the relation between sonority and sound spectrum

The character of a wave form is determined by its partials. We can make a diagram which represents those partials with their pitches and their relative intensities. Following our habits in the analysis of light, in which the different components together form a resultant colour, in sound we also speak of a spectrum. Such a diagram looks like the one below: a complex wave form on the left; the matching spectrum on the right.

diagram from John Backus' The Acoustical Foundations of Music.

Of a great number of musical instruments spectra have been made and analysed. You would expect that a certain instrument would produce a certain set of partials to characterize it and make it recognizable. This is not the case. An instrument – and certainly any other than a piano or an organ, in which the tone is produced by as it were a great many similar sub–instruments – does not show one and the same picture for its different pitches in which, say, the third partial has a certain intensity as comapred to the first one.
The two spectra above give witness to the case. They were taken from two neighbouring notes of a bassoon and show a markedly different composition. We can but make few sweeping statements as a result of spectra and the sonority we hear. We cannot reduce the characteristics of an instrument to an unambiguous spectrum. Even the mere presence of a partial over the range of an instrument doesn't show any constancy. The lower notes for example, that is to say notes that are produced by a greater tube length or by a longer string, tend to show a richer and more complex composition of (higher) partials than the higher ones. The acoustician John Backus (1911 - 1988), who played the bassoon as well, writes in his book The Acoustical Foundations of Music the following:

The low tones of a clarinet are deficient in even harmonics, but not all tones deficient in even harmonics sound like a clarinet; however, such tones are sometimes said to have a "hollow" sound. Tones with many high-frequency harmonics tend to sound "brighter", and those with fewer tend to sound "darker". This is about all that can be said; it is not possible to relate a tone of a given harmonic structure to a particular instrument.


Much more work needs to be done on the bassoon as well as the other woodwinds to work out the relationships between the structure of the instrument, the positions of the resonances, and the harmonic structure of the tone. At present we are not only ignorant of those relationships, but do not even know in physical terms the difference between a "good" and a "bad" tone.