Again, if different chords, in practice, vary in the frequency of their occurrence, this will be a sufficient reason for deviating from the system of equal temperament. Suppose, for example, that a given sum of temperament is to be divided between two Vths, one of which occurs in playing ten times as often as the other: there can be no doubt that the greater part of the temperament ought to be thrown upon the latter. Hence it becomes an important problem to ascertain, with some degree of precision, the relative frequency with which different consonances occur in practice. Before proceeding to a direct investigation of this problem, it may be observed, in general, that such a difference manifestly exists. In a given key, it cannot have escaped the most superficial observer, that the most frequent combination of sounds is the common chord on the tonic; that the next after this is that on the dominant, and the third, that on the subdominant. Perhaps scarcely a piece of music can be found, in which this order of frequency does not hold true. It is equally true that some signatures occur oftener than others. That of one sharp will be found to be more used, in the major mode, than any other; and, in general, the more simple keys will be found of more frequent occurrence than those which have more flats or sharps. These differences are not the result of accident. The tonic, dominant, and subdominant, are obviously the most prominent notes in the scale, and must always be the fundamental bases of more chords than either of the others; while the greater ease of playing on the simpler keys will always be a reason with composers for setting a larger part of their music on these, than on the more difficult keys. It is observable, that the greater part of musical compositions, whether of the major or minor mode, is reducible to two kinds: that in which the base chiefly moves between the tonic and its octave, and that in which the base moves between the dominant and subdominant of the key. The former class, in the major mode, are almost universally set on the key of one sharp; the latter, generally on the natural key, or that of two sharps. In the minor mode, the former class have usually the signature of two flats, or the natural key; the latter, that of one flat. Hence the three former keys will comprise the greater part of the music in the major mode, and the three latter, of that in the minor mode, in every promiscuous collection. But if we were even to suppose each of the chords in the same key, and each of the signatures, of equally frequent occurrence, some chords would occur much 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; while the Vth G♯D♯ forms a part only of the single key of four sharps. PROPOSITION IV.