Popular Science Monthly/Volume 16/February 1880/Imperfections of Modern Harmony

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By S. AUSTEN PEARCE, Mus. D., Oxon.

THE works of Helmholtz, and those of his English translator Ellis, have drawn attention to the fact that piano-fortes and instruments with similar keyboards are out of tune. The recent contribution to musical literature by Professor Pole having referred to the subject of intonation, it becomes a duty to the public to point out the misconceptions of these theorists, and to state that musical problems are far more complex than they believe.

Although professing to work scientifically, they allow their senses to deceive them. Professor Helmholtz in his elaborate work "On the Sensations of Tone as a Physiological Basis for the Theory of Music," says, for instance, "We often hear four musical amateurs, who have practiced much together, singing quartets in perfectly just intonation." He is deceived in this. It is a popular error that music for stringed instruments or for voices is or can be rendered in tune; and scientists writing upon the subject invariably cling to this fallacy.

An unaccompanied quartet of singers returning at the close to the exact pitch at which they began would thus most certainly prove that they had sung out of tune. This is a startling fact—stranger than the current fiction—and demands complete demonstration. The subject is abstruse, and difficult to present clearly to persons not practically acquainted with the points at issue, but for the benefit of all thoughtful readers the attempt is made.

All the great Oriental nations of antiquity were familiar with the difficulties to be overcome in establishing a tonal system. The results of their experiments are known to the musical historian. It is sufficient to say that the necessity of accurate definition was universally desired in the earliest ages of which we have any record by peoples who did not employ harmony. But our own use of chords makes questions of intonation much more intricate. We not only require a song or melody, but several songs or melodies to be given simultaneously, as in the ordinary church quartet or fugal chorus, where each singer demands the right to be provided with some important subject-matter, worthy the delivery of a feeling, acting, willing spirit—some "part" which fully occupies him. He will not be content with a mere accompaniment to some leading part. The harmonizing of these several melodies, that they may at every instant make recognized and well-proportioned combinations called chords, constitutes the modern science of harmony. This science is based upon the comparatively recent discoveries that Nature herself supplies a full chord, or cortége of harmonic sounds, to attend every single note, and others also to attend a union of two notes, and so on. The ancient Greeks, not being aquainted with these natural products, and misapplying certain mathematical laws, failed to obtain concords sufficiently true to satisfy their refined artistic perceptions, and therefore rejected them. The modern pagan nations, even those competent to produce perfect concords, refuse to adopt them; and therefore our modern harmony still remains either unknown or unappreciated outside Christendom. It may be that we are generally regarded in the East as Western barbarians. The Chinese, for instance, smile at the piano-forte as an ingeniously contrived arrangement for enabling the performer to produce many different sounds at once as required by Western harmony; but as a mere mechanical instrument, in which hammers are thrown against strings in an inartistic manner, illustrative of our insensibility. They formulate thirty-three ways of plucking a string with the finger, and therefore are more fastidious than ourselves respecting "touch" and the corresponding delicate variations of tone due to different modes of vibration.

It is quite evident that Eastern peoples have cultivated their perceptions in departments of musical art of which we are comparatively regardless, and it is fair to assume that they reject our harmony because of its inherent imperfections. The extremely sensitive ear of the ancient Hindoos, observed by all students of Sanskrit poetry, led them to make, in common with the Persians and modern Arabians, finer distinctions than we recognize in the musical scale. And it is certainly true that even our natural perceptions of perfect consonance are rendered less acute by habitually listening to and accepting as true harmonies that are systematically rendered untrue, and so far rough and discordant in conformity with our adopted scheme of slight deviations from strict accuracy.

Helmholtz directs us to remain satisfied no longer with this condition of things, and demands that our music be rendered in tune. Those writers who follow the lead of this great physicist take up this cry in common with others, and assume to teach musicians their art. Instrumentalists are advised to construct instruments having twenty-four, twenty-eight, thirty, forty, or more sounds within the octave, instead of allowing the ordinary twelve—the seven white and five black keys of the manual—to do duty for the whole. It is deliberately proposed that three rows of keys be provided, that a simple hymn-tune may be correctly rendered. Mr. Ellis thinks that a fourth or fifth organ or harmonium should be at hand to be used for exceptionally brilliant and novel combinations. Similar recommendations have often been made before. They are useless. For such instruments would be found too complicated for general artistic purposes, and yet would not be elaborate enough to obtain the desired perfection, because this is unattainable. It is unattainable, not on account of the incapacity of the musician, but from the nature of the case. It is impossible to unite melody and modern harmony, and retain for either its just proportions; nor can we set one or the other aside. No such sweeping change can be made in the art by the mere musician. He can not compose music worthy the attention of the world, under the direction of any theorist, interested only in some one principle of truth; nor can he greatly alter the character of his productions.

