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TABLES OF See also: Thus I represents tonic major, iv represents subdominant minor, and so on. A See also:flat or a See also:sharp after the figure indicates that the normal degree of the See also:standard See also: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, See also:express the position of the slow movement of Beethoven's See also:Sonata, Op. io6, which is written in F minor since Gb minor is beyond the See also:practical limits of notation. VI. Temperament and Enharmonic Changes.—As the facts of See also:artistic See also:harmony increased in complexity and range, the purely acoustic principles which (as See also:Helmholtz has shown) go so far to explain 16th-See also:century See also:aesthetics became more and more inadequate; and See also:grave practical obstacles to euphonious tuning began to assert themselves. The scientific (or natural) ratios of the diatonic scale were not interfered with by See also:art so See also:long as no discords were " fundamental "; but when discords began to assume See also:independence, one and the same See also:note often became assignable on scientific grounds to two slightly different positions in See also: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 See also:present in the dominant 7th), while the peculiarly dissonant See also: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 See also:time of See also: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 See also:close approximation to the natural ratios of the minor scale, while it owes its artistic possibility to the extremely See also:minute instinctive modification by which its dissonance becomes tolerable. As a See also:matter of fact, although we have shown here and in the See also:article See also:Music how artificial is the origin and nature of all but the very scantiest materials of the musical See also:language, there is no art in which the See also:element of practical See also:compromise is so minute and so hard for any but trained scientific observation to perceive. If a painter could have a scale of See also: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 See also:gallery would be a severe See also:strain on the strongest eyes, as See also: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 See also:law that artistic ideas must be realized, not in spite of, but by means of practical necessities. However See also: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 See also:float through the See also:harmonic See also:world adapting themselves, as best they can, to their environment, showing as much dependence upon the See also:stable See also:scheme of " just intonation " as a See also:crowd of metaphors and abstractions in language shows a dependence upon the rules of the See also:syllogism. As much and no more, but that is no doubt a See also:great dgal. 'Yet the See also:attempt to determine the point in See also:modern harmony where just intonation should end and the tempered scale begin, is as vexatious as the attempt to define in See also: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 See also:mystery and measuring a quantity hitherto unknown. Just intonation is a reality in so far as it emphasizes the contrast between See also:concord and discord; but when it forbids artistic interaction between harmony and See also:melody it is a chimera. It is sometimes said that Bach, by the example of his See also: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 See also:writing a See also:series of movements in every key of which thekeynote was present in the normal See also:organ and See also:harpsichord manuals of his and later times, he enforced the See also:system by which all facts of modern musical harmony are represented on keyed See also:instruments by dividing the See also:octave into twelve equal semitones, instead of tuning a few much-used keys as accurately as possible and sacrificing the euphony of all the See also: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 See also:ear instinctively supplies the See also:interpretation; as the observing See also: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 See also: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, See also:quiver from one meaning into the other. And even in the tempered scale the ear will interpret the See also: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 See also:vice versa, as in the passage already cited in the See also: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 See also: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 See also: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 See also:keyboard by the same note, this difference being the most impressive to the ear in " just intonation " and to the See also: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 See also: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 See also: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 See also:lower, becomes a perfectly diatonic modulation from C to E flat. But no transposition of the sixteen bars before the return of the See also:main theme in the See also: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 See also:original key by a See also: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 See also:ambiguity of the relationships between keys a tritone See also: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 See also: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 See also:Brahms's C Minor See also: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 (_ See also: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 See also: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 See also:bad See also:metaphysics and circle-squaring See also: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 See also:diagonal fits a wooden square. The following series of musical illustrations show the See also:genesis of typical harmonic resources of classical and modern music. ~vI *f —z;„ l Ex. 14.—See also:Ditto, with the further addition c L *I of a double suspension (*) and two passing < notes (ttI '_~ ~Ft T Ex. t5.—Ditto, with a See also:chromatic alteration of the second chord (*) and an "essential" discord (dominant 7th) at (t). See also:Definitions. (Intended to comprise the See also:general conceptions set forth in the above article.) 1. Musical sounds, or notes, are sensations produced by See also:regular periodical vibrations in the See also:air, sufficiently rapid to coalesce in a single continuous sensation, and not too rapid for the mechanism of the human ear to See also:respond. 2. The pitch of a note is the sensation corresponding to the degree of rapidity of its vibrations; being See also: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. See also: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 See also:laws of rhythm and melody. 7. The harmonic series is an See also:infinite series of notes produced by the subdivision of a vibrating See also:body or See also:column of air into See also:aliquot parts, such notes being generally inaudible except in the See also: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 See also: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 See also: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 See also: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 See also: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 See also: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 See also: 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 See also:differences between approximately equal See also: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 See also:bass (or, in the See also:case of an inverted pedal, in an upper See also:part) while the harmonies move independently. Unless the harmonies are some-times See also:foreign to the sustained note, it does not constitute a pedal. In modern music pedals take See also: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 See also:device is capable of very See also:free treatment, and has produced many very bold and rich harmonic effects in music since the earlier See also: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 See also:ancient and See also: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 See also:appoggiatura (*) prolonged till it seems to make a See also:strange foreign chord before it resolves on the See also: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 See also:resolution, but a chromatic passing-note. * —J~ WtONER. -rte ~i~ See also:WAGNER. —as ats-e-Oca,- Ex. tn.—Three concords (tonic, first See also:inversion of sub-dominant, and dominant of A minor, a possible 16th-century See also:cadence in the Phrygian mode). Ex. m.—The same chords varied by a See also:sus-See also:pension t*). III used a reiterated instrumental note as an See also:accompaniment above the melody. These primitive devices, though harmonic in the true modern sense of the word, are out of the See also: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 See also: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. 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