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Harmony In Music

( Originally Published Early 1900's )

We must begin by ascertaining exactly what harmony is, and this not in its general but in its technical sense. An answer to the question can be found in no better way than by recalling the discoveries of the scientists as a result of analyzing harmony as it appears in music, the art to the effects of which the term was first applied technically. In this art, through the use, among other methods, of resonators, so constructed as to enable one to detect the presence in a tone of any particular pitch, it has been found that notes which are harmonious are such as contain the same elements of pitch, or—what is the same thing—are notes in which effects of like pitch are repeated. For instance, when a string like that of a bass viol is struck, its note, if musical, is not single or simple: it is compound. Suppose that it produces the tone of the bass C—representing a sound-wave caused by the whole length of the string. This C is the main, or, as it is termed, the prime tone that we hear. But, at the same time, this same string usually divides at the middle, producing what is called a partial tone of the C above the base, representing a sound-wave caused by one half the string's length. It often produces, too, partial tones of the G above this, of the C above this, and of the E above the last C (etc.). . . . This C, G, C, and E of the major chord are in harmony with the lower bass C, because they are made up of effects that already enter into its composition. The chord as a whole, therefore, or any analogous development of it, is a result of putting like effects with like. Art in Theory, XII.

Glancing at the above, suppose that we were to sound the note C, and then to sound, either after or with it,—for the laws of harmony have to do with the methods of using notes both consecutively and conjointly,—notes whose partial tones connect them most closely with C, —what notes should we sound? We should sound F,—should we not?—of which C is the third partial, and G, which itself is the third partial of C. This would give us C—F—G—C. But these are the very tones accredited to the "lyre of Orpheus," which represented the earliest of the Greek scales.

Let us add to these notes those whose partial tones are the next nearly connected with C, F, or G. They are D the third partial of G, E the fifth partial of C, A the fifth of F, and B the fifth of G. This gives us C—D—EF—G—A—B—C, which is our own major scale, the main one that we use to-day; and is similar to one used by the Greeks after theirs had been expanded to seven notes. Essentials of Aesthetics, XVII.

Why is it necessary that tones should chord? Why does the mind or the ear demand concordance in the sounds used in music?—In answer to this we might begin by infer-ring a psychological reason. Sounds result from vibrations that cause oscillations in the air, and through it in the liquid within the inner labyrinth of the ear. There is a sense in which it may be said that the mind is conscious of these vibrations, for when it hears a certain number of them, per second, it invariably hears a sound of a certain pitch. Now if the vibrations causing two notes start together every second, third, fourth, fifth, or sixth time that they are made, as they do in the notes composing the musical concords, it is easy for the mind—on the supposition, of course, that in some subtle way it takes cognizance of vibrations—to perceive a unity in the result, because it can analyze the vibrations and perceive that they all form exact sub-divisions of certain definite wholes. But if the vibrations causing the tones start together at only long and irregular intervals, then any analysis or classification of the different constituent effects is impossible. Of course such a result cannot be else than confusing and unsatisfactory.

This explanation, which is the one given by Euler, has much to recommend it. We know how it is in the case of musical rhythm. Certain measures, to all of which an equal time is given, are filled with notes and rests that represent exact subdivisions of this time—the whole of it or a half, a quarter, an eighth, or more, as the case may be. When the musician composes or sings in rhythm, he beats time, mentally if not physically, and puts into each measure just the number of notes that will fill it. Why are we not justified in surmising that the principle which the mind applies consciously when it counts the beats that determine the relations of a note to rhythm, it applies unconsciously when it counts the beats or vibrations that deter-mine the relations of tone to pitch? The fundamental bass note of the chord represents a certain number of vibrations per second. These constitute, so to speak, the chord-measure, and only those notes can be used in the chord which represent the partial tones produced by exact subdivisions of this measure. In fact, there is ground enough for holding the theory that music is no more than an artistic adaptation of the laws of rhythm, of a part of which, as related to duration, the mind is conscious; but of another part of which, as related to pitch—i. e., to the rhythm resulting from tone-vibrations,—it is unconscious.

