65 Mr. Farey’s Letter on musical Intervals, §c. 
on Instruments which are perfect, as to their capability of 
yielding any degree of sound whatever, which either theory 
or the judgment of the ear might require. 
Since the period of the sublime invention of Logarithms, 
and their general diffusion in Tables of the present form, 
such have opened new and great facilities to the mathemat- 
ical calculators, on the subject of the musical Scale and its 
Temperaments ; but it has been almost in vain, that appeals 
have been made to the mere Musician or Tuner, on the 
utility and the easy application of these measures of Ratios, 
because the original difficulty has as often recurred, viz. the 
want of apparent and natural connection, between ratios and 
musical Intervals. I have on various occasions* attempted 
to remove this difficulty, by shewing, that the reciprocal 
common Logarithm of any Interval, correctly expresses 
the decimal relation which that Interval bears, to the concord 
called the maior Twentyfourth, or XXIV (or 3VHI-+ HI) 
whose Ratio is ;',, and its reciprocal logarithm is 1°0000000 ; 
but the difficulties with this class of Persons have in no de- 
gree been removed, by endeavouring to explain to them, 
» that the reciprocal of a logarithm answers to the substitution . 
of division in the place of multiplication ; (or vice versa;) and 
im the present case, that the change of or —9, as the tabular 
imdex of the common logarithm of the fraction 3., to 1-, 
answers completely to the tuning or considering of aX X1Vth 
downwards, instead of upwards, between its terminating 
sounds. 
When all the decimal places of figures beyond the fifth, 
had been arbitrarily rejected, and the recip. log. so abridged, 
multiplied by 100000 (as was done first, I believe, by Dr. 
Robison) in order to obtain measures for ‘the various Inter- 
vals of the scale, and its Temperaments, I have not found 
these further deviations from any visible natural connection, 
between the arbitrary numbers so obtained, and the musical 
Intervals they are made to represent, to have the least ten- 
dency towards gaining the attention and assent of the Mu- 
sicians and ‘Tuners whom I have conversed with; but the 
reverse of it, in more than one instance ; in one of which 
cases it has been urged to me, thus: “ If the logarithmic 
* See the “ Edinburgh Eneyclopeedia,” edited by Dr. Brewster, vol. VI. 
p. 31, vol. XI. p. 598, and in several other parts of that traly valuable Work. 
