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Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal.

But what if, instead, we started with solving the wave equation for a plucked string, which through d'Alembert's solution gives a nondifferentiable wave (inherently a more complicated object), and based music theory (and harmonics, etc) on that?

We could just as well pick some other basis of $L^2(\mathbb{R})$ instead of $\sin(nx)$, $\cos(nx)$, such as the above waves describing the motion of a plucked string at various frequencies. What effect would this have on the mathematics of harmony and resonance in music, if any? And what about completely different bases all together; does the standard $\sin(nx)$, $\cos(nx)$ basis have any musical importance, rather than just mathematical simplicity? Is the name "pure tone" really deserved? Could a spectrogram based on a decomposition into motions of plucked strings provide a different angle than just a Fourier transform decomposition?

Edit: A similar question arises when considering the vibrational modes of a circular membrane, see e.g. this Wikipedia article. This allows decomposition of any motion as a superposition of particular Bessel functions, which are no doubt chosen for mathematical simplicity.

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It's not clear to me what you mean by "musical importance," or rather, it's not clear to me how to interpret that as a mathematical question (as opposed to a biological or cultural one). –  Qiaochu Yuan Aug 18 '11 at 20:31
I'm not sure how a nondifferentiable function is supposed to solve a differential equation... but certainly solutions to the wave equation can differ from pure complex exponentials depending on the ambient conditions. –  anon Aug 18 '11 at 20:36
@Qiaochu I agree, it's not a particularly mathematical question. But I would have thought that the use of Fourier series in music theory would have more justifications than just their mathematical simplicity, some sort of explanation of why they should be particularly relevant. –  Will Aug 18 '11 at 21:21
@PAI If you are interested in the details of plucked strings, a good book on the "physics of musical instruments" is by Fletcher and Rossing ;) –  Alice Aug 18 '11 at 21:43
We decompose music into sine waves because our cochleas decompose music into sine waves. –  endolith Aug 19 '11 at 14:05

3 Answers 3

up vote 8 down vote accepted

It sounds like you really want to read Benson's Music: a Mathematical Offering (freely available at the link). I'm not completely sure what you're asking, but if it's anything like "why is it natural to think about music in terms of sine waves," this question is addressed right in the introduction:

...what's so special about sine waves anyway, that we consider them to be the "pure" sound of a given frequency? Could we take some other periodically varying wave and define it to be the pure sound of this frequency?

The answer to this has to do with the way the human ear works. First, the mathematical property of a pure sine wave that's relevant is that it is the general solution to the second order differential equation for simple harmonic motion. Any object that is subject to a returning force proportional to its displacement from a given location vibrates as a sine wave. The frequency is determined by the constant of proportionality. The basilar membrane inside the cochlea in the ear is elastic, so any given point can be described by this second order differential equation, with a constant of proportionality that depends on the location along the membrane.

The result is that the ear acts as a harmonic analyser [emphasis added]. If an incoming sound can be represented as a sum of certain sine waves, then the corresponding points on the basilar membrane will vibrate, and that will be translated into a stimulus sent to the brain.

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I have that book, and I can't find the answer to my question there. It does both solve the wave equation to find the motion of a plucked string, and of course discusses Fourier decomposition, but doesn't seem to relate them like I'm asking about. –  Will Aug 18 '11 at 20:37
@P.A.I.: Okay, then I officially don't have any idea what your question is. Is your question "what if, instead of picking a convenient basis to do computations in, we picked an inconvenient basis?" I'm not sure what the point of such a question is. –  Qiaochu Yuan Aug 18 '11 at 20:39
Inconvenient for the purpose of mathematics, sure (for instance solving the wave equation by separation of variables and superposition and guessing a "sum of sines" solution is less cumbersome than guessing a "sum of complicated functions" solution). But that's just a question of keeping the mathematics simple, and isn't necessarily going to give a good analysis in music theory. Justifying Fourier series and the such over other possible decompositions into periodic waves only through the simplicity seems a bit light. –  Will Aug 18 '11 at 21:18
@P.A.I.: again, I have no idea what your question is. In particular I have no idea what kind of answer you're expecting. What is a "good analysis in music theory"? –  Qiaochu Yuan Aug 18 '11 at 22:13

Hmm, sometimes confused are normal modes of a resonant system (string or tube of air) and periodic signals. Periodic signals can be decomposed into a Fourier series with integer harmonics. The vocal cords produce a nearly periodic signal (flaps going back and forth, but it's not a sine wave). Some musical instruments have normal modes with frequencies that are harmonic. A perfect string has harmonic overtones. Wind instruments are often designed so that modes blown have frequencies in integer ratios (e.g., the trumpet). But when you blow a trumpet and you get the air inside to resonate, you get essentially a nearly periodic signal out. Same goes for blowing into a flute or didgeridu. Why do we "like" periodic signals (or those with harmonic overtones) and tend to associate them with music? That's a perception question. Not to be confused with normal modes of a vibrating system or the Fourier transform of a nearly periodic signal. Our ears and brains seem to have a pretty sophisticated system of interpreting the voice. So maybe musical sounds are similar to voice and that's why we like them.

