Chapter IX.


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.

Beating in Slightly Different Keyboards

The top keyboard is a normal keyboard. The bottom keyboard plays every note at a frequency 4 Hertz higher than the top keyboard. Observe the beating created when playing the keyboards in unison. Adjust the shape of the wave played on the keyboards with the buttons below.

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.

[1]Thus Savart’s instrument, where a rotating rod strikes through a narrow slit, is totally unsuitable for making the lowest tone audible. The separate puffs of air are here very short in relation to the whole periodic time of the vibration, and consequently the upper partial tones must be very strongly developed, and the deepest tones, which are heard for 8 to 16 passages of the rod through the hole in a second, can be nothing but upper partials.
[2]It was a thin brass pianoforte string. The weight was a copper kreutzer piece, [pronounce kroitser; three kreutzers make a penny at Heidelberg, where the experiment was probably tried], pierced in the middle by a hole through which the wire passed, and then made to grip the wire immovably by driving a steel point between the hole in the kreutzer and the string.

Subsequently I obtained two large tuning-forks from Herr Koenig in Paris, with sliding weights on their prongs. By altering the position of the weights, the pitch was changed, and the corresponding number of vibrations was given on a scale which runs along the prongs. One fork gave 24 to 36, the other 35 to 61 vibrations. The sliding weight is a plate, 5 centimetres [nearly 2 inches] in diameter, and forms a mirror. On bringing the ear close to these plates the deep tones are well heard. For 30 vibrations I could still hear a weak drone, for 28 scarcely a trace, although this arrangement made it easily possible to form oscillations of 9 millimetres [about \(\tfrac13\) inch] in amplitude, quite close to the ear. Prof. W. Preyer has been thus able to hear down to 24 vib. He has also applied another method (Physiologische Abhandlungen, Physiological Treatises, Series 1, part 1, 'On the limits of the perception of tone,' pp. 1-17) by using very deep, loaded tongues, in reed pipes, which were constructed for this purpose by Herr Appunn of Hanau, and gave from 8 to 40 vib. These were set into strong vibration by blowing, and then on interrupting the wind, the dying off of the vibrations was listened to by laying the ear against the box. He states that tones were heard downwards as low as 15 vib. But the proof that the tones heard corresponded with the primes of the pipes depends only on the fact, that the pitch gradually ascended as they passed over into the tones of from 25 to 32 vib., which were more audible, but died off more rapidly. With extensive vibrations, however, the tongues may have very easily given their point of attachment longitudinal impulses of double the frequency, because when they reached each extremity of their amplitude they might drive back the point of attachment through their flexion, whereas in the middle of the vibration they would draw it forward by the centrifugal force of their weight. Since the power of distinguishing pitch for these deepest tones is extremely imperfect, I do not feel my doubts removed by the judgment of the ear when the estimates are not checked by the counting of beats.

[This check I am fortunately able to supply. A copy of the instrument used by Prof. Preyer is in the South Kensington Museum. It consists of an oblong box, in the lower part of which are the loaded harmonium reeds, not attached to pipes, but vibrating within the box, and governed by valves which can be opened at pleasure. On account of the beats between tongue and tongue taking place in strongly condensed air, they are accelerated, and the nominal pitch, obtained by counting the beats from reed to reed, is not quite the same as the actual pitch (see App. XX. sect. B. No. 6). The series of tones is supposed to proceed from 8 to 32 vib. by differences of 1 vib., from 32 to 64 by differences of 2 vib., and from 64 to 128 by differences of 4 vibs. In November, 1879, for another purpose, I determined the pitch of every reed by Scheibler’s forks (see App. XX. sect. B. No. 7), by means of the upper partials of the reeds. For Reeds 8, 9, 10, 11, I used from the 20th to the 30th partial, but I consider only Reed 11 as quite certain. I found it made 10.97 vib. by the 20th, and 10.95 by both the 21st and 24th partials. From Reed 11 upwards I determined every pitch, in many cases by several partials, the result only differing in the second place of decimals. I give the two lowest Octaves, the only pitches of interest for the present purpose, premising that I consider the three lowest pitches (for which the upper partials lay too close together) and the highest (which had a bad reed) to be very uncertain,

Nominal 8 9 10 11 12 13 14 15 16
Actual 7.91 8.89 9.81 10.95 11.90 12.90 13.93 14.91 15.91
Nominal 17 18 19 20 21 22 23 24 25
Actual 16.90 17.91 18.89 19.91 20.91 21.91 22.88 23.97 24.92
Nominal 26 27 28 29 30 31 32    
Actual 25.92 26.86 27.85 28.84 29.77 30.68 31.47    

There can therefore be no question as to the real pitch. At Prof. Preyer’s request I examined this instrument in Oct. 1877, and he has printed my notes in his Akustische Untersuchungen, pp. 6-8. From these I extract the following : —

R means Reed, and R 21\(\cdot\cdot\)25 means that the two reeds 21 and 25 were sounded together and gave beets.

