such emphasis and proportion as shall leave upon the mind the impression that the whole is in the primary key and that the foreign keys have been as artistically grouped around it as its own local harmonies. The true principles on which keys are related proved so elastic in the hands of Beethoven that their results utterly outstripped the earlier theory which adhered desperately to the limitations of the 16th century; and so vast is the range of key which Beethoven is able to organize in a convincing scheme of relationship, that even modern theory, dazzled by the true harmonic possibilities, is apt to come to the conclusion, more lame and impotent than any ancient pedantry, that all keys are equally related. A vague conception, dubbed "the unity of the chromatic scale," is thus made to explain away the whole beauty and power of Wagner's no less than Beethoven's harmonic system. We have not space to dispute the matter here, and it must suffice to state dogmatically and statistically the classical facts of key-relationship, including those which Beethoven established as normal possibilities on the suggestion of Haydn, in whose works they appear as special effects. mere survival of ancient customs, at least where great masters | should at last return to the primary key, to remain there with are concerned. (This final major chord is known as the Tierce de Picardie.) The effect of the minor mode is thus normally plaintive because it centres round the artificial concord instead of the natural; and, though the keynote bears this minor artificial triad, the ear nevertheless has an expectation (which may be intensified into a powerful emotional effect) that the final conclusion of the harmonic scheme may brighten out into the more sonorous harmonic system of major chords. Let us once more recall those ecclesiastical modes of which the 3rd degree is normally minor. We have seen how they may be regarded as the more oblique of the various cross-sections of the 16th-century harmonic scheme. Now, the modern minor mode is too firmly rooted in its minor tonic chord for the 16th-century feeling of an oblique harmonic scheme to be of more than secondary importance, though that feeling survives, as the discussion of key-relationships will show us. But it is constantly thrust into the background by the new possibility that the minor tonic chord with its attendant minor harmonies may give place to the major system round the same tonic, and by the certainty that if any change is made at the conclusion of the work it will be upon the same tonic and not have reference to some other harmonic centre. In other words, a major and minor key on the same tonic are felt as identical in everything but expression (a point in which the Tonic Sol Fa system, as hitherto practised, with its identification of the minor key with its "relative" instead of its tonic major, shows a most unfortunate confusion of thought). The characteristics of the major and minor modes may of course be modified by many artistic considerations, and it would be as absurd to develop this account into a scheme of pigeon-holed passions as to do the same for the equally obvious and closely parallel fact that in drama a constant source of pathos is the placing of our sympathies in an oblique relation to the natural sequence of events or to the more universal issues of the subject. V. Key-Relationships.-On the modern sense of the identity of the tonic in major and minor rests the whole distinctive character of modern harmony, and the whole key-system of the classical composers. The masters of the 16th century naturally found it necessary to make full closes much more frequently than would be desirable if the only possible close was that on the final of the mode. They therefore formed closes on other notes, but they formed them on these exactly as on a final. Thus, a close on the second degree of the Ionian mode was identical with a Dorian final close. The notes, other than the final, on which closes could be made were called modulations. And what between the three "regular modulations" (known as the dominant, mediant, and participant) and the " conceded modulations," of which two were generally admitted in each mode simply in the interests of variety, a composer was at liberty to form a full close on any note which did not involve too many extraneous sharps or flats for its correct accomplishment. But there was a great difference between modal and modern conceptions of modulation. We have said that the close on the second degree of the Ionian mode was Dorian, but such a modulation was not regarded as a visit paid to the Dorian mode, but merely as the formation of a momentary point of repose on the second degree of the Ionian mode. When therefore it is said that the modulations of 16th-century music are "purposeless and shifting," the criticism implies a purpose in change of key which is wholly irrelevant. The modal composers' purpose lay in purely local relationships of harmony, in various degrees of refinement which are often crowded out of the larger and more coarse-grained scheme of modern harmony, but which modern harmony is perfectly capable of employing in precisely the same sense whenever it has leisure. Modulation, in the modern sense of the term, is a different thing. The modern sense of tonality is so firm, and modern designs so large, that it is desirable that different portions of a composition should be arranged round different harmonic centres or keys, and