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Originally appearing in Volume V13, Page 10 of the 1911 Encyclopedia Britannica.
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TABLES OF KEY-RELATIONSHIPS A. From Major Tonic I Direct Relationships IV V vi i i Indirect through both , ; i 1 and the second key , iv v , r , Indirect, through i Indirect through the Ifb vs second key I'll VI Doubly indireict through the ~\ former indirect keys iiib vi- ~. Artificial, direct VII & vii \ \ \ Artificial, indirect' \ \1 t ',sIet& ivfl=V6 &vb \,s k III &viib B. From Minor Tonic 3 i Direct Relationships III iv y VI ViI A t I i 11 Indirect 'through both I and the second key . IJ' V ' Indirect, through I D¢ubly indirect III# Vitt 2 Very rare, but the slow movement of Schubert's C major String Quintet demonstrates it magnificently. 3 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. See 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. Unrelated Contradictory Indirect through the second key Contradictory' 1I II viib Unrelated \ IV# & iv* =IV' & vQ Artificial, direct Artificial, indirect' I Ib VII4/& vii# i Alt 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. io6, 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 dgal. '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 mcdern scale. This is true practically, but not aesthetically. By writing a series of movements in every key of which thekeynote 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 Ex. ii. b 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 196). But the special musical meaning of the word " enharmonic " is restricted to the difference between such pairs of sharps with fiats 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. fro. 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 semi-tone. 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. 3r, No. 3, will get rid of the fact that the diminished 7th (G Bb DI? Ed), on the dominant of F minor, must have changed into G Bb Di? 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 arid 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# [=Gbl 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 keys of C and B# after a wide series of modulations. The difference would be perceptible enough in immediate juxtaposition, but after some interval of time the memory will certainly accept two keys so near in pitch as identical, whether in "just intonation " or not. And hence the enharmonic circle of fifths is a conception of musical harmony by which infinity is at once rationalized and avoided, just as some modern mathematicians are trying to rationalize the infinity of space by a non-Euclidian space so curved in the fourth dimension as to return upon itself. A similar enharmonic circle progressing in major 3rds is of frequent occurrence and of very rich effect. For example, the keys of the movements of Brahms's C Minor Symphony are C minor, E major, Ab major (= G), and C ( = B#). And the same circle occurs in the opposite direction in the first movement of his Bhird Symphony, where the first subject is in F, the transition passes directly to Db and thence by exactly the same step to A (_ Bbb). The exposition is repeated, which of course means that in " just intonation " the first subject would begin in Gbb and then pass through a transition in Ebbb to the second subject in Cbbb. As the development contains another spurious enharmonic modulation, and the recapitulation repeats in another position the first spurious enharmonic modulation of the exposition, it would follow that Brahms's movement began in F and ended in C sextuple-flat! So much, then, for the application of bad metaphysics and circle-squaring mathematics to the art of music. Neither in mathematics nor in art is an approximation to be confused with an imperfection. Brahms's movement begins and ends in F much more exactly than any wooden diagonal fits a wooden square. The following series of musical illustrations show the genesis of typical harmonic resources of classical and modern music. ~vI *f —z;„ l Ex. 14.—Ditto, with the further addition c L *I of a double suspension (*) and two passing < notes (ttI '_~ ~Ft T Ex. t5.—Ditto, with a chromatic alteration of the second chord (*) and an "essential" discord (dominant 7th) at (t). 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 properties interact with laws of rhythm and melody. 7. The harmonic series is an infinite series of notes produced by 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. to. The perfect concords and perfect intervals are those comprised within the first four members of the harmonic series, namely, the octave, as between numbers 1 and 2 of the series (see Ex. i above) ; the 5th, as between Nos. 2 and 3; and the 4th, as between Nos. 3 and 4. 11. All notes exactly one or more octaves apart are regarded as harmonically identical. t2. 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. t3. 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. t6. 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 some-times 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 Ex. t6.—Ditto, with chromatic passing notes (f i and appoggiaturas (tf). Ex. 17. —The last two chords of Ex. 16 attacked unexpectedly, the first appoggiatura (*) prolonged till it seems to make a strange foreign chord before it resolves on the short note at t, while the second appoggiatura (t) is chromatic. Ex. 18.—The same en-harmonically transformed so as to become a variation of the "dominant ninth" of C minor. The GS at * is redly A?, and t is no longer a note of resolution, but a chromatic passing-note. * —J~ WtONER. -rte ~i~ WAGNER. —as ats-e-Oca,- Ex. tn.—Three concords (tonic, first inversion of sub-dominant, and dominant of A minor, a possible 16th-century cadence in the Phrygian mode). Ex. m.—The same chords varied by a sus-pension t*). III used a reiterated instrumental note as an accompaniment above the melody. These primitive devices, though harmonic in the true modern sense of the word, are out of the line of harmonic development, and did not help it in any definite way. 27. The fundamental bass of a harmonic passage is an imaginary bass consisting of the roots of the chords. 28. A figured bass, or continuo, is the bass of a composition supplied with numerals indicating the chords to be filled in by the accompanist. Thorough-bass (Ger. Generalbass) is the art of interpreting such figures. (D. F. T.)
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