Chapter IX.

DEEP AND DEEPEST TONES.

Beats give us an important means of determining the limit of the deepest tones, and of accounting for certain peculiarities of the transition from the sensation of separate pulses of air to a perfectly continuous musical tone, and to this inquiry we now proceed.

The question: what is the smallest number of vibrations in a second which can produce the sensation of a musical tone? has hitherto received very contradictory replies. The estimates of different observers fluctuate between 8 (Savart) and about 30. The contradiction is explained by the existence of certain difficulties in the experiments.

In the first place it is necessary that the strength of the vibrations of the air for very deep tones should be extremely greater than for high tones, if they are to make as strong an impression on the ear. Several acousticians have occasionally started the hypothesis that, caeteris paribus, the strength of tones of different heights is directly proportional to the vis viva of the motion of the air, or, which comes to the same thing, to the amount of the mechanical work applied for producing it. But a simple experiment with the siren shews that when equal amounts of mechanical work are applied to produce deep and high tones under conditions otherwise alike, the high tones excite a very much more powerful sensation than the deep ones. Thus, if the siren is blown by a bellows, which makes its disc revolve with increasing rapidity, and if we take care to keep up a perfectly uniform motion of the bellows by raising its handle by the same amount the same number of times in a minute, so as to keep its bag equally filled, and drive the same amount of air under the same pressure through the siren in the same time, we hear at first, while the revolution is slow, a weak deep tone, which continually ascends, but at the same time gains strength at an extraordinary rate, till when the highest tones producible on my double siren (near to \(a''\), with 880 vibrations in a second) are reached, their strength is almost insupportable. In this case by far the greatest part of the uniform mechanical work is applied to the generation of sonorous motion, and only a small part can be lost by the friction of the revolving disc on its axial supports, and the air which it sets into a vortical motion at the same time; and these losses must even be greater for the more rapid rotation than for the slower, so that for the production of the high tones less mechanical work remains applicable than for the deep ones, and yet the higher tones appear to our sensation extraordinarily more powerful than the deep ones. How far upwards this increase may extend, I have as yet been unable to determine, for the rapidity of my siren cannot be further increased with the same pressure of air.

The increase of strength with height of tone is of especial consequence in the deepest part of the scale. It follows that in compound tones of great depth, the upper partial tones may be superior to the prime in strength, even though in musical tones of the same description, but of greater height, the strength of the prime greatly predominates. This is readily proved on my double siren, because by means of the beats it is easy to determine whether any partial tone which we hear is the prime, or the second or third partial tone of the compound under examination. For when the series of 12 holes are open in both windboxes, and the handle, which moves the upper windbox, is rotated once, we shall have, as already shewn, 4 beats for the primes, 8 for the second partials, and 12 for the third partials. Now we can make the disc revolve more slowly than usual, by allowing a well-oiled steel spring to rub against the edge of one disc with different degrees of pressure, and thus we can easily produce series of puffs which correspond to very deep tones, and then, turning the handle, we can count the beats. By allowing the rapidity of the revolution of the discs to increase gradually, we find that the first audible tones produced make 12 beats for each revolution of the handle, the number of puffs being from 36 to 40 in the second. For tones with from 40 to 80 puffs, each revolution of the handle gives 8 beats. In this case, then, the upper Octave of the prime is the strongest tone. It is not till we have 80 puffs in a second that we hear the four beats of the primes.

It is proved by these experiments that motions of the air, which do not take the form of pendular vibrations, can excite distinct and powerful sensations of tone, of which the pitch number is 2 or 3 times the number of the pulses of the air, and yet that the prime tone is not heard through them. Hence, when we continually descend in the scale, the strength of our sensation decreases so rapidly that the sound of the prime tone, although its vis viva is independently greater than that of the upper partials, as is shewn in higher positions of a musical tone of the same composition, is overcome and concealed by its own upper partials. Even when the action of the compound tone on the ear is much reinforced, the effect remains the same. In the experiments with the siren the uppermost plate of the bellows is violently agitated for the deep tones, and when I laid my head on it, my whole head was set into such powerful sympathetic vibration that the holes of the rotating disc, which vanish to an eye at rest, became again separately visible, through an optical action similar to that which takes place in stroboscopic discs. The row of holes in action appeared to stand still, the other rows seemed to move partly backwards and partly forwards, and yet the deepest tones were no more distinct than before. At another time I connected my ear by means of a properly introduced tube with an opening leading to the interior of the bellows. The agitation of the drumskin of the ear was so great that it produced an intolerable itching, and yet the deepest tones remained as indistinct as ever.

