In the first part of this book we had to enunciate and constantly apply the proposition that oscillatory motions of the air and other elastic bodies, produced by several sources of sound acting simultaneously, are always the exact sum of the individual motions producible by each source separately. This law is of extreme importance in the theory of sound, because it reduces the consideration of compound cases to those of simple ones. But it must be observed that this law holds strictly only in the case where the vibrations in all parts of the mass of air and of the sonorous elastic bodies are of infinitesimally small dimensions; that is to say, only when the alterations of density of the elastic bodies are so small that they may be disregarded in comparison with the whole density of the same body; and in the same way, only when the displacements of the vibrating particles vanish as compared with the dimensions of the whole elastic body. Now certainly in all practical applications of this law to sonorous bodies, the vibrations are always very small, and near enough to being infinitesimally small for this law to hold with great exactness even for the real sonorous vibrations of musical tones, and by far the greater part of their phenomena can be deduced from that law in conformity with observation. Still, however, there are certain phenomena which result from the fact that this law does not hold with perfect exactness for vibrations of elastic bodies, which, though almost always very small, are far from being infinitesimally small. One of these phenomena, with which we are here interested, is the occurrence of Combinational Tones, which were first discovered in 1745 by Sorge, a German organist, and were afterwards generally known, although their pitch was often wrongly assigned, through the Italian violinist Tartini (1754), from whom they are often called Tartini’s tones.
These tones are heard whenever two musical tones of different pitches are sounded together, loudly and continuously. The pitch of a combinational tone is generally different from that of either of the generating tones, or of their harmonic upper partials. In experiments, the combinational are readily distinguished from the upper partial tones, by not being heard when only one generating tone is sounded, and by appearing simultaneously with the second tone. Combinational tones are of two kinds. The first class, discovered by Sorge and Tartini, I have termed differential tones, because their pitch number is the difference of the pitch numbers of the generating tones. The second class of summational tones, having their pitch number equal to the sum of the pitch numbers of the generating tones, were discovered by myself.
On investigating the combinational tones of two compound musical tones, we find that both the primary and the upper partial tones may give rise to both differential and summational tones. In such cases the number of combinational tones is very great. But it must be observed that generally the differential are stronger than the summational tones, and that the stronger generating simple tones also produce the stronger combinational tones. The combinational tones, indeed, increase in a much greater ratio than the generating tones, and diminish also more rapidly. Now since in musical compound tones the prime generally predominates over the partials, the differential tones of the two primes are generally heard more loudly than all the rest, and were consequently first discovered. They are most easily heard when the two generating tones are less than an octave apart, because in that case the differential is deeper than either of the two generating tones. To hear it at first, choose two tones which can be held with great force for some time, and form a justly intoned harmonic interval. First sound the low tone and then the high one. On properly directing attention, a weaker low tone will be heard at the moment that the higher note is struck; this is the required combinational tone. For particular instruments, as the harmonium, the combinational tones can be made more audible by properly tuned resonators. In this case the tones are generated in the air contained within the instrument. But in other cases, where they are generated solely within the ear, the resonators are of little or no use.
The following table gives the first differential tones of the usual harmonic intervals:—
|Intervals||Ratio of vibrational numbers||Difference of the same||The combinational tone is deeper than the deeper generating tone by|
or in ordinary musical notation, the generating tones being written as minims and the differential tones as crotchets—
When the ear has learned to hear the combinational tones of pure intervals and sustained tones, it will be able to hear them from inharmonic intervals and in the rapidly fading notes of a pianoforte. The combinational tones from inharmonic intervals are more difficult to hear, because these intervals beat more or less strongly, as we shall have to explain hereafter. The combinational tones arising from such as fade rapidly, for example those of the pianoforte, are not strong enough to be heard except at the first instant, and die off sooner than the generating tones. Combinational tones are also in general easier to hear from the simple tones of tuning-forks and stopped organ pipes than from compound tones where a number of other secondary tones are also present. These compound tones, as has been already said, also generate a number of differential tones by their harmonic upper partials, and these easily distract attention from the differential tones of the primes. Combinational tones of this kind, arising from the upper partials, are frequently heard from the violin and harmonium.
Example. — Take the major Third \(c'\)\(e'\), ratio of pitch numbers 4 : 5. First difference 1, that is \(C\). The first harmonic upper partial of \(c'\) is \(c''\), relative pitch number 8. Ratio of this and \(e'\), 5 : 8, difference 3, that is \(g\). The first upper partial of \(e'\) is \(e''\), relative pitch number 10; ratio for this and \(c'\), 4 : 10, difference 6, that is g'. Then again \(c''\) \(e''\) have ratio 8 : 10, difference 2, that is \(c\). Hence, taking only the first upper partials we have the series of combinational tones \(1\), \(3\), 6, 2 or \(C\), \(g\), \(g'\), \(c\). Of these the tone 3, or \(g\), is often easily perceived.
