HELMHOLTZ ON THE SCIENCE OF MUSIC.* THE new science of
music, as distinguished from the rudimentary knowledge of twenty years ago, is in large measure the creation of a single mind. The great work of Helmholtz, published in 1862, at once conquered for science a domain which had belonged almost exclusively to art. The rules of harmony have ceased to be merely empirical precepts for guiding the artist's ear. They now find their rational explanation, partly in physics and partly in physiology. The rules of melody have resisted more obstinately, and in great part acknowledge no sway but that of aesthetic fancy. But a beginning of the invasion has been made, for the governing principle of tune has been fairly brought under the same laws which regulate its harmonic accompaniment. We may wonder that a German book of such mark as the Lehre der Tonemp- findungen should have had to wait thirteen years before appearing in English. But it may be said in excuse that the combination of musical and scientific knowledge which enabled Helmholtz to make his own discoveries and incorporate them with those of others into a scientific whole was also in some degree needed in • On the Sensations of Tone as a Physiological Basis for the Theory of Music. By Hermann L. F. Helmholtz. Translated from the Third German Edition, with Additional Notes and Appendix, by Alexander J. Ellis. London: Loegmane and Co. his English translator, and there are as yet few men in Europe who have this double competence. At last, one of the few, Mr. A. J. Ellis, undertook the task, and has carried it out success- fully. It is not our office to discuss technically a theory which only a high class of trained students can thoroughly enter into. But its leading principles can be popularly explained, so far, at least, as to introduce a certain rationality into the music- lessons of the future, and we are concerned to show how this may be done.
It has long been known that the "pitch" of musical notes depends on the greater or less rapidity of their vibrations. But notes of the same pitch, if sounded on different instruments, such as the pianoforte, cornet, flute, and voice, produce remarkably unlike effects on the ear. Such notes differ in something, and that something is called their musical "quality." The first problem before us is to explain what " quality " consists in. This may, perhaps, be easiest done by appealing to the eye. If we take a cord (a child's skipping-rope will do), fasten it by one end, and swing or whirl the other, we see that it will sometimes oscillate as a whole, and sometimes divide itself into two, three, four, or more parts, each swinging like a separate string. The half- strings swing twice as fast as the whole, the third-parts three times as fast, and so on. If now the skipping-rope were sounding its vibrations like a violin-string, the whole swings would corre- spond to the fundamental tone, and the fractional swings to what are called its upper harmonics, whose rates of vibration are twice, thrice, four times, &c., as fast. If the fundamental note were C, the first harmonic would be its octave C, and the others ascend- ing the scale thus, U, C, E, U, B flat, C, &c. In practice, the movement which a violin-string executes when played on is a compound movement, made up of a mixture of these kinds of vibration. The sound it produces is accordingly a mixture of the pure fundamental tone and its upper harmonics, and it is on the proportion in which these upper tones are mingled with the fundamental or pitch-tone that the peculiar quality of the resulting note depends.
This idea, that the notes which musicians treat as simple are in fact compound sounds, is as yet unfamiliar. To bring it clearly before our minds, it is worth while to visit the interior of a church- organ. This should be one of the completest build, with many stops,—that is to say, really consisting of several organs of various musical qualities, so that the organist may play on one organ separately, or may alternate or combine several, according as he arranges his stops. Let us first confine our attention to a row of wide stopped wooden pipes, and listen to their tone as the organist plays on them only. The single notes are soft, but the musical effect of the whole is so indefinite, dull, and tiresome as to be unbearable for long. Now these tones of wide stopped organ-pipes, which resemble the sound of tuning-forks or of our vowel oo, are nearly pure or simple musical tones, corresponding with the simple oscillation of the violin-string or skipping-rope. Having listened to them for a few minutes, we discover that pure or simple tones are not by themselves suited for practical music, which demands mixed tones of a more exciting and brilliant quality. Looking next to other ranges of the organ-pipes, we observe a particular kind which is constructed to produce a particular quality of tone, and does this in a most instructive way. These pipes belong to the so-called " mixture " stops. To produce each single note, not -one organ-pipe only, but a combi- nation of several pipes, is employed,—one large pipe to give the primary tone, and two or more small pipes tuned to produce its upper harmonics, which blend with the fundamental so as to give it new quality without altering its pitch. The result is as different as possible from the sweet, dull tone of the wide stopped pipes ; it is, in fact, stirring and noisy to excess. Here, then, we have a perfect illustration of the way in which musical quality is given to a pure, fundamental tone, by mixing with it a moderate proportion of its upper harmonica. Now, what the mixture-stop does un- disguisedly, other organ-stops, and indeed musical instruments in general, do disguisedly. They produce compound tones ready mixed, the proportion of the mixture being, of course, different in each sort of instrument. One of the most remarkable points in the whole matter is, that our ears have become so used from childhood to these compound tones, that we do not recognise them as compound, but listen to the blare of the trumpet or the skirl of the bagpipe as though its notes were simple sounds. It is true that musicians of delicate ear have long been conscious of the presence of the upper tones under some circumstances, but the importance of the fact was not realised. Any one, however, may hear them by striking a low note on a grand piano, and listening to the sound dying away, for some of the component
tones last after others cease, and may be heard to spring out almost separately.
