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.

  • Hermann von Helmholtz
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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    consisting of a sounding board and box 011 which a single String was strctched with a scale below, so as to set the bridge correctly. * It was not tili mach later that, through the investigations of Galileo (1638), Newton, Euler (1729), and Daniel Bernouilli (1771), the law governing the motions of strings became known, and it was thus found that the simple ratios of the lengths of the strings existed also for the pitch numbers of the tones they pro- duced, and that they consequently belonged to the musical intervals of the tones of all instrumentSj and were not confined to the lengths of strings through which the law had beeil first discovered. This relation of whole numbers to musical consoiiances was from all time looked upon as a wonderful mystery of deep significance. The Pythagoreans themselves made use of it in their speculations on the harmony of the spheres. From that time it remained partly the goal and partly the starting point of the strängest and most venturesome, fantastic or pliilosophic combinations, tili in *! modern times the majority of investigators adopted the notion accepted by Euler liimself, that the human mind had a peculiar pleasure in simple ratios, because it could better understand them and comprehend their bearings. But it remained uninvestigated how the mind of a listener not versed in physics, who perhaps was not even aware that musical tones depended on periodical vibrations, contrived to recognise and compare these ratios of the pitch numbers. To show what pro- cesses taking place in the ear, render sensible the difference between consonance and dissonance, will be one of the principal problems in the second part of this work. CaLCULATION OF THE PlTCH NüMBERS FOR ALL THE TONES OF THE Musical Scale. By means of the ratios of the pitch numbers already assigned for the consonant intervals, it is easy, by pursuing these intervals throughout, to calculate the ratios H for the whole extent of the musical scale. The major triad or chord of tliree tones, consists of a major Third and a Fiftli. Hence its ratios are : C : E : G 1 : I : I or 4:5:6 If we associate with this triad that of its dominant G : B : D, and that of its sub-dominant F : A : C, each of which has one tone in common with the triad of the tomc C : E : G, we obtain the complete series of tones for the major scale of C, with the following ratio of the pitch numbers : C 1 D : E : F : G : A : B : jL' 8 4 ¥ [or 24 : 27 : 30 : 32 : 36 : 40 : 45 : 48] U In Order to extond the calculation to other octaves, we shall adopt the followino- notation of musical tones, marking the higher octaves by accents, as is usual in Lormany,+ as follows: 1. I he unaccented or small octave (the 4-foot. octave on the OrganU : bi: — ö—— d * [As the monochord is very liable to error these results were happy generalisations from necessarily imperfect experiments.—Trans- tator.] 1 [English works uso strokes above and / (J below tho letters, which are typographicallv mconvement. Hence the Germanhotation is retamed.—Translator. ] + [The note G in the small octave was once omitted by an organ pipe 4 fect in longth: