Mr. Farey’s Letter on musical Intervals, &c. 67 
excited in the air, for yielding different sounds 5 because it 
is the ratios, only, of these, that can be applied to the com- 
paring or calculating of musical Intervals; involving, in 
all such cases, the unnatural and laborious substitution 
of the multiplication of vulgar Fractions, in the place of 
simple addition, and the substitution of division of vulgar 
Fractions, in the place of simple subtraction, of the Inter- 
vals under consideration : a consequence of which is, that 
the smaller the Intervals are, the larger do the numbers ex- 
pressing them become, and the more difficult of couception 
and the more laborious, does the expressing or calculating 
of them become ; and hence it can excite no wonder, that 
nearly all who may not have been induced to cultivate 
some acquaintance with Mathematics, for its own sake, have, 
as Musicians or Tuners, been so bewildered and disgusted, 
at the very outset of their attempts to understand this im- 
portant and fundamental pari of their subject, as to have 
given up the pursuit; being content to remain ignorant of 
that which was presented to them by the professed Writers 
on the subject, in so unnatural and forbidding a form. 
It is observable, that the small Intervals above alluded to, 
as occasioning the chief stumbling block, are not merely 
such as curiosity only, and not utility, requires to be 
brought into review, but they concern each and every one of 
the Intervals which are considered, when we attempt to 
speak of the Temperaments of the Musical Seale : and hence, 
it has been next to impossible, that the mere Arithmetician, 
who proceeded to add and subtract Intervals according to 
the unnatural plan above mentioned,* could complete the 
calculation, or understand the true nature, of any one of the 
various modes in which the musical Scale may be attemper- 
ed, or even comprehend the untempered Scale itself, in so 
much of its generality as the same is now actually exhibit- 
ed, on the Euharmonic organs of Mr. Liston, and always 
has, although almost unperceived, been practiced, by the 
correct Singer, the Violinist, and a few other Practitioners 
* That most indefatigable Calculator, the late Mr. Marmaduke Overend, 
proceeded in this way, and brought his labours to no useful conclusions, ex- 
cept in the discovery of three smaller Intervals than any that had before 
been mentioned by Authors, and of some few other new Intervals, which 
are somewhat larger, as I have fully explained in Mr. Tilloch’s Philosophical 
Magazine, in Vol. 28. p. 140. 
