72 Mr. Farey’s Letter on musical Intervals, §c. 
Fraction) is of the diatonic value—9 T+7 t+5 5, and its 
ratio is 397+ 25452; and my third Term m (or the most 
Minute) is =—21'T+10t+422 8, and its ratio is 3°* x 
5 12 — a 161 § y 
Complicated and appalling as these diatonic expressions 
and ratios may appear at first sight, to many, the Intervals 
z,fand mare, 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 increasing series, throughout, in each of its 
terms, as the Intervals increase in magnitude, which are 
thereby expressed ; and yet, without negatiwe 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 612: 
+12f+53m, the major Twelfth (or VITI+V) or 4, is= 
970=-+4+19f4+84m, and the major Seventeenth (or 2VIII+ 
HIT) or}, is =14212428f4+123m: which three expres- 
sions, in terms of =, f and m, answer to the three prime 
integers 2, 3and 5, and will therefore serve for reducing 
any diatonic Interval whose Ratio is given, into this nota- 
tion, by merely adding either of these expressions, as often 
as its corresponding integer is multiplied into the denomz- 
nator of the Fraction (or largest number of the Ratio) and 
subtracting such expressions, as often as such integers, re- 
spectively, are found multiplied in the numerator of the 
fraction. The following examples will, | hope, make the 
application of this rule easy to any one. 
Ist. If the ratio given, be that of the major Fifth, or 3, 
we have only to take 9702+19f+84m, and deduct from 
it 61254+12f+53m, and the remainder, or 358 5+ 7f+ 
31m, is the notation of V, as required. 2nd. If the major 
Third or 2X2 be given, we must take 142124 28f+-123m, 
and from it deduct the double of the first expression, or 
122454.24f4103m, which leaves 1972+4f+17m, for 
the notation of Hf. 3rd. If the major Comma be given, 
its ratio is 24, or 24 x 5~34, and we must first take 4 times 
the second expression, or 3880=+76f+ 336m ; and next, 
