Musical timbre and quantum theory

In what sense does the same note, played by different musical instruments, make a different sound? Why, for example, does C played on a piano, differ from C played on a saxophone?

Well, a musical sound is uniquely characterised by three things: pitch, loudness, and timbre. The pitch is the fundamental frequency of the oscillatory vibration, the loudness is the average amplitude of the vibration, and the timbre is what characterises the difference between different instruments.

To understand timbre it's necessary to understand that a musical note played on a particular instrument does not consist of a single frequency of vibration. Rather, it consists of a superposition of different frequencies. The pitch of a note uniquely specifies the fundamental frequency, but the sound produced will be a superposition of the fundamental frequency and numerous multiples of it ('overtones'). The superposition which defines the sound is specified by the amplitude of each constituent frequency, and this is the timbre of the sound. Mathematically, if one expresses the superposition as a Fourier series, then the amplitudes are simply the Fourier coefficients.

Now, the metaphysically interesting point is that the note made by a particular musical instrument is a quite definite sound, yet it can be decomposed into numerous parts, each of which would correspond to an equally definite sound, were it to be produced in isolation. It's exactly the same sense in which a prism can decompose white light into it's component parts, the colours of the spectrum. As philosopher of physics Richard Healey has pointed out, the prism simply performs a physical Fourier decomposition.

Which brings us back to certain versions of the Many-Worlds Interpretation of quantum theory, which propose a radical compositional metaphysics in which the universe consists of numerous mutually interfering branches. It's well established that quantum superpositions cannot be treated exactly like the superpositions of sound or light waves in classical physics, yet the basic mereological scheme (i.e., the relations between parts and wholes) may still be similar, with the universe consisting of mutually interfering branches just as the note produced by a musical instrument consists of numerous mutually interfering frequencies.