26 On Musical Temperament. 



oftener, as forming an essential part of the harmony «f moi'e 

 keys than others. The Vth DA, for example, forms one of 

 the essential chords of six different keys ; while the Vt" 

 G:*D* 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 instrument to be 

 tuned, sufl&ciently 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 

 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 harmoni- 

 ous in its kind, with the V-f 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 |, &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. 



It will be evident to all who are much conversant with mu- 

 iiical compositions, that the internal structure of all pieces in 



