1868.] On Musical Scales. 339 
tance of the finger in every part of the scale and in every octave, 
higher or lower)—gets over this difficulty, and is accordingly often 
resorted to in treatises on music. In this, however, as in most 
other things, the proper choice of a unit is a matter of much im- 
portance. Every subject of mensuration has its natural unit: in 
angular measure, the circumference of the circle; in geodesy, the 
earth’s polar axis; in time, the length of the day; and in music, 
the octave. The ordinary tabular logarithms, however, which give 
0-30103, or (striking off the last two figures as unimportant, and 
multiplying by 1,000) 301 as the logarithm of 2, labour under two 
ereat disadvagtages, vz. Ist, that of assuming an awkward incom- 
mensurable or prime number as the measure and representative 
of this natural unit; and 2ndly, that, in different octaves, the same 
note (by name) will come to be represented by a totally different and 
unrecognizable set of figures. ‘Thus, for instance, taking the key- 
note Do as 0 and its octave do as 3801, Sol* will be expressed by 
176; while so/ (the same note in the next higher octave) will be 
expressed by 477 (176+ 301), m the next by 778 (176 + 602), 
and so on—figures which no way recall or suggest one another, and 
serve only uselessly to burden the memory. The same objection 
applies to the division of the octave into 1,200 equal parts, which, 
on the system of what is called “mean temperament,” would assign 
100 as the representative of a mean semitone; or into 600 (tempt- 
ingly near to 2x 301), which would give 100 for the expression 
of a mean tone. 
Both these disadvantages are avoided by altering all the 
logarithms proportionally, so as to make 1,000 the logarithm of 2; 
or, speaking practically and without reference to any theory or to 
any mathematical phraseology, to regard the octave as divided into 
1,000 equal intervals, each singly denoting a difference of tone so 
small as to be undistinguishable by the nicest ear, as the thousandth 
part of an inch would be to any ordinary eyesight. On this con- 
vention the numbers of such minute equal intervals, contained 
respectively in a Fifth (Vth), a Fourth (I1Vth), a Major Third 
(11rd), a Minor Third (8rd) ; a Major and a Minor tone, a Limma, 
and a Comma, are 585, 415, 322, 263, 170, 152, 93, and 18. 
The two first together make up an exact octave (585 + 415 = 
1000), and are therefore complementary to each other, while the 
last is the difference between a Major and a Minor tone, the 
smallest interval recognized as such in musical language, and one 
which it requires a nice ear to discriminate. The way in which 
* Tadhere throughout to the good old system of representing by Do, Re, Mi, 
Fa, &c., the seale of natural notes in any key whatever, taking Do for the key-note, 
whatever that may be, in opposition to the practice lately introduced (and soon, I 
hope, to be exploded) of taking Do to express one fixed tone C—the greatest retro- 
grade step, in my opinion, ever taken in teaching music, or any other branch of 
knowledge, 
