36 On Musical Temperament. 
oftener, as forming an essential part of the harmony of more 
keys, than others. The Vth DA, for example, forms one of 
the essential chords of six different keys 5 while the VthG#DZ 
forms a part only of the single key of four sharps. 
Proposition IV. 
To find a set of numbers, expressing the ratio of the probable 
number of times that each of the different consonances in 
the scale will occur, in any set of musical compositions. 
This can be done only by investigating their actual frequency 
of occurrence in a collection of pieces for the imstrument to be. 
tuned, sufficiently extensive and diversified to serve as a spe- 
cimen of music for the same instrument in general. This may 
appear, at first view, an endless task ; and it would be really 
such, were we to take music promiscuously, and count all the 
censonances which the base makes with the higher parts, and- 
the higher parts with each other. But it appears, from Prop. 
I. Cor. that all the positions and inversions of a chord, when 
the octaves are kept perfect, are equally harmonious with the 
chord itself. The Vth, for example, which makes one of the 
consonances in a common harinonic triad, is equally harmoni- 
ous, in its kind, with the V+ VIII, which takes its place in the 
$d_ position of this triad; and with the 4th in its 2d inversion. 
Hence, instead of counting single consonances, we have only 
to count chords ; and this is done with the greatest ease, by 
means of the figures of the’ thorough base. The labour will — 
be still farther abridged by reducing the derivative chords, 
such as the 6, the 9%, &c. to their proper roots, as they are — 
taken down. But even after these reductions, the labour of 
numbering the different chords in a sufficiently extensive set of — 
compositions, to establish, with any degree of certainty, the 
relative frequency of the different signatures, would be vely = 
irksome. A method, however, presents itself, which renders — 
it sufficient to’examine the chords in such a set of pieces only — 
as will give their chance of occurrence in two keys—a major, 
and its relative minor. 
It will be evident to all who are much conversant with mu-— 
sieat compositions, that the internal structure of all pieces if 
