1868. ] On Musical Seales. 345 
$=678) which satisfy 8— 8 = 415. And lastly, (@=229, «= 814) 
and (8 = 263, « = 848): so that either of the two systems— 
(VIL) 
a=59, B=229, y=492, 5=644, «=814 
a=93, B=263, y=492, 5=678, e—848 
will satisfy the differential relations 6 — 2 = 585, 6 — 8 = 415, 
and e — 8 = 585, and will thus give us three more implied Fifths, 
without sacrificing either of the five Thirds which they imply, or 
the independent Fifth secured by taking y = 492. 
On the other hand, any one of the systems of values in (VI.) 
will give us five Fifths, but their adoption necessarily sacrifices four 
out of the five Thirds given by (V.); that dependent on y = 492 
being the only one retained. Here, however, our attention is natu- 
rally drawn to the singularly close coincidence of all the six values 
of a, 8, y, &c., in these systems (the direct consequence, be it ob- 
served, of that remarkable relation ITI. + 8 V.= 5000 + 2 we have 
already pointed out). And it will not fail to strike us that if we 
consent to overlook so very minute a deviation from absolute per- 
fection in the value of ¢ as a single unit (a thousandth part of an 
octave, or 1-18th part of a comma, an interval no human ear is nice 
enough to distinguish), and adopt the series of values, 
a=77, B=247, y=492, 3=662, «=831, 
we shall obtain sez Fifths, four of which are perfect, and two de- 
fective only by the infinitesimal error above mentioned. 
Thus, then, we have arrived at three scales, or chromatic sub- 
divisions of the octave, which alone can be considered as having any 
distinct claim to preference among the innumerable systems which 
might be proposed, wz. :— 
Do Re Mi Fa Sol La Si do 
(A) 0, 93, 170, 263, 322, 415, 492, 585, 678, 737, 848, 907, 1000. 
B) 0, 59, 170, 229, 322, 415, 492, 585, 644, 737, 814, 907, 1000. 
(C) 0, 77, 170, 247, 322, 415, 492, 585, 662, 737, 831, 907, 1000. 
The first of these (A), when translated into the language of 
ratios and fractions, will be found to coincide with that given by 
Cavallo (Phil. Trans., vol. Ixxvui., p. 239) as the then received har- 
monic subdivision of the octave into twelve intervals. The second 
(B) is one proposed by Euler ; * while the third (C) coincides very 
nearly, so far as the accidental notes (the sharps or flats) are con- 
cerned, with either of the two tempered scales propounded by Dr. 
* T derive my knowledge of it from the Abbé Moigno’s Journal, ‘ Annuaire du 
Cosmos,’ 1859, part ii., p. 200. As there given, however, it is full of misprints. 
The fractions expressing the ratios of vibrations, evidently intended by Euler, are— 
NSS. OR OEE Sue Se PSB DS SR ARE. Be 2 
> Oa) 6 Gas «ae 39 3S? 99 Tao 3 ‘Vas? Bb? 
