An Index for the Relative Quality among Musical Instruments
L. Knopoff · Ethnomusicology · 1963
n a class of abstractions that has an objective definition, a scale of values may exist. The question of whether no scale, only one scale, or more than one scale of values exists depends upon the definition of the class. As will be seen below, the definition of the quality of a musical instrument, as given in this paper, is sufficiently ambiguous that several solutions for a scale of values may be given; the solution offered here is not unique. Consider two instruments whose musical qualities are to be compared. The usual statement that we make is that the instrument with the greater quality is the one with the greater number of strong higher partials in its overtone spectrum. This statement may be put quantitatively and will form the basis for our index of instrumental quality. Evidently it is possible, by means of the definition, to compare two instruments sounding the same fundamental merely by comparing the overtone structures. How then are we to compare the quality of a piccolo with that of a bass clarinet when the normal registers are so markedly different? One way in which this can be done is to play back a recording of the piccolo sound sufficiently slowly that the fundamentals of the clarinet and piccolo are the same; conversely, the comparison may be made between the two instruments by speeding up a recording of the clarinet sound. In fact, the comparison may be made at any common fundamental pitch to which both instruments are artificially forced to sound, by means of playback at accelerated or decelerated speeds, as appropriate. Practically these techniques are cumbersome and are to be avoided. This technical difficulty can be avoided in at least one way. We fix our attention on the relative shape of the overtone spectrum of the sound or sounds in question but do not concern ourselves with the absolute frequencies that are involved. For example, in the hypothetical overtone spectrum of Figure la we draw the envelopel of the spectrum as indicated by the dotted line. For purposes of discussion, we define the point of inflection of this envelope as the place where the envelope changes its curvature from concave downward to concave upward; a certain frequency, designated by F on the diagram, will correspond to the point of inflection of the envelope. A real overtone need not exist at this frequency; the change of curvature may occur between spectral lines as well as at a specific line. Let the frequency F be compared with some reference frequency f, the fundamental for example, by taking the ratio; thus we obtain a number that will be characteristic of the sound. As a corollary to the above, we require that the definition of instrumental quality be such that the result is independent of playback speed for a given instrumental sound. Consider the imaginary instrument producing the sound whose spectrum is shown in Figure la. We further imagine that we play back a tape recording of this sound at a speed such that instrument