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Rh the latter) show very clearly the origin of so many primitive scales in the interval of the downward fourth. That interval (e.g. from C to G) is believed to be the earliest melodic relationship which the ear learnt to fix; and most of the primitive scales were formed by the accretion of auxiliary notes at the bottom of this interval, and the addition of a similar interval, with similar accretions, below the former. In this way a pentatonic scale, like that of so many Scotch melodies, can easily be formed (thus, C, A, G; F, D, C); and though some primitive scales seem to have been on the nucleus of the rising fifth, while the Siamese now use two scales of which not a single note within the octave can be accounted for by any known principle, still we may consider that for general historic purposes the above example is typical. The Greeks divided their downward fourth into four notes, called a tetrachord; and by an elaborate system of linking tetrachords together they gave their scale a compass of two octaves. The enharmonic tetrachord, being the most ancient, gathered the lower three notes very closely to the bottom, leaving the second note no less than a major third from the top, thus—C, A♭, G′, G; (where G′ stands for a note between A♭ and G). The chromatic tetrachord was C, B♭♭, A♭, G; and the diatonic tetrachord was C, B♭, A♭, G. It is this last that has become the foundation of modern music, and the Greeks themselves soon preferred it to the other genera and found a scientific basis for it. In the first place they noticed that its notes (and, less easily, the notes of the chromatic scale) could be connected by a series of those intervals which they recognized as concordant. These were, the fourth; its converse, or inversion, the fifth; and the octave. The notes of the enharmonic tetrachord could not be connected by any such series. In the articles on and  account is given of the historic and scientific foundations of the modern conception of concord; and although this harmonic conception applies to simultaneous notes, while the Greeks concerned themselves only with successive notes, it is nevertheless permissible to regard the Greek sense of concord in successive notes as containing the germ of our harmonic sense. The stability of the diatonic scale was assured as early as the 6th century when Pythagoras discovered (if he did not learn from Egypt or India) the extremely simple mathematical proportions of its intervals. And this discovery was of unique importance, as fixing the intervals by a criterion that could never be obscured by the changes of taste and custom otherwise inevitable in music that has no conscious harmonic principles to guide it. At the same time, the foundation of a music as yet immature and ancillary to drama, on an acoustic science ancillary to a priori mathematics, was not without disadvantage to the art; and it is arguable that the great difficulty with which during the medieval beginnings of modern harmony the concords of the third and sixth were rationalized may have been increased by the fact that the Pythagorean system left these intervals considerably out of tune. In preharmonic times mathematics could not direct even the most observant ear to the study of those phenomena of upper partials of which Helmholtz, in 1863, was the first to explain the significance; and thus though the Greeks knew the difference between a major and minor tone, on which half the question depended, they could not possibly arrive at the modern reasons for adding both kinds of tone in order to make the major third. (See .)

Here we must digress in order to illustrate what is implied by our modern harmonic sense; for the difference that this makes to our whole musical consciousness is by no means universally realized. Music, as we now understand it, expresses itself in the interaction of three elements—rhythm, melody and harmony. The first two are obviously as ancient as human consciousness itself. Without the third a musical art of permanent value and intelligibility has not been known to attain independent existence. With harmony music assumes the existence of a kind of space in three dimensions, none of which can subsist without at least implying the others. When we hear an unaccompanied melody we cannot help interpreting it in the light of its most probable harmonies. Hence, when it does not .imply consistent harmonies it seems to us quaint or strange; because, unless it is very remote from our harmonic conceptions, it at least implies at any given moment some simple harmony which in the next moment it contradicts. Thus our inferences as to the expression intended by music that has not come under European influence are unsafe, and the pleasure we take in such music is capricious. The effort of thinking away our harmonic preconceptions is probably the most violent piece of mental gymnastics in all artistic experience, and furnishes much excuse for a sceptical attitude as to the artistic value of preharmonic music, which has at all events never become even partially independent of poetry and dance. Thus the rhythm of classical Greek music seems to have been entirely identical with that of verse, and its beauty and expression appreciated in virtue of that identity. From the modern musical point of view the rhythm of words is limited to a merely monotonous uniformity of flow, with minute undulations which are musically chaotic (see ). The example of Greek tragedy, with the reports of its all-pervading music (in many cases, as in that of Aeschylus, composed by the dramatist himself) could not fail to fire the imaginations of modern pioneers and reformers of opera; and Monteverde, Gluck and Wagner convinced themselves and their contemporaries that their work was, amongst other things, a revival of Greek tragedy. But all that is known of Greek music shows that it represents no such modern ideas, as far as their really musical aspect is concerned. It represents, rather, an organization of the rise and fall of the voice, no doubt as elaborate and artistic as the organization of verse, no doubt powerful in heightening the emotional and dramatic effect of words and action, but in no way essential to the understanding or the organization of the works which it adorned. The classical Greek preference for the diatonic scale indicates a latent harmonic sense and also that temperance which is at the foundation of the general Greek sense of beauty; but, beyond this and similar generalities, all the research in the world will not enable us to understand the Greek musician’s mind. Non-harmonic music is a world of two dimensions, and we must now inquire how men came to rise from this “flatland” to the solid world of sound in which Palestrina, Bach, Beethoven and Wagner live.

3. Harmonic Origins.—Although the simultaneous blending of different sounds was never seriously contemplated by the Greeks, yet in classical times they were fond of singing with high and low voices in octaves. This was called magadizing, from the name of an instrument on which playing in octaves was rendered easy by means of a bridge that divided the strings at two-thirds of their length. While the practice was esteemed for the beauty of the blending of different voices, it was tolerated only because of the peculiar effect of identity furnished by the different notes of the octave, and no other interval was so used by the Greeks. In the article on the degrees of identity-in-difference which characterize the simpler harmonic intervals are analysed, and the main steps are indicated by which the more complicated medieval magadizing uses of the fourth and fifth (the symphonia, diaphonia or organum of Hucbald) gave way (partly by their own interchange and partly through experiments in the introduction of ornaments and variety) to the modern conception of harmony as consisting of voices or parts that move independently to the exclusion of such parallel motion. In The Oxford History of Music, vols. i. and ii., will be found abundant examples of every stage, of the process, which begins with the organum or diaphony that prevailed until the death of Guido of Arezzo (about 1050) and passes through the discant, or measured music, of the 13th century, in which rhythm is first organized on a sufficiently firm basis to enable voices to sing contrasted rhythms simultaneously, while the new harmonic criterion of the independence of parts more and more displaces and shows its opposition to the old criterion of parallelism.

The most extraordinary example of these conflicting principles is the famous rota “Sumer is icumen in,” a 13th-century round in four parts on a canonic ground-bass in two. Recent researches