probably for this reason that the key of the Subdominant is unsatisfactory as a balance or complementary key of a movement, since in progressing to its Dominant to verify the tonality, the mind of an intelligent listener recognises the original Tonic again, and thus the force of the intended contrast is weakened. This, as has been above indicated, is frequently found in works of the early harmonic period, while composers were still searching for the scale which should give them a major Dominant chord, and the effect of such movements is curiously wandering and vague. The use of the Dominant as the complementary key becomes frequent in works of the latter portion of the 17th century, as in Corelli; and early in the next, as in Bach and Handel, it is recognised as a matter of course; in the time of Haydn and Mozart so much strain was put upon it as a centre, that it began to assume the character of a conventionalism and to lose its force. Beethoven consequently began very early to enlarge the range of harmonic bases of the key by the use of chords which properly belonged to other nearly related keys, and on his lines composers have since continued to work. The Tonic and Dominant centres are still apparently inevitable, but they are supplemented by an enlarged range of harmonic roots giving chromatic combinations which are affiliated on the original Tonic through their relations to the more important notes of the scale which that Tonic represents, and can be therefore used without obscuring the tonality. As examples of this may be taken the minor seventh on the tonic, which properly belongs to the nearly allied key of the subdominant; a major concord on the supertonic, with the minor seventh superimposed, which properly belong to the Dominant key; the major chord on the mediant, which properly belongs to the key of the relative minor represented by the chord of the submediant, and so on.
Bach's use of harmony was a perfect adaptation to it of the principles of polyphony. He resumed the principle of making the harmony ostensibly the sum of the independent parts, but with this difference from the old style, that the harmonies really formed the substratum, and that their progressions were as intelligible as the melodies of which they seemed to be the result. From such a principle sprang an immense extension of the range of harmonic combinations. The essential fundamental chords are but few, and must remain so, but the combinations which can be made to represent them on the polyphonic principle are almost infinite. By the use of chromatic passing and preliminary notes, by retardations, and by simple chromatic alterations of the notes of chords according to their melodic significance, combinations are arrived at such as puzzled and do continue to puzzle theorists who regard harmony as so many unchangeable lumps of chords which cannot be admitted in music unless a fundamental bass can be found for them. Thus the chord of the augmented sixth is probably nothing more than the modification of a melodic progression of one or two parts at the point where naturally they would be either a major or minor sixth from one another, the downward tendency of the one and the upward tendency of the other causing them to be respectively flattened and sharpened to make them approach nearer to the notes to which they are moving. In the case of the augmented sixth on the flat second of the key, there is only one note to be altered; and as that note is constantly altered in this fashion in other combinations—namely by substituting the flattened note for the natural diatonic note, as D♭ for D in the key of C, by Carissimi, Bach, Beethoven, Chopin, in all ages of harmonic music—it seems superfluous to consider whether or no it is a chord with a double root as theorists propose, in which one note is the minor ninth of one root, and the other the major third of another. The way in which ideas become fixed by constant recurrence has already (p. 678) been indicated in the case of a figure which was very characteristic of the polyphonic school, and in that of the subdominant seventh with the early harmonists; in like manner modifications, such as the augmented sixth, and the sharp fifth (which is merely the straining upwards of the upper note of a concord in its melodic progression to the next diatonic note), become so familiar by constant recurrence, that they are accepted as facts, or rather as representatives, by association, of the unmodified intervals, and are used to all intents as essential chords; and moreover being so recognised, they are made liable to resolutions and combinations with other notes which would not have been possible while they were in the unaltered condition; which is not really more to be wondered at than the fact that Bach and his contemporaries and immediate predecessors habitually associated tunes originally cast in the old ecclesiastical modes with harmonies which would have been impossible if those modes had not been superseded by the modern system of scales. The inversion of the above-mentioned augmented sixth as a diminished third is remarkable for two reasons. In the first place, because when used with artistic purpose it is one of the most striking chords in modern music, owing to the gradual contraction towards the resolution—as is felt in the employment of it by both Bach and Beethoven to the words 'et sepultus est' in the 'Crucifixus' of their masses in B minor and D respectively; and in the second, because a distinguished modern theorist (whose work is in many respects very valuable) having discovered that the augmented sixth is a double rooted chord, says that it 'should not be inverted, because the upper note, being a secondary harmonic, and capable of belonging only to the secondary root, should not be beneath the lower, which can only belong to the primary root.' It must not be forgotten, however, in considering the opinions of theorists on the origin of chords such as these, that their explanations are not unfrequently given merely