Chapter 5b
Modulation and Key-relationship
For paragraphs § 1 to § 8 see: Chapter 5a
§ 9
We must now examine those keys which stand in the second degree of relationship. Taking first the major keys, they will be those in which the tonics are still consonant, but in which the consonance is imperfect, instead of perfect. Taking, as before, C as our center, the keys will be those at the distance of a major or minor third above and below it – E, E flat, A, and A flat. Here the test of relationship we have hitherto applied fails us altogether, for there is not a single common chord which, as a diatonic chord, belongs to C and to anyone of the other keys we have just named. Evidently we must now look for some other bond of connection.§ 10
Fortunately we have not to go far in order to find what we require. In major keys, we shall always find some triads which are diatonic in one of the two keys and chromatic in the other. For example, the triads on the tonic, subdominant, and submediant of C major are all of them chromatic triads (notice the asterisks) in E major.
Obviously the relation between the keys of A flat and C will be the same as between C and E, the tonics in both cases being a major third apart, and the chromatic triads on the minor second, the subdominant, and the minor sixth of C will be diatonic in A flat.
§ 11
If we now take keys whose tonics are a minor third apart – C and A major, or C and E flat, we shall find similar connections. The chords of the supertonic and subdominant of C are the chromatic chords on the subdominant and submediant of A, and the diminished triad on the leading note of C (B natural) is the upper part of the dominant minor ninth of A. On the other hand, the diatonic chord on the subdominant of A is the chromatic chord on the supertonic of C. It will be evident that, as the relation between C and E flat is the same as that between A and C, we can work out a similar connection of chords between these two key also.
§ 12
We remember that with nearly related keys, the major keys of the tonic and subdominant bring their relative minors along with them into the circle of near relationship. But this is only partially the case with the relative minors of keys distant a third from what we may term the central tonic. In the group which we have been discussing, in which C is the center, and the related major keys of the second degree are E, E flat, A, and A flat, it will be evident at once that the relative minors of these keys will not stand upon the same footing. Only the key of C minor has so many chords common with C major that it is doubtful whether it ought not to be included among the most nearly related keys; while the key of F minor has several of its most important chords (tonic, chromatic supertonic, dominant, and submediant) appearing either as diatonic or chromatic chords in the key of C.
On the other hand, the connection between the key of C and those of F sharp minor and C sharp minor is very remote, for the few triads common to the keys are among those which, either in the one key or the other, are the least frequently used. We arrive therefore at this result – that only those relative minors of keys in the second degree of relationship are themselves related to the central tonic which contain more flats in the signature than the tonic (C minor with 3 flats, and F minor with 4).
§ 13
We have already seen (Chapter 5 § 8) that among the nearly related keys to a minor key the minors of its dominant and subdominant were much less nearly related to the tonic than was the case with the dominant and subdominant of a major key. In fact, all minor keys are more loosely related to each other than major keys, and the rule given above, that two major keys are related to one another when their tonics are consonant, does not apply at all to two minor keys in the second degree of relationship.
§ 14
Let us test this statement with the minor keys at a distance of a major and minor third above and below A minor. These will be C minor and C sharp minor above, F minor and F sharp minor below. The only triads in A minor which are common to it and to either of the 4 keys we have just named are the two diminished triads on the second and seventh degree of the scale. But in no case we find a complete chord; and this slight point of contact is not enough to establish a relationship between the keys.
§ 15
While, however, these minor keys are not themselves related to the minor key whose tonic is at a distance of a third from them, we shall nevertheless find that two of their relative majors are related. A major and E major will be just as closely related as we saw that the keys of C minor and F minor were related to C major. The relationship is in fact identical.
§ 16
It is an interesting point, and worth noticing as we pass, that, in the relations we have just noticed, the minor key is the exact converse of the major. In this second group of related keys, the minors which are related to any tonic major are the tonic minor and the subdominant minor – in other words, those minor keys whose signature contains 3 and 4 flats more than the original key; while in the same group the related major keys to any tonic minor are the tonic major and the dominant major – the major keys whose signature contains 3 and 4 sharps more than the original key. It will be seen that in all cases of relationship in the second degree between a major and a minor key, the tonics of the two key will be consonant.
§ 17
It should further be observed that we do not include the dominant minor of a major key, nor the subdominant major of a minor key, among the related keys, because the immediate juxtaposition of these keys produces a disturbing effect on the tonality. This will be clearly seen if we put the two keys next to one another, with one intermediate chord to make the modulation.
Here the modulation is effected at (a), but the effect is unsatisfactory; for it suggests at the second and third bars either a major subdominant chord in G minor (+), or a minor dominant chord in C (*). If we make the converse modulation, from G minor to C, we shall have the same unpleasant effect.
§ 18
All other keys, whether major or minor, excepting those already spoken of, are said to be unrelated keys. In the case of two major keys, the want of relationship arises from the fact of the tonics being dissonant; while the connection of minor keys is, as we have already seen, so much looser than that of major ones, that in all other keys than those above considered the points of contact are too slight to allow us to regard the keys as related.
§ 19
We conclude this chapter with a table of all related major and minor keys – not now taking any one key as a center, but expressing the relationship of the tonics to one another according to their intervals.
Cfr. Prout, Musical form (Augener ed.), Ch. 4