corresponds to a tone in music. It is a dissonant combination. Two series, which number respectively fifteen and sixteen orifices, make the interval of a semitone; it is a very sharp and grating dissonance.*

of which each portion consists of two | the two rates of vibration, the more perfect is lines, one rising gently to a certain height the consonance of the two sounds. The most like an inclined plane, the other falling perfect consonance is the unison I: I; next abruptly down to the horizontal line again. comes the octave, I: 2; after that the fifth, As such a line in the main represents the 2:3; then the fourth, 3:4; then the major motion of a point in a string excited by a We can also open two series, numbering rethird, 45; and finally the minor third, 5: 6. violin-bow, we may at once fully compre-spectively eight and nine orifices; this interval hend the difference in the form of the sonorous wave between the tuning-fork and the violin-bow, both producing the same note of equal intensity. We shall have two waves, in which the crests have equal height, in which the distance from crest to crest is the same; the only difference - that of quality or timbre-being the form of each vibration, one being a wave in which ridge and hollow are gently rounded off, equally broad and symmetrical so that if we inverted the curve the ridges would exactly fit into the hollows, and conversely; while the other consists of straight slopes, gently ascending on one side and abruptly descending on the other, with a sharp ridge at each crest and a corresponding sharp angle in cach hollow.

accordance with definite well-established

is this: whence does this arise; why The question most obvious at this stage should the smaller ratio express the more perfect consonance? In order to answer this question it is absolutely necessary to proceed to a more refined analysis of sound than that with which we have so far become acquainted. Let us for this purpose first suppose that we have two metallic wires, precisely equal in every respect, stretched along a sounding-board. It is well known that wires of this kind, when plucked with the finger or excited by a violin-bow, will generate a musical note, the pitch depending on the length, tension, and thickness of the wire, in physical laws. By our hypothesis both wires are precisely equal, and if they are sounded simultaneously and in the same manner, we shall expect that two notes of the same pitch will be produced independently, that the effect of each sound taken singly will be increased considerably by the effect of the other, that consequently the sound of both will be louder than the sound of each when heard alone, but that if both tones are really in unison the sound must have a uniform intensity throughout. But let us suppose that the pitch of the two notes is only very nearly the same. What will happen in this case is an obvious consequence of the fundamental facts in the production of sound. Since the rate of vibration is not exactly the same for both sounds, the condensations up by Professor Tyndall with his usual and rarefactions of air which are produced compactness and grasp of facts, in the fol- by the two sonorous bodies cease to take lowing manner, after describing the experi- place at the same time. After a short ments made with a siren, in which the time the condensation produced by one number, of holes opened could be varied at will with the requirements of each ex-duced by the other body, and vice versâ; body coincides with the rarefaction properiment:

Let us next glance at some effects of combinations of musical tones. By, producing different notes simultaneously we shall at once discover that some combinations produce a much more pleasing effect than others. The most pleasing result is attained when one note is just an octave above the other. In this case the ratio of the vibrational numbers is I: 2. Such a combination of two musical sounds which make an agreeable impression is called a "concord," or consonance. Next to the octave the most pleasing concords are produced by notes, the ratios of whose numbers of vibrations are those given by the numbers 4: 5: 6. Three Such notes form a harmonic triad, and if sounded with a fourth note, which is the octave of the first of the triad, the whole constitutes in music the "major chord." It is unnecessary for our purpose to enter into numerical details of the series of consonances. The whole has been summed

These experiments amply illustrate two things-firstly, that a musical interval is determined, not by the absolute number of vibrations of the two combining notes, but by the ratio of their vibrations; secondly and this is of the utmost significance. that the smaller the two numbers which express the ratio of

both sounds mutually destroy one another, and this happens clearly when one body has performed just half a vibration more than the other. If one body is in advance of the other by a whole vibration, the condensations and rarefactions again take

* Sound, p. 362.

place at the same time, and the intensity | excited by our fingers or a violin-bow, but of the sound is again increased. These if the experiment is really performed an alternations in the intensity are termed unexpected and remarkable event will take beats, and they indicate a difference in the pitch of two notes, which is the greater the more frequent the beats.

