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).