I. Concord and Discord.-By means of harmony modern music has attained the dignity of an independent art. In ancient times, as at the present day among nations that have not come under the influence of European music, the harmonic sense was, if not altogether absent, at all events so obscure and undeveloped as to have no organizing power in the art. The formation by the Greeks of a scale substantially the same as that which has received our harmonic system shows a latent harmonic sense, but shows it in a form which positively excludes harmony as an artistic principle. The Greek perception of certain successions of sounds as concordant rests on a principle identifiable with the scientific basis of concord in simultaneous sounds. But the Greeks did not conceive of musical simultaneity as consisting of anything but identical sounds; and when they developed the practice of magadizing-i.e. singing in octaves-they did so because, while the difference between high and low voices was a source of pleasure, a note and its octave were then, as now, perceived to be in a certain sense identical. We will now start from this fundamental identity of the octave, and with it trace the genesis of other concords and discords; bearing in mind that the history of harmony is the history of artistic instincts and not a series of progressive scientific theories.

The unisonous quality of octaves is easily explained when we examine the "harmonic series " of upper partials (see SOUND). Every musical sound, if of a timbre at all rich (and hence pre-eminently the human voice), contains some of these upper partials. Hence, if one voice produce a note which is an upper

if a note and its octave be sung together, the upper octave is itself No. 2 in the harmonic series of the lower, No. 2 of its own series is No. 4 of the lower, and its No. 3 is No. 6, and so on. The impression of identity thus produced is so strong that we often find among people unacquainted with music a firm conviction that a man is singing in unison with a boy or an instrument when he is really singing in the octave below. And even musical people find a difficulty in realizing more than a certain brightness and richness of single tone when a violinist plays octaves perfectly in tune and with a strong emphasis on the lower notes. Doubling in octaves therefore never was and never will be a process of harmonization.

Now if we take the case of one sound doubling another in the 12th, it will be seen that here, too, no real addition is made by the higher sound to the lower. The 12th is No. 3 of the harmonic series, No. 2 of the higher note will be No. 6 of the lower, No. 3 will be No. 9, and so on. But there is an important difference between the 12th and the octave. However much we alter the octave by transposition into other octaves, we never get anything but unison or octaves. Two notes two octaves apart are just as devoid of harmonic difference as a plain octave or unison. But, when we apply our principle of the identity of the octave to the 12th, we find that the removal of one of the notes by an octave may produce a combination in which there is a distinct harmonic element. If, for example, the lower note is raised by an octave so that the higher note is a fifth from it, No. 3 of the harmonic series of the higher note will not belong to the lower note at all. The 5th is thus a combination of which the two notes are obviously different; and, moreover, the principle of the identity of octaves can now operate in a contrary direction and transfer this positive harmonic value of the 5th to the 12th, so that we regard the 12th as a 5th plus an octave, instead of regarding the 5th as a compressed 12th. At the same time, the relation between the two is quite close enough to give the 5th much of the feeling of harmonic poverty and reduplication that characterizes the octave; and hence when medieval musicians

Ex. 2.

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The limited compass of voices soon caused modifications in the medieval parallelisms of 4ths and 5ths, and the introduction of independent ornaments into one or more of the voices increased to an extent which drew attention to other intervals. It was long, however, before the true criterion of concord and discord was attained; and at first the notion of concord was purely acoustic, that is to say, the ear was sensitive only to the difference in roughness and smoothness between combinations in themselves. And even the modern researches of Helmholtz fail to represent classical and modern harmony, in so far as the phenomena of beats are quite independent of the contrapuntal nature of concord and discord which depends upon the melodic intelligibility of the motion of the parts. Beats give rise to a strong physical sense of discord akin to the painfulness of a flickering light (see SOUND). Accordingly, in the earliest experiments in harmony, the car, in the absence of other criteria, attached much more importance to the purely acoustic roughness of beats than our ears under the experience of modern music. This, and the circumstance that the imperfect concords (the 3rds and 6ths) long remained out of tune owing to the incompleteness of the Pythagorean system of harmonic ratios, sufficiently explain the medieval treatment of these combinations as discords differing only in degree from the harshness of 2nds and 7ths. In the earliest attempts at really contrapuntal writing (the astonishing 13th and 14th-century motets, in which voices are made to sing different melodies at once, with what seems to modern ears a total disregard of sound and sense) we find that the method consists in a kind of rough-hewing by which the concords of the octave, 5th and 4th are provided at most of the strong accents, while the rest of the harmony is left to take care of itself. As the art advanced the imperfect concords began to be felt as different from the discords; but as their true nature appeared it brought with it such an increased sense of the harmonic poverty of octaves, 5ths and 4ths, as ended in a complete inversion of the earliest rules of harmony.

