CHAPTER IX

Combination of Musical Sounds—The smaller the Two Numbers

which express the Ratio of their Rates of Vibration, the

more perfect is the Harmony of Two Sounds—Notions of

the Pythagoreans regarding Musical Consonance—Euler’s

Theory of Consonance—Theory of Helmholtz—Dissonance

due to Beats—Interference of Primary Tones and of

Over-tones—Mechanism of Hearing—Schultze’s Bristles—The

Otoliths—Corti’s Fibres—Graphic Representation of

Consonance and Dissonance—Musical Chords—The Diatonic

Scale—Optical Illustration of Musical Intervals—Lissajous’s

Figures—Sympathetic Vibrations—Various Modes of illustrating

the Composition of Vibrations

§ 1. _The Facts of Musical Consonance_

The subject of this day’s lecture has two sides, a physical and

an æsthetical. We have to-day to study the question of musical

consonance—to examine musical sounds in definite combination with each

other, and to unfold the reason why some combinations are pleasant and

others unpleasant to the ear.

Pythagoras made the first step toward the physical explanation of the

musical intervals. This great philosopher stretched a string, and then

divided it into three equal parts. At one of its points of division he

fixed it firmly, thus converting it into two, one of which was twice

the length of the other. He sounded the two sections of the string

simultaneously, and found the note emitted by the short section to be

the higher octave of that emitted by the long one. He then divided

his string into two parts, bearing to each other the proportion of

2:3, and found that the notes were separated by an interval of a

fifth. Thus, dividing his string at different points, Pythagoras

found the so-called consonant intervals in music to correspond with

certain lengths of his string; and he made the extremely important

discovery that _the simpler the ratio of the two parts into which

the string was divided, the more perfect was the harmony of the two

sounds_. Pythagoras went no further than this, and it remained for

the investigators of a subsequent age to show that the strings act in

this way in virtue of the relation of their lengths to the number of

their vibrations. Why simplicity should give pleasure remained long an

enigma, the only pretence of a solution being that of Euler, which,

briefly expressed, is, that the human soul takes a constitutional

delight in simple calculations.

The double siren (Fig. 163) enables us to obtain a great variety of

combinations of musical sounds. And this instrument possesses over all

others the advantage that, by simply counting the number of orifices

corresponding respectively to any two notes, we obtain immediately

the ratio of their rates of vibration. Before proceeding to these

combinations I will enter a little more fully into the action of the

double siren than has been hitherto deemed necessary or desirable.

[Illustration: FIG. 163.]

The instrument, as already stated, consists of two of Dove’s sirens, C′

and C, connected by a common axis, the upper one being turned upside

down. Each siren is provided with four series of apertures, numbering

as follows:

Upper siren Lower siren

Number of apertures Number of apertures

1st Series 16 18

2d Series 15 12

3d Series 12 10

4th Series 9 8

The number 12, it will be observed, is common to both sirens. I

open the two series of 12 orifices each, and urge air through the

instrument; both sounds flow together in perfect unison; the unison

being maintained, however the pitch may be exalted. We have, however,

already learned (Chapter II.) that by turning the handle of the upper

siren the orifices in its wind-chest C′ are caused either to meet those

of its rotating disk, or to retreat from them, the pitch of the upper

siren being thereby raised or lowered. This change of pitch instantly

announces itself by beats. The more rapidly the handle is turned, the

more is the tone of the upper siren raised above or depressed below

that of the lower one, and, as a consequence, the more rapid are the

beats.

Now the rotation of the handle is so related to the rotation of the

wind-chest C′ that when the handle turns through half a right angle

the wind-chest turns through one-sixth of a right angle, or through

the one-twenty-fourth of its whole circumference. But in the case

now before us, where the circle is perforated by 12 orifices, the

rotation through one-twenty-fourth of its circumference causes the

apertures of the upper wind-chest to be closed at the precise moments

when those of the lower one are opened, and _vice versa_. It is plain,

therefore, that the intervals between the puffs of the lower siren,

which correspond to the rarefactions of its sonorous waves, are here

filled by the puffs, or condensations, of the upper siren. In fact,

the condensations of the one coincide with the rarefactions of the

other, and the absolute extinction of the sounds of both sirens is the

consequence.

I may seem to you to have exceeded the truth here; for when the handle

is placed in the position which corresponds to absolute extinction,

you still hear a distinct sound. And, when the handle is turned

continuously, though alternate swellings and sinkings of the tone

occur, the sinkings by no means amount to absolute silence. The reason

is this: The sound of the siren is a highly composite one. By the

suddenness and violence of its shocks, not only does it produce waves

corresponding to the number of its orifices, but the aërial disturbance

breaks up into secondary waves, which associate themselves with the

primary waves of the instrument, exactly as the harmonics of a string,

or of an open organ-pipe, mix with their fundamental tone. When the

siren sounds, therefore, it emits, besides the fundamental tone, its

octave, its twelfth, its double octave, and so on. That is to say,

it breaks the air up into vibrations which have twice, three times,

four times, etc., the rapidity of the fundamental one. Now, by turning

the upper siren through one-twenty-fourth of its circumference, we

extinguish utterly the fundamental tone. But we do not extinguish its

octave.[75] Hence, when the handle is in the position which corresponds

to the extinction of the fundamental tone, instead of silence we have

the full first harmonic of the instrument.

Helmholtz has surrounded both his upper and his lower siren with

circular brass boxes, B, B′, each composed of two halves, which can

be readily separated (one-half of each box is removed in the figure).

These boxes exalt by their resonance the fundamental tone of the

instrument, and enable us to follow its variations much more easily

than if it were not thus reinforced. It requires a certain rapidity of

rotation to reach the maximum resonance of the brass boxes; but when

this speed is attained, the fundamental tone swells out with greatly

augmented force, and, if the handle be then turned, the beats succeed

each other with extraordinary power.

Still, as already stated, the pauses between the beats of the

fundamental tone are not intervals of absolute silence, but are

filled by the higher octave; and this renders caution necessary when

the instrument is employed to determine rates of vibration. It is

not without reason that I say so. Wishing to determine the rate of

vibration of a small singing-flame, I once placed a siren at some

distance from it, sounded the instrument, and after a little time

observed the flame dancing in synchronism with audible beats. I took

it for granted that unison was nearly attained, and, under this

assumption, determined the rate of vibration. The number obtained was

surprisingly low—indeed not more than half what it ought to be. What

was the reason? Simply this: I was dealing, not with the fundamental

tone of the siren, but with its higher octave. This octave and the

flame produced beats by their coalescence; and hence the counter of

the instrument, which recorded the rate, not of the octave, but of

the fundamental, gave a number which was only half the true one. The

fundamental tone was afterward raised to unison with the flame. On

approaching unison beats were again heard, and the jumping of the flame

proceeded with an energy greater than that observed in the case of the

octave. The counter of the instrument then recorded the accurate rate

of the flame’s vibration.

The tones first heard in the case of the siren are always overtones.

These attain sonorous continuity sooner than the fundamental, flowing

as smooth musical sounds while the fundamental tone is still in

a state of intermittence. The siren is, however, so delicately

constructed that a rate of rotation which raises the fundamental tone

above its fellows is almost immediately attained. And if we seek, by

making the blast feeble, to keep the speed of rotation low, it is at

the expense of intensity. Hence the desirability, if we wish to examine

the overtones, of devising some means by which a strong blast and slow

rotation shall be possible.

Helmholtz caused a spring to press as a light brake against the disk

of the siren. Thus raising by slow degrees the speed of rotation, he

was able deliberately to notice the predominance of the overtones at

the commencement, and the final triumph of the fundamental tone. He did

not trust to the direct observation of pitch, but determined the tone

by the number of beats corresponding to one revolution of the handle

of the upper siren. Supposing 12 orifices to be opened above and 12

below, the motion of the handle through 45° produces interference, and

extinguishes the fundamental tone. The coincidences of that tone occur

at the end of every rotation of 90°. Hence, for the fundamental tone,

there must be _four_ beats for every complete rotation of the handle.

Now Helmholtz, when he made the arrangement just described, found that

the first beats numbered, not 4, but 12, for every revolution. They

were, in fact, the beats, not of the fundamental tone, not even of the

first overtone, but of the second overtone, whose rate of vibration is

three times that of the fundamental. These beats continued as long as

the number of air-shocks did not exceed 30 or 40 per second. When the

shocks were between 40 and 80 per second, the beats fell from 12 to

8 for every revolution of the handle. Within this interval the first

overtone, or the octave of the fundamental tone, was the most powerful,

and made the beats its own. Not until the impulses exceeded 80 per

second did the beats sink to 4 per revolution. In other words, not

until the speed of rotation had passed this limit was the fundamental

tone able to assert its superiority over its companions.

