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 instrument to be tuned, sufficiently extensive and diversified to serve as a specimen 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 consonances 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 harmonic triad, is equally harmonious in its kind, with the V + VIII, which takes its place in the 3d position of this triad, and with the 4th in its second 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 6/4, &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 very 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.