Bauhaus-Universität Weimar

On the sensations of tone as a physiological basis for the study of music. Translated with the author's sanction from the third German edition, with additional notes and an additional appendix by Alexander J. Ellis
Helmholtz, Hermann von
retical researches both kinds of tones must be kept distinct, 
although in modern music they are practically confused. 
The idea of this notation belongs to Hauptmann, but as the 
capital and small letter which he uses, and which I also, in conse¬ 
quence, employed in the first edition of this book, have a different 
meaning in our method of writing tones, I now introduce a slight 
modification of his notation. 
Let G be the initial tone, and write 1 its Fifth 6r, the Fifth of 
this Fifth D, and so on. In the same way let the Fourth of G he 
F7, the Fourth of this Fourth B\>, and so on. In this way we have a 
series of Tones, here written with simple capitals, all distant from 
each other by a perfect Fifth or a perfect Fourth : 
B\>-F-C-G-D-A-E9 &c. 
The pitch of every tone in the whole series is, therefore, known when 
that of any one is known. 
The major Third of G, on the other hand, will be expressed by 
E, that of F by A, and so on. Hence the series of tones 
B-D-F-A-C-E-G-B-D-F$-A, &c., 
is a series of alternate major and minor Thirds. It is therefore 
clear that the Tones 
n-A-E-B-F^, &c., 
also form a series of perfect Fifths. 
We have already found that the tone F), that is the minor 
Third below or major Sixth above F, is lower in pitch than the 
tone 1), which would be reached by a series of Fifths from F, and 
that the difference of pitch is that known as a comma, the nume¬ 
rical value of which is f^-, or musically about the tenth part of a 
whole Tone.2 * Since, then, B — A and D — A are both perfect 
Fifths, A must be also a comma higher that A, and so also every 
letter with a stroke below it will represent a tone which is a comma 
lower in pitch than that represented by the same letter with no 
stroke below it, as is easily seen by carrying on the series. 
1 ‘ Die Natur der Harmonik und Metrik,’ Leipzig, 1853, pp. 26 and following. I 
cannot but join with C. E. Naumann in expressing my regret that so many delicate 
musical apperceptions as this work contains, should have been needlessly buried under 
the abstruse terminology of Hegelian dialectics, and hence have been rendered in¬ 
accessible to any large circle of readers. 
2 [The logarithm of an equally tempered tone is '050 1717, of a major Tone or f is 
■051 1525, of a minor Tone or %0 is -045 7575, and of a comma f£ is ’005 3950, so that 
a comma is more nearly the ninth part of a tempered Tone.—Translator .]


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