Bauhaus-Universität Weimar

On sound and atmospheric vibrations, with the mathematical elements of music
Airy, George Biddell
Now if we begin (as the most advantageous place) 
from one of the notes marked F, as F = 160, we have 
the following succession of perfect Fifths (the propor¬ 
tion of vibrations being always 2 : 3) : 
From 160 to 240, F to C ; 
From 240 to 360, C to Gr. 
From 360 to 540, G to D ; 
From 540 to 810, D to a note beyond a, 
vibrations = of those 
Thus we find that, even beginning at the most advan¬ 
tageous place, we cannot have more than three consecu¬ 
tive perfect Fifths ; and, generally speaking, we cannot 
have so many. If we could put up with an error 
represented by the factor — (either in the last interval, 
or distributed through the threaisptervals) we might 
proceed further. It will be seerie nat we must either 
diminish the Fifth interval a little below - (“flatten the 
Fifths ”) or increase the Octave interval above 2 
(‘‘sharpen the Octave”); no compositor for the piano 
dares to recommend the latter course. TVe are inclined 
to think that it is best also to leave the Fifths un¬ 
In the use of the violin, however, the temptation 
to retain the perfect Fifths is very strong. In the 
process of tuning, every interval of the fundamental 


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