This premised, we will combine the tones in definite order, while the

cultivated ears here present shall judge of their musical relationship.

The flow of perfect unison when the two series of 12 orifices each are

opened has been already heard. I now open a series of 8 holes in the

upper and of 16 in the lower siren. The interval you judge at once to

be an octave. If a series of 9 holes in the upper and of 18 holes in

the lower siren be opened, the interval is still an octave. This proves

that the interval is not disturbed by altering the absolute rates of

vibration, so long as the _ratio_ of the two rates remains the same.

The same truth is more strikingly illustrated by commencing with a

low speed of rotation, and urging the siren to its highest pitch; as

long as the orifices are in the ratio of 1:2, we retain the constant

interval of an octave. Opening a series of 10 holes in the upper and of

15 in the lower siren, the ratio is as 2:3, and every musician present

knows that this is the interval of a fifth. Opening 12 holes in the

upper and 18 in the lower siren does not change the interval. Opening

two series of 9 and 12, or of 12 and 16, we obtain an interval of a

fourth; the ratio in both these cases being as 3:4. In like manner two

series of 8 and 10, or of 12 and 15, give us the interval of a major

third; the ratio in this case being as 4:5. Finally, two series of 10

and 12, or of 15 and 18, yield the interval of a minor third, which

corresponds to the ratio 5:6.

These experiments amply illustrate two things: First, 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 the two rates of vibration, the

more perfect is the consonance of the two sounds. The most perfect

consonance is the unison 1:1; next comes the octave 1:2; after that

the fifth 2:3; then the fourth 3:4; then the major third 4:5; and

finally the minor third 5:6. We can also open two series numbering,

respectively, 8 and 9 orifices: this interval corresponds to _a tone_

in music. It is a dissonant combination. Two series which number

respectively 15 and 16 orifices make the interval of a _semi-tone_: it

is a very sharp and grating dissonance.

§ 2. _The Theory of Musical Consonance. Pythagoras and Euler_

Whence, then, does this arise? Why should the smaller ratio express the

more perfect consonance? The ancients attempted to solve this question.

The Pythagoreans found intellectual repose in the answer “All is number

and harmony.” The numerical relations of the seven notes of the musical

scale were also thought by them to express the distances of the planets

from their central fire; hence the choral dance of the worlds, the

“music of the spheres,” which, according to his followers, Pythagoras

alone was privileged to hear. And might we not in passing contrast this

glorious superstition with the grovelling delusion which has taken

hold of the fantasy of our day? Were the character which superstition

assumes in different ages an indication of man’s advance or

retrogression, assuredly the nineteenth century would have no reason to

plume itself, in comparison with the sixth B.C. A more earnest attempt

to account for the more perfect consonance of the smaller ratios was

made by the celebrated mathematician, Euler, and his explanation,

if such it could be called, long silenced, if it did not satisfy,

inquirers. Euler analyzes the cause of pleasure. We take delight in

_order_; it is pleasant to us to observe means “co-operant to an end.”

But then, the effort to discern order must not be so great as to weary

us. If the relations to be disentangled are too complicated, though we

may see the order, we cannot enjoy it. The simpler the terms in which

the order expresses itself, the greater is our delight. Hence the

superiority of the simpler ratios in music over the more complex ones.

Consonance, then, according to Euler, was the satisfaction derived from

the perception of order without weariness of mind.

But in this theory it was overlooked that Pythagoras himself, who first

experimented on the musical intervals, knew nothing about rates of

vibration. It was forgotten that the vast majority of those who take

delight in music, and who have the sharpest ears for the detection

of a dissonance, are in the condition of Pythagoras, knowing nothing

whatever about rates or ratios. And it may also be added that the

scientific man, who is fully informed upon these points, has his

pleasure in no way enhanced by his knowledge. Euler’s explanation,

therefore, does not satisfy the mind, and it was reserved for an

eminent German investigator of our own day, after a profound analysis

of the entire question, to assign the physical cause of consonance and

dissonance—a cause which, when once clearly stated, is so simple and

satisfactory as to excite surprise that it remained so long without a

discoverer.

Various expressions employed in our previous lectures have already, in

part, forestalled Helmholtz’s explanation of consonance and dissonance.

Let me here repeat an experiment which will, almost of itself, force

this explanation upon your attention. Before you are two jets of

burning gas, which can be converted into singing-flames by inclosing

them within two tubes (represented in Fig. 118). The tubes are of the

same length, and the flames are now singing in unison. By means of

a telescopic slider I lengthen slightly one of the tubes; you hear

deliberate beats, which succeed each other so slowly that they can

readily be counted. I augment still further the length of the tube.

The beats are now more rapid than before: they can barely be counted.

It is perfectly manifest that the shocks of which you are now sensible

differ only in point of rapidity from the slow beats which you heard

a moment ago. There is no breach of continuity here. We begin slowly,

we gradually increase the rapidity, until finally the succession of

the beats is so rapid as to produce that particular grating effect

which every musician that hears it would call _dissonance_. Let us now

reverse the process, and pass from these quick beats to slow ones. The

same continuity of the phenomenon is noticed. By degrees the beats

separate from each other more and more, until finally they are slow

enough to be counted. Thus these singing-flames enable us to follow the

beats with certainty, until they cease to be beats and are converted

into dissonance.

This experiment proves conclusively that dissonance _may_ be produced

by a rapid succession of beats; and I imagine this cause of dissonance

would have been pointed out earlier, had not men’s minds been thrown

off the proper track by the theory of “resultant tones” enunciated by

Thomas Young. Young imagined that, when they were quick enough, the

beats ran together to form a resultant tone. He imagined the linking

together of the beats to be precisely analogous to the linking together

of simple musical impulses; and he was strengthened in this notion by

the fact already adverted to, that the first difference-tone, that is

to say, the loudest resultant tone, corresponded, as the beats do, to

a rate of vibration equal to the difference of the rates of the two

primaries. The fact, however, is that the effect of beats upon the ear

is altogether different from that of the successive impulses of an

ordinary musical tone.

§ 3. _Sympathetic Vibrations_

But to grasp, in all its fulness, the new theory of musical consonance

some preliminary studies will be necessary. And here I would ask you to

call to mind the experiments (in Chapter III.) by which the division of

a string into its harmonic segments was illustrated. This was done by

means of little paper riders, which were unhorsed, or not, according

as they occupied a ventral segment or a node upon the string. Before

you at present is the sonometer, employed in the experiments just

referred to. Along it, instead of one, are stretched two strings, about

three inches asunder. By means of a key these strings are brought into

unison. And now I place a little paper rider upon the middle of one

of them, and agitate the other. What occurs? The vibrations of the

sounding string are communicated to the bridges on which it rests, and

through the bridges to the other string. The individual impulses are

very feeble, but, because the two strings are in unison, the impulses

can so accumulate as finally to toss the rider off the untouched string.

Every experiment executed with the riders and a single string may be

repeated with these two unisonant strings. Damping, for instance, one

of the strings, at a point one-fourth of its length from one of its

ends, and placing the red and blue riders formerly employed, not on the

nodes and ventral segments of the damped string, but at points upon the

second string exactly opposite to those nodes and segments, when the

bow is passed across the shorter segment of the damped string, the five

red riders on the adjacent string are unhorsed, while the four blue

ones remain tranquilly in their places. By relaxing one of the strings,

it is thrown out of unison with the other, and then all efforts to

unhorse the riders are unavailing. That accumulation of impulses,

which unison alone renders possible, cannot here take place, and the

consequence is that, however great the agitation of the one string may

be, it fails to produce any sensible effect upon the other.

The influence of synchronism may be illustrated in a still more

striking manner, by means of two tuning-forks which sound the same

note. Two such forks mounted on their resonant supports are placed upon

the table. I draw the bow vigorously across one of them, permitting

the other fork to remain untouched. On stopping the agitated fork, the

sound is enfeebled, but by no means quenched. Through the air and

through the wood the vibrations have been conveyed from fork to fork,

and the untouched fork is the one you now hear. When, by means of a

morsel of wax, a small coin is attached to one of the forks, its power

of influencing the other ceases; the change in the rate of vibration,

if not very small, so destroys the sympathy between the two forks as to

render a response impossible. On removing the coin the untouched fork

responds as before.