These multiple combinational tones cannot in general be distinctly heard, except when the generating compound tones contain audible harmonic upper partials. Yet we cannot assert that the combinational tones are absent, where such partials are absent; but in that case they are so weak that the ear does not readily recognise them beside the loud generating tones and first differential. In the first place theory leads us to conclude that they do exist in a weak state, and in the next place the beats of impure intervals, to be discussed presently, also establish their existence. In this case we may, as Hallstroem suggests, consider the multiple combinational tones to arise thus: the first differential tone, or combinational tone of the first order, by combination with the generating tones themselves, produce other differential tones, or combinational tones of the second order; these again produce new ones with the generators and differentials of the first order, and so on.
Example. — Take two simple tones \(c'\) and \(e'\), ratio 4 : 5, difference 1, differential tone of the first order \(C\). This with the generators gives the ratios 1 : 4 and 1 : 5, differences 3 and 4, differential tones of the second order \(g\), and \(c'\) once more. The new tone 3, gives with the generators the ratios 3 : 4 and 3 : 5, differences 1 and 2, giving the differential tones of the third order \(C\) and \(c\), and the same tone 3 gives with the differential of the first order 1, the ratio 1 : 3 difference 2, and hence as a differential of the fourth order \(c\) once more and so on. The differential tones of different orders which coincide when the interval is perfect, as it is supposed to be in this example, no longer exactly coincide when the generating interval is not pure; and consequently such beats are heard, as would result from the presence of these tones. More on this hereafter.
The differential tones of different orders for different intervals are given in the following notes, where the generators are minims, the combinational tones of the first order crotchets, of the second quavers, and so on. The same tones also occur with compound generators as combinational tones of their upper partials.
The series are broken off as soon as the last order, of differentials furnishes no fresh tones. In general these examples shew that the complete series of harmonic partial tones 1, 2, 3, 4, 5, &c., up to the generators themselves, is produced.
The second kind of combinational tones, which I have distinguished as summational, is generally much weaker in sound than the first, and is only to be heard with decent ease under peculiarly favourable circumstances on the harmonium and polyphonic siren. Scarcely any but the first summational tone can be perceived, having a vibrational number equal to the sum of those of the generators. Of course summational tones may also arise from the harmonic upper partials. Since their vibrational number is always equal to the sum of the other two, they are always higher in pitch than either of the two generators. The following notes will shew their nature for the simple intervals:—
In relation to music I will here remark at once that many of these summational tones form extremely inharmonic intervals with the generators. Were they not generally so weak on most instruments, they would give rise to intolerable dissonances. In reality, the major and minor Third, and the minor Sixth, sound very badly indeed on the polyphonic siren, where all combinational tones are remarkably loud, whereas the Octave, Fifth, and major Sixth are very beautiful. Even the Fourth on this siren has only the effect of a tolerably harmonious chord of the minor Seventh.
It was formerly believed that the combinational tones were purely subjective, and were produced in the ear itself. Differential tones alone were known, and these were connected with the beats which usually result from the simultaneous sounding of two tones of nearly the same pitch, a phenomenon to be considered in the following chapters. It was believed that when these beats occurred with sufficient rapidity, the individual increments of loudness might produce the sensation of a new tone, just as numerous ordinary impulses of the air would, and that the frequency of such a tone would be equal to the frequency of the beats. But this supposition, in the first place, does not explain the origin of summational tones, being confined to the differentials; secondly, it may be proved that under certain conditions the combinational tones exist objectively, independently of the ear which would have had to gather the beats into a new tone; and thirdly, this supposition cannot be reconciled with the law confirmed by all other experiments, that the only tones which the ear hears, correspond to pendular vibrations of the air.
And in reality a different cause for the origin of combinational tones can be established, which has already been mentioned in general terms (p. 152c). Whenever the vibrations of the air or of other elastic bodies which are set in motion at the same time by two generating simple tones, are so powerful that they can no longer be considered infinitely small, mathematical theory shews that vibrations of the air must arise which have the same frequency as the combinational tones. 
