340 On Musical Scales. [July, 
these numbers stand connected with each other is as follows :— 
the Octave being 1,000, the Fifth, V. = 585, and the Third,* 
IIT. = 322; then, denoting the Major tone by T, the Minor by ¢, 
the Limma by 7, and the Comma by ¢, the [Vth will be expressed 
by 1,000—V., the Minor Third (263) by V.—III.; while T, ¢, JZ, 
and ¢ will be respectively expressed by T = 2V. — 1,000, =1,000+- 
IIL. —2V., 7=1,000 — V.—IIL, and e=4V.—III.—2,000; equa- 
tions which, musically interpreted, express that the Minor Third 
is obtained from the fundamental or key note, by tuning upwards 
a Fifth, and thence, downwards, a Major Third ; a Major tone, by 
tuning upwards two Fifths, and thence, downwardg, an octave ; 
and so of the rest. 
As regards the numerical values of notes in higher or lower 
octaves, they are formed by adding in the former case, or subtracting 
in the latter, 1,000, 2,000, 3,000, &c., according to the number of 
octaves. Thus Sol being represented by 585, its representative 
numbers in the higher octaves will be 1,585, 2,585, &c., and in the 
lower by 585 — 1,000, 585 — 2000, &c., which, in order to preserve 
the same terminal figures 585, may be written (in analogy with what 
is usually done in ordinary logarithmic’calculation) 1,585, 2,585, &., 
the superscript negative sign applying only to the index figures 
1, 2, &c., and the three terminal figures being regarded as always 
positive. 
To those who study music simply as music, without troubling 
themselves with ratios, logarithms, or vibrations, it will save some 
trouble and bewilderment to accept these numbers as they stand, 
and to regard the interval called the Octave as made up of the seven 
successive intervals, T, ¢, 7, T,¢, T, 7, constituting what is called 
the Diatonic Scale in its ordinary acceptation (though, as will be 
presently shown, the order Ty ¢, of the two first intervals may 
be reversed, so as to give the scale ¢, T, 7, T, ¢, T, 2, without mate- 
rially altering its character). It is thus that we accept the division 
of the year into twelve months of unequal lengths, and of these 
again into four weeks of seven days in each, with a greater or less 
number of supernumerary days. ‘There is this difference, however, 
viz.: that the latter division is purely arbitrary, whereas the 
former is founded in the nature of harmony; and it will therefore 
not be amiss, before proceeding farther, to explain the rationale 
on which the scale is thus, as it were, built up, and how its elements 
come to succeed each other in this manner. It is recognized, then, 
as a matter of experience, that the intervals designated as an Octave, 
a Fifth, and a Major Third (1,000, 585, 322), and also the com- 
* Whenever a “Third,” without any adjunct, is spoken of, a Major Third is 
hereafter understood. 