The composer is the child of his time and nation, and can not free himself from the conditions under which he works. Yet he obeys a blind necessity no more than other men, equally powerless to turn back the tide of modern civilization. It proceeds by virtue of a force which is incontrollable, and at most can only be slightly diverted. Music is one of its art-products. It broke in upon the darkness of mediæval ages, and was a factor in the general illumination that dispelled the gloom, when the Western world arose refreshed as from a deep sleep. It appealed to the sense of hearing—the last to sink to rest, always the first to awaken.

Counterpoint and the science of harmony aided in the formation of this new art, which has earned the distinction of being recognized as the only classic art of the nineteenth century. It is a glorious fabric, and will endure, although scientific purists wish it to be destroyed and another one erected in its stead on the principles of abstract truth. They propound a destructive theory, and yet give no practicable mode of procedure. They do not even remove difficulties to be overcome at the very outset. The musician is invited to attempt the impossible, namely, to make equal things that are unequal—to make the melodic scale agree with the harmonic scale, their proportions being dissimilar. He is compelled to make compromises, either acknowledged and defined, as in the systems of tuning piano-fortes, or unacknowledged and undefined, as in performances on violins. In no other manner can he proceed.

It would be easy for him to employ a few simple chords with the most perfect proportions, but he is required to produce a long-extended and complicated web of harmony—some thousands of combinations in a symphony or other similar work of art. It would be still easier for him to present chords singly as isolated columns, that they may be contemplated "as they stand"—each resting on a fundamental base or bass; but he must exhibit them connectedly—"as they move." Only by passing on and on, from chord to chord in ever-changing forms, from sweetest consonance to most brilliant dissonance, from exciting successions and combinations to calming ones, is harmonic texture provided for musical compositions. Sometimes, also, the melody remains unchanged when the harmonies are greatly altered, so that its expression may be varied, as the features of the same countenance may express varied emotions, and yet be always recognized.

Similarly, architectural purists, previous to the year 1837, made perfectly cylindrical columns, straight lines, and plane surfaces, and, proceeding to build on these principles of truth, made imitations of Greek art that are regarded with indifference in London, Paris, and Berlin. The Parthenon and Theseum were to be contemplated as wholes, and therefore all apparently straight lines and plane surfaces were made respectively curved and convex, that the effect might be smoother—less hard and dry. There are, however, these notable differences in the two cases: It was possible for the architectural purist to make line meet line, but for the musician it would prove impossible. For he, working in a tonal system that is a slowly ascending infinite spiral, can not make his lines return into themselves without making delicate modifications.

Whichever way he proceeds he is led out into infinity;[1] but, without PSM V16 D537 Ancient greek concepts applied to melodic harmony.jpg passing the fourth step, he is compelled to make modifications to reconcile melody with harmony. And whereas the Greek architects made slight variations from abstract truth that the effect on the beholder might be heightened, knowing well the peculiarities of the human eye, the musician trusts in the inability of the ear to detect the modifications he is bound to make, and which he hides with subtile devices.

Five modes of proceeding are now to be exemplified to prove that it is impossible to observe just proportions, and that the course taken by musicians is the best known:

1. The errors may be acknowledged and defined, by the employment of some one fixed "temperament," to determine the nature and extent of the deviations. Whether this be Pythagorean or Helmholtzian, cyclic or non-cyclic, skhismo-cyclic or mesotonic, equal or unequal, it will have as a matter of course its own peculiar inherent imperfections. The "equal" temperament is the one universally employed now for piano-fortes and organs.

2. The errors may be unacknowledged and undefined. In such cases all is left to chance or vague feeling, and the performance may be an ignorant floundering in an open sea of tone, rather than the mathematically accurate rendering as supposed by Helmholtz in the quotation given above. It is true that performers who have practiced long together, who sympathize with one another and with the aims of the composer, and who are deeply imbued with the spirit of his work, give artistically good renderings. The soul-state recorded by the composer is impressed upon the auditors. This is the purport of the music, and nothing more seems desired. It transcends all formal, cold measurements. If the performers tried to be true to these, the gist of the story, as it were, would be lost in the telling. The performance, therefore, may be psychologically true and yet mathematically false. There is a natural tendency to raise notes under the influence of extremely excited, passionate phrases and to depress them in sinking to a state of repose. But, irrespective of such changes, errors must certainly be made by singers and violinists of the nature of those now to be defined, however they may be glossed over and hidden.

Mr. Ellis illustrates his system of "Duodenes" by the first line of the hymn "My country, 'tis of thee," which is better known to German and Anglo-Saxon peoples as "God save the Queen." It will be employed here, for it is not only familiar and very simple, but usually has equally simple harmonies that may be conveniently illustrated with low numbers and a few fractions.

