42 Mr. Farey’s Letter on musical Intervals, &. 
Fraction) is of the diatonic value—9 T+7 t-+5 S, and its 
ratio is 37+ 2°* x 5°; and my third Term m (or the most 
—* . =—21 T+10t+22 S, and its ratio is 3°4 x 
fren 
Scniphiened and appalling as these diatonic expressions 
and ratios may appear at first sight, to many, the Intervals 
x, fand m are, nevertheless, strictly founded in Nature, 
and will as truly and as correctly represent musical Intervals, 
in every possible case, as the Ratios composed of the prime 
integers 2, 3 and 5, or any notation by Intervals, can do: 
and with the important advantage, in no other way so well 
attainable, of an encreasing series, throughout, in each of its 
terms, as the Intervals increase in magnitude, which are 
thereby expressed ; and yet, without negative signs, in any 
case that can be of the least use. They have other material 
advantages over any other notation by means of Intervals 
that has been proposed: yet these I shall not here enlarge 
on, but proceed briefly to mention, as follows : 
The Octave, or 3, is in this notation, of ‘the value 6125 
+12f4+53m, the major Twelfth ah VITI+V) or 
970 +19f+84m, and the major or 2VIT 
dia’ 
tion, ay eartiy adding either of these expressions, as often 
as its corresponding integer is multiplied _ & denom- 
nator of the Fraction (or largest number of the Ratio) and 
subtracting such expressions, as often as such Shenae re- 
spectively, are found multiplied in the numerator of the 
fraction. The following examples wee I hope, make the 
eo eror of this rule easy to any on 
Ist. If the ratio given, be that of ahs major Fifth, or 3, 
we have only to take 9705+ 19f484m, and deduct from 
it 61224 12f+ 53m, and the a =~ 358z+ 7f+ 
31m, is the notation of V, as required. . If the major 
Third or 922 be given, we must take aetek. os 
and he it deduct the double of the first expression, 0 
122454 24f+103m, which leaves 197244 +17, for 
the notation of IH. 3rd. Hf the major Comma be given, 
its ratio is $¢, or 24 x 5+3', and we must first take 4 times 
the second expression, or 388024-76f+ 336m ; and next, 