The harmonic system of the later 15th century, which culminated in the "golden age" of the 16th-century polyphony, may be described as follows: Imagine a flux of simultaneous independent melodies, so ordered as to form an artistic texture based not only on the variety of the melodies themselves, but also upon gradations between points of repose and points in which the roughness of sound is rendered interesting and beautiful by means of the clearness with which the melodic sense in each part indicates the convergence of all towards the next point of repose. The typical point of repose owes its effect not only to the acoustic smoothness of the combination, but to the fact that it actually It is at least probable that this is one of the several rather obscure reasons for the peculiar instability of the 4th in modern harmony, which is not yet satisfactorily explained.

The perfect concords are the octave, unison, 5th and 4th. Other diatonic combinations, whether concords or discords, are called imperfect.

of the harmonic series. The major 3rd has thus in this scheme asserted itself as a concord, and the fundamental principle of the identity of octaves produces the result that any combination of a bass note with a major 3rd and a perfect 5th above it, at any distance, and with any amount of doubling, may constitute a concord available even as the final point of repose in the whole composition. And by degrees the major triad, with its major 3rd, became so familiar that a chord consisting of a bare 5th, with or without an octave, was regarded rather as a skeleton triad without the 3rd than as a concord free from elements of imperfection. Again, the identity of the octave secured for the combination of a note with its minor 3rd and minor 6th a place among concords; because, whether so recognized by early theorists or not, it was certainly felt as an inversion of the major triad. The fact that its bass note is not the fundamental note (and therefore has a series of upper partials not compatible with the higher notes) deprives it of the finality and perfection of the major triad, to which, however, its relationhsip is too near for it to be felt otherwise than as a concord. This sufficiently explains why the minor 6th ranks as a concord in music, though it is acoustically nearly Ex. 3. as rough as the discord of the minor 7th, and considerably rougher than that of the 7th note of the harmonic series, which has not become accepted in our musical system at all.

But the major triad and its inversion are not the only concords that will be produced by our flux of melodies. From time to time this flux will arrest attention by producing a combination which, while it does not appeal to the ear as being a part of the harmonic chord of nature, yet contains in itself no elements not already present in the major triad. Theorists have in vain tried to find in "nature" a combination of a note with its minor 3rd and perfect 5th; and so long as harmony was treated unhistorically and unscientifically as an a priori theory in which every chord must needs have a" root," the minor triad, together with nearly every other harmonic principle of any complexity, remained a mystery. But the minor triad, as an artistic and not purely acoustic phenomenon, is an inevitable thing. It has the character of a concord because of our intellectual perception that it contains the same elements as the major triad; but its absence of connexion with the natural harmonic series deprives it of complete finality in the simple system of 16th-century harmony, and at the same time gives it a permanent contrast with the major triad; a contrast which is acoustically intensified by the fact that, though its intervals are in themselves as concordant as those of the major triad, their relative position produces decidedly rough combinations of "resultant tones." By the time cur flux of melodies had come to include the major and minor triads as concords, the notion of the independence of parts had become of such paramount importance as totally to revolutionize the medieval conception of the perfect concords. Fifths and octaves no longer formed an oasis in a desert of cacophony, but they assumed the character of concord so nearly approaching to unison that a pair of consecutive 5ths or octaves began to be increasingly felt as violating the independence of the parts. And thus it came about that in pure 16th-century counterpoint (as indeed at the present day whenever harmony and counterpoint are employed in their purest significance) consecutive 5ths and octaves are strictly forbidden. When we compare our laws of counterpoint with those of medieval discant (in which consecutive 5ths and octaves are the rule, while consecutive 3rds and 6ths are strictly forbidden) we are sometimes tempted to think that the very nature of the human ear has changed. But it is now generally recognized that the process was throughout natural and inevitable, and the above account aims at showing that consecutive 5ths are forbidden by our harmonic system for the very reason which inculcated them in the system of the 12th century.

II. Tonality. As soon as the major and minor triad and their first inversions were well-defined entities, it became evident that

the successions of these concords and their alternations with discord involved principles at once larger and more subtle than those of mere difference in smoothness and artificiality. Not only was a major chord (or at least its skeleton) necessary for the final point of repose in a composition, but it could not itself sound final unless the concords as well as the discords before it showed a well-defined tendency towards it. This tendency was best realized when the penultimate concord had its fundamental note at the distance of a 5th or a 4th above or below that of the final chord. When the fundamental note of the penultimate chord is a 5th above or (what is the same thing) a 4th below that of the final chord, we have an "authentic" or "perfect" cadence, and the relation between the two chords is very clear. While the contrast between them is well marked, they have one note in common-for the root of the penultimate chord is the 5th of the final chord; and the statement of this common note, first as an octave or unison and then as a 5th, expresses the first facts of harmony with a force which the major 3rds of the chords can only strengthen, while it also involves in the bass that melodic interval of the 4th or the 5th which is now known to be the germ of all melodic scales. The relation of the final note of a scale with its upper 5th or lower 4th thus becomes a fundamental fact of complex harmonic significance-that is to say, of harmony modified by melody in so far as it concerns the succession of sounds as well as their simultaneous combination. In our modern key-system the final note of the scale is called the tonic, and the 5th above or 4th below it is the dominant. (In the 16th century the term "dominant " has this meaning only in the "authentic" modes other than the Phrygian, but as an aesthetic fact it is present in all music, though the theory here given would not have been intelligible to any composers before the 18th century). Another penultimate chord asserts itself as the converse of the dominant-namely, the chord of which the root is a 5th below or a 4th above the final. This chord has not that relationship to the final which the dominant chord shows, for its fundamental note is not in the harmonic series of the final. But the fundamental note of the final chord is in its harmonic series, and in fact stands to it as the dominant stands to the final. Thus the progression from subdominant, as it is called, to tonic, or final, forms a full close known as the 'plagal cadence,” second only in importance to the "perfect" or "authentic cadence." In our modern

Ex. 4.