Particular instruments give very powerful combinational tones. The condition for their generation is that the same mass of air should be violently agitated by two simple tones simultaneously. This takes place most powerfully in the polyphonic siren, in which the same rotating disc contains two or more series of holes which are blown upon simultaneously from the same windchest. The air of the windchest is condensed whenever the holes are closed; on the holes being opened, a large quantity of air escapes, and the pressure is considerably diminished. Consequently the air in the windchest, and partly even that in the bellows, as can be easily felt, comes into violent vibration. If two rows of holes are blown, vibrations arise in the air of the windchest corresponding to both tones, and each row of openings gives vent not to a stream of air uniformly supplied, but to a stream of air already set in vibration by the other tone. Under these circumstances the combinational tones are extremely powerful, almost as powerful, indeed, as the generators. Their objective existence in the mass of air can be proved by vibrating membranes tuned to be in unison with the combinational tones. Such membranes are set in sympathetic vibration immediately upon both generating tones being sounded simultaneously, but remain at rest if only one or the other of them is sounded. Indeed, in this case the summational tones are so powerful that they make all chords extremely unpleasant which contain Thirds or minor Sixths. Instead of membranes it is more convenient to use the resonators already recommended for investigating harmonic upper partial tones. Resonators are also unable to reinforce a tone when no pendular vibrations actually exist in the air; they have no effect on a tone which exists only in auditory sensation, and hence they can be used to discover whether a combinational tone is objectively present. They are much more sensitive than membranes, and are well adapted for the clear recognition of very weak objective tones.
The conditions in the harmonium are similar to those in the siren. Here, too, there is a common windchest, and when two keys are pressed down, we have two openings which are closed and opened rhythmically by the tongues. In this case also the air in the common receptacle is violently agitated by both tones, and air is blown through each opening which has been already set in vibration by the other tongue. Hence in this instrument also the combinational tones are objectively present, and comparatively very distinct, but they are far from being as powerful as on the siren, probably because the windchest is very much larger in proportion to the openings, and hence the air which escapes during the short opening of an exit by the oscillating tongue cannot be sufficient to diminish the pressure sensibly. In the harmonium also the combinational tones are very clearly reinforced by resonators tuned to be in unison with them, especially the first and second differential and the first summational tone. Nevertheless I have convinced myself, by particular experiments, that even in this instrument the greater part of the force of the combinational tone is generated in the ear itself. I arranged the portvents in the instrument so that one of the two generators was supplied with air by the bellows moved below by the foot, and the second generator was blown by the If reserve bellows, which was first pumped full and then cut off by drawing out the so-called expression-stop, and I then found that the combinational tones were not much weaker than for the usual arrangement. But the objective portion which the resonators reinforce was much weaker. The noted examples given above (pp. 154-5-6) will easily enable any one to find the digitals which must be pressed down in order to produce a combinational tone in unison with a given resonator.
On the other hand, when the places in which the two tones are struck are entirely separate and have no mechanical connection, as, for example, if they come from two singers, two separate wind instruments, or two violins, the reinforcement of the combinational tones by resonators is small and dubious. Here, then, there does not exist in the air any clearly sensible pendular vibration corresponding to the combinational tone, and we must conclude that such tones, which are often powerfully audible, are really produced in the ear itself. But analogously to the former cases we are justified in assuming in this case also that the external vibrating parts of the ear, the drumskin and auditory ossicles, are really set in a sufficiently powerful combined vibration to generate combinational tones, so that the vibrations which correspond to combinational tones may really exist objectively in the parts of the ear without existing objectively in the external air. A slight reinforcement of the combinational tone in this case by the corresponding resonator may, therefore, arise from the drumskin of the ear communicating to the air in the resonator those particular vibrations which correspond to the combinational tone. Now it so happens that in the construction of the external parts of the ear for conducting sound, there are certain conditions which are peculiarly favourable for the generation of combinational tones. First we have the unsymmetrical form of the drumskin itself. Its radial fibres, which are externally convex, undergo a much greater alteration of tension when they make an oscillation of moderate amplitude towards the inside, than when the oscillation takes place towards the outside. For this purpose it is only necessary that the amplitude of the oscillation should not be too small a fraction of the minute depth of the arc made by these radial fibres. Under these circumstances deviations from the simple superposition of vibrations arise for very much smaller amplitudes than is the case when the vibrating body is symmetrically constructed on both sides. 
But a more important circumstance, as it seems to me, when the tones are powerful, is the loose formation of the joint between the hammer and anvil (p. 133b). If the handle of the hammer is driven inwards by the drumskin, the anvil and stirrup must follow the motion unconditionally. But that is not the case for the subsequent outward motion of the handle of the hammer, during which the teeth of the two ossicles need not catch each other. In this case the ossicles may click. Now I seem to hear this clicking in my own ear whenever a very strong and deep tone is brought to bear upon it, even when, for example, it is produced by a tuning fork held between the fingers, in which there is certainly nothing that can make any click at all.