place, which has not only supplied Helmholtz with experimental means for the most refined analysis of sound, but brings Let us as our second step of analysis us to the clearest insight into the physiosuppose that one of the wires only is logical phenomena presented by the most excited close to one extremity. It will delicate structures of the internal ear, and vibrate and produce a definite note, its that connection of these structures with "fundamental note,” and it is well known sounds on which the modern theory of that by lightly touching the wire in the harmony rests. The facts which we shall middle the note produced will be the observe are these. Let us suppose that octave of the fundamental note; the wire both wires are at starting perfectly in will be seen to vibrate on both sides of unison, and one wire is now excited: the the middle point in two "ventral seg- other wire will immediately begin to viments," while the middle point itself re-brate without being touched. Close to mains at rest, and forms a "node." In a the wire these vibrations are easily obsimilar manner the string may be divided served; at a distance they may be reninto 3, 4, 5, 6, and many more ventral dered visible by little paper "riders" segments, and each new division will placed on the second wire, which will be clearly generate a new note. The great thrown off as soon as the first wire is set fact to be learned from these experiments in vibration; or immediately after the first is that a body is thus capable of produ- wire is sounded it may be touched in sevcing notes which are higher than its fun-eral places by the fingers; its vibrations damental tone. These higher notes are termed "overtones" by Professor Tyndall, but Professor Ellis has preferred to use different terms for this phenomenon, which will be best comprehended from the manner in which he renders Helmholtz's introduction to this part of the subject, which is of primary importance in the present theory of harmony:

both wires are supposed to be stretched;

will thus be stopped, so that it can no longer produce sound; but as the note is still heard, it is clear that the second wire is sounding. Vibrations communicated in this manner by one body to another are called sympathetic vibrations, and when the communicated vibrations produce sound the whole phenomenon is called resonance. Whenever it happens On exactly and carefully examining the that a body capable of performing indeeffect produced on the ear by different forms pendent sonorous vibrations is reached by of vibration, as, for example, that correspond- the sound-waves of a tone of the same ing nearly to a violin-string, we meet with a pitch as that which it would itself emit, strange and unexpected phenomenon, long resonance is produced. In our experiknown indeed to individual musicians and ment the motion of one wire is transphysicists, but commonly regarded as a mere ferred to the other by the intervening solid curiosity, its generality and its great signifi-particles of the wooden box upon which cance for all matters relating to musical tones not having been recognized. The ear, when its attention has been properly directed to the effect of the vibrations which strike it, does not hear merely that one musical tone whose pitch is determined by the period of the vibrations in the manner already explained, but in addition to this it becomes aware of a whole series of higher musical tones, which we will call the harmonic_upper partial tones, and sometimes simply the upper partials of that musical tone, in contradistinction to the first tone, the fundamental or prime partial tone, or simply the prime, which is the lowest and generally the loudest of all, and by whose pitch we judge of the pitch of the whole compound musical tone, or simply the compound.* We have assumed in our ideal experiment that one of the wires only has been

* Sensations of Tone, p. 33.

but transference of sonorous vibrations may also be effected, and resonance produced, by the mere undulations of the air itself.

Gently touch one of the keys of a pianoforte without striking the string, so as to raise the damper only, and then sing a note of the corresponding pitch, forcibly directing the voice against the strings of the instrument. On ceasing to sing the note will be echoed back from the piano. It is easy to discover that this echo is caused by the string which is in unison with the note, for directly the hand is removed from the key and the damper is The sympaallowed to fall, the echo ceases. thetic vibration of the string is still better shown by putting little paper riders upon it, which are jerked off as soon as the string vibrates. The more exactly the singer hits

tion.

In this experiment the sounding-board of the instrument is first struck by the vibrations of the air excited by the human voice.*

Sympathetic vibrations act frequently as a disturbing element in musical perceptions. When a piece of music is played on the piano some particular note is often unpleasantly accompanied by the jingling of some object of glass or metal which is in the room. If we find out what body it is that jingles, and strike it so as to make it sound, it will be found to give out the same note as that which, when played on the piano, causes it to chime in. The jingling sound is caused by its striking against neighboring bodies as soon as it begins to vibrate.

the pitch of the string the more strongly it | composed. The principle of the resonavibrates. A very little deviation from the tor, or sound-analyzer, is of special interexact pitch fails in exciting sympathetic vibra- est, as it is founded on sympathetic vibrations. A volume of air contained in an when caused to vibrate tends to yield a open vessel for example, a bottlecertain note, easily produced by blowing across the mouth of the bottle, and when that note is actually sounded in its neighborhood it will strengthen it by its own sympathetic vibrations. A resonator is thus essentially a glass globe furnished with two openings, one of which is turned towards the origin of the sound, and the other is, by means of an india-rubber tube, applied to the ear. If the tone proper to the resonance-globe exists among the upper partials of the compound tone that is sounded, it is strengthened by the globe, and thereby rendered distinctly audible. Since the note proper to a given globe, The first question which these facts other things being the same, depends on suggest is this: what is the cause of the the diameter of the globe and that of the various forms of vibration to which, as we uncovered opening, it follows that by have seen, the different quality or timbre means of a series of such globes the of notes is due? Can it be possible that whole series of upper partials in a given the "color" of sound is due to the vary-compound tone can be rendered distinctly ing upper partials which accompany the audible, and their existence put beyond a same prime on different instruments? If doubt. we strike any note on a piano, and then sound the same note on a flute, an organ, a violin, or utter it with the voice, the difference in quality is undoubtedly partly due to some accidental accompaniments of the particular mode of producing the sound, as, for instance, the slight sound of rushing air which accompanies the blowing of the flute, or to the circumstance that the note struck may either rapidly decrease in intensity, as happens with the sounds of the piano, or may be distinguished by maintaining a uniform intensity, as in the case of the organ. But the main cause of the difference of quality is the production of "overtones," or upper partials, which accompany the fundamental tone. These upper partials not only differ in various sounding bodies, but differ even in the same body if it is sounded in different ways.

