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, A♭ major (= G♯), and C (= B♯). And the same circle occurs in the opposite direction in the first movement of his Third Symphony, where the first subject is in F, the transition passes directly to D♭ and thence by exactly the same step to A (= B♭♭). The exposition is repeated, which of course means that in “just intonation” the first subject would begin in G♭♭ and then pass through a transition in E♭♭♭ to the second subject in C♭♭♭. 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.
| Ex. 12.—Three concords (ionic, first inversion of subdominant, and dominant of A minor, a possible 16th-century cadence in the Phrygian mode). |  |
| Ex. 13.—The same chords varied by a suspension (*). |  |
| Ex. 14.—Ditto, with the further addition of a double suspension (*) and two passing notes (††). |  |
| Ex. 15.—Ditto, with a chromatic alteration of the second chord (*) and an “essential” discord (dominant 7th) at (†). |  |
| Ex. 16.—Ditto, with chromatic passing notes (**) and appoggiaturas (††). |  |
| 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 ‡, while the second appoggiatura (†) is chromatic. |  |
| Ex. 18.—The same enharmonically transformed so as to become a variation of the “dominant ninth” of C minor. The G♯ at * is really A♭, and ‡ is no longer a note of resolution, but a chromatic passing-note. |  |
(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.
10. 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. 1 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.
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
