that of the other. Thus, if we take any G and the next C above it, we find the number of vibrations in the G series during any interval of time to be just three-fourths that of the C series in the same time.

Having thus, under the guidance of the physicist, arrived at the threshold of the ear, we may ask the physiologist to conduct us through the unknown mazes of this organ, and to unveil for us those fine processes which form the continuation of aërial waves, and are the immediate antecedents of our sensations. From his descriptions we find that the air-waves pass into vibrations of tympanum, bone and fluid, until, in an intricate part of the inner ear (the scala media of the cochlea), these fine movements reach the extremities of the nerve-fibres, and are transformed into those obscure processes of the nervous substance that underlie all our sensations. The terminations of the nerve-fibres, the so-called fibres or appendages of Corti, are arranged in a very curious manner, and constitute a kind of key-board, each distinct filament being supposed to answer to some one set of aërial waves, and so to subserve the production of one particular tone. In the case of a combination of simultaneous sounds, one has to suppose that the compounded undulations before described are resolved by the fibres of Corti into their constituent series, so that the compound series has precisely the same effect on the nervous fibres that the simple ones would have had if they had entered the ear singly. Of the exact nature of the nervous process but little is as yet known. Most physiologists are, however, agreed that it is a mode of molecular vibration more or less analogous to the physical processes of light, electrical phenomena, and so on. If this be so, one may conjecture that each distinct series of sound-waves is transformed into a corresponding series of molecular movements. That is to say, the vibrations in the nervous fibre affected by a rapid series of aërial undulations would also be rapid; those in another fibre affected by a slower series would be relatively slow; and, in the case of harmonious notes, the distinct sets of vibrations going on in the various fibres concerned would bear to each other a ratio similar to that borne by the two aërial series of undulations.

As the result of this physical and physiological teaching, we should have the following basis of musical pleasure. The pleasing effect of tone, as contrasted with mere noise, arises from the even regularity of the sequent molecular movements of a nervous fibre.

The delight of harmony and of melody is connected with a simple variation in the mode of this regular sequence in two or more fibres, the molecular vibrations in each fibre acted upon being continuous and equal, but varying in their absolute rapidity in a simple numerical ratio. In order to explain any further why such modes of nervous excitation are pleasurable, rather than others, it would be necessary to learn more about the nature of pleasurable nervous stimulation in general, and to connect this particular mode of exciting pleasure with other modes observable in the processes of vision and so on.

Such appears to be the view of the organic foundations of tone and harmony commonly adopted by recent physiologists. In opposition to this, a new theory of musical sensation has for some time been propounded by one of the most eminent of contemporary physiologists. The researches of Professor Helmholtz into the basis of musical pleasure have naturally, perhaps, excited much more attention in his own land, so fruitful of music and of science, than in our country. They imply, no doubt, mathematical details which are apt to repel the non-scientific reader. Nevertheless, their principal results are easily apprehended, while the fact that they profess to account for much of the mysterious influence of musical impres sion, should make them a matter of importance to every reflective lover of the art.*

The substance of Helmholtz's principal contributions to musical theory is implicitly contained in his system of upper-tones. It has long been known, as a physical fact, that a string or wire when made to vibrate, swings to and fro, not only throughout its whole length, but also in its half length, quarter length, and so on. Similarly, too, the column of air in certain wind-instruments vibrates in several different series of waves, similarly related to each other in length. Out of these simultaneous sets of vibrations there arise concurring series of waves in the connecting medium of the air, which answer,

* The full exposition of the author's theory is given in his great work Die Lehre von den Tonempfindungen. A very succinct and clear sketch of the doctrine may also be found in a lecture, Ueber die physiologischen Ursachen der musikalischen Harmonie (see Populäre wissenschaftliche Vorträge, Brunswick, 1865). A brief account of the physical side of the theory is supplied by Professor Tyndall in the last chapter of his valuable Lectures on Sound. Finally it may be added that, since the first appearance of this essay, Mr. Sedley Taylor, in his able treatise, Sound and Music, has supplied English students with a lucid statement of the bearings of the doctrine on the elementary impressions of music.

in their several rapidities, to a principal deep or fundamental tone, and to feeble upper-tones, which are simple multiples of the same. Thus, when a G wire of a piano is struck, there are propagated faint undulations corresponding to the next octave, others to the fifth above the octave, and so on. The order of these upper-tones, in the case of any given note, may be seen by a glance at the following example, in which the ground-tone is marked 1 :—

The higher members of the series are very feeble, and need not be taken into account as affecting our sensibility. It is found that the peculiar quality of timbre, or richness of tone, is due to the number of upper-tones present in the note. Following Professor Tyndall's example, I may adopt the German word and say that clang, as distinguished from simple tone, owes its superior quality to the presence of upper-tones. For example, a note sung by the human voice, or struck on a violin, is found to be much fuller and finer in quality than one uttered by a flute; and this difference exactly corresponds to the variation in the number of the upper-tones present. When the note is nearly destitute of upper-tones, as happens in the case of the sounds of a stopped organ-pipe, it is thin and poor, and does not minister the proper enjoyment of clang. Why these partial tones should not be discoverable as distinct ingredients of a note is to be explained, according to our author, by the fact of habitual inattention. He considers that after close observation, a fine ear may soon come to detect their separate existence.

