofollow">stochastic approaches in algorithmic composition as used by Xenakis</a>:</p> <blockquote> <p>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.</p> </blockquote> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101425#101425 Answer by David Feldman for Music: mathematical point of view (revised) David Feldman 2012-07-05T18:22:40Z 2012-07-05T18:22:40Z <p>I have to disagree strongly with Gerald Edgar.</p> <p>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</p> <p><a href="http://en.wikipedia.org/wiki/Guerino_Mazzola" rel="nofollow">http://en.wikipedia.org/wiki/Guerino_Mazzola</a></p> <p>and probably in need a refinement.</p> <p>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.</p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101426#101426 Answer by Margaret Friedland for Music: mathematical point of view (revised) Margaret Friedland 2012-07-05T18:52:24Z 2012-07-05T18:52:24Z <p>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):</p> <p><a href="http://www-acad.sheridanc.on.ca/~degazio/pdf/Musical%20Aspects.pdf" rel="nofollow">http://www-acad.sheridanc.on.ca/~degazio/pdf/Musical%20Aspects.pdf</a></p> <p>It includes e.g. a reference to "the study of music as a scaling (1/f) noise".</p> <p>One can find a lot of "fractal music" on the web, but the connection with mathematics is usually very remote. </p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101428#101428 Answer by Denis Serre for Music: mathematical point of view (revised) Denis Serre 2012-07-05T19:20:00Z 2012-07-05T19:20:00Z <p>Mathematics is a toolbox (the <em>unreasonable efficiency of Mathematics in natural Sciences</em> 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. </p> <p>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 <strong>CAT scan</strong>. It is not appropriate when designing softwares for financial trade.</p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101429#101429 Answer by Dick Palais for Music: mathematical point of view (revised) Dick Palais 2012-07-05T19:32:51Z 2012-07-05T19:32:51Z <p>I guess that this is a much an additional question as a response, but here is a question that I have frequently wondered about:</p> <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101430#101430 Answer by Lee Mosher for Music: mathematical point of view (revised) Lee Mosher 2012-07-05T19:40:42Z 2012-07-05T19:40:42Z <p>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". </p> <p>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. </p> <p>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, <a href="http://www.amazon.com/The-Physics-Psychophysics-Music-Introduction/dp/0387943668" rel="nofollow">Introduction to the physics and psychophysics of music</a>.</p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101446#101446 Answer by Joseph O'Rourke for Music: mathematical point of view (revised) Joseph O'Rourke 2012-07-05T23:43:11Z 2012-07-05T23:43:11Z <p>You may be interested in Dimitri Tymoczko's 2011 book <em>A Geometry of Music</em> (<a href="http://dmitri.tymoczko.com/geometry-of-music.html" rel="nofollow">author link</a>; <a href="http://www.oup.com/us/catalog/general/subject/Music/MusicTheoryAnalysisComposition/?view=usa&amp;ci=9780195336672" rel="nofollow">Oxford link</a>): 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 href="http://dmitri.tymoczko.com/ChordGeometries.html" rel="nofollow">a video of Chopin's E minor prelude</a> "as it travels through a slice of the four-dimensional space containing seventh chords." <br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="http://cs.smith.edu/~orourke/MathOverflow/TrichordPrism.jpg" alt="Trichord Prism"></p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/101450#101450 Answer by Mahdi Majidi-Zolbanin for Music: mathematical point of view (revised) Mahdi Majidi-Zolbanin 2012-07-06T00:40:59Z 2012-10-08T22:16:45Z <p>Mathematics itself is music. See <a href="http://www.amazon.com/Mathematics-Music-Reason-Jean-Dieudonne/dp/3642080987/ref=sr_1_1?ie=UTF8&amp;qid=1341535160&amp;sr=8-1&amp;keywords=Mathematics+the+music+of+reason" rel="nofollow">Mathematics, the Music of Reason</a>. The point I am making here is that to "define (aspects of) music" is <em>not</em> 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 <em>show</em> others that mathematics is useful and can <em>even</em> define aspects of music. What is special about <em>music</em> 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.</p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/109168#109168 Answer by Gene Ward Smith for Music: mathematical point of view (revised) Gene Ward Smith 2012-10-08T18:55:17Z 2012-10-08T18:55:17Z <p>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: <a href="http://xenharmonic.wikispaces.com/Mathematical+Theory" rel="nofollow">http://xenharmonic.wikispaces.com/Mathematical+Theory</a> 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.</p> http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised/109189#109189 Answer by ex0du5 for Music: mathematical point of view (revised) ex0du5 2012-10-08T22:21:38Z 2012-10-08T22:21:38Z <p>If you define</p> <p>$$Discretise_n(x) = \{ \frac{m}{n} | \min_{m \in \mathbb{Z}}|x - \frac{m}{n}| \}$$</p> <p>then you can calculate</p> <p>$$Dissonance (x) = \sum_{j=1}^{\infty} ( x - Discretise_j (x) )^2$$</p> <p>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.</p>