This communication of vibrations through wood and air may be obtained

when the forks, mounted on their cases, stand several feet apart. But

the vibrations may also be communicated through the air alone. Holding

the resonant case of a vigorously vibrating fork in my hand, I bring

one of its prongs near an unvibrating one, placing the prongs back

to back, but allowing a space of air to exist between them. Light as

is the vehicle, the accumulation of impulses, secured by the perfect

unison of the two forks, enables the one to set the other in vibration.

Extinguishing the sound of the agitated fork, that which a moment

ago was silent continues sounding, having taken up the vibrations of

its neighbor. Removing one of the forks from its resonant case, and

striking it against a pad, it is thrown into strong vibration. Held

free in the air, its sound is audible. But, on bringing it close

to the silent mounted fork, out of the silence rises a full mellow

sound, which is due, not to the fork originally agitated, but to its

sympathetic neighbor.

Various other examples of the influence of synchronism, already

brought forward, will occur to you here; and cases of the kind might

be indefinitely multiplied. If two clocks, for example, with pendulums

of the same period of vibration, be placed against the same wall, and

if one of the clocks is set going and the other not, the ticks of the

moving clock, transmitted through the wall, will act upon its neighbor.

The quiescent pendulum, moved by a single tick, swings through an

extremely minute arc; but it returns to the limit of its swing just in

time to receive another impulse. By the continuance of this process,

the impulses so add themselves together as finally to set the clock

a-going. It is by this timing of impulses that a properly-pitched voice

can cause a glass to ring, and that the sound of an organ can break a

particular window-pane.

§ 4. _Sympathetic Vibration in Relation to the Human Ear_

If I dwell so fully upon this object, it is for the purpose of

rendering intelligible the manner in which sonorous motion is

communicated to the auditory nerve. In the organ of hearing, in man,

we have first of all the external orifice of the ear, closed at the

bottom by the circular tympanic membrane. Behind that membrane is the

drum of the ear, this cavity being separated from the space between it

and the brain by a bony partition, in which there are two orifices, the

one round and the other oval. These orifices are also closed by fine

membranes. Across the drum stretches a series of four little bones. The

first, called the _hammer_, is attached to the tympanic membrane; the

second, called the _anvil_, is connected by a joint with the hammer; a

third little round bone connects the anvil with the _stirrup-bone_, the

base of which is planted against the membrane of the oval orifice just

referred to. This oval membrane is almost covered by the stirrup-bone,

a narrow rim only of the membrane surrounding the bone being left

uncovered. Behind the bony partition, and between it and the brain,

we have the extraordinary organ called the _labyrinth_, filled with

water, over the lining membrane of which are distributed the terminal

fibres of the auditory nerve. When the tympanic membrane receives a

shock, it is transmitted through the series of bones above referred

to, being concentrated on the membrane against which the base of the

stirrup-bone is fixed. The membrane transfers the shock to the water of

the labyrinth, which, in its turn, transfers it to the nerves.

The transmission, however, is not direct. At a certain place within

the labyrinth exceedingly fine elastic bristles, terminating in sharp

points, grow up between the terminal nerve-fibres. These bristles,

discovered by Max Schultze, are eminently calculated to sympathize with

such vibrations of the water as correspond to their proper periods.

Thrown thus into vibration, the bristles stir the nerve-fibres which

lie between their roots. At another place in the labyrinth we have

little crystalline particles called _otolites_—the Hörsteine of the

Germans—imbedded among the nervous filaments, which, when they vibrate,

exert an intermittent pressure upon the adjacent nerve-fibres. The

otolites probably serve a different purpose from that of the bristles

of Schultze. They are fitted, by their weight, to accept and prolong

the vibrations of evanescent sounds, which might otherwise escape

attention, while the bristles of Schultze, because of their extreme

lightness, would instantly yield up an evanescent motion. They are, on

the other hand, eminently fitted for the transmission of continuous

vibrations.

Finally, there is in the labyrinth an organ, discovered by the

Marchese Corti, which is to all appearance a musical instrument, with

its chords so stretched as to accept vibrations of different periods,

and transmit them to the nerve-filaments which traverse the organ.

Within the ears of men, and without their knowledge or contrivance,

this lute of 3,000 strings[76] has existed for ages, accepting the

music of the outer world and rendering it fit for reception by the

brain. Each musical tremor which falls upon this organ selects from

the stretched fibres the one appropriate to its own pitch, and throws

it into unisonant vibration. And thus, no matter how complicated the

motion of the external air may be, these microscopic strings can

analyze it and reveal the constituents of which it is composed. Surely,

inability to feel the stupendous wonder of what is here revealed would

imply incompleteness of mind; and surely those who practically ignore,

or fear them, must be ignorant of the ennobling influence which such

discoveries may be made to exercise upon both the emotions and the

understanding of man.

§ 5. _Consonant Intervals in Relation to the Human Ear_

This view of the use of Corti’s fibres is theoretical; but it comes to

us commended by every appearance of truth. It will enable us to tie

together many things, whose relations it would be otherwise difficult

to discern. When a musical note is sounded its corresponding Corti’s

fibre resounds, being moved, as a string is moved by a second unisonant

string. And when two sounds coalesce to produce beats, the intermittent

motion is transferred to the proper fibre within the ear. But here it

is to be noted that, for the same fibre to be affected simultaneously

by two different sounds, it must not be far removed in pitch from

either of them. Call to mind our repetition of Melde’s experiments (in

Chapter III.). You then had frequent occasion to notice that, even

before perfect synchronism had been established between the string and

the tuning-fork to which it was attached, the string began to respond

to the fork. But you also noticed how rapidly the vibrating amplitude

of the string increased, as it came close to perfect synchronism

with the vibrating fork. On approaching unison the string would open

out, say to an amplitude of an inch; and then a slight tightening or

slackening, as the case might be, would bring it up to unison, and

cause it to open out suddenly to an amplitude of six inches.

So also in reference to the experiment made a moment ago with the

sonometer; you noticed that the unhorsing of the paper riders was

preceded by a fluttering of the bits of paper; showing that the

sympathetic response of the second string had begun, though feebly,

prior to perfect synchronism. Instead of two strings, conceive three

strings, all nearly of the same pitch, to be stretched upon the

sonometer; and suppose the vibrating period of the middle string to

lie midway between the periods of its two neighbors, being a little

higher than the one and a little lower than the other. Each of the side

strings, sounded singly, would cause the middle string to respond.

Sounding the two side strings together they would produce beats; the

corresponding intermittence would be propagated to the central string,

which would beat in synchronism with the beats of its neighbors. In

this way we make plain to our minds how a Corti’s fibre may, to some

extent, take up the vibrations of a note, nearly, but not exactly,

in unison with its own; and that when two notes close to the pitch

of the fibre act upon it together, their beats are responded to

by an intermittent motion on the part of the fibre. This power of

sympathetic vibration would fall rapidly on both sides of the perfect

unison, so that on increasing the interval between the two notes, a

time would soon arrive when the same fibre would refuse to be acted on

simultaneously by both. Here the condition of the organ, necessary for

the perception of audible beats, would cease.

In the middle region of the pianoforte, with the interval of a

semitone, the beats are sharp and distinct, falling indeed upon the

ear as a grating dissonance. Extending the interval to a whole tone,

the beats become more rapid, but less distinct. With the interval of a

minor third between the two notes, the beats in the middle region of

the scale cease to be sensible. But this smoothening of the sound is

not wholly due to the augmented rapidity of the beats. It is due in

part to the fact, for which the foregoing considerations have prepared

us, that the two notes here sounded are too far removed from that of

the intermediate Corti’s fibre to affect it powerfully. By ascending

to the higher regions of the scale we can produce, with a narrower

interval than the minor third, the same, or even a greater, number

of beats, which are sharply distinguishable because of the closeness

of their component notes. In the very highest regions of the scale,

however, the beats, when they become very rapid, cease to appeal as

roughness to the ear.

Hence both the rapidity of the beats, and the width of the interval,

enter into the question of consonance. Helmholtz judges that in the

middle and higher regions of the musical scale, when the beats reach 33

per second, the dissonance reaches its maximum. Both slower and quicker

beats have a less grating or dissonant effect. When the beats are very

slow, they may be of advantage to the music; and, when they reach 132

per second, their roughness is no longer discernible.