R 21\(\cdot\cdot\)25, beat 4 in 1 sec., counted for 20 sec. Hence both of their lowest partials must have been effective.

R 20\(\cdot\cdot\)24, beat 4 in 1 sec., counted for 10 sec.

R 19\(\cdot\cdot\)23, beat 4 in 1 sec., counted for 20 sec.

R 17\(\cdot\cdot\)21, same beats.

R 16\(\cdot\cdot\)20, same beats quite distinctly.

R 15\(\cdot\cdot\)19, at first I lost the beats, but afterwards by getting R 15 well into action before R 19 was set on, and keeping on pumping, I got out the 4 in a second quite distinctly. Hence the lowest partial of R 15 was effective.

R 15\(\cdot\cdot\)17, here also I once heard 4 in a sec., but this must have been from the Octaves.

R 14\(\cdot\cdot\)16, I was quite unable to distinguish anything in the way of beats, but volleys like a feu de joie about a second in length, but impossible to count accurately; they may have been 2 in a sec. and I counted double. At the same time I seemed occasionally to hear a low beat, so low and gentle that I could not count it, and the great exertion of pumping the bellows full enough to keep these two low reeds in action, prevented accurate observation.

R 15 decidedly seemed flatter than R 13, so that I could have only heard the lowest partial of R 15 and the Octave of R 13.

On sounding R 14 and R 15 separately, I seemed to hear from each a very low tone, in quality more like a differential tone than anything else. This could also be heard even with R 13 and R 12, below the thumps, and even in R 11.

At R 8 I heard only the sishing of the escape of wind from the reed, 8 times in a second, as well as I could count, and I also heard beats evidently arising from the higher partials, and also 8 in a second.

At R 9 there was the same kind of sishing and equally rapid beats. But in addition I seemed to hear a faint low tone.

At R 10 there was no mistake as to the existence of such a musical tone.

At R 11 and R 12 it was still more distinct.

At R 13 the tone was very distinct and was quite a good musical tone at R 14, but the sish was still audible. Was this the lowest partial or its Octave?

R 16 gave quite an organ tone, nothing like a hum or a differential, but the sish and beats remain. I must have heard the lowest partial, and by continual pumping I was able to keep it in my ear.

R 18\(\cdot\cdot\)20 gave beats of 2 in a sec. very distinctly.

Up to R 25 the sish could be heard at the commencement, but it rapidly disappeared. It feels as if the tone were getting gradually into practice. This effect continued up to R 22, after which the sish was scarcely brought out at all. In fact long before this the sish was made only at the first moment, and was rather a bubble than a sish.

In listening to the very low beats, the beats of the lowest partials as such could not be separated from the general mass of beats, but the 4 in a sec. were quite clear from R 15\(\cdot\cdot\)19. The lowest pair in which I was distinctly able to hear the bell-like beat of the lowest partials distinct from the general crash was R 30\(\cdot\cdot\)34. But I fancied I heard it at R 28\(\cdot\cdot\)82.

Prof. Preyer also, in the same place, details his experiments with two enormous tuning-forks giving 13.7 and 18.6 vib. The former gave no musical tone at all, though the vibrations were visible for 3 min. and were distinctly separable by touch. The latter had 'an unmistakable dull tone, without droning or jarring.' He concludes: 'Less than 15 vib. in a sec. give no musical tone. At from 16 to 24, say then 20 in the sec. the series of aerial impulses begins to dissolve into a tone, assuming that there are no pauses between them. Above 24 begins the musical character of these bass tones. Herr Appunn,' adds Prof. Preyer, 'informed me that the differential tone of 27.85 vib., generated by the two forks of 111.3 and 83.45 vib., was "surprisingly beautiful" and had a "wondrous effect."' — Translator.]

[4][The student should now perform the experiments on the Harmonical indicated on p. 22d, note. — Translator.]
[5][This is particularly noticeable on Appunn's Reed pipes of 32 and 64 vib. in the South Kensington Museum. Their musical character is quite destroyed by the loud thumping of the upper partials. — Translator.]
[6][In Prof. Tyndall’s paper 'On the Atmosphere as a Vehicle of Sound,' read before the Royal Society, Feb. 12, 1874, in trying the distance at which intense sounds could be heard at sea, he says (Philosophical Transactions for 1874, vol. clxiv. p. 189), 'The influence of "beats" was tried on June 3 [1873] by throwing the horns slightly out of unison; but though the beats rendered the sound characteristic, they did not seem to augment the range.' — Translator.]
[7][On the compass of voices see App. XX. sect. N. No. 1. — Translator.]
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