But it has not yet been shown here that the mind actually does count or compare vibrations. It may do this, but is there any proof of it? We may best begin an answer to this question by going back of the action of the mind to that of the ear that occasions it, and ask, is there any proof of a physical requirement in the ear underlying an operation analogous to comparison as made in the realm of consciousness?

There is proof of such a requirement. If we sound at the same time two very low notes of an organ separated from each other on the scale by only half a tone,—C and C# for instance,—we shall hear, not a consecutive tone, but a succession of throbs or beats. Knowing that all sounds are caused by vibrations, and that a difference in pitch is caused by a difference in the time of vibrations, it is easy to understand how these beats are produced. Sup-pose that one of the notes is a result of fifty vibrations in a second, and the other of fifty-one. At the end of the twenty-fifth vibration in the first of the tones, there will have been, in the second, twenty-five and one half vibrations. But as each vibration necessitates a movement in one direction half the time, and in a contrary direction the other half the time, the vibrations in the first tone will move from the twenty-fifth to the fiftieth in an opposite direction from those in the second tone. For this reason the vibrations causing the two tones will tend to suppress and to still one another, just as is the case where two waves of nearly equal size but contrary motions come together at the mouth of a river. However, at the fiftieth vibration in the first tone, and at the fifty-first in the second, the vibrations in the two will again move in the same direction, and tend to reinforce one another. A difference between two notes, therefore, corresponding to one vibration in a second, will cause one suppressed period and one reinforced period of sound,—or one beat in a second; a difference of two vibrations, two beats in a second, and so on. In a difference of this kind between low notes caused by a limited number of vibrations in a second, these beats are perceptible, as has been said, and are easily counted; but this is not the case when produced by high notes. Then one of two results follows. The beats either become so numerous as to form vibrations causing an entirely new tone, or else they continue to exist as beats which the ear cannot distinguish, but feels to be disagreeable.

Why does the ear find these beats disagreeable? For this reason. They are interruptions in the continuity or regularity of the vibrations. On page 194 attention was directed to the fact that a musical sound, and therefore all the pleasure derivable from it as such, is due to the rapid periodic, or—what means the same—the regularly recurring motion of the sonorous body; and a noise to its non-periodic, or irregularly recurring motion.

When beats occur that interfere with harmony, there-fore, there is noise instead of music. But noise in music not only violates the artistic principle which requires that like amid varied effects be put with like, but it communicates to the auditory nerves a series of shocks, conveying an intermittent, irregular, disordered excitation; whereas it is natural to suppose that, in all agreeable excitations of the nerves, the thrill and glow that are pleasurable are characterized by the elasticity and freedom accompanying non-interference. We may infer this from the fact that in nature all movements are regular and rhythmical. The leaves and limbs of a twig, for instance, vibrate, when struck by a blow, as regularly as does a pendulum. The same must be true of the oscillations in . the auditorium of the ear. At any rate, we know that only regularly recurring vibrations can produce the sensations in the auditory nerves which render music enjoyable.

In conclusion, we may blend the physiological and psychological reasons for the effects of music, thus : The ear has become habituated through long experience to search for unity of effect in sounds. When it hears musical chords, it recognizes, after a few vibrations, that all the sounds are exact subdivisions of some one note,—in other words, that what is heard results from a succession of like amid varied effects. At other times, when the mind cannot recognize that this is the case, it is natural to suppose that there is an endeavor to recognize the fact, and, owing to this endeavor, that there is a positive effort on the part of the organs of sensation in the ear to adjust themselves to the new conditions and to discover elements of unity and likeness that do not exist. That the ear is sometimes successful in doing this, is proved by its acceptance of the slight variations from true harmony that are found in the temperate scale. In decided discords, however, nothing can make the sounds seem to compare, and the nerves and muscles are wearied by the effort of trying to do it, just as they would be, were they listening intently for sounds or footsteps which they failed to hear. Of course, the nerves of hearing, strained, and on the alert, but without success, give the ear pain, not pleasure. Pleasure in connection with sound, esthetic satisfaction in connection with tone, is experienced by mind or ear in the degree only in which the result is perceived to be a unity obtained from the apparent variety of unlike complex wholes by putting together those that have like partial effects.—Rhythm and Harmony in Poetry and Music, XVI.