Thanks to Helmholtz, whose views I have here sought to express in the

briefest possible language, we are now in a condition to grapple with

the question of musical intervals, and to give the reason why some

are consonant and some dissonant to the ear. Circumstanced as we are

upon earth, all our feelings and emotions, from the lowest sensation

to the highest æsthetic consciousness, have a mechanical cause: though

it may be forever denied to us to take the step from cause to effect;

or to understand why the agitations of nervous matter can awaken the

delights which music imparts. Take, then, the case of a violin. The

fundamental tone of every string of this instrument is demonstrably

accompanied by a crowd of overtones; so that, when two violins are

sounded, we have not only to take into account the consonance or

dissonance of the fundamental tones, but also those of the higher tones

of both. Supposing two strings sounded whose fundamental tones, and

all of whose partial tones, coincide, we have then absolute unison;

and this we actually have when the ratio of vibration is 1:1. So

also when the ratio of vibration is accurately 1:2, each overtone of

the fundamental finds itself in absolute coincidence with either the

fundamental tone or some higher tone of the octave. There is no room

for beats or dissonance. When we examine the interval of a fifth, with

a ratio of 2:3, we find the coincidence of the partial tones of the

two so perfect as almost, though not wholly, to exclude every trace of

dissonance. Passing on to the other intervals, we find the coincidence

of the partial tones less perfect, as the numbers expressing the ratio

of the vibrations become more large. Thus, the dissonance of intervals

whose rates of vibration can only be expressed by large numbers, is

not to be ascribed to any mystic quality of the numbers themselves,

but to the fact that the fundamental tones which require such numbers

are inexorably accompanied by partial tones whose coalescence produces

beats, these producing the grating effect known as dissonance.

§ 6. _Graphic Representation of Consonance and Dissonance_

Helmholtz has attempted to represent this result graphically, and

from his work I copy, with some modification, the next two diagrams.

He assumes, as already stated, the maximum dissonance to correspond

to 33 beats per second; and he seeks to express different degrees of

dissonance by lines of different lengths. The horizontal line _c′ c″_,

Fig. 164, represents a range of the musical scale in which _c″_ is our

middle C, with 528 vibrations, and _c′_ the lower octave of _c″_. The

distance from any point of this line to the curve above it represents

the dissonance corresponding to that point. The pitch here is supposed

to ascend continuously, and not by jumps. Supposing, for example, two

performers on the violin to start with the same note _c′_, and that,

while one of them continues to sound that note, the other gradually

and continuously shortens his string, thus gradually raising its pitch

up to the octave _c″_. The effect upon the ear would be represented

by the irregular curved line in Fig. 164. Soon after the unison,

which is represented by contact at _c′_, is departed from, the curve

suddenly rises, showing the dissonance here to be the sharpest of all.

At _c′_, the curve approaches the straight line _c′ c″_, and this

point corresponds to the major third. At _f′_ the approach, is still

nearer, and this point corresponds to the fourth. At _g′_ the curve

almost touches the straight line, indicating that at this point, which

corresponds to the fifth, the dissonance almost vanishes. At _a′_ we

have the major sixth; while at _c″_, where the one note is an octave

above the other, the dissonance entirely vanishes. The _e s′_ and the

_a s′_, of this diagram are the German names of a third and a flat

sixth.

[Illustration: FIG. 164.]

Maintaining the same fundamental note _c′_, and passing through the

octave above _c″_, the various degrees of consonance and dissonance

are those shown in Fig. 165. That is to say, beginning with the octave

_c′-c″_, and gradually elevating the pitch of one of the strings till

it reaches _c″′_, the octave of _c″_, the curved line represents the

effect upon the ear. We see, from both these curves, that dissonance

is the general rule, and that only at certain definite points does the

dissonance vanish, or become so decidedly enfeebled as not to destroy

the harmony. These points correspond to the places where the numbers

expressing the ratio of the two rates of vibration are small whole

numbers. It must be remembered that these curves are constructed on the

supposition that the beats are the cause of the dissonance; and the

agreement between calculation and experience sufficiently demonstrates

the truth of the assumption.[77]

[Illustration: FIG. 165.]

You have thus accompanied me to the verge of the Physical portion of

the science of Acoustics, and through the æsthetic portion I have not

the knowledge of music necessary to lead you. I will only add that, in

comparing three or more sounds together, that is to say, in choosing

them for _chords_, we are guided by the principles just mentioned.

We choose sounds which are in harmony with the fundamental sound and

with each other. In choosing a series of sounds for combination two by

two, the simplicity alone of the ratios would lead us to fix on those

expressed by the numbers 1, 5/4, 4/3, 3/2, 5/3, 2; these being the

simplest ratios that we can have within an octave. But, when the notes

represented by these ratios are sounded in succession, it is found that

the intervals between 1 and 5/4, and between 5/3 and 2, are wider than

the others, and require the interpolation of a note in each case. The

notes chosen are such as form chords, not with the fundamental tone,

but with the note _3/2_ regarded as a fundamental tone. The ratios of

these two notes with the fundamental are 9/8 and 15/8. Interpolating

these, we have the eight notes of the natural or diatonic scale,

expressed by the following names and ratios:

Names C. D. E. F. G. A. B. C′.

Intervals 1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th.

Rates of vibration 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2.

Multiplying these ratios by 24, to avoid fractions, we obtain the

following series of whole numbers, which express the relative rates of

vibration of the notes of the diatonic scale:

24, 27, 30, 32, 36, 40, 45, 48.

The meaning of the terms third, fourth, fifth, etc., which we have so

often applied to the musical intervals, is now apparent; the term has

reference to the position of the note in the scale.

§ 7. _Composition of Vibrations_

In our second lecture I referred to, and in part illustrated, a method

devised by M. Lissajous for studying musical vibrations. By means of a

beam of light reflected from a mirror attached to a tuning-fork, the

fork was made to write the story of its own motion. In our last lecture

the same method was employed to illustrate optically the phenomenon

of beats. I now propose to apply it to the study of the composition

of the vibrations which constitute the principal intervals of the

diatonic scale. We must, however, prepare ourselves for the thorough

comprehension of this subject by a brief preliminary examination of the

vibrations of a common pendulum.

Such a pendulum hangs before you. It consists of a wire carefully

fastened to a plate of iron at the roof of the house, and bearing a

copper ball weighing 10 lbs. I draw the pendulum aside and let it go;

it oscillates to and fro almost in the same plane.

I say “almost,” because it is practically impossible to suspend a

pendulum without some little departure from perfect symmetry around

its point of attachment. In consequence of this, the weight deviates

sooner or later from a straight line, and describes an oval more or

less elongated. Some years ago this circumstance presented a serious

difficulty to those who wished to repeat M. Foucault’s celebrated

experiment, demonstrating the rotation of the earth.

Nevertheless, in the case now before us, the pendulum is so carefully

suspended that its deviation from a straight line is not at first

perceptible. Let us suppose the amplitude of its oscillation to be

represented by the dotted line _a b_, Fig. 166. The point _d_, midway

between _a_ and _b_, is the pendulum’s point of rest. When drawn aside

from this point to _b_, and let go, it will return to _d_, and in

virtue of its momentum will pass on to _a_. There it comes momentarily

to rest, and returns through _d_ to _b_. And thus it will continue to

oscillate until its motion is expended.

The pendulum having first reached the limit of its swing at _b_, let us

suppose a push in a direction perpendicular to _a b_ imparted to it;

that is to say, in the direction _b c_. Supposing the time required by

the pendulum to swing from _b_ to _a_ to be one second,[78] then the

time required to swing from _b_ to _d_ will be half a second. Suppose,

further, the force applied at _b_ to be such as would carry the bob,

if free to move in that direction alone, to _c_ in half a second, and

that the distance _b c_ is equal to _b d_, the question then occurs,

where will the bob really find itself at the end of half a second? It

is perfectly manifest that both forces are satisfied by the pendulum

reaching the point _e_, exactly opposite the centre _d_, in half a

second. To reach this point, it can be shown that it must describe the

circular arc _b e_, and it will pursue its way along the continuation

of the same arc, to _a_, and then pass round to _b_. Thus, by the

rectangular impulse the rectilinear oscillation is converted into a

rotation, the pendulum describing a circle, as shown in Fig. 167.

[Illustration: FIG. 166.]

[Illustration: FIG. 167.]

If the force applied at _b_ be sufficient to urge the weight in half

a second through a greater distance than _b c_, the pendulum will

describe an ellipse, with the lines _a b_ for its smaller axis; if,

on the contrary, the force applied at _b_ urge the pendulum in half a

second through a distance less than _b c_, the weight will describe an

ellipse, with the line _a b_ for its greater axis.

