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
And in this way we obtain a final settlement of the proper in- 
tonation of the supplementary tones of the scale for the first four 
modes. Hauptmann, with whom I agree, considers the tone D 
alone to be the essential constituent of both the major and minor 
modes of G. This D forms an imperfect (Pythagorean) minor 
Third with F, so that the chord D — F — A must be considered as 
dissonant. This chord thus intoned is in reality most decidedly 
dissonant to the ear. On the other hand, Hauptmann admits a 
major mode which inclines to the subdominant, and uses D in 
place of D. I consider this conception to be a very happy expres¬ 
sion of the real state of things. When the consonant chord 
JD — F — A occurs in any composition it is impossible to return 
immediately, without any transitional tone, to the tonic chord 
G—F—G. The result would be felt as an harmonic leap without 
adequate notice. Hence it is a correct expression of the state of 
affairs to look upon the use of this chord as the beginning 
of a modulation beyond the boundaries of the key of G major, 
that is, beyond the limits of direct relationship to its tonic 
chord. In the minor mode this would correspond to a modulation 
into the chord of D[)—F—Älp. Of course in the modern tempered 
intonation the consonant chord H — F — A is not distinguished 
from the dissonant D—F—A, and hence the feeling of musicians 
has not been sufficiently cultivated to make them appreciate this 
difference on which Hauptmann insists.1 
As regards the other supplementary tone hip, which may occur 
in the chords ëjp —g—hip and g— bip — d', I have already shewn in 
the last chapter that even in homophonie music it is almost always 
replaced by b. Harmonic considerations likewise favour the use of 
b, independently of melodic progression. It has been already 
shewn that when the two tones of the scale which are but distantly 
related to the tonic, make their appearance as constituents of the 
dominant, they enter into close relation to the tonic. Now this 
can only be the case with the compound tones of the major chord 
g — b — d, and not with those of the minor chord g—bip — d. Con¬ 
sidered independently, the tones bip and d are quite as closely 
related to c as the tones b and d. But by regarding the two latter 
as constituents of the compound tone g, we connect them with c by 
the same closeness of relationship as g is itself connected with c. 
[This was referred to in p. 455, note 1.— Translator.']


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