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II. The second is the reason why the division of the notes 
in the ordinary diatonic scale, artificial as it manifestly is, has 
become the universal division of Kuropean music, and appears 
so simple and natural. 
_ Upon these two points I propose to offer some suggestions in 
this paper, premising to professional musicians (if the paper 
should fall under the eye of such) that musically I write only 
as an amateur. ‘This, however, is perhaps of no great import- 
ance, as the question is one rather of numbers and mathematics 
than of technical musical knowledge. 
I. It is well known to every one acquainted with music. 
that the ordinary musical keys, having for their signatures 
respectively no sharps, one sharp, two sharps, &e. are formed 
each from the other by sharpening the subdominant of the scale, 
and so bringing it within half a tone of the dominant to which 
in the next scale it becomes the leading note. Thus we pass 
simply from any key to the key of the dominant, and the 
transition is so easy that even in the least complicated com- 
positions, as for example in hymn-tunes, the modulation con- 
stantly takes place. The tonics or key-notes in the scales thus 
formed are C, G, D, A, E, B. 
Now it is not very easy to see the manner in which these 
Successive scales are related to each other, nor why the system 
is so complete as it appears to be. But the relation may be 
exhibited to the eye, and the symmetry of the system con- 
sequently made plain, in the following manner. 
Let us regard the semitone intervals of the chromatic scale 
as being all equal, which, though not true upon any theoretical 
principle, is true according to that system of temperament, upon 
which pianofortes have long been tuned and upon which (as 
I understand upen good authority) it is now becoming the 
practice to tune organs. Then the musical interval between 
each note and its octave will be divided into twelve equal 
intervals, and these intervals may be conveniently represented 