[Illustration: Fig. 167.]

Let us now inquire what occurs when the rectangular impulse is applied

at the moment the ball is passing through its position of rest at _d_.

Supposing the pendulum to be moving from _a_ to _b_, Fig. 168, and that

at _d_ a shock is imparted to it sufficient of itself to carry it in

half a second to _c_; it is here manifest that the resultant motion

will be along the straight line _d g_ lying between _b d_ and _d c_.

The pendulum will return along this line to _d_, and pass on to _h_. In

this case, therefore, the pendulum will describe a straight line, _g

h_, oblique to its original direction of oscillation.

Supposing the direction of motion at the moment the push is applied to

be from _b_ to _a_, instead of from _a_ to _b_, it is manifest that the

resultant here will also be a straight line oblique to the primitive

direction of oscillation; but its obliquity will be that shown in Fig.

169.

[Illustration: FIG. 168.]

[Illustration: FIG. 169.]

When the impulse is imparted to the pendulum neither at the centre nor

at the limit of its swing, but at some point between both, we obtain

neither a circle nor a straight line, but something between both. We

have, in fact, a more or less elongated ellipse with its axis oblique

to _a b_, the original direction of vibration. If, for example, the

impulse be imparted at _d′_, Fig. 170, while the pendulum is moving

toward _b_, the position of the ellipse will be that shown in Fig. 170;

but if the push at _d′_ be given when the motion is toward _a_, then

the position of the ellipse will be that represented in Fig. 171.

[Illustration: FIG. 170.]

[Illustration: FIG. 171.]

[Illustration: FIG. 172.]

By the method of M. Lissajous we can combine the rectangular vibrations

of two tuning-forks, a subject which I now wish to illustrate before

you. In front of an electric lamp, L, Fig. 172, is placed a large

tuning-fork, T′, fixed in a stand horizontally, and provided with a

mirror, on which a narrow beam of light, L T′, is permitted to fall.

The beam is thrown back, by reflection. In the path of the reflected

beam is placed a second upright tuning-fork, T, also furnished with

a mirror. By the horizontal fork, when it vibrates, the beam is

tilted laterally; by the vertical fork, vertically. At the present

moment both forks are motionless, the beam of light being reflected

from the mirror of the horizontal to that of the vertical fork, and

from the latter to the screen, on which it prints a brilliant disk. I

now agitate the upright fork, leaving the other motionless. The disk

is drawn out into a fine luminous band, 3 feet long. On sounding the

second fork, the straight band is instantly transformed into a white

ring _o p_, Fig. 172, 36 inches in diameter. What have we done here?

Exactly what we did in our first experiment with the pendulum. We have

caused a beam of light to vibrate simultaneously in two directions, and

have accidentally hit upon the phase when one fork has just reached

the limit of its swing and come momentarily to rest, while the beam is

receiving the maximum impulse from the other fork.

That the _circle_ was obtained is, as stated, a mere accident; but it

was a fortunate accident, as it enables us to see the exact similarity

between the motion of the beam and that of the pendulum. I stop both

forks, and, agitating them afresh, obtain an ellipse with its axis

oblique. After a few trials we obtain the straight line, indicating

that both the forks then pass simultaneously through their positions of

equilibrium. In this way, by combining the vibrations of the two forks,

we reproduce all the figures obtained with the pendulum.

When the vibrations of the two forks are, in all respects, absolutely

alike, whatever the figure may be which is first traced upon the

screen, it remains unchanged in form, diminishing only in size as

the motion is expended. But the slightest difference in the rates of

vibration destroys this fixity of the image. I endeavored before the

lecture to reader the unison between these two forks as perfect as

possible, and hence you have observed very little alteration in the

shape of the figure. But by moving a small weight along the prong of

either fork, or by attaching to either of them a bit of wax, the unison

is impaired. The figure then obtained by the combination of both passes

slowly from a straight line into an oblique ellipse, thence into a

circle; after which it narrows again to an ellipse with an opposed

obliquity, it then passes again into a straight line, the direction of

which is at right angles to the first direction. Finally, it passes, in

the reverse order, through the same series of figures to the straight

line with which we began. The interval between two successive identical

figures is the time in which one of the forks succeeds in executing one

complete vibration more than the other. Loading the fork still more

heavily, we have more rapid changes; the straight line, ellipse, and

circle being passed through in quick succession. At times the luminous

curve exhibits a stereoscopic depth, which renders it difficult to

believe that we are not looking at a solid ring of white-hot metal.

[Illustration: FIG. 173.]

By causing the mirror of the fork, T, to rotate through a small arc,

the steady circle first obtained is drawn out into a luminous scroll

stretching right across the screen, Fig. 173. The same experiment

made with the changing figure, obtained by throwing the forks out of

unison, gives us a scroll of irregular amplitude, Fig. 174.[79]

[Illustration: FIG. 174.]

[Illustration: FIG. 175.]

We have next to combine the vibrations of two forks, one of which

oscillates with twice the rapidity of the other; in other words, to

determine the figure corresponding to the combination of a note and

its octave. To prepare ourselves for the mechanics of the problem,

we must resort once more to our pendulum; for it also can be caused

to oscillate in one direction twice as rapidly as in another. By a

complicated mechanical arrangement this might be done in a very perfect

manner, but at present simplicity is preferable to completeness. The

wire of our pendulum is therefore permitted to descend from its point

of suspension, A, Fig. 175, midway between two horizontal glass rods,

_a b_, _a′ b′_, supported firmly at their ends, and about an inch

asunder. The rods cross the wire at a height of 7 feet above the bob of

the pendulum. The whole length of the pendulum being 28 feet, the glass

rods intercept one-fourth of this length. On drawing the pendulum aside

in the direction of the rods, _a b_, _a′ b′_, and letting it go, it

oscillates freely between them. I bring it to rest and draw it aside

in a direction perpendicular to the last; a length of 7 feet only can

now oscillate, and by the laws of oscillation a pendulum 7 feet long

vibrates with twice the rapidity of a pendulum 28 feet long.

I wish to show you the figure described by the combination of these two

rates of vibration. Attached to the copper ball, _p_, is a camel’s-hair

pencil, intended to rub lightly upon a glass plate placed on black

paper and over which is strewed white sand. Allowing the pendulum to

oscillate as a whole, the sand is rubbed away along a straight line

which represents the amplitude of the vibration. Let _a b_, Fig. 176,

represent this line, which, as before, we will assume to be described

in one second. When the pendulum is at the limit, _b_, of its swing,

let a rectangular impulse be imparted to it sufficient to carry it to

_c_ in one-fourth of a second. If this were the only impulse acting

on the pendulum, the bob would reach _c_ and return to _b_ in half a

second. But under the actual circumstances it is also urged toward _d_,

which point, through the vibration of the whole pendulum, it ought also

to reach in half a second. Both vibrations, therefore, require that the

bob shall reach _d_ at the same moment; and to do this it will have to

describe the curve _b c′ d_. Again, in the time required by the long

pendulum to pass from _d_ to _a_, the short pendulum will pass _to and

fro_ over the half of its excursion; both vibrations must therefore

reach _a_ at the same moment, and to accomplish this the pendulum

describes the lower curve between _d_ and _a_. It is manifest that

these two curves will repeat themselves at the opposite sides of _a b_,

the combination of both vibrations producing finally a figure of 8,

which you now see fairly drawn upon the sand before you.

The same figure is obtained if the rectangular impulse be imparted when

the pendulum is passing its position of rest, _d_.

[Illustration: FIG. 176.]

[Illustration: FIG. 177.]

[Illustration: FIG. 178.]

I have here supposed the time occupied by the pendulum in describing

the line _a b_ to be one second. Let us suppose three-fourths of the

second exhausted, and the pendulum at _d′_, Fig. 177, in its excursion

toward _b_; let the rectangular impulse then be imparted to it,

sufficient to carry it to _c_ in one-fourth of a second. Now the long

pendulum requires that it should move from _d′_ to _b_ in one-fourth of

a second; both impulses are therefore satisfied by the pendulum taking

up the position _c′_ at the end of a quarter of a second. To reach

this position it must describe the curve _d′ c′_. It will manifestly

return along the same curve, and at the end of another quarter of a

second find itself again at _d′_. From _d′_ to _d_ the long pendulum

requires a quarter of a second. But at the end of this time the short

pendulum must be at the lower limit of its swing: both requirements are

satisfied by the pendulum being at _e_. We thus obtain one arm, _c′

e_, of a curve, which repeats itself to the left of _e_; so that the

entire curve, due to the combination of the two vibrations, is that

represented in Fig. 165. This figure is a parabola, whereas the figure

of 8 before obtained is a lemniscata.