For the purpose of experimentally proving the presence of overtones as distinct tones, Professor Helmholtz has not only analyzed and decomposed sounds into their constituents, but he has verified the result of his analysis by performing the reverse operation, the synthesis; that is, he has reproduced a given sound by combining the individual sounds of which his " resonators " had shown that it was

* Sensations of Tone, p. 61.

By means of such analytical and synthetical researches it has been placed now beyond any doubt that differences in musical quality of tone depend solely on the presence and strength of partial tones, and in no respect, as has been supposed, on the differences in the phase of vibra tion under which these partial tones enter into composition.

It must be here observed that we are speaking only of musical quality as previously defined. When the musical tone is accompanied by unmusical noises, such as jarring, scratching, soughing, whizzing, hissing, these motions are either not to be considered as periodic at all, or else correspond to high upper partials of nearly the same pitch, which consequently form strident dissonances.*

Many interesting facts are connected with the results which have established the fundamental law which governs the quality of musical sounds. The following are among the most important. Simple tones, as those produced by a tuning-fork with a resonance-box, and by wide covered pipes, are soft and agreeable without any roughness, but weak and in the deeper notes dull. Musical sounds which are accompanied by a series of upper partials, up to a certain limit, in moderate strength

* Sensations of Tone, p. 187.

are full and musical. In comparison with | string of a piano with a nervous fibre in such simple tones they are grander, richer, and a manner that this fibre would be excited and more sonorous. Such are the sounds of experience a sensation every time the string vibrated. open organ-pipes, of the pianoforte, etc. Then every musical tone which imIf only the uneven partials are present, as know to be really the case in the ear, a series pinged on the instrument would excite, as we in the case of narrow covered pipes, of of sensations exactly corresponding to the pianoforte strings struck in the middle, pendular vibrations into which the original clarionets, etc., the sound becomes indis- motions of the air had to be resolved. By tinct, and when a greater number of par- this means, then, the existence of each partial tials are audible, the sound acquires a tone would be exactly so perceived, as it really nasal character. Again, if the upper par- is perceived by the ear. The sensations extials beyond the sixth and seventh are cited by the different higher partials would, very distinct, the sound becomes sharp under the supposed conditions, fall to the lot and rough. If less strong, the partials of different nervous fibres, and here be proare not prejudicial to the musical useful- Now, as a matter of fact, later microscopic duced perfectly, separately, and independently. ness of the notes. On the contrary, they discoveries respecting the internal construcare useful as imparting character and im- tion of the ear, lead to the hypothesis, that pression to the music. Of this kind are arrangements exist in the ear similar to those most stringed instruments, and most pipes which we have imagined.* furnished with tongues. Sounds in which upper partial tones are particularly strong acquire thereby a peculiarly penetrating character; such are those yielded by brass instruments.

We proceed now to consider the part played by the ear in the apprehension of quality of tone, and in the perception of harmony or dissonance. Like the complex systems of waves, each passing over others, and undisturbedly pursuing its own path, which may be observed from the parapet of any bridge spanning a river, or from a cliff beside the sea, in the same way we must conceive the air of a concert-hall traversed in every direction, and not merely on the surface, by a variegated crowd of intersecting wave systems. Each voice, each rustle of a dress, each instrument in the orchestra emits its peculiar waves which expand spherically from their respective centres, dart through each other, are reflected from the walls of the room, and thus rush backwards and forwards; and although this spectacle is veiled from the material eye, we have another organ of sense which reveals it to our mental perception. The ear analyzes this seemingly labyrinthic intersection of sound-waves, far more confused than that of waves of water; it separates the several tones which compose it, and distinguishes the voices even of individuals, the peculiar qualities of tone given out by each instrument, the rustling of the dresses, the footfall of the walkers, and so on. By what physiological apparatus is this astounding result effected? Professor Helmholtz begins his answer to this question with the following hypothe

sis:

Suppose we were able to connect every

The essential parts of our organ of hearing, on either side of the head, consist, substantially, of two peculiarly-formed membranous bags, called respectively the "membranous labyrinth," and the "scala media" of the cochlea. Both these bags are lodged in cavities, situated in the midst of a dense and solid mass of bone, which forms part of the temporal bone. Each bag is filled with a fluid, and is also supported in a fluid which fills the cavity in which it is lodged. In the interior of each bag certain small mobile, hard bodies are contained; and the ultimate filaments of the auditory nerves are so distributed upon the walls of the bags, that their terminations must be affected by the vibrations of these small hard bodies, should anything set them in motion. It is also quite possible that the vibrations of the fluid contents of the sacs may themselves suffice to excite the filaments of the auditory nerve; but however this may be, any such effect must be greatly intensified by the co-operation of the solid particles; just as in bathing in a tolerably smooth sea, on a rocky shore, the movement of the little waves as they run backwards and forwards is hardly felt, while on a sandy and gravelly beach the pelting of the showers of little stones and sand, which are raised and let fall by cach wavelet, makes a very different impression on the nerves of the skin. In like manner the membrane on which the ends of the auditory nerves are spread out is virtually a sensitive beach, and waves, which by themselves would not be felt, are readily perceived when they raise and let fall hard particles.

Sensations of Tone, p. 190.

.

205

Under this view the scala media of the

Both these membranous bags are lined | nerves with a peculiar auditory apparatus, by an epithelium, and the auditory nerve partly elastic, partly firm, which may be put after passing through the dense bone of into sympathetic vibration under the influence the skull is distributed to certain regions of external vibration, and will then probably of each bag, where its ultimate filaments agitate and excite the mass of nerves.* come into peculiar connection with the epithelial lining. The epithelium itself too at these spots becomes specially modified. In certain parts of the membranous labyrinth, for instance, the epithelium connected with the terminations of the auditory nerve is produced "into long, stiff, slender, hair-like processes," ject into the fluid filling the bag, and which therefore are readily affected by any vibration of that fluid, and communicate the impulse to the ends of the nerve. certain other parts of the same labyrinth these hairs are scanty or absent, but their place is supplied by minute angular particles of calcareous sand, called "otoliths," lying free in the fluid of the bag; these, driven by the vibrations of that fluid, strike the epithelium and so affect the auditory nerve.

"*which pro

In

In the scala media of

the cochlea the lower wall is very elastic, and on it rest the fibres of Corti, named after their discoverer, the Marchese Corti: they are minute, rod-like bodies, and modifications of the epithelial lining of the scala media. Each fibre is composed of two filaments joined at an angle. An

immense number of these filaments are

set side by side, with great regularity, throughout the whole length of the scala media, "so that this organ presents almost the appearance of a keyboard." t The ends of the nerves probably come into close relation either with these fibres, or with the modified epithelium-cells lying close to them, which are capable of being agitated by the slightest impulse. These are then Helmholtz's conclusions:

cochlea resembles a keyboard, in function as well as in appearance, the fibres of Corti being the keys, and the ends of the nerves representing the strings which the keys strike. If it were possible to irritate each of these nerve-fibres experimentally, we should be able to produce any musical · tone, at will, in the sensorium of the person experimented upon, just as any note on a piano is produced by striking the appropriate key. Now experiment proves that bodies like tuning-forks, which when once struck go on sounding for a long time, are susceptible of sympathetic vibrathe difficulty of putting their mass in motions in a high degree, notwithstanding tion, because they admit of a long accumulation of impulses in themselves minute, produced in them by each separate vibration of the existing tone. And precisely for this reason there must be the exactest. agreement between the pitches of the proper tone of the fork, and of the existing tone, because otherwise subsequent impulses given by the motion of the air could not constantly recur in the same phase of the vibration, and thus be suitable for increasing the subsequent effect of the preceding impulses.

If we can suppose that of a set of tuning-forks-and it is suggested that the the functions of such tuning-forks-tuned fibres of Corti are competent to perform of a note in the scale, one were thus conto every note and distinguishing fractions nected with the end of every fibre of the cochlear nerve, then any vibration commuOn reviewing the whole arrangement, there nicated to the perilymph would affect the can be no doubt that Corti's organ is an ap-tuning-fork, which would vibrate with it, paratus adapted for receiving the vibrations of the membrana basilaris, and for vibrating of itself, but our present knowledge is not sufficient to determine with accuracy the manner in which these vibrations take place. For this purpose we require to estimate the stability of the several parts, and the degree of tension and flexibility with more precision than can be deduced from such observations as have hitherto been made on isolated parts, as they casually group themselves under the microscope.

The essential result of our description of the ear consequently consists in the constant connection of the termination of the auditory

Huxley's "Physiology," p. 199. † Ibid., p. 204.

while the rest would be absolutely, or rel atively, indifferent to that vibration. In other words, the vibration would give rise to the sensation of one particular tone, and no other, and every musical interval would be represented by a distinct impression on the sensorium. To the auditory apparatus of the cochlea must thus be assigned the function of discriminating with exactness the pitch and quality of tones, while the perception of intensity has been suggested as a function of the membranous labyrinth; the nerve-fibres terminating in it tell us that sounds are

• Sensations of Tone, p. 211.