The reader will not fail to notice that the effects of harmony as thus described are, in important regards, analogous to those of rhythm, and yet of a rhythm so finely grained that it is impossible that the mind should be conscious of its constituent elements. . . . It is sometimes said that, as the mind consciously counts the beats in determining rhythm, so, in some subtle way, it unconsciously counts the vibrations in determining harmony. But is it necessary to suppose this? When influenced by tones that seem consonant we are certainly not conscious of counting. Are we conscious of doing it even when influenced by the effects of rhythm? Are we conscious of anything except of certain accentuations of tone that are equally subdivided into other accentuations—all of which, in some way, are so related that they exactly fit, the smaller into the larger and all into the largest? And if we need not count the accents in rhythm, why should we do it in harmony? Why need we do more than experience certain throbs or thrills of sound equally subdivided into other thrills, all of which are so related that they exactly fit, the smaller into the larger and all into the largest? As a result of experiencing these, every part of the auditory organism, under any influence of sound, is under the same influence,—as much so as is every part of a still pool when we have thrown a single stone into it, infinitely varied as may be the sizes of different waves that in remote places circle into ripples. The result, inasmuch as all the sound-waves represent a single impulse, is an unimpeded, free, regularly recurrent vibratory glow of the whole auditory apparatus. But if, on the contrary, the effect resemble that upon the waters of a pool when more than one stone is thrown into it, i. e., if the sound-waves do not coalesce, if the smaller do not fit into the larger, and all together into the largest, then nothing ensues but a broken, impeded, constrained, irregular series of jolts or jars. The difference in the ear between the sensation of harmony and of a lack of it, is the physical difference between thrilling or glowing and jolting or jar-ring. Notice, too, that this illustration applies to notes when sounding not only, as in one chord, simultaneously, but, as in different chords, successively. Two things related to the same thing cannot fail, in some way, to be related to each other; and two chords, each containing sets of vibrations for which there is a common multiple, and both containing one set of vibrations (i. e., one tone) which is the same, must both be entirely composed of vibrations for which there is some common multiple. This common multiple, moreover, for the vibrations of a first and second chord may be different from that of the vibrations of the second and third chord. It is possible, therefore, for a series of chords, each in part repeating the same tones as the last sounded, and in part introducing new tones, to change, very soon, the whole character of the general vibratory effect; and yet if this be done with sufficient gradualness, the auditory apparatus will experience no jolt or jar, while, at the same time, it will be conscious of a constant progress and so of relief from anything resembling monotony.—Pro portion and Harmony of Line and Color, XX.

Of course, the early musicians could not have explained exactly why they selected certain notes and put them into a musical scale, and from these began to develop that which has now come to be our elaborated system of melody and harmony. Those artists followed merely the instincts of their aesthetic nature. This prompted them, in constructing forms, to select sounds that would naturally go together; and to use these and these only. But what connection is there, it may be asked, between sounds that naturally go together, and those that go together because certain of their effects are alike? None, perhaps, so far as the first musicians were aware. They judged merely by the results that they heard, and had only a limited knowledge of the causes of these. Nevertheless, as will be shown presently from an examination of the discoveries of modern science, their ears guided them aright. All the notes of the scale and all the methods of musical harmony owe their origin to a literal fulfilment of the art-principle declared in " The Genesis of Art-Form" to be of universal applicability. This principle is that, in order to receive an impression of unity, the mind groups complex wholes by putting those together that produce like partial efects.—Rhythm and Harmony in Poetry and Music, XII.

Harmony, like rhythm and proportion, often involves very intricate arrangements and developments, but through them all can be detected the presence of this one underlying principle. The following, for instance, represents a common way of accomplishing the result which is termed "making the circuit" of all the major keys. Those unacquainted with music will understand sufficiently what is meant when it is said that the chords of one key are often discordant with those of another key unless, in some such way as is indicated in this music, an artificial connection has been made between the two.—The Essentials of Aesthetics, XVII.

