On the sensations of tone as a physiological basis for the theory of music / by Hermann L.F. Helmholtz ; translated, thoroughly revised and corrected, rendered conformable to the 4th (and last) German edition of 1877, with numerous additional notes and a new additional appendix bringing down information to 1885, and especially adapted to the use of musical students, by Alexander J. Ellis.

  • Helmholtz, Hermann von, 1821-1894.
Date:
1895
Catalogue details

Licence: Public Domain Mark

Credit: On the sensations of tone as a physiological basis for the theory of music / by Hermann L.F. Helmholtz ; translated, thoroughly revised and corrected, rendered conformable to the 4th (and last) German edition of 1877, with numerous additional notes and a new additional appendix bringing down information to 1885, and especially adapted to the use of musical students, by Alexander J. Ellis. Source: Wellcome Collection.

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    vibrational curves A and B bear to eacli other, or displacing the curve B by other distances to the right or left, than those already selected in tlie figures. By these simplest possible cxamples of such compositions, the reader will be able to form some idea of the enormous variety of forme which would result from using more than two simple forms of Vibration, each form representing an upper partial tone of the same prime, and hence, on addition, always producing fresh periodic cnrves. We shonld be able to make the heights of each single simple vibrational curve greater or smaller at pleasure, and displace each one separately by auy amount in respect to the prime,—or, in physical language, we shoidd be able to alter their amplitudes and the difterence of their phases; and each such alteration of ampli- tude and difference of phase in each one of the simple vibrations would produce a fresh change in the resulting composite vibrational form. [See App. XX. sect. M. No. 2.] The multiplicity of vibrational forms which can be thus produced by the com- II position of simple pendular vibrations is not merely extraordinarily great: it is so great that it cannot be greater. The French mathematician Fourier has proved the correctness of a mathematical law, which in reference to our present subject may be thus enunciated: Any given regulär periodic form of Vibration can always be produced by the addition of simple vibrations, having pitch numbers which are once, twice, thrice, four times, &c., as great as the pitch numbers of the given motion. The amplitudes of the elementary simple vibrations to which the height of our wave-curves corresponds, and the difference of phase, that is, the relative amount of horizontal displacement of the wave-curves, can always be found in every given case, as Fourier has shown, by peculiar methods of calculation (which, however, do not admit of any populär explanation), so that any given regularly periodic motion can always be exliibited in one single way, and in no other way whatever, as the sum of a certain number of pendular vibrations. 11 Since, according to the results already obtained, any regularly periodic motion corresponds to some mnsical tone, and any simple pendular Vibration to a simple musical tone, these propositions of Fourier may be thus expressed in acoustical terms: Any vibrational motion of the air in the entrance to the ear, corresponcling to a musical tone, may be always, and for each case only in one single way, exhibited as the sum of a number of simple vibrational motions, corresponding to the partials of this musical tone. Since, according to these propositions, any form of Vibration, no matter what shape it may take, can be expressed as the sum of simple vibrations, its analysis into such a sum is quite independent of the power of the eye to perceive, by looking at its representative curve, whether it contaius simple vibrations or not, and if it does, what they are. I am obliged to lay stress upon this point, because 1 ha\e by no mcans unfrequently found even physicists Start on the false hypothesis, that the U vibrational form must exhibit little waves corresponding to the several audible upper partial tones. A mere inspection of the figs. 11 and 12 (p. 30/> and p. 32/q will suffice to show that although the composition can be casily traced in the parts where the curve of the prime tone is dotted in, this is quite impossible in those parts of the curves C and D in each figure, where no such assistance has bcen provided. Or, if we suppose that an observer who had rendered himself thoroughly familiär with the curves of simple vibrations imagined that he could trace the com- position in these easy cases, he would certainly utterly fail on attempting to dis- cover by bis eye alono the composition of such curves as are shown in tigs. and 9 (p. 21c). In these will be found straight lines and acute augles. Pcrhaps it will be asked how it is possible by compounding such srnooth and uniform y rounded curves as those of our simple vibrational forms A and B in tigs. 11 an 12, to generate at one time straight lines, and at another acute angles. 1 10 answer is, that an infinite number of simple vibrations are required to geneiate curves with such discontinuities as are there shown. But wlien a gieat many