Chapter 5a
Modulation and Key-relationship
§ 1
So far we have only concerned ourselves with Rhythm; we must now approach the subject of Form. Rhythm is concerned with the construction of musical sentences. Form has to do with the way in which musical sentences or melodies are combined so as to construct movements. The chief relation which binds melodies together into a movement is Key. But what is meant by the key of a piece of music? For example, we say that Beethoven’s Sonata Op. 53 “Waldstein” is in C major. Does it mean that the whole Sonata is in this key? Of course not; but it means that this is the central key of the whole, that it begins in C and finally returns to and ends in C, and that all other keys used in the composition have some key-relationship to this central key.§ 2
In the preceding chapters we have explained the nature of a musical sentence, or period, and have shown how it can be divided into phrases, sections, and motives. We have seen that in many sentences, even in those that are quite short, modulations are employed, and are indeed often useful. Before proceeding to show how musical sentences are constructed, it will therefore be advisable to deal with the whole question of modulation – a very large subject, as will be seen, but a basic knowledge of which is indispensable.
§ 3
By Modulation is simply meant a change of key – that is, the temporary disestablishment of the original tonic, and the substitution of a new one in its place. If the substitution of the new tonic is only momentary, the modulation is said to be a “transition”.
Here we see at (a) a modulation into D minor, returning in the next bar to the original key. At (b) is a modulation to A minor, and at (c) to F major; but in no case are there more than two chords in the new key. As no single chord can ever definite a key, the modulations here are the briefest possible; they are therefore “transitions”.
§ 4
Here the reader may naturally ask: how can we tell that at (c) there are only two chords in the key of F major? For the B flat is not contradicted by a B natural till later. Why may not the third and fourth chords from the end be also considered to belong to the key of F, as they contain no notes foreign to the signature of that key?
§ 5
The answer to this question illustrates the importance of considering the division of a sentence into the motives spoken of in the last chapter. It is quite clear that the third and fourth chords from the end form part of the same motive, and will therefore be in the same key. It is true that they might both be in the key of F. But if the [6 4] chord on G were in the key of F, it would be the second inversion of the dominant chord, which cannot be used followed by another chord on the same bass note or its octave. It is evident from what follows that this chord is here in the key of C; consequently we regard the unaccented chord of the same motive as being also in that key, and so there are only two chords in the key of F.
§ 6
The first question with which we have to deal is that of key-relationship.
Two major keys are said to be related to one another when their tonics are consonant
; and the more perfect the consonance the nearer the relationship. Further more nearly the keys are related, more chords they will have in common.
It is evident that the nearest related major keys to C are G and F, because their tonics form perfect consonances with C, (G is the fifth above, F the fifth below); and, as each key has only one note of its diatonic scale different from that of the scale of C, (F sharp for G and B flat for F), it is clear that each of these keys will have four diatonic triads common to itself and the key of C. For the same reason, A minor, having six of the seven notes of its harmonic scale common to itself and C, and therefore four diatonic triads in common, is a nearly related key to C; while the relative minors of G and F (E minor and D minor), though less closely connected with the key of C, are also included among the nearly related keys by reason of their close relationship to its dominant and subdominant.
§ 7
We have just seen that, though the five keys G major, F major, A minor, E minor and D minor are all nearly related to C major, they are not all equally nearly related. The exact degree of relationship depends on the number of chords which they have in common. We find that, of diatonic triads (of which alone we are speaking at present), G major, F major, and A minor have four each chords in common with C (see Example 2), while E minor has only three, and D minor only one.
By the addition of discords and chromatic harmonies, the number of chords in common will be considerably increased in each case; but D minor will still be the least nearly related of the “attendant keys” of C, as this circle of related keys is often called, and E minor will come next in ascending order. This is probably the reason why a modulation from any major key to its supertonic minor (II grade) is less often met with, especially as a first modulation, than one to any other of the nearly related keys. It must be understood that we are not referring to such merely transient modulations as those shown in our Example 1, but to those more pronounced modulations where a phrase or sentence ends in the new key.
§ 8
Let us now examine the nearly related keys to a minor key, and we shall obtain some different results. We make our circle of attendant keys on the same plan as with the major key – that is, we take the minor keys a perfect fifth above and below our tonic, and the relative majors of these three keys. But if, in order to determine the respective nearness of relationship of the different keys, we apply the same test as with the major keys, we shall find some striking differences in the results now obtained. Let us take A minor as our central tonic, because the whole circle of nearly related keys will be the same as with C major. But whereas the keys a perfect fifth above and below C major have four chords each in common with it, the keys of E minor and D minor have but one chord each in common with A minor. The only key which has four chords in common with A minor is its relative major, C; this is therefore its most nearly related key. Next comes its submediant major (VI grade), F – the relative major of its subdominant – with three chords in common; while G major (the relative major of the dominant, E minor) has, like D minor and E minor, only one chord common to itself and A minor.
Again our theory as to nearness of relationship depending upon the number of chords common to the two keys agrees with the practice of composers; for we find that the most frequent modulations, especially as first modulations, from any minor key are to its relative major, or to its submediant major.
Cfr. Bertenshaw, Elements of Music (Longmans ed. 1896), Ch. LXV; Cfr. Prout, Musical form (Augener ed.), Ch. 4
For nomenclature: perfect consonances are unison, perfect fifth, and octave. Imperfect consonances are third and sixth major; third and sixth minor.