The main result . . . is secured through using such methods as those of interchange, gradation, and transition, which, nevertheless, cause all the divergent parts of a composition to assimilate. Because, too, all the methods in the chart (see page 89 of this volume) are, more or less, connected, music, at times, reveals traces of the influence of every one of these.—Idem.

HARMONY OF COLOR (see also different paragraphs under BEAUTY, COMPARISON, and VIBRATORY).

Like tone-harmony, this was developed, at first, by artists of exceptional taste, knowing little and caring less about the scientific reasons underlying their choice of combinations. But, after art has developed to a certain extent, scientists always make a study of its effects. That which they discover increases not only the knowledge and the appreciation of art on the part of the general public, but also adds not a little to the resources of the artist and to his ability to make further progress.

Nor must it be supposed that color-harmony, so far as it has been developed from the contributions of science, has been based upon the relations between vibrations in the eye in the same way in which tone-harmony has been based upon the relations between vibrations in the ear. The numbers of the latter vibrations can be and have been definitely determined. The numbers of vibrations causing the colors have not been determined except approximately. For this reason, and very wisely, the principles of color-harmony have been developed from facts which, though related to those of vibration, have, unlike them, been definitely ascertained. The different stages of development have been somewhat as follows:

The discoveries with reference to the complementary colors,' as described on page 370, led to the natural sup-position that the eye takes pleasure in seeing these two together; and as, in all cases, the two were found to make white, it led to the supposition that any two or more colors making white would cause harmony. Not long after, too, it led to the supposition that these colors must be introduced into a painting in just such proportions as to make white. . . . A law of this kind, however, though it might be applied to decoration, would evidently interfere with one of the first requisites of the art of painting, namely, that it should represent nature. In how many landscapes can we find the blue of the sky, or the green of the foliage, or the bluish gray of a lowery day, exactly mingled in such proportions with the warmer and lighter yellows, reds, or browns?

On the face of it, therefore, this theory did not seem tenable. Modern artists universally reject it. They tell us that the slightest spot of crimson against the green of a forest, or of yellow against the blue of the sky, is all that is needed in order to bring out the brilliancy of the complementary coloring. . . . But when it, is added that these effects are owing to merely a suggestion given to the mind, one must demur. Those who say it have forgotten a very important principle in aesthetics. That is, that psychological effects (see Chapter II.) must harmonize with physiological, and, as the latter come first in the order of time, it is not logical either to overlook them or to fail to consider them first.

