2On Miisles ooe27 175 
6th, 9th, or 10th columns of the above table (rejecting or 
borrowing 1° when necessary) the logarithms of «c,d, b dy 
&c. in the next octave above will be obtained: for example; 
if to .9030900 the diatonic E, .6989700 be added, we have 
‘€ = .6020600; which ought to be a true Vth to A, and 
such we accordingly findit, on adding the logarithm ofa Vth, 
or .8239087, to that of .7781513 = A. © The true, or 
harmonic 3d, is § = .9208188; this deducted from E = 
.9030900, gives x C = .9822719, differing 53950, or a 
comma, = £1, from * Coin the etidée The true 6th is.$ 
= +7958800, which added to bE = 9311187 gives B= 
#7269987, as we find’ it to be in the table. In this: manner 
may the curious reader’ examine all the harmonies, or any 
particular ones, in the systems contained |in. the. above or 
any similar tables, and decide on the pretensions to truth 
and exclusive advantages claimed, by the advocates for each 
system. 
Mr. Hawkes, in the work before us, has left his i: 
without any directions for tuning fifths + of a comma flatter 
than perfect, as-required in the tuning of his system, except 
the judgment of the ear, which is incompetent to the purpose. 
To those who wish to adopt or try this, or indeed any other 
tempered system, I recommend a careful study of that excel- 
lent work Dr. Smith’s Harmonics, and the article Tempera- 
ment in the Supplement to the third edition of the Encyclope- 
dia Britannica, where the correct and elegant method of tun- 
ing any system by Leafs, is thoroughly and clearly explained. 
I am somewhat surprised to find a nobleman of lord 
Stanhope’s degree of information, laying so much stress 
upon four false intervals, which he calls wolves, oecasioned 
by the particular series of perfect i Jifths, which he recom- 
mends to follow each other in the tuning, without duly 
considering, whether any perfect fifils can be introduced 
between the twelve sounds of a tempered system for our 
present keyed instruments, without doing mischief, and 
particularly, by occasioning transitions during performance 
from a better to a worse harmony; which Dr. Smith, and I 
think justly, considers as the principal cause of the disa- 
greeable effects which nice ears experience, in our best con- 
8 certs 