moreover that the relation between these keys and the primary key should be felt, and the whole design a. Direct Relationships. The first principle on which two keys are considered to be related is a strengthening of that which determined the so-called modulations of the 16th-century modes. Two keys are directly related when the tonic chord of the one is among the common chords of the other. Thus, D minor is related to C major because the tonic chord of D minor is the common chord on the supertonic of C (see Ex. 6). In the same way the four other related keys to C major are E minor the mediant, F major the subdominant, G major the dominant and A minor the submediant.. This last key-relationship is sometimes called the "relative " minor, partly because it is usually expressed by the same keysignature as the tonic, but probably more justifiably because it is the point of view from which to reckon the key-relationships of the minor tonic. If we take the minor scale in its "harmonic" form (i.e. the form deducible from its chords of minor tonic, minor subdominant and major dominant, without regard to the exigencies of melody in concession to which the "melodic " minor scale raises the 6th in ascent and flattens the 7th in descent), we shall find it impossible to build a common chord upon its mediant (Ex. 10). But we have. seen that A minor is related to C major; Ex. 10. therefore it is absurd to suppose that C major is not related to A minor. Clearly then we must deduce some of the relationships of a minor tonic as the converse of those of a major tonic. Thus we may read Ex. 6 backwards and reason as follows: A minor is the submediant of C major; therefore C major is the mediant or relative major of A minor. D minor is the supertonic of C major; therefore C major is related to D minor and may be called its flat 7th. Taking A minor as our standard key, G major is then the flat 7th to A minor. The remaining major keys (C major to E minor = F major to A minor) may be traced directly as well as conversely; and the subdominant, being minor, does not involve an appeal to the major scale at all. But with the dominant we find the curious fact that while the dominant chord of a minor key is major it is impossible to regard the major dominant key as directly related to the minor tonic, since it does not contain the minor tonic chord at all; e.g. the only chord of A in E major is A major. But the dominant minor key contains the tonic chord of the primary minor key clearly enough as subdominant, and therefore when we modulate from a minor tonic to a minor dominant we feel that we have a direct key-relationship and have not lost touch with our tonic. Thus in the minor mode modulation to the dominant key is, though frequent and necessary, a much more uphill process than in the major mode, because the naturally major dominant chord has first to be contradicted. On the other hand, a contrast between minor tonic and major dominant key is very difficult to work on a large scale (as, for example, in the complementary key for second subjects of sonata movements) because, while the major dominant key behaves as if not directly related to the minor tonic, it also gives a curious sensation of | have always been connected by every composer with a wide being merely on the dominant instead of in it; and thus we find that in the few classical examples of a dominant major second subject in a minor sonata-movement the second subject either relapses into the dominant minor, as in Beethoven's Kreutzer Sonata and the finale of Brahms's Third Symphony, or begins in it, as in the first movement of Brahms's Fourth Symphony. The effect of a modulation to a related key obviously depends upon the change of meaning in the chords common to both keys, and also in the new chords introduced. Thus, in modulating to the dominant we invest the brightest chord of our first key with the finality and importance of a tonic; our original tonic chord becomes comparatively soft in its new position as subdominant; and a new dominant chord arises, surpassing in brilliance the old dominant (now tonic) as that surpassed the primary tonic. Again, in modulating to the subdominant the softest chord of the primary key becomes tonic, the old tonic is comparatively bright, and a new and softer subdominant chord appears. We have seen the peculiarities of modulation to the dominant from a minor tonic, and it follows from them that modulation from a minor tonic to the subdominant involves the beautiful effect of a momentary conversion of the primary tonic chord to major, the poetic and often dramatically ironical power of which is manifested at the conclusion of more than half the finest classical slow movements in minor keys, from Bach's Eb minor Prelude in the first book of the Forty-eight to the slow movement of Brahms's G major String Quintet, Op. 111. ་་ The effect of the remaining key-relationships involves contrasts between major and minor mode; but it is otherwise far less defined, since the primary tonic chord does not occupy a cardinal position in the second key. These key-relationships are most important from a minor tonic, as the change from minor to major is more vivid than the reverse change. The smoothest changes are those to relative" minor, 46 relative major (C to A minor; C minor to Eb); and mediant minor and submediant major (C to E minor; C minor to Ab). The change from major tonic to supertonic minor is extremely natural on a small scale, i.e. within