In order, then, to discover the limit of deepest tones, it is necessary not only to produce very violent agitations in the air but to give these the form of simple pendular vibrations. Until this last condition is fulfilled we cannot possibly say whether the deep tones we hear belong to the prime tone or to an upper partial tone of the motion of the air.^[1] Among the instruments hitherto employed the wide-stopped organ pipes are the most suitable for this purpose. Their upper partial tones are at least extremely weak, if not quite absent. Here we find that even the lower tones of the 16-foot octave, \(C_{\prime}\), to \(E_{\prime}\), begin to pass over into a droning noise, so that it becomes difficult for even a practised musical ear to assign their pitch with certainty; and, indeed, they cannot be tuned by the ear alone, but only indirectly by means of the beats which they make with the tones of the upper octaves. We observe a similar effect on the same deep tones of a piano or harmonium; they form drones and seem out of tune, although their musical character is on the whole better established than in the pipes, because of their accompanying upper partial tones. In music, as artistically applied in an orchestra, the deepest tone used is, therefore, the \(E_{\prime}\) of the [4-stringed German] double bass, with \(41\tfrac14\) vibrations in a second [see p. 18c, note], and I think I may predict with certainty that all efforts of modem art applied to produce good musical tones of a lower pitch must fail, not because proper means of agitating the air cannot be discovered, but because the human ear cannot hear them. The 16-foot \(C\), of the organ, with 33 vibrations in a second, certainly gives a tolerably continuous sensation of drone, but does not allow us to give it a definite position in the musical scale. We almost begin to observe the separate pulses of air, notwithstanding the regular form of the motion. In the upper half of the 32-foot octave, the perception of the separate pulses becomes still clearer, and the continuous part of the sensation, which may be compared with a sensation of tone, continually weaker, and in the lower half of the 32-foot octave we can scarcely be said to hear anything but the individual pulses, or if anything else is really heard, it can only be weak upper partial tones, from which the musical tones of stopped pipes are not quite free.

I have tried to produce deep simple tones in another way. Strings which are weighted in their middle with a heavy piece of metal, on being struck give a compound tone consisting of many simple tones which are mutually inharmonic. The prime tone is separated from the nearest upper partials by an interval of several Octaves, and hence there is no danger of confusing it with any of them; besides, the upper tones die away rapidly, but the deeper ones continue for a very long time. A string of this kind^[2] was stretched on a sounding-box having a single opening which could be connected with the auditory passage, so that the air of the sounding box could escape nowhere else but into the ear. The tones of a string of customary pitch are under these circumstances insupportably loud. But for \(D_{\prime}\), with \(37\tfrac18\) vibrations in a second, there was only a very weak sensation of tone, and even this was rather jarring, leading to the conclusion that the ear began even here to feel the separate pulses separately, notwithstanding their regularity. At \(B_{\prime\prime}\flat\), with \(29\tfrac13\) vibrations in a second, there was scarcely anything audible left. It appears, then, that those nerve fibres which perceive such tones begin as early as at this note to be no longer excited with a uniform degree of strength during the whole time of a vibration, whether it be the phases of greatest velocity or the phases of greatest deviation from their mean position in the vibrating formations in the ear which effect the excitement.^[3]

Hence although tones of 24 to 28 vib. have been heard, notes do not begin to have a definite pitch till about 40 vibrations are performed in a second. These facts will agree with the hypothesis concerning the elastic appendages to the auditory nerves, on remembering that the deeply intoned fibres of Corti may be set in sympathetic vibration by still deeper tones, although with rapidly decreasing strength, so that sensation of tone, but no discrimination of pitch, is possible. If the most deeply intoned of Corti’s fibres lie at greater intervals from each other in the scale, but at the same time their damping power is so great that every tone which corresponds to the pitch of a fibre, also pretty strongly affects the neighbouring fibres, there will be no safe distinction of pitch in their vicinity, but it will proceed continuously without jumps, while the intensity of the sensation must at the same time become small.

Whilst simple tones in the upper half of the 16-foot octave are perfectly continuous and musical, yet for aerial vibrations of a different form, for example when compound tones are used, discontinuous pulses of sound are still heard even within this octave. For example, blow the disc of the siren with gradually increasing speed. At first only pulses of air are heard; but after reaching 36 vibrations in a second, weak tones sound with them, which, however, are at first upper partials. As the velocity increases the sensation of the tones becomes continually stronger, but it is a long time before we cease to perceive the discontinuous pulses of air, although these tend more and more to coalesce. It is not till we reach 110 or \(117\tfrac13\) vibrations in a second (\(A\) or \(B\flat\) of the great octave) that the tone is tolerably continuous. It is just the same on the harmonium, where, in the cor anglais stop, \(c\) with 132 vibrations in a second still jars a little, and in the bassoon stop we observe the same jarring even in \(c'\) with 264 vibrations in a second. Generally the same observation can be made on all cutting, snarling, or braying tones, which, as has been already mentioned, are always provided with a very great number of distinct upper partial tones.

