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
App. XIX. B. 
When the pitch of the two notes is not precisely fixed by italicising 
the letters, the,difference of an Octave is not noticed, but it is supposed 
that the first-named note has the lower pitch. But when the precise 
Octave is fixed by italic letters, after proceeding from the first by the 
intervals pointed out, we must ascend or descend the requisite number of 
Octaves to reach the second. Thus c X g is a proper Fifth, but for c x G, 
after ascending the Fifth c X g, we must descend an Octave, so that c x G 
is really a descending Fourth. Similarly c + E is a descending minor 
Sixth, and e — G a descending major Sixth. But c x g' is an ascending 
Twelfth ; and so on. 
To shew that an interval is repeated, put an exponent above the 
symbol of the interval, thus c x2 d' is made up of c X g X domitting 
the auxiliary g, and using x 2 for xx. But c x 2 d shows that after 
ascending to d' we have descended an Octave, and have hence reached a 
major Second d. 
To shew that an interval is taken downwards, write the exponent 
below. Even in the case of a single descent the exponent must then be 
written. Thus c + j fffb shows that we descend a major Third from 
c to fffb ; but c +1 \cù> shows that we afterwards ascend an Octave, and 
hence that the interval is a minor Sixth. 
In Tables IY. and Y. the tones in every horizontal line proceed regu¬ 
larly in major Thirds up (or minor Sixths down) from left to right, and 
those in every vertical column proceed in Fifths up (or Fourths down) 
from bottom to top. Hence we have for the horizontal line 5m in Table Y. 
fFb + fAb + C + E + JG-ÎT + fBB 
and |Bît -j~i ^05 -j-jE -f-x C ft-1 ■j'Ab ft-} ■j'Fb 
and for the middle section of the vertical column III. 
Eb x Bb xFxCxGxDxfAxfE 
and fE X x fA xtD Xj G X x C XjF xxBb xxEb. 
We can evidently proceed from any tone in this-table to any other by 
going up and down by Fifths (or Fourths), and to the left or right by 
major Thirds (or minor Sixths). Hence any interval can be expressed 
and noted by means of Fifths and major Thirds. Thus C to ft is 
G X G X d + ft, or omitting the auxiliary tones C x X + ft, that 
is C X2 ft- ft. 
The ratio of the interval, supposing both tones to lie within the same 
Octave, is then calculated by putting x = §, + = f, multiplying them 
together, and subsequently multiplying or dividing the result by 2 as 
often as is necessary to make the result greater than 1 (which would 
indicate a Unison), and less than 2 (which would indicate an Octave). 
Thus for G to Ft we have x2+= (f)2 X f = ff> and as this is greater


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