We have here supposed that, at the moment when the rectangular impulse

was applied, the motion of the pendulum was _toward_ _b_: if it were

toward _a_ we should obtain the inverted parabola, as shown in Fig. 178.

Supposing, finally, the impulse to be applied, not when the pendulum

is passing through its position of equilibrium, nor when it is passing

a point corresponding to three-fourths or one-fourth of the time of

its excursion, but at some other point in the line, _a b_, between its

end and centre. Under these circumstances we should have neither the

parabola nor the perfectly symmetrical figure of 8, but a distorted 8.

And now we are prepared to witness with profit the combined vibration

of our two tuning-forks, one of which sounds the octave of the other.

Permitting the vertical fork, T, Fig. 172, to remain undisturbed

in front of the lamp, we can oppose to it a horizontal fork, which

vibrates with twice the rapidity. The first passage of the bow across

the two forks reveals the exact similarity of this combination, and

that of our pendulum. A very perfect figure of 8 is described upon

the screen. Before the lecture the vibrations of these two forks were

fixed as nearly as possible to the ratio of 1:2, and the steadiness

of the figure indicates the perfection of the tuning. Stopping both

forks, and again agitating them, we have the distorted 8 upon the

screen. A few trials enable me to bring out the parabola. In all these

cases the figure remains fixed upon the screen. But if a morsel of

wax be attached to one of the forks, the figure is steady no longer,

but passes from the perfect 8 into the distorted one, thence into

the parabola, from which it afterward opens out to an 8 once more. By

augmenting the discord, we can render those changes as rapid as we

please.

When the 8 is steady on the screen, a rotation of the mirror of the

fork, T, produces the scroll shown in Fig. 179.

[Illustration: FIG. 179.]

Our next combination will be that of two forks vibrating in the ratio

of 2:3. Observe the admirable steadiness of the figure produced by

the compounding of these two rates of vibration. On attaching a

four-penny-piece with wax to one of the forks the steadiness ceases,

and we have an apparent rocking to and fro of the luminous figure.

Passing on to intervals of 3:4, 4:5, and 5:6, the figures become more

intricate as we proceed. The last combination, 5:6, is so entangled

that to see the figure plainly a very narrow band of light must be

employed. The distance existing between the forks and the screen also

helps us to unravel the complication.

[Illustration: FIG. 180.]

And here it is worth noting that, when the figure is fully developed,

the loops along the vertical and horizontal edges express the ratio of

the combined vibrations. In the octave, for example, we have two loops

in one direction, and one in another; in the fifth, two loops in one

direction, and three in another. When the combination is as 1:3, the

luminous loops are also as 1:3. The changes which some of these figures

undergo, when the tuning is not perfect, are extremely remarkable. In

the case of 1:3, for example, it is difficult at times not to believe

that you are looking at a solid link of white-hot metal. The figure

exhibits a depth, apparently incompatible with its being traced upon a

plane surface.

[Illustration: FIG. 181.]

Fig. 180 (page 445) is a diagram of these beautiful figures, including

combinations from 1:1 to 5:6. In each case, the characteristic phases

of the vibration are shown; and through all of these each figure passes

when the interval between the two forks is not pure. I also add here,

Fig. 181, two phases of the combination 8:9.

[Illustration: FIG. 182. 1:2.]

[Illustration: FIG. 183. 2:3.]

To these illustrations of rectangular vibrations I add two others,

Figs. 182 and 183, from a very beautiful series obtained by Mr. Herbert

Airy with a compound pendulum. The experiments are described in

“Nature” for August 17 and September 7, 1871. As their loops indicate,

the figures are those of an octave and a twelfth.

[Illustration: FIG. 184. 2:3.]

[Illustration: FIG. 185. 3:4.]

But the most instructive apparatus for the compounding of rectangular

vibrations is that of Mr. Tisley. Figs. 184 and 185 are copies of

figures obtained by him through the joint action of two distinct

pendulums; the rates of vibration corresponding to these particular

figures being 2:3 and 3:4 respectively. The pen which traces the

figures is moved simultaneously by two rods attached to the pendulums

above their places of suspension. These two rods lie in the two planes

of vibration, being at right angles to the pendulums, and to each

other. At their place of intersection is the pen. By means of a ball

and socket, of a special kind, the rods are enabled to move with a

minimum of friction in all directions, while the rates of vibration are

altered, in a moment, by the shifting of movable weights. The figures

are drawn either with ink on paper, or, when projection on a screen is

desired, by a sharp point on smoked glass. When the pendulums, having

gone through the entire figure, return to their starting-point, they

have lost a little in amplitude. The second excursion will, therefore,

be smaller than the first, and the third smaller than the second. Hence

the series of fine lines, inclosing gradually-diminishing areas, shown

in these exquisite figures.[80] Mr. Tisley’s apparatus reflects the

highest credit upon its able constructor.

[Illustration: FIG. 186.]

Sir Charles Wheatstone devised, many years ago, a small and very

efficient apparatus for the compounding of rectangular vibrations.

A drawing, Fig. 186, and a description of this beautiful little

instrument, for both of which I am indebted to its eminent inventor,

may find a place here: _a_ is a steel rod polished at its upper end so

as to reflect a point of light; this rod moves in a ball-and-socket

joint at _b_, so that it may assume any position. Its lower end is

connected with two arms _c_ and _d_, placed at right angles to each

other, the other ends of which are respectively attached to the

circumferences of the two circular disks _e_ and _f_. The axis of

the disk _e_ carries at its opposite end another large disk _g_,

which gives motion to the small disk _h_, placed on the axis which

carries the disk _f_; and, according as this small disk _h_ is placed

nearer to or further from the centre of the disk _g_, it communicates

a different relative motion to the disk _f_. The nut and screw _i_

enable the disk _h_ to be placed in any position between the centre,

and circumference of the larger disk _g_; and by means of the fork

_j_ the disk _f_ is caused to revolve, whatever may be the position

of the disk _h_. By this arrangement, while the wheel _k_ is turned

regularly, the rod _a_ is moved backward and forward by the disk _e_

in one direction, and by the disk _f_, with any relative oscillatory

motion, in the rectangular direction. The end of the rod is thus made

to describe and to exhibit optically all the beautiful acoustical

figures produced by the composition of vibrations of different periods

in directions rectangular to each other. A lever _l_, bearing against

the nut _i_, indicates, on a scale _m_, the numerical ratio of the two

vibrations.[81]

I close these remarks on the combination of rectangular vibrations with

a brief reference to an apparatus constructed by Mr. A. E. Donkin, of

Exeter College, Oxford, and described in the “Proceedings of the Royal

Society,” vol. xxii., p. 196. In its construction great mechanical

knowledge is associated with consummate skill. I saw the apparatus as

a wooden model, before it quitted the hands of its inventor, and was

charmed with its performance. It is now constructed by Messrs. Tisley

and Spiller.

SUMMARY OF CHAPTER IX

By the division of a string Pythagoras determined the consonant

intervals in music, proving that, the simpler the ratio of the

two parts into which the string was divided, the more perfect is

the harmony of the sounds emitted by the two parts of the string.

Subsequent investigators showed that the strings act thus because of

the relation of their lengths to their rates of vibration.

With the double siren this law of consonance is readily illustrated.

Here the most perfect harmony is the unison, where the vibrations are

in the ratio of 1:1. Next comes the octave, where the vibrations are

in the ratio of 1:2. Afterward follow in succession the fifth, with a

ratio of 2:3; the fourth, with a ratio of 3:4; the major third, with a

ratio of 4:5; and the minor third, with a ratio of 5:6. The interval

of a tone, represented by the ratio 8:9, is dissonant, while that of a

semitone, with a ratio of 15:16, is a harsh and grating dissonance.

The musical interval is independent of the absolute number of the

vibrations of the two notes, depending only on the _ratio_ of the two

rates of vibration.

The Pythagoreans referred the pleasing effect of the consonant

intervals to number and harmony, and connected them with “the music of

the spheres.” Euler explained the consonant intervals by reference to

the constitution of the mind, which, he affirmed, took pleasure in

simple calculations. The mind was fond of order, but of such order as

involved no weariness in its contemplation. This pleasure was afforded

by the simpler ratios in the case of music.