the compass of a single melody, as may be seen in countless openings of classical sonatas. But on a large scale the identity of primary dominant with secondary subdominant confuses the harmonic perspective, and accordingly in classical music the supertonic minor appears neither in the second subjects of first movements nor as the key for middle movements. But since the key-relationships of a minor tonic are at once more obscure harmonically and more vivid in contrast, we find that the converse key-relationship of the flat 7th, though somewhat bold and archaic in effect on a small scale, has once or twice been given organic function on a large scale in classical movements of exceptionally fantastic character, of which the three great examples are the ghostly slow movement of Beethoven's D major Trio, Op. 70, No. 1, the scherzo of his Ninth Symphony, and the finale of Brahms's D minor Violin Sonata (where, however, the C major theme soon passes permanently into the more orthodox dominant minor). Thus far we have the set of key-relationships universally recognized since the major and minor modes were established, a relationship based entirely on the place of the primary tonic chord in the second key. It only remains for us to protest against the orthodox description of the five related keys as being the "relative" minor or major and the dominant and subdominant with their "relative" minors or majors; a conception which expresses the fallacious assumption that keys which are related to the same key are related to one another, and which thereby implies that all keys are equally related and that classical composers were fools. It cannot be too strongly insisted that there is no foundation for key-relationship except through a tonic, and that it is through the tonic that the most distant keys range of modulation, from Haydn to Brahms and (with due Beethoven rationalized every step in the whole possible range of key-relationship by such harmonic means as are described in the article BEETHOVEN. & Haydn's favourite key-relationships he used for the complementary key in first movements; and he at once discovered that the use of the major mediant as complementary key to a major tonic implied at all events just as much suggestion of the submediant major in the recapitulation as would not keep the latter half of the movement for too long out of the tonic. The converse is not the case, and where Beethoven uses the submediant major as complementary key in a major first movement he does not subsequently introduce the still more remote and brilliant mediant in the recapitulation. The function of the complementary key is that of contrast and vividness, so that if the key is to be remote it is as well that it should be brilliant rather than sombre; and accordingly the easier key-relationships obtainable through transforming the tonic into minor do not appear as complementary keys until Beethoven's latest and most subtle works, as the Quartet in Bb, Op. 130 (where we again note that the flat submediant of the exposition is temporarily answered by the flat mediant of the recapitulation). c. Artificial Key-relationships.-Early in the history of the minor mode it was discovered that the lower tetrachord could be very effectively and naturally altered so as to resemble the upper (thus producing the scale C Db E‡ F, G Aь B‡ C). This produces a flat supertonic (the chord of which is generally presented in its first inversion, and is known as the Neapolitan 6th, from its characteristic use in the works of the Neapolitan school which did so much to establish modern tonality) and its origin, as just described, often impels it to resolve on a major tonic chord. Consequently it exists in the minor mode as a phenomenon not much more artificial than the mode itself; and although the keys it thus connects are extremely remote, and the effect of their connexion very surprising, the connexion is none the less real whether from a major or a minor tonic, and is a crucial test of a composer's sense of key-perspective. Thus Philipp Emanuel Bach in a spirit of mere caprice puts the charming little slow movement of his D major Symphony into Eb and obliterates all real relationship by chaotic operatic connecting links. Haydn's greatest pianoforte sonata (whicn, | There now remains only one pair of keys that have never been Converse harmonic relationships are, as we have seen, always weaker than their direct forms. And thus the relation of C major to B major or minor (as shown in the central episode of the slow movement just mentioned) is rare. Still more rare is the obtaining of indirect artificial relationships, of which the episode in the first movement just mentioned is an illustration in so far as it enhances the effect of the slow movement, but is inconclusive in so far as it is episodic. For with remote key-relationships everything depends upon whether they are used with what may be called cardinal function (like complementary keys) or not. Even a near key may occur in the course of wandering modulations without producing any effect of relationship at all, and this should always be borne in mind whenever we accumulate statistics from classical music. d. Contrary and Unconnected Keys.