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ON THE QUALITY OF MUSICAL SOUNDS. 75
Before stating the various methods by which this proposition has been proved, it may be advisable to explain its meaning a little more fully. In the first place, the proposition asserts, that the quality of a tone varies with the number of its component partials; thus if one tone consists of three partials, another of four, and another of six; then, each of these three tones will have a different quality from the other two. In the second place, the proposition declares that the quality of a tone varies with the order of its con­stituent partials; for example, suppose we have three tones, the first consisting, say, of the 1st, 2nd, and 3rd partials, the second of the 1st, 3rd, and 5th, and the third of the 1st, 3rd, and 6th, then each of these three tones will have a different quality from the other two. In the above cases, we have supposed the partials to be of the same relative intensities in each case. If, however, the relative intensities vary, the proposition affirms that the quality will vary also. Thus to take a simple case, suppose we have two tones each consisting of the 1st and 2nd partials, and that the two fundamentals are of the same intensity ; then if the second partial of the one differs in intensity from the second partial of the other, the proposition asserts, that the quality of the one tone will differ from that of the other.
On reading the above propositions the following question at once suggests itself. Is the alleged cause, viz., the variation in the number, order, and relative intensities of the partials, sufficient to account for the observed effect, viz., the variation in quality ? The variations in quality of tone are infinite; therefore, if the proposition be true, the variations in the number, order, and relative intensities must be infinite also. Now the number of variations in number and order of partials although very great are practically limited; but it is obvious that the relative intensities of the partials may vary in an infinite number of ways, and thus the above question must be answered in the affirmative. In the next place, it is easy to see that if the proposition be true, simple tones can have no particular quality at all, they must all resemble one another in this respect, from whatever source they come. On trial this will be found to be the case. We have already observed that these tones can be approximately obtained from tuning-forks mounted on appropriate resonance boxes, wide-stopped organ pipes and flutes gently blown, and the highest notes of the pianoforte. The tones from these four sources cannot be compared together very well, for the reason already referred to: the first mentioned being almost