The researches of Helmholtz prove the rapid succession of beats to be

the real cause of dissonance in music.

By means of two singing-flames, the pitch of one of them being

changeable by the telescopic lengthening of its tube, beats of any

degree of slowness or rapidity may be produced. Commencing with beats

slow enough to be counted, and gradually increasing their rapidity, we

reach, without breach of continuity, downright dissonance.

But, to grasp this theory in all its completeness, we must refer to the

constitution of the human ear. We have first the tympanic membrane,

which is the anterior boundary of the drum of the ear. Across the drum

stretches a series of little bones, called respectively the _hammer_,

the _anvil_, and the _stirrup-bone_; the latter abutting against a

second membrane, which forms part of the posterior boundary of the

drum. Beyond this membrane is the labyrinth filled with water, and

having its lining membrane covered with the filaments of the auditory

nerve.

Every shock received by the tympanic membrane is transmitted through

the series of bones to the opposite membrane; thence to the water of

the labyrinth, and thence to the auditory nerve.

The transmission is not direct. The vibrations are in the first place

taken up by certain bodies, which can swing sympathetically with

them. These bodies are of three kinds: the otolites, which are little

crystalline particles; the bristles of Max Schultze; and the fibres of

Corti’s organ. This latter is to all intents and purposes a stringed

instrument, of extraordinary complexity and perfection, placed within

the ear.

As regards our present subject, the strings of Corti’s organ probably

play an especially important part. That one string should respond, in

some measure, to another, it is not necessary that the unison should

be perfect; a certain degree of response occurs in the immediate

neighborhood of unison.

Hence each of two strings, not far removed from each other in

pitch, can cause a third string, of intermediate pitch, to respond

sympathetically. And if the two strings be sounded together, the beats

which they produce are propagated to the intermediate string.

So, as regards Corti’s organ, when single sounds of various pitches, or

rather when vibrations of various rapidities, fall upon its strings,

the vibrations are responded to by the particular string whose period

coincides with theirs. And when two sounds, close to each other in

pitch, produce beats, the intermediate Corti’s fibre is acted on by

both, and responds to the beats.

In the middle and upper portions of the musical scale the beats are

most grating and harsh when they succeed each other at the rate of 33

per second. When they occur at the rate of 132 per second, they cease

to be sensible.

The perfect consonance of certain musical intervals is due to the

absence of beats. The imperfect consonance of other intervals is due

to their existence. And here the overtones play a part of the utmost

importance. For, though the primaries may sound together without any

perceptible roughness, the overtones may be so related to each other as

to produce harsh and grating beats. A strict analysis of the subject

proves that intervals which require large numbers to express them are

invariably accompanied by overtones which produce beats; while in

intervals expressed by small numbers the beats are practically absent.

The graphic representation of the consonances and dissonances of

the musical scale, by Helmholtz, furnishes a striking proof of this

explanation.

The optical illustration of the musical intervals has been effected in

a very beautiful manner by Lissajous. Corresponding to each interval is

a definite figure, produced by the combination of its vibrations.

The compounding of vibrations has, of late years, been beautifully

illustrated by apparatus constructed by Sir C. Wheatstone, Mr. Herbert

Airy, and Mr. A. E. Donkin; and by the beautiful pendulum apparatus of

Mr. Tisley, of the firm of Tisley and Spiller.

The pressure which, on a former occasion, prevented me from adding a

“summary” to this chapter, was also the cause of hastiness, and partial

inaccuracy, in its sketch of the theory of Helmholtz. That the sketch

needed emendation I have long known, but I did not think it worth while

to anticipate the correction here made; as the chapter, imperfect as

it was, had been published, without comment, in Germany, by Helmholtz

himself.

APPENDICES

APPENDIX I

ON THE INFLUENCE OF MUSICAL SOUNDS ON THE FLAME OF A JET OF COAL-GAS.

BY JOHN LE CONTE, M.D.[82]

A short time after reading Prof. John Tyndall’s excellent article

“On the Sounds produced by the Combustion of Gases in Tubes,”[83] I

happened to be one of a party of eight persons assembled after tea

for the purpose of enjoying a private musical entertainment. Three

instruments were employed in the performance of several of the grand

trios of Beethoven, namely, the piano, violin, and violoncello. Two

“_fish-tail_” gas-burners projected from the brick wall near the piano.

Both of them burned with remarkable steadiness, the windows being

closed and the air of the room being very calm. Nevertheless, it was

evident that _one_ of them was under a pressure nearly sufficient to

make it _flare_.

Soon after the music commenced, I observed that the flame of the

last-mentioned burner exhibited pulsations in height which were

_exactly synchronous_ with the audible beats. This phenomenon was very

striking to every one in the room, and especially so when the strong

notes of the violoncello came in. It was exceedingly interesting

to observe how perfectly even the _trills_ of this instrument were

reflected on the sheet of flame. _A deaf man might have seen

the harmony_. As the evening advanced, and the diminished consumption

of gas in the city _increased the pressure_, the phenomenon became

more conspicuous. The _jumping_ of the flame gradually increased,

became somewhat irregular, and finally it began to flare continuously,

emitting the characteristic sound indicating the escape of a greater

amount of gas than could be properly consumed. I then ascertained

by experiment that the phenomenon _did not_ take place unless the

discharge of gas was so regulated that the flame approximated to the

condition of _flaring_. I likewise determined by experiment that the

effects _were not_ produced by jarring or shaking the floor and walls

of the room by means of repeated concussions. Hence it is obvious that

the pulsations of the flame _were not_ owing to _indirect_ vibrations

propagated through the medium of the walls of the room to the burning

apparatus, but must have been produced by the _direct_ influence of the

aërial sonorous pulses on the burning jet.

In the experiments of M. Schaffgotsch and Prof. J. Tyndall, it is

evident that “the shaking of the singing-flame within the glass

tube,” produced by the voice or the siren, was a phenomenon perfectly

analogous to what took place under my observation _without the

intervention of a tube_. In my case the discharge of gas was so

regulated that there was a tendency in the flame to flare, or to emit a

“_singing-sound_.” Under these circumstances, strong aërial pulsations

occurring at _regular intervals_ were sufficient to develop synchronous

fluctuations in the height of the flame. It is probable that the

effects would be more striking when the tones of the musical instrument

are _nearly_ in unison with the sounds which would be produced by the

flame under the slight increase in the rapidity of discharge of gas

required to manifest the phenomenon of flaring. This point might be

submitted to an experimental test.

As in Prof. Tyndall’s experiments on the jet of gas burning within

a tube, clapping of the hands, shouting, etc., were ineffectual in

converting the “silent” into the “singing-flame,” so, in the case under

consideration, _irregular_ sounds did not produce any perceptible

influence. It seems to be necessary that the impulses should

_accumulate_, in order to exercise an appreciable effect.

With regard to the mode in which the sounds are produced by the

combustion of gases in tubes, it is universally admitted that the

explanation given by Prof. Faraday in 1818 is essentially correct.

It is well known that he referred these sounds to the successive

explosions produced by the periodic combination of the atmospheric

oxygen with the issuing jet of gas. While reading Prof. J. Plateau’s

admirable researches (third series) on the “Theory of the Modifications

experienced by Jets of Liquid issuing from Circular Orifices when

exposed to the Influence of Vibratory Motions,”[84] the idea flashed

across my mind that the phenomenon which had fallen under my

observation was nothing more than a _particular case_ of the effects

of sounds on _all kinds of fluid jets_. Subsequent reflection has only

served to fortify this first impression.

The beautiful investigations of Felix Savart, on the influence of

sounds on jets of water, afford results presenting so many points of

analogy with their effects on the jet of burning gas, that it may be

well to inquire whether both of them may be referred to a common cause.

In order to place this in a striking light, I shall subjoin some of the

results of Savart’s experiments. Vertically-descending jets of water

receive the following modifications under the influence of vibrations:

1. The continuous portions become shortened; the vein resolves itself

into separate drops nearer the orifice than when _not_ under the

influence of vibrations.

2. Each of the masses, as they detach themselves from the extremity of

the continuous part, becomes flattened alternately in a vertical and

horizontal direction, presenting to the eye, under the influence of

their translatory motion, regularly-disposed series of maxima and

minima of thickness, or ventral segments and nodes.