-There remain only two pairs of keys that classical music has not brought into connexion, a circumstance which has co-operated with the utter vagueness of orthodox theories on the subject to confirm the conventionally progressive critic in his conviction that all modulations are alike. We have seen how the effect of modulation from major tonic to minor supertonic is, on a large scale, obscured by the identity of the primary dominant with the secondary subdominant, though the one chord is major and the other minor. Now when the supertonic becomes major this difference no longer obviates the confusion, and modulation from C major to D major, though extremely easy, is of so bewildering effect that it is used by classical composers only in moments of intensely dramatic surprise, as, for example, in the recapitulation of the first subject of Beethoven's Eroica Symphony, and the last variation (or coda) of the slow movement of his Trio in Bb, Op. 97. And in both cases the balance is restored by the converse (and equally if not more contradictory) modulation between major tonic and major flat 7th, though in the slow. movement of the Bb Trio the latter is represented only by its dominant chord which is "enharmonically" resolved into quite another key. The frequent attempts made by easy-going innovators to treat these key-contrasts on another footing than that of paradox, dramatic surprise or hesitation, only show a deficient sense of tonality, which must also mean an inability to see the intensely powerful effect of the true use of such modulations in classical music, an effect which is entirely independent of any ability to formulate a theory to explain it. 'Many theorists mistake the usual extreme emphasis on the dominant chord of the dominant key, in preparation for second subjects, for a modulation to the major supertonic, but this can deceive no one with any sense of tonality. A good practical test is to see what becomes of such passages when translated into the minor mode. Illusory modulation to the flat 7th frequently occurs as a bold method of throwing strong emphasis on to the subdominant at the outset of a movement, as in Beethoven's Sonata, Op. 31, No. 1. Indirect, through I A. From Major Tonic Doubly indirect through the Artificial, direct Direct Relationships IV V vi ་ Indirect through both 1 iv . VII & vii IVE & (va Vb & vb 1 VIID & viib Very rare, but the slow movement of Schubert's C major String Quintet demonstrates it magnificently. All the indirect relationships from a minor tonic are distinctly strained and, except in the violently contrasted doubly indirect keys, obscure as being themselves minor. But the direct artificial modulation is quite smooth, and rich rather than remote. Sec Beethoven's C# minor Quartet. No classical example, though the clearer converse from a major tonic occurs effectively. 'Not (with the exception of II) so violent as when from major tonic. Bach, whose range seldom exceeds direct key-relationships, is not afraid to drift from D minor to C minor, though nothing would induce him to go from D major to C major or minor. when minor. Thus I represents tonic major, iv represents subdominant minor, and so on. A flat or a sharp after the figure indicates that the normal degree of the standard scale has been lowered or raised a semitone, even when in any particular pair of keys it would not be expressed by a flat or a sharp. Thus vib would, from the tonic of Bb major, express the position of the slow movement of Beethoven's Sonata, Op. 106, which is written in F# minor since Gb minor is beyond the practical limits of notation. VI. Temperament and Enharmonic Changes.-As the facts of artistic harmony increased in complexity and range, the purely acoustic principles which (as Helmholtz has shown) go so far to explain 16th-century aesthetics became more and more inadequate; and grave practical obstacles to euphonious tuning began to assert themselves. The scientific (or natural) ratios of the diatonic scale were not interfered with by art so long as no discords were "fundamental "; but when discords began to assume independence, one and the same note often became assignable on scientific grounds to two slightly different positions in pitch, or at all events to a position incompatible with even tolerable effect in performance. Thus, the chord of the diminished 7th is said to be intolerably harsh in " just intonation," that is to say, intonation based upon the exact ratios of a normal minor scale. In practical performance the diminished 7th contains three minor 3rds and two imperfect 5ths (such as that which is present in the dominant 7th), while the peculiarly dissonant interval from which the chord takes its name is very nearly the same as a major 6th. Now it can only be said that an intonation which makes nonsense of chords of which every classical composer from the time of Corelli has made excellent sense, is a very unjust intonation indeed; and to anybody who realizes the universal relation between art and nature it is obvious that the chord of the diminished 7th must owe its naturalness to its close approximation to the natural ratios of the minor scale, while it owes its artistic possibility to the extremely minute instinctive modification by which its dissonance becomes tolerable. As a matter of fact, although we have shown here and in the article Music how artificial is the origin and nature of all but the very scantiest materials of the musical language, there is no art in which the element of practical compromise is so minute and so hard for any but trained scientific observation to perceive. If a painter could have a scale of light and shade as nearly approaching nature as the practical intonation of music approaches