The influence in a painting of very slight quantities of complementary coloring seems to suggest the importance of the method of interpretation indicated on pages 375 to 378. If we may suppose that a color associated with its complementary produces in the eye an agreeable effect because, for the vibrations causing both colors, there is a common multiple, then we may also suppose that these colors influence, at the same time, the organs of the same retina without producing any sensation of jolting or jarring. All the vibrations are variations of the same unity in that they are partial effects of the same single impulse or set of impulses, resulting in a free, unrestrained vibratory thrill or glow. The quantity of color, therefore, makes no difference with the harmony of the effect. All that is necessary is that the form of vibration causing the one color, be it much or little, should exactly coalesce with the form of vibration causing the other color. It could coalesce in this way, of course, in several different circumstances. First of all, it could do so when there was one predominating color. . . . Thus, in a scene representing moonlight or twilight, or even a storm, especially if at sea, there would necessarily be one pervading color, in some cases banishing almost the suggestion of other colors. . . . Such paintings are said to be characterized by tone, and, as this quality, is usually understood, it is difficult to perceive why it does not fulfil a different law of harmony from that which is fulfilled through a use of great variety in coloring. Indeed, it is often represented that it does; as if the theory that harmony of coloring is produced by uniformity of coloring were antagonistic to the theory that it is produced by variety. . . . But why cannot an identical law be perceived to be operative in both cases? Differences in tints and shades of the same hue, while they involve differences in the intensity of the sight-waves, do not necessarily involve differences in their rates or shapes. Therefore uniformity of coloring is fitted to cause all the vibrations of the same retina to coalesce, i. e., to cause all to be exact subdivisions of some common multiple. But the same effect is produced by the use of one predominating color with its various tints and shades, enlivened . by an occasional introduction of some tint or shade of its complementary color; and it is produced also when both complementary colors are used in almost equal proportions. In fact, color-harmony may result from the use of any colors whatsoever, if only they can be made in some way to produce in the organs of color-apprehension an effect of unity. This effect follows whenever all the vibrations of the retina that are near together are multiples of some common unit, as is the case when adjoining tints and shades in a painting are of the same hue, or of hues that form complementaries, or for some reason allied to this, as indicated on pages 370 to 374, are fitted to go together. If, in connection with these hues, others must be used requiring what may be termed conflicting forms of vibration, these others must, in the painting, be remote from the first, and be connected with them in accordance with methods of securing partial consonance like those of interchange, gradation, and transition. . . . Why this should be the case, may be surmised by recalling that a single vibration is to the whole retina about what a single wave is to an ocean. On an ocean, divergent forms of waves would not be recognized to be conflicting were they widely separated, or were they changed from one form into another with great graduality; and were thus made—to apply the term of physiological psychology —to assimilate. . Color-harmony, to be successful, must be a result of an application of the same endeavor after unity of effect which, starting with the principle of putting like with like wherever possible, leads to a careful study and embodiment of all such requirements as those of variety, complement, principality, subordination, balance, parallelism, repetition, alternation, symmetry, massing, interchange, continuity, consonance, gradation, transition, and progress. This fact is developed in the author's " Proportion and Harmony of Line and Color in Painting, Sculpture, and Architecture. "—The Essentials of Esthetics, XVIII.

The third method of arriving at the principles underlying the joining of colors is advocated by those who hold that as in music the ratios between the numbers of vibrations per second producing the different notes determine which should go together, so, in painting, the ratios between the numbers of vibrations per second producing the different colors should determine this. As a rule, physicists have had little respect for any advocate of this theory, because he has usually started out with the hypothesis that there is some absolute and necessary connection between the seven colors of the spectrum and the seven notes of the musical scale. As was shown, however, in Chapter XIV. of "Rhythm and Harmony in Poetry and Music, " these seven notes happen to be used merely as a matter of convenience. There have been scales extensively used of four and six notes, and possibly our own might be improved by the addition of two more. As it is, it contains not seven but twelve distinct intervals. There is a principle, however, underlying the formation of all musical scales, as well as of all melody and harmony, which depends upon the relative numbers of vibrations. One cannot refrain from feeling, there-fore, that it is logical to suppose that this same principle should be exemplified in that which causes colors to harmonize.

It does not allay this feeling, to remind one that between, say, the 400 trillions of vibrations causing extreme red and the 750 causing extreme violet, the differences in vibration are not sufficient for those of a single octave. . As it is, we have in the colors all the range of intervals corresponding to those of one octave if containing no note be-longing to another. Moreover, the possibility of producing variations in a single color is much greater than that of doing the same in a single sound. Indeed, when we consider the innumerable shades and tints not merely of one color but of all other colors in connection with which this one may pro-duce mixed effects, we are forced to recognize that the range both of single colors and of those that are exactly complementary to these is practically infinite, and thus far more than sufficient to make up for the absence in the color-scale of more than one octave.

So much for the theory; now for the facts confirming it. Let us take the ratios of the numbers of vibrations producing the sounds, not of all the scale, but of those that harmonize, and apply these ratios to the numbers of vibrations producing the different colors, and notice what colors they cause to go together. As the numbers of vibrations producing the colors are exceedingly great, and the difficulty in the spectrum of determining just where one color leaves off and another begins is also great, we must content our-selves with approximate measurements, but even with these we can attain our object. Proportion and Harmony of Line and Color, XXIII.


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