Having thus analysed musical tones, we ask why some combi- nations of them are harmonious, while others are discordant. The answer to this, the second great problem of music, is to be found in the theory of interference of waves. A musical vibration in_ the air consists of swings forward and backward. If two such vibrations come together in the air, they tend to strengthen one another when they are in the same direction, but to destroy one another when they are in opposite directions, just as the waves from the wake of two steamboats may be seen to swell into a higher wave where they rise and fall together, but both to disappear where one rising meets the other falling. How two sounds can thus de- stroy one another may be well heard by turning a tuning-fork round near the ear, its tone becoming almost lost when the vibrations from. the two legs counteract each other, then swelling out and again sinking. Such an alternate swelling and sinking is what musicians. call a "beat." Whenever two notes of different pitch are sounded together, each time that the higher and more quickly vibrating tone catches up and leaves behind the lower and slower, it first strengthens and then weakens it, and a beat is the result. To hear beats most perfectly, they should be produced from:the organ, but the pianoforte will do, as when such an interval as a third in the bass is struck and carefully listened to. Now, slow- beats, recurring only a few times in a second, are not unpleasant to the ear, while fast beats, recurring hundreds of times in a second, as when octaves or fifths are sounded in pure high tones, are not perceived at all. But there is an intermediate state of things, when the beats are too fast to be heard separately, but too slow to disappear in the general tone. They then produce the- annoying, rough sensation which is called dissonance or discord, and which may be heard at its worst when two notes a semitone apart are struck together. Here, then, is the physical cause of musical discord, and it becomes a mere matter of calculation to find the musical intervals of notes which, when sounded together, shall produce (with their fundamental and upper tones together) the smallest amount of discord. These prove to be the intervals long since discovered by ear, and estimated according to their comparative pleasantness,—the octave, fifth, fourth, major and minor thirds, &c. The science of harmony, it need hardly be said, has its basis in this distinction of harmonious and inharmonious intervals.
Harmony is a lately invented art, hardly reaching back beyond the middle-ages ; but melody is as old as articulate speech, whose every phrase, in its emphatic rise and fall of tone, shapes itself into the rudiment of a tune, which takes more definite form in the religious intoning of church or conventicle and in the recita- tive of the opera. All developed melody follows a scale of tones. But why does it so ? Why does the ear require the notes com- bined with one another in a tune to belong to a set related to each other by proportionate rates of vibration? Musical scales had been found out by musicians ages before Pythagoras dis- covered the arithmetical ratios among their notes. Pan did not cut up poor Syrinx with a measuring-scale, nor Apollo vanquiiih- Marsyas by closer calculation of vibrations. There is some relation among the sequent notes in a melody which the ear can perceive,. and for which a rational cause must exist. Helmholtz finds in his law of harmonic tones such an explanation. When a note is struck on the piano and followed by its octave, we discern a simi- larity between them, and no wonder, for a part of the tones which make up the higher octave have already been heard among the partial tones of the lower octave. Thus, there is a strong real connection, a sort of mental bridge, between a note and its octave, and a somewhat weaker one between a note and its fifth. In this way, a scale of notes may be arrived at which are related to the key-note, directly or indirectly, by community of partial tones. This natural division of notes seems first to lead to the kind of simple scales known as pentatonic, having only five notes in the octave. These are common in savage and barbaric music, a familiar example being found in genuine old Scotch airs, which can be played on the five black keys of the pianoforte. From. this simple stage the fuller scale used in modern music was
derived. It is true that this explanation of melodic scales has neither the fullness nor certainty of the explanation of harmonic intervals, but, as has been already intimated, it for the first time brings a scientific hypothesis to bear on the problem. And there is this important point in its favour, that it gives a reason why playing on the wide organ-pipes is as colourless in melody as in. harmony. The notes, being almost pure tones, want the upper harmonics to connect them into tune.
It may strike readers already initiated in Helmholtz's doctrines,. that these are here stated without the reservations and additions necessary to correct them. This is true, but these qualifications can hardly be made clear except by study of the work itself. By the way, the attention of intending students may be called beforehand to a mistake which may mystify them. Fig. 6 is intended to re- present the picture of a simple vibration, such as may be prettily obtained by drawing the corner of a struck tuning-fork across a piece of paper smoked over a candle. Fig. 58 gives a picture of a beat, showing its alternate rise and fall of vibration. In the last German edition, a printer, who did not see the vital difference between these two, inserted the woodcut of the beat in both places, and this unlucky blunder has been repeated in the present English translation, carefully corrected as it seems to be elsewhere. In finally commending this book to musicians, it remains to say that those who find it too abstruse will do well to approach it through Tyndall's Lectures on Sound, and afterwards to go through all they can master, leaving the rest.
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