These upper-tones, moreover, have a bearing not only on the phenomenon of timbre or clang, but also on that of harmony and of melody. It is obvious at once that, by supposing several subordinate tones to be present, though blending indistinguishably, in a musical note, we very much complicate the problem of harmony. For when two notes are sounded together, there must be agreeable unison, not only between the two ground-tones, but also between the ground-tone of the one note and the upper-tones of the other, and, finally, between the several upper-tones of the two notes. Unless all these elements combine in a pleasing mode of sensation, it seems impossible to obtain the pure enjoyment of harmony. Helmholtz distinctly recognises this consequence of his theory, and seeks, with the help of it, to establish a new principle of harmony,

which, if accepted, must be regarded as an important contribution to the physiology of music.

In order to understand this principle, it will be necessary to return for an instant to the physical explanation of sound. When two series of air-waves of widely different lengths simultaneously travel to the ear, they combine, as we have already seen, in the form of a single series of compound waves, and affect the two corresponding nerve-fibres just as if they entered the ear apart. But if the undulations of the two series are very nearly equal in length, no such even effect is produced by their conjunction. As the one set is only slightly in advance of the other, their several phases tend now to strengthen, now to neutralize, one another. Thus, if we take the phase of an undulation which represents condensed particles of air, the effect will be now to double this density, now to reduce it to the average level. The result of these aërial interruptions on the ear is a series of alternate swellings and dyings of sound, having the character of abrupt shocks of tone (Tonstösse). Such unpleasant sensations are frequently produced by a piano when out of tune, since in this case the two or three wires which concur in producing a note are no longer exactly of one pitch. When these beats are very slow they are by no means disagreeable, but have a certain impressiveness. Hence they are actually sought after in slow church music, since both the organ and the harmonium have a stop (tremolo) supplied with two sets of pipes or of vibrating tongues which are fitted to produce these beats. The greater the difference in the length of the waves of the two notes, provided only that they are sufficiently approximate to produce the effect at all, the more frequently do the beats recur. The number of beats which arise in a given time is equal to the difference in the numbers of vibrations of the two notes during this period. When they are not more than four or six in a second the ear can readily follow their successive phases. When they are from twenty to thirty in a second, the ear still recognises them as beats, though it is of course unable to follow them with the same distinctness as before. The mental effect is now the unpleasant feeling of a jarring sound. It resembles the impression of the guttural R, which is known to be due to rapid intermissions and sudden explosions of sound. By means of the double sirene, it is possible to produce a series of rapid intermittent sounds, which have precisely the same character of roughness and harshness. When these beats reach a certain rapidity, above 132 in a second, they are no longer

recognisable, and the tones lose their harsh character. At the same time it is found that there is another limit to the production of this effect, besides that of the number of beats per second. Thus, for example, the semitone B-C produces beats just as distinctly in the higher regions of the diatonic scale, where the number of beats is 132 per second, as it does in the lower octave, where the number of beats is 66, and as it does two octaves below, where the number of beats is only 33. The ear recognises a far more rapid series of beats if they are produced by high tones than if they are produced by low tones; and the maximum number recognisable at different points of the scale varies roughly as the number of vibrations of the tones, that is to say, in a geometrical ratio. In other words, the greatest number of beats per second perceptible at any given point, is double that of the same interval an octave below, and so on. Thus, there is a limit of interval in the producing tones, as well as a limit of rapidity in the beats produced, each of which serves to confine the effect of these intermittent shocks on our sensibility. It is ascertained, moreover, that the most piercing harshness of tone arises from a series numbering from 30 to 40 beats per second, in the upper regions of the scale. Finally, the character of this disagreeable effect is found to vary according to the rapidity of the beats which produce it. While slower beats give a coarser sort of roughness to tones, more rapid successions produce a sharper mode of harshness.

Helmholtz seeks to supply us with physiological reasons for these disturbing shocks. He supposes that their effect on our auditory sensibility is produced by means of the joint action of two adjacent tones on the same fibre of Corti. When two tones produce appreciable beats, they are conceived as setting up vibrations in one and the same nervous appendage in the ear, which vibrations undergo the same changes of increase and decrease of intensity as those of the air. Now it is known that all rapid and intermittent stimulation is disagreeable; as we may see in the effects of flickering light, tickling, etc. While continuous and even stimulation tends to exhaust the nerve, and so to lose its effect, intermittent stimulation, by giving brief intervals for nervous restoration, continues to be effective, and by its excessive intensity produces a feeling of pain. To speak in the language of Sir W. Hamilton or of Mr. Spencer, one may say that pain is here the concomitant of an excessive employment of the nervous energies. On this supposition, one may see how it is that both the rapidity of the beat and the width of the interval serve to determine the range of this painful