It will be evident to all who are much conversant with musical compositions, that the internal structure of all pieces in the same mode, whatever be their signature, is much the same. There is scarcely more difference, for example, in the relative frequency of different chords in the natural key, and in that of two sharps, or two flats, than there is in different pieces on the same key. If the Vth CG on the tonic has to the Vth EB on the mediant in the natural key, any given ratio of frequency m : n, the relative frequency of the Vth DA on the tonic, and the Vth F

C

on the mediant in the key of two sharps, will not sensibly differ from that of m : n. Hence, if we examine a sufficient number of pieces to establish the relative frequency of the different consonances in one major and its relative minor key, and, by a much more extensive investigation, ascertain the relative frequency of occurrence of the different signatures, it is evident, that by multiplying this last series of numbers into the first, and adding those products which belong to chords terminated by the same letters, we shall have a series of numbers expressing the chance of occurrence in favour of each of the consonances of the scale, when all the keys are taken into view.

It was judged that 200 scores, taken promiscuously from all the varieties of music for the organ,[7] would afford a set of numbers expressing, with sufficient accuracy, the chance that a given consonance will occur in a single major, and its relative minor key. Accordingly 200 scores were examined, 150 in the major, and 50 in the minor mode, (as it will appear hereafter that this is nearly the ratio of their frequency) of the various species of music for the organ, comprising a proper share both of the simpler and of the more rapid and chromatic movements. As the selecting and reducing to their proper keys all the occasional modulations which occur in the same piece would render the labour of ascertaining the relative frequency of different signatures very tedious, it was thought best to consider all those modulations which are too transient to be indicated by a new signature, as belonging to the same key. This will account for the occurrence of the chords in the following table, which are affected by flats and sharps.

The minim, or the crotchet, was taken for unity, according to the rapidity of the movement. Bases of greater or less length had their proper values assigned them; although mere notes of passage, which bore no proper harmony, were generally disregarded. The scores were taken promiscuously from all the different keys; and were reduced, when taken down, to the same tonic; the propriety of which will evidently appear from the foregoing remarks. The following table contains the result of the investigation.

TABLE I.

Bases.Common Chords.Flat Fifths.7ths.9-sevenths.
MajorMinorMajor.Minor.Major.Minor.Major.Minor.
mode.mode.
B III58——7———
B3—1635511172—
B44——————
A VII——————3—
A III198——72——
A166588212652—
G——338————
G 31815——————
G96593——178153—
F——464112——
F35260——111273
E III26271——125——
E3225518—14
D III——21————
D———4————
D III294——497——
D120129——551861
C——241———
C 32———————
C1769275——5141