the acoustic facts it really involves, a visit to a picture gallery would be a severe strain on the strongest eyes, as Ruskin constantly points out. Yet music is in this respect exactly on the same footing as other arts. It constitutes no exception to the universal law that artistic ideas must be realized, not in spite of, but by means of practical necessities. However independent the treatment of discords, they assert themselves in the long run as transient. They resolve into permanent points of repose of which the basis is natural; but the transient phenomena float through the harmonic world adapting themselves, as best they can, to their environment, showing as much dependence upon the stable scheme of "just intonation" as a crowd of metaphors and abstractions in language shows a dependence upon the rules of the syllogism. As much and no more, but that is no doubt a great deal. Yet the attempt to determine the point in modern harmony where just intonation should end and the tempered scale begin, is as vexatious as the attempt to define in etymology the point at which the literal meaning of a word gives places to a metaphorical meaning. And it is as unsound scientifically as the conviction of the typical circle-squarer that he is unravelling a mystery and measuring a quantity hitherto unknown. Just intonation is a reality in so far as it emphasizes the contrast between concord and discord; but when it forbids artistic interaction between harmony and melody it is a chimera. It is sometimes said that Bach, by the example of his forty-eight preludes and fugues in all the major and minor keys, first fixed the modern scale. This is true practically, but not aesthetically. By writing a series of movements in every key of which the | keynote was present in the normal organ and harpsichord manuals of his and later times, he enforced the system by which all facts of modern musical harmony are represented on keyed instruments by dividing the octave into twelve equal semitones, instead of tuning a few much-used keys as accurately as possible and sacrificing the euphony of all the rest. This system of equal temperament, with twelve equal semitones in the octave, obviously annihilates important distinctions, and in the most used keys it sours the concords and blunts the discords more than unequal temperament; but it is never harsh; and where it does not express harmonic subtleties the ear instinctively supplies the interpretation; as the observing faculty, indeed, always does wherever the resources of art indicate more than they express. Now it frequently happens that discords or artificial chords are not merely obscure in their intonation, whether ideally or practically, but as produced in practice they are capable of two sharply distinct interpretations. And it is possible for music to take advantage of this and to approach a chord in one significance and quit it with another. Where this happens in just intonation (in so far as that represents a real musical conception) such chords will, so to speak, quiver from one meaning into the other. And even in the tempered scale the ear will interpret the change of meaning as involving a minute difference of intonation. The chord of the diminished 7th has in this way four different meanings and the chord of the augmented 6th, when accompanied by the fifth, may become a dominant 7th or vice versa, as in the passage already cited in the coda of the slow movement of Beethoven's Bb Trio, Op. 97. Such modulations are called enharmonic. We have seen that all the more complex musical phenomena involve distinctions enharmonic in the sense of intervals smaller than a semitone, as, for instance, whenever the progression D E in the scale of C, which is a minor tone, is identified with the progression of D E in the scale of D, which is a major tone (differing from the former as from). But the special musical meaning of the word "enharmonic "is restricted to the difference between such pairs of sharps with flats or naturals as can be represented on a keyboard by the same note, this difference being the most impressive to the ear in "just intonation" and to the imagination in the tempered scale. Not every progression of chords which is, so to speak, spelt enharmonically is an enharmonic modulation in itself. Thus a modulation from D flat to E major looks violently enharmonic on paper, as in the first movement of Beethoven's Sonata, Op. 110. But E major with four sharps is merely the most convenient way of expressing F flat, a key which would need six flats and a double flat. The reality of an enharmonic modulation can be easily tested by transporting the passage a semitone. Thus, the passage just cited, put a semitone lower, becomes a perfectly diatonic modulation from C to E flat. But no transposition of the sixteen bars before the return of the main theme in the scherzo of Beethoven's Sonata in Eb, Op. 31, No. 3, will get rid of the fact that the diminished 7th (G Bb Db EF), on the dominant of F minor, must have changed into G Bb Db Fb (although Beethoven does not take the trouble to alter the spelling) before it could resolve, as it does, upon the dominant of Ab. But though there is thus a distinction between real and apparent enharmonic modulations, it frequently happens that a series of modulations perfectly diatonic in themselves returns to the original key by a process which can only be called an enharmonic circle. Thus the whole series of keys now in practical use can be arranged in what is called the circle of fifths (C G D A E B F# [=Gb] Db Ab Bb F C, from which series we now see the meaning of what was said in the discussion of key-relationships as to the ambiguity of the relationships between keys a tritone fourth apart). Now no human memory is capable of distinguishing the difference of pitch between the The keys of C and B# after a wide series of modulations. The following series of musical illustrations show the genesis of typical harmonic resources of classical and modern music. WAGNER. WAGNER. Definitions. (Intended to comprise the general conceptions set forth in the above article.) 