Question

Suppose you do a Fourier analysis of the sounds from a trumpet and a guitar, each playing the same note. What would be the same in the two analyses and what would be different?

Answer 1

When an instrument produces sound (as a result of a standing wave condition) it not only produces a frequency f_o but many other frequencies which are integral multiples of this fundamental frequency f_o. These are called harmonics of the fundamental frequency.

f_n = nf_o

Sound produced by the these frequencies can be represented in time-domain as a sum of series of sine and cosine functions:

where, a_o, a_n and b_n are the amplitudes of the nth harmonic.

If a guitar and a trumpet play the same note (same fundamental frequency f_o), they would have the same higher harmonics. However, the amplitudes of these harmonics will be different which qualitatively means that the energy in each harmonic is different for the two instruments (in musical terms, the two instruments have different 'timbres'). This implies that the Fourier spectrum will look quite different for the same note played on a guitar and a trumpet: both would have the same frequencies in its Fourier spectrum but the amplitudes for them would be different for the two instruments.