This peculiar feeling of mechanical tingling in the ear had long ago struck me when two clear and powerful soprano voices executed passages in Thirds, in which case the combinational tone comes out very distinctly. If the phases of the two tones are so related that after every fourth oscillation of the deeper and every fifth of the higher tone, there ensues a considerable outward displacement of the drumskin, sufficient to cause a momentary loosening in the joint between the hammer and anvil, a series of blows will be generated between the two bones, which would be absent if the connection were firm and the oscillation regular, and these blows taken together would exactly generate the first differential tone of the interval of a major Third. Similarly for other intervals.
It must also be remarked that the same peculiarities in the construction of a sonorous body which makes it suitable for allowing combinational tones to be heard when it is excited by two waves of different pitch, must also cause a single simple tone to excite in it vibrations corresponding to its harmonic upper partials; the effect being the same as if this tone then formed summational tones with itself.
This result ensues because a simple periodical force, corresponding to pendular vibrations, cannot excite similar pendular vibrations in the elastic body on which it acts, unless the elastic forces called into action by the displacements of the excited body from its position of equilibrium, are proportional to these displacements themselves. This is always the case so long as these displacements are infinitesimal. But if the amplitude of the oscillations is great enough to cause a sensible deviation from this proportionality, then the vibrations of the exciting tone are increased by others which correspond to its harmonic upper partial tones. That such harmonic upper partials are occasionally heard when tuning-forks are strongly excited, has been already mentioned (p. 54d). I have lately repeated these experiments with forks of a very low pitch. With such a fork of 64 vib. I could, by means of proper resonators, hear up to the fifth partial. But then the amplitude of the vibrations was almost a centimetre [.3937 inch]. When a sharp-edged body, such as the prong of a tuning-fork, makes vibrations of such a length, vortical motions, differing sensibly from the law of simple vibrations, must arise in the surrounding air. On the other hand, as the sound of the fork fades, these upper partials vanish long before their prime, which is itself only very weakly audible. This agrees with our hypothesis that these partials arise from disturbances depending on the size of the amplitude.
Herr R Koenig, with a series of forks having sliding weights by which the pitch might be gradually altered, and provided also with boxes giving a good resonance and possessing powerful tones, has investigated beats and combinational tones, and found that those combinational tones were most prominent which answered to the difference of one of the tones from the partial tone of the other which was nearest to it in pitch; and in this research partial tones as high as the eighth were effective (at least in the number of beats), He has unfortunately not stated how far the corresponding upper partials were separately recognised by resonators.
Since the human ear easily produces combinational tones, for which the principal causes lying in the construction of that organ have just been assigned, it must also form upper partials for powerful simple tones, as is the case for tuning-forks and the masses of air which they excite in the observations described. Hence we cannot easily have the sensation of a powerful simple tone, without having also the sensation of its harmonic upper partials. 
The importance of combinational tones in the construction of chords will appear hereafter. We have, however, first to investigate a second phenomenon of the simultaneous sounding of two tones, the so-called beats.
Octave, notes 4 : 8. Diff. 8-4 = 4.
Fifth, notes 4 : 6. Diff. 6-4 = 2.
2nd order, 4-2 = 2, 6-2 = 4.
Fourth, notes 6 : 8. Diff. 8-6 = 2.
2nd order, 8-2 = 6, 6-2 = 4.
3rd order, 6-4 = 2, 6-2 = 4.
Major Third, notes 4 : 5. Diff. 5-4=1.
2nd. 4-1 = 3, 5-1 = 4.
3rd. 4-3 = l, 5-3 = 2.
4th. 4-2 = 2, 4-1 = 8.
Minor Third, notes 5 : 6. Diff. 6-5=1.
2nd. 5-1 = 4, 6-1 = 5.
3rd. 5-4 = 1, 6-4 = 2.
4th. 4-1 = 3, 6-2 = 4.
5th. 6-4 = 2, 6-8 = 3.
Major Sixth, notes 6 : 10. Diff. 10-6=4.
2nd. 10-4 = 6, 6-4 = 2.
3rd. 10-2 = 8, 6-2 = 4.
4th. 6-4 = 2.
Minor Sixth, notes 5 : 8. Diff. 8-5 = 3.
2nd. 5-3 = 2, 8-3 = 5.
3rd. 5-2 = 3, 8-2 = 6.
4th. 3-2 = 1, 5-3 = 2.
5th. 5-1 = 4, 8-1 = 7.
6th. 8-7 = 5-4=l, 4-2 = 2, 8-4 = 4.