The cause of this phenomenon must be looked for in the beats produced by the high upper partials of such compound tones, which are too nearly of the same pitch. If the 15th and 16th partials of a compound tone are still audible, they form the interval of a Semitone, and naturally produce the cutting beats of this dissonance. That it is really the beats of these tones which cause the roughness of the whole compound tone, can be easily felt by using a proper resonator. If \(G_{\prime}\) is struck, having \(49\tfrac12\) vibrations in a second, the 15th partial is \(f''\sharp\), the 16th \(g''\), and the 17th \(g''\sharp\) [nearly], &c. Now when I apply the resonator \(g''\), which reinforces \(g''\) most, and \(f''\sharp\), \(g''\sharp\) somewhat less, the roughness of the tone becomes extremely more prominent, and exactly resembles the piercing jar produced when \(f''\sharp\) and \(g''\) are themselves sounded. This experiment succeeds on the pianoforte, as well as on both stops of the harmonium. It also distinctly succeeds for higher pitches, as far as the resonators reach. I possess a resonator for \(g'''\), and although it only slightly reinforces the tone, the roughness of \(G\), with 99 vibrations in a second, was distinctly increased when the \(f'''\sharp\) and \(g'''\) resonator was applied.^[4]

Even the 8th and 9th partials of a compound tone, which are a whole Tone apart, cannot but produce beats, although they are not so cutting as those from the higher upper partials. But the reinforcement by resonators does not now succeed so well, because the deeper resonators at least are not capable of simultaneously reinforcing the tones which differ from each other by a whole Tone. For the higher resonators, where the reinforcement is slighter, the interval between the tones capable of being reinforced is greater, and thus by means of the resonators \(g''\) and \(g'''\) I succeeded in increasing the roughness of the tones \(G\) to \(g\) (having 99 and 198 vibrations in a second respectively), which is due to the 7th, 8th and 9th partial tones (\(f''\), \(g''\), \(a''\), and \(f'''\), \(g'''\), \(a'''\) respectively). On comparing the tone of \(G\) as heard in the resonators with the tone of the dissonances \( f''\) \(g'' \) and \( g''\) \(a'' \) as struck directly, the ear felt their close resemblance, the rapidity of intermittence being nearly the same.

Hence there can no longer be any doubt that motions of the air corresponding to deep musical tones compounded of numerous partials, are capable of exciting at one and the same time a continuous sensation of deep tones and a discontinuous sensation of high tones, and become rough or jarring through the latter.^[5] Herein lies the explanation of the fact already observed in examining qualities of tone, that compound tones with many high upper partials are cutting, jarring, or braying; and also of the fact that they are more penetrating and cannot readily pass unobserved, for an intermittent impression excites our nervous apparatus much more powerfully than a continuous one, and continually forces itself afresh on our perception.^[6] On the other hand simple tones, or compound tones which have only a few of the lower upper partials, lying at wide intervals apart, must produce perfectly continuous sensations in the ear, and make a soft and gentle impression, without much energy, even when they are in reality relatively strong.

We have not yet been able to determine the upper limit of the number of intermittences perceptible in a second for high notes, and have only drawn attention to their becoming more difficult to perceive, and making a slighter impression, as they became more numerous. Hence even when the form of vibration, that is the quality of tone, remains the same, while the pitch is increased, the quality of tone will generally appear to diminish in roughness. The part of the scale adjacent to \(f'''\sharp\),

for which the ear is peculiarly sensitive, as I have already remarked (p. 116a), must be particularly important, as dissonant upper partials which lie in this neighbourhood cannot but be especially prominent. Now \(f''''\sharp\) is the 8th partial of \(f'\sharp\) with \(366\tfrac23\) vibrations in a second, a tone belonging to the upper tones of a man’s and the lower tones of a woman’s voice, and it is the 16th partial of the unaccented \(f\sharp\), which lies in the middle of the usual compass of men's voices.^[7] I have already mentioned that when human voices are strained these high notes are often heard sounding with them. When this takes place in the deeper tones of men’s voices, it must produce cutting dissonances, and in fact, as I have already observed, when a powerful bass voice is trumpeting out its notes in full strength, the high upper partial tones in the four-times-accented octave are heard, in quivering tinkles (p. 116c). Hence jarring and braying are much more usual and more powerful in bass than in higher voices. For compound tones above \(f'\sharp\), the dissonances of the higher upper partials in the four-times-accented octave, are not so strong as those of a whole Tone, and as they occur at so great a height they can scarcely be distinct enough to be clearly sensible.

In this way we can explain why high voices have in general a pleasanter tone, and why all singers, male and female, consequently strive to touch high notes. Moreover in the upper parts of the scale slight errors of intonation produce many more beats than in the lower, so that the musical feeling for pitch, correctness, and beauty of intervals is much surer for high than low notes.

According to the observations of Prof. W. Preyer the difference in the qualities of tone of tuning-forks and reeds entirely disappears when they reach a height of \(c^{\backprime}{'}\) 4224, doubtless for the reason he assigns, namely that the upper partials of the reeds fall in the seventh and eighth accented octave, which are scarcely audible.