An additional question might be why do we prefer sounds with musical intervals (harmony) or complex tones (say a well designed bell) that have overtones that are in integer ratios (musical intervals called fifths, thirds, sixths which are tones in ratios of 3:2 5:4 6:5 etc.)? This is harder to answer but may be related to our preference for periodic signals. If you combine two nearly periodic signals that have periodicity in a ratio of 3:2 (a fifth interval) then the overtones beat if they aren't in tune. If there is a beat frequency and it's noticeable then it's either pleasing (like a vibrato) or annoying like two out of tune violins playing simultaneously.

To make things more complicated it turns out that we actually don't really like to listen to exactly periodic signals. They sound electronic. As a flute player I can tell you that I adjust "volume" not by playing softer but reducing my vibrato so that people pay less attention to my sound. So slight variations in voice and differences between exact periodicity are really important for musical sounds.

You might ask do we like to listen to pure sine waves or pure tones? Not really, this also sounds electronic, and actually pretty dull too (unless it's high frequency in which case it would be pretty annoying). As Qiaochu mentions above our ear separates different frequencies into different regions on the basal membrane. That means that more neurons fire if the signal is richer (and has a wide range of spectral frequencies). This gives us more information. The spectral information is used for example to identify formants in speech (these are bright spectral bands) which vary for different vowel sounds. So maybe we really like nearly periodic signals that have rich spectral information (or lots of harmonics).

To answer your question: Is there a better way to expand musical signals than using a Fourier series? Given that we "like" nearly periodic signals, Fourier series is probably the way to go. However wavelets give alternative types of bases and are used in seismology, for example, where you have rapidly changing signals (rather than continuous nearly periodic ones).

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I don't see why you put such an emphasis on periodicity, that was never the question, as one could pick different bases of functions that are still periodic but might be more closely related to other musical phenomena than just sine waves (which additionally are not particularly pleasant to listen to on their own). The spectral information doesn't have to come in a decomposition into sine waves - it could be a decomposition into another family of periodic waves. –  Will Aug 18 '11 at 21:25
Periodic sine waves can be decomposed into a series of square waves, too, no? I think the question is "What makes one decomposition more natural than the other?" But I think that's already been answered with simple harmonic motion and the action of the ear. –  endolith Aug 19 '11 at 3:17
An interesting thing about our hearing is that we are insensitive to phase information; this is known as "Ohm's law" as I remember. That means that if you Fourier decompose a wave and shift phases you will not be able to detect the difference. I think other decompositions (not using Fourier) might have trouble pulling out the amplitude dependence. –  Alice Aug 19 '11 at 13:19
@Alice: en.wikipedia.org/wiki/Ohm%27s_acoustic_law, but it's not actually true? –  endolith Aug 19 '11 at 14:04
@endolith I think true at high frequencies but at low frequencies we use phase for localization (figuring out where the noise makers are). Correlations between neurons firing and phase I think primarily seen at low frequencies (experiments not in humans). Lost of experiments done ... this field is vast ... –  Alice Aug 19 '11 at 14:22

Most of the answer has already been given with the reference to the auditory system. It dissolves the sound into a certain base functions. I don't know of a good description of their behaviour, but sine waves are a good approximation. So their choice provides – though historically inspired – a good description of the transition from air into brain.

You are asking whether there are other bases, that are useful as well. Your answer concerning acoustics is: yes. So there is no reason not to use them.

But you must always consider: Acoustics is not music. Even a pure harmonic sound is still not music. Music can not easily described by mathematical formulae. It is dependent on acoustics, physics, cultural background, your mood and many other factors.

William Sethares describes in his book “Tuning, Timbre Spectrum Scale” how scales for instruments with non-harmonic spectra can be constructed. In this book he provides examples where the harmonic analysis fails from a musical point of view. The resulting music can be judged differently. As it has to pass the bottleneck of our auditory system it will never sound like a perfect organ pipe or flute (both can produce nearly perfect sine waves).

If you want to construct a guitar then you should consider both descriptions: the plucked string and a convenient model for our auditory system, thus the harmonic dissolution.

Nevertheless our modern society has agreed to the equal temperament in a certain degree. As described in the other answers this coincides with non of the usual spectra in music. Thus, accuracy is not the only paradigm. On the other hand history has shown that not every mathematical model is suitable for music, even if it describes a certain aspect with a high precision.

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