3. The foregoing modifications become much more developed and regular

when a note, in unison with that which would be produced by the shock

of the discontinuous part of the jet against a stretched membrane, is

sounded in its neighborhood. The continuous part becomes considerably

shortened, and the ventral segments are enlarged.

4. When the note of the instrument is _almost_ in unison, the

continuous part of the jet is alternately lengthened and shortened

and the beats which coincide with these variations in length _can be

recognized by the ear_.

5. Other tones act with less energy on the jet, and some produce no

sensible effect.

When a jet is made to ascend _obliquely_, so that the discontinuous

part appears scattered into a kind of _sheaf_ in the same vertical

plane, M. Savart found:

_a._ That, under the influence of vibrations of a determinate period,

this sheaf may form itself into _two_ distinct jets, each possessing

regularly-disposed ventral segments and nodes; sometimes with a

different node the sheaf becomes replaced by _three_ jets.

_b._ The note which produces the greatest shortening of the continuous

part always reduces the whole to a _single_ jet, presenting a perfectly

regular system of ventral segments and nodes.

In the last memoir of M. Savart—a posthumous one, presented to the

Academy of Sciences of Paris, by M. Arago, in 1853[85]—several

remarkable acoustic phenomena are noticed in relation to the musical

tones produced by the efflux of liquids through short tubes. When

certain precautions and conditions are observed (which are minutely

detailed by this able experimentalist), the discharge of the liquid

gives rise to a succession of musical tones of great intensity and of a

peculiar quality, somewhat analogous to that of the human voice. That

these notes were not produced by the descending drops of the liquid

vein was proved by permitting it to discharge itself into a vessel of

water, while the orifice was below the surface of the latter. In this

case the jet of liquid must have been _continuous_, but nevertheless

the notes were produced. These unexpected results have been entirely

confirmed by the more recent experiments of Prof. Tyndall.[86]

According to the researches of M. Plateau, all the phenomena of

the influence of vibrations on jets of liquid are referable to the

conflict between the vibrations and the _forces of figure_ (“_forces

figuratrices_”). If the physical fact is admitted—and it seems to be

indisputable—that a liquid cylinder attains a _limit of stability_ when

the proportion between its length and its diameter is in the ratio

of twenty-two to seven, it is almost a _physical necessity_ that the

jet should assume the constitution indicated by the observations of

Savart. It likewise seems highly probable that a liquid jet, while

in a transition stage to discontinuous drops, should be exceedingly

sensitive to the influence of all kinds of vibrations. It must be

confessed, however, that Plateau’s beautiful and coherent theory does

not appear to embrace Savart’s last experiment, in which the musical

tones were produced by a jet of water issuing under the surface of the

same liquid. It is rather difficult to imagine what agency the “forces

of figure” could have, under such circumstances, in the production of

the phenomenon. This curious experiment tends to corroborate Savart’s

original idea, that the vibrations which produce the sounds must

take place in the glass reservoir itself, and that the cause must be

inherent in the phenomenon of the flow.

To apply the principles of Plateau’s theory to gaseous jets, we are

compelled to abandon the idea of the _non-existence of molecular

cohesion in gases_. But is there not abundant evidence to show that

cohesion _does exist_ among the particles of gaseous masses? Does not

the deviation from rigorous accuracy, both in the law of Mariotte and

Gay-Lussac—especially in the case of condensable gases, as shown by the

admirable experiments of M. Regnault—clearly prove that the hypothesis

of the non-existence of cohesion in aëriform bodies is fallacious? Do

not the expanding rings which ascend when a bubble of phosphuretted

hydrogen takes fire in the air indicate the existence of some cohesive

force in the gaseous product of combustion (aqueous vapor), whose

outlines are marked by the opaque phosphoric acid? In short, does not

the very _form_ of the flame of a “fish-tail” burner demonstrate that

cohesion _must exist_ among the particles of the issuing gas? It is

well known that in this burner the single jet which issues is formed by

the union of _two oblique jets_ immediately before the gas is emitted.

The result is a perpendicular _sheet of flame_. How is such a result

produced by the mutual action of two jets, unless the force of cohesion

is brought into play? Is it not obvious that such a fanlike flame must

be produced by the same causes as those varied and beautiful forms of

aqueous sheets, developed by the mutual action of jets of water, so

strikingly exhibited in the experiments of Savart and of Magnus?

If it be granted that gases possess molecular cohesion, it seems to be

physically certain that jets of gas must be subject to the same laws as

those of liquid. Vibratory movements excited in the neighborhood ought,

therefore, to produce modifications in them analogous to those recorded

by M. Savart in relation to jets of water. Flame or incandescent gas

presents gaseous matter in a _visible_ form, admirably adapted for

experimental investigation; and, _when produced by a jet_, should be

amenable to the principles of Plateau’s theory. According to this view,

the pulsations or _beats_ which I observed in the gas-flame when under

the influence of musical sounds, are produced by the conflict between

the aërial vibrations and the “forces of figure” (as Plateau calls

them) giving origin to periodical fluctuations of intensity, depending

on the sonorous pulses.

If this view is correct, will it not be necessary for us to modify

our ideas in relation to the agency of tubes in developing musical

sounds by means of burning jets of gas? Must we not look upon all

burning jets—as in the case of water-jets—as _musically inclined_; and

that the use of tubes merely places them in a condition favorable for

developing the tones? It is well known that burning jets frequently

emit a _singing-sound_ when they are perfectly _free_. Are these

sounds produced by successive explosions analogous to those which take

place in glass tubes? It is very certain that, under the influence

of molecular forces, any cause which tends to elongate the flame,

without affecting the velocity of discharge, must tend to render it

discontinuous, and thus bring about that mixture of gas and air which

is essential to the production of the explosions. The influence of

tubes, as well as of aërial vibrations, in establishing this condition

of things, is sufficiently obvious. Was not the “beaded line” with its

succession of “luminous stars,” which Prof. Tyndall observed when a

flame of olefiant gas, burning in a tube, was examined by means of a

moving mirror, an indication that the flame became _discontinuous_,

precisely as the continuous part of a jet of water becomes _shortened_,

and resolved into isolated drops, under the influence of sonorous

pulsations? But I forbear enlarging on this very interesting subject,

inasmuch as the accomplished physicist last named has promised

to examine it at a future period. In the hands of so sagacious a

philosopher, we may anticipate a most searching investigation of the

phenomena in all their relations. In the meantime I wish to call the

attention of men of science to the view presented in this article, in

so far as it groups together several classes of phenomena under one

head, and may be considered a partial generalization.—From Silliman’s

“American Journal” for January, 1858.

APPENDIX II

ON ACOUSTIC REVERSIBILITY[87]

On the 21st and 22d of June, 1822, a commission, appointed by the

Bureau des Longitudes of France, executed a celebrated series of

experiments on the velocity of sound. Two stations had been chosen,

the one at Villejuif, the other at Montlhéry, both lying south of

Paris, and 11·6 miles distant from each other. Prony, Mathieu, and

Arago were the observers at Villejuif, while Humboldt, Bouvard, and

Gay-Lussac were at Montlhéry. Guns, charged sometimes with two pounds

and sometimes with three pounds of powder, were fired at both stations,

and the velocity was deduced from the interval between the appearance

of the flash and the arrival of the sound.

On this memorable occasion an observation was made which, as far as

I know, has remained a scientific enigma to the present hour. It was

noticed that while every report of the cannon fired at Montlhéry was

heard with the greatest distinctness at Villejuif, by far the greater

number of the reports from Villejuif failed to reach Montlhéry. Had

wind existed, and had it blown from Montlhéry to Villejuif, it would

have been recognized as the cause of the observed difference; but the

air at the time was calm, the slight motion of translation actually

existing being from Villejuif toward Montlhéry, or against the

direction in which the sound was best heard.

So marked was the difference in transmissive power between the two

directions, that on June 22d, while every shot fired at Montlhéry was

heard _à merveille_ at Villejuif, but one shot out of twelve fired at

Villejuif was heard, and that feebly, at the other station.

With the caution which characterized him on other occasions, and

which has been referred to admiringly by Faraday,[88] Arago made no

attempt to explain this anomaly. His words are: “Quant aux différences

si remarquables d’intensité que le bruit du canon a toujours

présentées suivant qu’il se propageait du nord au sud entre Villejuif

et Montlhéry, ou du sud au nord entre cette seconde station et la

première, nous ne chercherons pas aujourd’hui à l’expliquer, parce que

nous ne pourrions offrir au lecteur que des conjectures denuées de

preuves.”[89]