1. Musical sounds, or notes, are sensations produced by regular periodical vibrations in the air, sufficiently rapid to coalesce in a single continuous sensation, and not too rapid for the mechanism of the human ear to respond. 2. The pitch of a note is the sensation corresponding to the degree of rapidity of its vibrations; being low or grave where these are slow, and high or acute where they are rapid. 3. An interval is the difference in pitch between two notes. 4. Rhythm is the organization, in a musical scheme, of sounds in respect of time. 5. Melody is the organization, in a musical scheme, of rhythmic notes in respect of pitch. 6 Harmony is the organization, in a musical scheme, of simultaneous combinations of notes on principles whereby their acoustic 7. The harmonic series is an infinite series of notes produced by properties interact with laws of rhythm and melody. the subdivision of a vibrating body or column of air into aliquot parts, such notes being generally inaudible except in the form of the timbre which their presence in various proportions imparts to the fundamental note produced by the whole vibrating body or air-column. 8. A concord is a combination which, both by its acoustic smoothness and by its logical origin and purpose in a musical scheme, can form a point of repose. 9. A discord is a combination in which both its logical origin in a musical scheme and its acoustic roughness show that it cannot form a point of repose. 10. The perfect concords and perfect intervals are those comprised within the first four members of the harmonic series, namely, the the 5th, as between Nos. 2 and 3; and the 4th, as between Nos. octave, as between numbers 1 and 2 of the series (see Ex. I above); 3 and 4. 11. All notes exactly one or more octaves apart are regarded as harmonically identical. 12. The root of a chord is that note from which the whole or the most important parts of the chord appear (if distributed in the right octaves) as members of the harmonic series. 13. A chord is inverted when its lowest note is not its root. 14. The major triad is a concord containing three different notes which (octaves being disregarded) are identical with the first, third and fifth members of the harmonic series (the second and fourth members being negligible as octaves). 15. The minor triad is a concord containing the same intervals as the major triad in a different order; in consequence it is artificial, as one of its notes is not derivable from the harmonic series. 16. Unessential discords are those that are treated purely as the phenomena of transition, delay or ornament, in an otherwise concordant harmony. 17. Essential discords are those which are so treated that the mind tends to regard them as definite chords possessing roots. 18. A key is an harmonic system in which there is never any doubt as to which note or triad shall be the final note of music in that system, nor of the relations between that note or chord and the other notes or chords. (In this sense the church modes are either not keys or else they are subtle mixtures of keys,) 19. This final note of a key is called its tonic. 20. The major mode is that of keys in which the tonic triad and the two other cardinal triads are major. 21. The minor mode is that of keys in which the tonic triad and one other cardinal triad are minor. 22. A diatonic scale is a series of the notes essential to one major or minor key, arranged in order of pitch and repeating itself in other octaves on reaching the limit of an octave. 23. Modulation is the passing from one key to another. 24. Chromatic notes and chords are those which do not belong to the diatonic scale of the passage in which they occur, but which are not so used as to cause modulation. 25. Enharmonic intervals are minute intervals which never occur in music as directly measured quantities, though they exist as differences between approximately equal ordinary intervals, diatonic or chromatic. In an enharmonic modulation, two chords differing by an enharmonic quantity are treated as identical. 26. Pedal or organ point is the sustaining of a single note in the bass (or, in the case of an inverted pedal, in an upper part) while the harmonies move independently. Unless the harmonies are sometimes foreign to the sustained note, it does not constitute a pedal. In modern music pedals take place on either the tonic or the dominant, other pedal-notes being rare and of complex meaning. Double pedals (of tonic and dominant, with tonic below) are not unusual. The device is capable of very free treatment, and has produced many very bold and rich harmonic effects in music since the earlier works of Beethoven. It probably accounts for many so-called essential discords." In the form of drones the pedal is the only real harmonic device of ancient and primitive music. The ancient Greeks sometimes |