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A lot of people claim that music is just math and I don't understand why. Is there any facts behind this claim? It angers me when people make this claim and when I ask them to explain, even when they can't they don't see they are wrong.

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If you can read spanish, there is a book called "Matemática, maestro!" from the collection "Ciencia que Ladra" that explains it very nicely and in layman terms. Here it is. – Pedro Tamaroff Jul 29 '13 at 23:25
Also, could you please define music? Do you mean notes on a page, or the sound waves generated by an instrument? Or, do you perhaps mean the feeling you have when listening to a recording? – apnorton Jul 29 '13 at 23:29
Music can be represented by math, but music is NOT math. I believe both mathematician and musician would have agree on "$\text{Music}\neq \text{Math}$". Especially there are emotions in both the composer's work and the interpreters's interpretation (conductor, orchestra, etc...), can you exactly measure emotions by math? maybe quantitatively by tempo, and that's a maybe. If music IS math, I would say Mahler is a pretty good mathematician. – Shuhao Cao Jul 29 '13 at 23:32
up vote 1 down vote accepted

There are many mathematical aspects of music, but there also many non-mathematical aspects that are inherently cultural.

As an example of a "math aspect", take a look at harmonics: we like hearing sounds that produce the "same" frequencies, and these are just integer multiples of the basic frequency that is being played. Another example would be that of equal temperament, which creates a semi-optimal distance between notes, such that the possible harmonies are maximized.

That being said, the actual number of tones in an octave as well as the choice of scale are completely culture dependent - Western ears are used to 12 tone octaves and certain scales but not others, in what seems to be a rather arbitrary choice. So I think it's fair to say that music is definitely not "just math" - it's very strongly tied to the culture we were brought up in rather than to some mathematical formula.

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Additionally to he two answers given so far: I think the main link between maths and music is hidden behind the scenes:

  • Rules: Composition as well as Improvisation heavily relies on a reduced decision space as humans cannot deal with all possible alternatives at once. This reduction is usually done by exercising certain rules. One of the tasks of modern music theory is to find out the rules that are used in a certain composition or define a certain style. Students in music theory or composition courses learn the rules that underlie certain compositional styles such as Gregorian music, Counterpoint or Bach chorals. On the other hand rules have been used to compress music such as figured bass or chord symbols in modern songs.
  • Intuition: Both mathematics and music heavily rely on intuition. Intuition enables humans to follow or analyse rules implicitly. This is the case when a mathematician formulates a conjecture or when a composer has some kind of feeling how a composition should be done. In my music theory course the lecturer told the students to learn the rules, practice them and forget them afterwards in favour of the intuition they got during practising. The same idea is helpful in solving mathematical problems.
  • Fantasy: Both mathematics and music rely on fantasy. For music this is a well-known fact. The most well-known fact is that transformation of some inspiration into a score is linked with fantasy. But also on the technical level of composition music lives from a small amount of surprise, i.e. breaking the rules that have been established during the past of a piece. Surprise is not a paradigm in mathematics. But breaking the rules is an important as well as hidden feature in mathematics. It is done mainly in three ways: Using a duality to switch over to another branch of mathematics, introducing a newly defined object that has properties that are not directly deduced from the used formalism. One of the important ways to solve mathematical problems is to imagine some solution, explore its properties and solve the problem backwards.[1][1] Sometimes this imagination comes from the problem itself, sometimes people use an analogue from the “real world”. The fantasy is usually invisible as soon as a proof has been written down.
  • Creativity: Some mathematicians consider mathematics as an art like lyrics. The constraints that limit mathematical methods are usually as wide as in the arts. In comparison: Engineers usually deal with much more constraints of technical or economic nature that can not be widened temporarily. Some more ideas can be found in a book about musical creativity [2]

    [1] G. Polya: How to Solve it?, 1945, Princeton University Press [2] G. Mazzola, J. Park, and F. Thalmann: Musical Creativity, 2011, Springer

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If you want to have a full picture of what is music, you have to keep in mind that we are humans listening and creating this music. Therefore, you have to include cognitive science to get the whole picture. As far as I know, cognitive science is not "just math."

That said, there are some really cool connections between math and music, like the harmonic series and equal temperament, chords and groups, and some more involved things like the book "The Topos of Music" by Mazzola.

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