METHODS OF MODULATION The commonest way of establishing a change of key is, as we have seen, to emphasize the dominant chord of the new key until only the new tonic can be expected. This we will call dominant modulation; leaving out of consideration how the new dominant is reached. (It was probably surrounded by its own dominant of-dominant which could be reached from various other direc tions.) A more interesting type of modulation begins with Beethoven, arising out of hints given by Haydn and Mozart. It may be called functional modulation, and consists in placing indirectly related keys into positions which make their exact relation appear vividly. If the first chord of the second key is a dominant, the relation will still appear in high relief ; but any further decoration of that dominant will reduce the result to an ordinary modula tion. (Compare bars 22-23 of the first movement of Beethoven's Waldstein sonata with the drastic process of bars 37-38 in the first movement of op. io6.) Functional modulation might well be called "natural" if that term had not been commonly assigned to modulation within the five directly-related keys, irrespective of method.
Mere juxtaposition of tonics will suffice for the purposes of a functional modulation. If Beethoven had wished to explain the presence of F sharp minor (vib) in the scheme of op. io6, the natural (or functional) process would consist of the following four chords: Closely akin to this method is Beethoven's dramatic way of reducing a chord to a single note, and then building up there from a quite remote chord. (See opp. 90 and 8ra.) All such devices show the listener what is really happening. The object of enharmonic modulation is frankly to mystify. It is pop ularly supposed to belong specially to tempered scales; but it really presupposes just intonation. All discords, as we have seen, set up a conflict in their intonation ; and an enharmonic modu lation is merely a conflict so coarse-grained that it appears in the notation by some such mark as a change from G# to Ab. An ill-motived enharmonic modulation is like a bad pun ; a great enharmonic modulation is a sublime mystery. Here is the com monest pivot of enharmonic changes, the diminished 7th, with its four vastly different resolutions: designs seldom stretch without break over 15 minutes, and always show their purport within five. But take, for example, the con flict between two major keys a tone apart. The jealous Fricka did hope (in F major) that the domestic comforts of Walhalla would induce Wotan to settle down. Wotan, gently taking up her theme in E flat, dashes her hopes by this modulation more effec tively than by any use of his artillery of tubas and trombones.
But the most distinctive feature of Wagner's harmony is his use of long auxiliary notes in such a way as to suggest immensely remote keys, which vanish with the resolution. (Chopin antici pates Wagner in what Sir Henry Hadow finely describes as "chromatic iridescence.") Here is the evolution of the wonderful opening of Tristan and Isolde: Of course, these are really four different chords. If the true theory of just intonation demanded that the minor scale should be rigid, a chord of the diminished 7th would be much harsher than the tempered scale makes it. But what really happens in just intona tion is that two notes of the minor scale become so unstable in the stress of discord that it becomes a small matter to shift the strains to whichever notes you please. Even with a limited key board the ear imagines a change of intonation when the unex pected resolution appears. This is why chromatic intervals are difficult to sing; the singer loses confidence when he has to aim at a note which will not stand still.
Not every change of notation represents a genuine enharmonic modulation. Modulate diatonically from A to F : and transpose your modulation down a semitone. You will start in A flat, but if you have much to say in the second key you will probably pre fer to write it as E, instead of F flat. Sad nonsense has been written by many commentators on the most ordinary harmonies disguised by convenience of notation.
Nevertheless, a merely notational change may eventually have an enharmonic result, for it may be part of an enharmonic circle. If the harmonic world is round, why should just intonation be plane? Adjustments infinitely smaller than those of temperament will suffice to make the ends of an enharmonic circle meet in the course of a long composition. The first movement of Brahms's F major symphony, played with its repeat, goes four times round an enharmonic circle of major 3rds (F, Db, Bbb=A, F). Every time the key written as F returns it identifies itself by the open ing theme. If the pitch rose to Gbb we would scarcely notice the fact after the intervening passages, and when the pitch had risen noticeably we should complain. Temperament keeps the pitch; but just intonation could do so by an even distribution of infinitely smaller adjustments.
It now becomes clear why keys a tritone 4th apart cannot be come related. That interval (which modal musicians identified with the devil) constitutes the kink in musical space. It sets up an enharmonic short-circuit ; a modulation from C to F sharp is exactly the same as one from G flat back to C; and which ever key you start from, the other will sound like the dominant of a Neapolitan key instead of asserting its own rights. No sensi ble person forbids the modulation; its effect may be excellent, but it is not the effect of a key-relation.