Whatever numbers may by chosen for convenience, the proper proportions of the melodic notes are 15 : 16 : 18. Those of the chords in columns 1, 4, 5, 6, are in the ratio of 4, 5, 6, and those in columns 2, 3 in the ratio of 10 : 12 : 15. It is understood that the doubling or halving any of these chordal numbers will merely represent the note in another octave, above or below, it will not change its character so as to affect the investigation.

3. When the melodic proportions are true, the chords are untrue.

1 2 3 4
C 48 C 48 D 54 B 45
G 36 A 40 A 40 G 36
E 30 E 30 F 322/5 D 27
C 24 A 20 D 27 G 18

At the third note of the melody the chord is out of proportion, for the interval D : A, 27 : 40, should be D : A, 27 : 401/2 to be in the required ratio of 2 : 3 or 10 : 15.

4. When the chords are true, the melody is temporarily out of proportion.

1 2 3 4 5 6
C 128 C 1293/5 D 144 B 120 C 128 D 144
G 96 A’ 108 A’ 108 G 96 G 96 G 96
E 80 E’ 81 F 86 3/5 D 72 E 80 B 120
C 64 A’ 54 D 72 G 48 C 64 G 48

In this case the melody leaves the scale, but returns to it again, as shown by the notes marked ( ) which are raised. This method of altering the melody to obtain correct harmony is almost impossible to performers. It being understood that a note repeated or sustained is to be repeated or sustained at the same pitch, that it may become a pivot (ligature) for the harmonies to turn on, and form a standard of measurement. The errors would therefore more often be as follows:

5. When the chords are true, the melody is permanently out of proportion.

1 2 3 4 5 6
8 C 129 35 24 C 129 35 8 D 108 10 B 120 8 C 128 12 D 124
6 G 97 15 20 A 108 15 A 108 8 G 96 6 G 96 8 G 96
5 E 81 15 E 81 12 F 86 25 6 D 72 5 E 80 5 B 60
4 C 64 45 10 A 54 10 D 72 4 G 48 4 C 64 4 G 48

Here the ratios of each chord are prefixed to the letters, representing the musical notes, that the harmonies may be readily verified. At the fifth chord the key-note is seen at once to be changed, and the melody therefore to be untrue. Viewed vertically, all is correct; viewed horizontally, errors appear in all four lines. Such music can not be made correct from both points of view.

No idea is more firmly rooted in the minds of musicians than that of a fixed key-note. Whenever the pitch is changed the belief is universal that the chords have been out of tune. Even Helmholtz and other scientists are unaware of the fact that perfect harmony requires a moving key-note. It will probably surprise them as much as it would have surprised Ptolemy Philadelphus to learn that the sun is moving in the direction of the constellation Hercules.

For the solar system to be, as it were, in tune, the sun must move; for the harmonic system to be in tune, the key-note must move. In the last illustration the pitch of the key-note (C) was depressed in the ratio of 129 3/5 : 128. There would be three such depressions made in the first half of the melody, and by the same chords. There is no method by which the sum of the errors made in this direction may be atoned for by errors in the opposite direction. If, on repeating this half, the composer were to adopt the following harmonies, the key-note (C) would rise in the ratio of 63 : 64.

1 2 3 4 5 6
C 252 C 252 D 288 B 240 C 256 D 288
G 189 F-sharp 180 F-sharp 180 G 192 G 192 G 192
E 157 12 D 144 C 126 D 144 E 160 B 120
C 126 A 108 D 144 G 96 C 128 G 96

After exploring the whole known field of harmony, and calculating the elevations and depressions consequent on using more elaborate chords, it is asserted that the exact pitch could not be regained.

The formulated results need not be stated here; it is sufficient to give the conclusions to which they point. But assuming that the composer could succeed in so planning his chords that the second half of this melody would so correct the eccentricities of the key-note in the first half that at its completion the composition would be rounded off at the true pitch, it is easy to see that if the first strain were repeated, and the second left unrepeated, or any such ordinary change made, all his elaborate calculations would be of no avail.

Mr. Ellis, who proposes a system with 117 notes within the octave, is thus shown that an infinite number of notes is required, for there is no synonymity in any system when the key-note moves. At each change of pitch the whole series is changed. Mr. Bosanquet, with 53 notes to the octave, offers to provide musicians with materials for 84 scales; and thus we are more reminded of the musical formulæ of the ancient Hindoos—their 16,000 keys—than informed how the above simple melody may be correctly rendered.

It is somewhat amusing to find Mr. Ellis seriously proposing to employ three harmoniums, the three players having to touch the notes that happen to fall to their respective instruments, not only because, as shown above, the music would still be out of tune, but because no performer would play a complete melody by himself, but a note here, another there, unconnectedly. For, however neatly this might be managed, expression or artistic rendering would be unattainable. Yet it is remarked, "The performers would merely require a little drilling and practice together."