Ex. 5.

key-system these three chords, the tonic, the dominant and the subdominant, form a firm harmonic centre in reference to which all other chords are grouped. The tonic is the final in which everything ultimately resolves: the dominant stands on one side of it as a chord based on the note harmonically most closely related to the tonic, and the subdominant stands on the other side as the converse and opposite of the dominant, weaker than the dominant because not directly derived from the tonic. The other triads obtainable from the notes of the scale are all minor, and of less importance; and their relationship to each other and to the tonic is most definite when they are so grouped that their basses rise and fall in 4th and 5ths, because they then tend to imitate the relationship between tonic, dominant and subdominant.

Here are the six common chords of the diatonic scale. The triad on the 7th degree or "leading-note" (B) is a discord, and is therefore not given here.

Now, in the 16th century it was neither necessary nor desirable that chords should be grouped exclusively in this way. The relation between tonic, dominant and subdominant must necessarily appear at the final close, and in a lesser degree at

1 The submediant is so-called because if the subdominant is taken a 5th below the tonic, the submediant will come midway between it and the tonic, as the mediant comes midway between tonic and dominant,

3

subordinate points of repose; but, where no harmonies were dwelt on as stable and independent entities except the major and minor triads and their first inversions, a scheme in which these were confined to the illustration of their most elementary relationship would be intolerably monotonous, It is therefore neither surprising nor a sign of archaism that the tonality of modal music is from the modern point of view often very indefinite. On the contrary, the distinction between masterpieces and inferior works in the 16th century is nowhere more evident than in the expressive power of modal tonality, alike where it resembles and where it differs from modern. Nor is it too much to say that that expressive power is based on the modern sense of key, and that a description of modal tonality in terms of modern key will accurately represent the harmonic art of Palestrina and the other supreme masters, though it will have almost as little in common with 16th-century theory and inferior 16thcentury practice as it has with modern custom. We must conceive modal harmony and tonality as a scheme in which voices move independently and melodiously in a scale capable of bearing the three chords of the tonic, dominant and subdominant, besides three other minor triads, but not under such restrictions of symmetrical rhythm and melodic design as will necessitate a confinement to schemes in which these three cardinal chords occupy a central position. The only stipulation is that the relationship of at least two cardinal chords shall appear at every full close. At other points the character and drift of the harmony is determined by quite a different principle-namely, that, the scale being conceived as indefinitely extended, the voices are agreed in selecting a particular section of it, the position of which determines not only the melodic character of each part but also the harmonic character of the whole, according to its greater or less remoteness from the scale in which major cardinal chords occupy a central position. Historically these modes were derived, with various errors and changes, from the purely melodic modes of the Greeks. Aesthetically they are systems of modern tonality adapted to conditions in which the range of harmony was the smallest possible, and the necessity for what we may conveniently call a clear and solid key-perspective incomparably slighter than that for variety within so narrow a range. We may thus regard modal harmony as an essentially modern scheme, presented to us in cross-sections of various degrees of obliquity, and modified at every close so as either to take us to a point of view in which we see the harmony sym

metrically (as in those modes of which the final chord is normally major, namely the Ionian, which is practically our major scale, the Mixolydian and the Lydian, which last is almost invariably turned into Ionian by the systematic flattening of its 4th degree) or else to transform the mode itself so that its own notes are flattened and sharpened into suitable final chords (as is necessary in those modes of which the triad on the final is normally minor, namely, the Dorian, Phrygian and Aeolian). In this way we may describe Mixolydian tonality as a harmonic scheme in which the keys of G major and C major are so combined that sometimes we feel that we are listening to harmony in C major that is disposed to overbalance towards the dominant, and sometimes that we are in G major with a pronounced leaning towards the subdominant. In the Dorian mode our sensations of tonality are more confused. We seem to be wandering through all the key-relationships of a minor tonic without defining anything, until at the final close the harmonies gather strength and bring us, perhaps with poetic surprise, to a close in D with a major chord. In the Phrygian mode the difficulty in forming the final close is such that classical Phrygian compositions actually end in what we feel to be a half-close, an impression which is by the great masters rendered perfectly artistic by the strong feeling that all such parts of the composition as do not owe their expression to the variety and inconstancy of their harmonic drift are on the dominant of A minor.

It cannot be too strongly insisted that the expression of modal music is a permanent artistic fact. Its refinements may be crowded out by the later tonality, in which the much greater

See PLAIN SONG.