Logarithms may be piled and compiled to define scales, but it is not so easy to reconcile the conflicting principles that appear in actual composition. The musician baffles the mathematician, who fails to follow him in his operations, as proved by the hitherto unnoticed discrepancies between melodic and harmonic proportions herein demonstrated. Although the composer's notation is not an exact statement, the performers do not experience practical difficulties: the intention is known, and the intonation is made as perfect as may be, according to the nature of the instruments employed.

The method of tuning the piano-forte may be stigmatized as reducing music to a mere game of permutations and combinations of twelve tones, but no better method is offered by mathematicians and physicists, whose schemes for music prove them more visionary than musicians themselves, who within the limits of their art must be acknowledged to be practical. They are art-workers as a rule, not talkers. Writers on music are generally amateurs, occupied with some one principle, apparently forgetful of the fact that many principles have to be regarded in the production of an art-work—sometimes one, sometimes another having the ascendancy. Therefore, false ideas readily gain currency, for the public can more easily comprehend one or two ideas, put forth with literary skill, than a multiplicity of considerations requiring technical definition, and only correctly estimated by persons practically acquainted with their relative value. Well-written treatises on the plastic arts are frequently found suited to the use of the public, engraved illustrations being more immediately understood than musical quotations, for comparatively few persons can read, and imagine in silence, written harmonies. And, besides, the forms being original, neither geometrical nor taken from nature, no appeal to experience can be made.

Music appears as something quite apart, as though it held aloof from the realities of daily life. Yet, on closer inspection, it is seen to be connected so closely with art and life as to make its classification difficult.

Its rhythmic forms transcend any found in poetry and dancing. Its melodies are not merely grammatically correct constructions, but are felicitous expressions of the highest kind of rhetorical eloquence, which spring up as happy thoughts, and may endure from age to age with wonderful vitality as the national songs of a whole people. It is not merely dramatic, it is preeminently dramatic, many parts being employed not only consecutively but simultaneously. It simulates the gestures indicated in sculptured groups, not as fixed, but in motion, and with such ability as to create in some persons an almost irresistible desire to make corresponding movements. Its forms are original and independent of words, and are not copied like those of painting, which is still dependent on drawing.

It not only resembles Gothic architecture, in the sense of parts depending upon parts for the stability of the whole, so that a cathedral may be aptly spoken of as "petrified music,"; but is more like celestial architecture, in which the base is not an immovable foundation, but moves itself; and, in the due observance of distances (intervals) and speeds (time), the balance is preserved—as, for instance, in the choruses of Bach and Handel.

Its science of acoustics allies it with optics. It can be expressed in algebraic terms or simple numbers, as the above illustrations prove. It transfigures the spoken word in song. For its performance gymnastic exercises are required. Its expressions are like words, in being either conventional or imitative, or partly both; and, unlike words, in that their meaning can not change. It does not describe soul-states or cause their formation after reflection, like poetry, but reveals and induces them immediately, and so surely that Beethoven's sonatas are so many psychologic records.

The composer is more bound by natural laws than other artists, and yet is so free that his productions more nearly resemble actual creations. Music, in its threefold nature, appeals to man in his threefold nature. With great splendor of manifestation an orchestra engages the ear, and sometimes powerfully affects the nervous system; whatever is surveyable in the music occupies the intellect, and its signification affects the soul. It is not so much art-calculated as science inspired.

Here is ample evidence that a mere "physiological basis" is insufficient for the artist, and the advice that he should form a new art, less dependent upon gorgeous harmonies, is equally futile. For, although a composer exercises greater power over music than the philologist over language (who can only explain and classify roots already existing, being powerless to provide a new one), yet still the course of music is propelled by forces that can not be long or successfully opposed. No one affects to believe that steam, electricity, etc., will be set aside at the bidding of Mr. Ruskin.

Modern compositions are the natural expression of our time. Even the music of Mozart and Haydn seems to be truly Arcadian, compared with that of Beethoven and Schumann. It is comparatively artless, cheerful, and free from sighs. The works of these later writers rise to loftier heights and sink to deeper depths, reveling in a larger scale of human passion than those of their predecessors. Here aspirations, longings, strivings, are portrayed with a vividness that mirrors the spirit of the age. This music is not, like Tennyson's "Sleeping Beauty," "a perfect form in perfect rest," but is as in a state of evolution. It wears not so much the expression of Raphael's Madonnas—of the peaceful faith of the cloister—as that of strong, earnest men, exercised with honest doubts in the battle of the creeds. We can not turn back, or remain still; the cry is "Onward!" and, for good or for evil, we must proceed. Our art, side by side with the civilization it represents, will continue to grow, and then perhaps begin to decay, and finally give place to another still more glorious.

  1. See article on "The Modern Piano-forte" in "Popular Science Monthly" for October, 1877, p. 701.