Chapter XI.

BEATS DUE TO COMBINATIONAL TONES.

When two or more compound tones are sounded at the same time beats may arise from the combinational tones as well as from the harmonic upper partials. In Chapter VII. it was shewn that the loudest combinational tone resulting from two generating tones is that corresponding to the difference of their pitch numbers, or the differential tone of the first order. It is this combinational tone, therefore, which is chiefly effective in producing beats. Even this loudest combinational tone is somewhat weak, unless the generators are very loud; the differential tones of higher orders, and the summational tones, are still weaker. Beats due to such weak tones as those last mentioned cannot be observed unless all other beats which would disturb the observer are absent, as, for instance, in sounding two simple tones, which are entirely free from upper partials. On the other hand the beats of the first differential tones [owing to difference of pitch and quality] can be heard very well at the same time as those due to the harmonic upper partials of compound tones, by an ear accustomed to hear combinational tones.

The differential tones of the first order alone, and independently of the combinational tones of higher orders, are capable of causing beats (1) when two compound tones sound together, (2) when three or more simple or compound tones sound together. On the other hand beats generated by combinational tones of higher orders have to be considered when two simple tones sound together.

We commence with the differential tones of compound tones. In the same way that the prime tones in such cases develop combinational tones, any pair of upper partials of the two compounds will also develop combinational tones, but such tones will diminish very rapidly in intensity as the upper partials become weaker. When one or more of these combinational tones nearly coincide with other combinational tones, or the primes or upper partials of the generators, beats ensue. Let us take as an example a slightly incorrectly tuned Fifth, having the pitch numbers 200 and 301, in place of 200 and 300, as in a justly intoned Fifth. We calculate the vibrational numbers of the upper partials by multiplying those of the primes by 1, 2, 3, and so on. We find the vibrational numbers of the differential tones of the first order, by subtracting these numbers from each other, two and two. The following table contains in the first horizontal line and vertical column the vibrational numbers of the several partials of the two compound tones, and in their intersections the differences of those numbers, which are the pitch numbers of the differential tones due to them.

If we arrange these tones by pitch we find the following groups : —

The number 2 is too small to correspond to a combinational tone. It only shews the number of beats due to the two upper partials 600 and 602.^[1] In all the other groups, however, tones are found whose vibrational numbers differ by 2, 4, or 6, and hence produce respectively 2, 4, and 6 beats in the same time that the two first-named partials produce 2 beats. The two strongest combinational tones are 101 and 99, and these also are well distinguished from the rest by their low pitch.

We observe in this example that the slowest beats due to the combinational tones are the same in number as those due to the upper partials [600 and 602]. This is a general rule and applies to all intervals.^[2]

Further it is easy to see that if in our example we replaced 200 and 301, by the numbers 200 and 300 belonging to the perfect Fifth, all the numbers in our table would become multiples of 100, and hence all the different combinational and upper partial tones which now beat would become coincident and not generate any beats. What is here shewn to be the case in this example for the Fifth is also true for all other harmonic intervals.^[3]

The first differential tones of compounds cannot generate beats, except when the upper partials of the same compounds generate them, and the rapidity of the beats is the same in both cases, supposing that the series of upper partials is complete. Hence the addition of combinational tones makes no essential difference in the results obtained in the last chapter on investigating the beats due to the upper partials only. There can be only a slight increase in the strength of the beats.^[4]

But the case is essentially different when two simple tones are sounded together, so that there are no upper partials to consider. If combinational tones were not taken into account, two simple tones, as those of tuning-forks or stopped organ pipes, could not produce beats unless they were very nearly of the same pitch, and such beats are strong when their interval is a minor or major Second, but weak for a Third and then only recognisable in the lower parts of the scale (p. 171d), and they gradually diminish in distinctness as the interval increases, without shewing any special differences for the harmonic intervals themselves. For any larger interval between two simple tones there would be absolutely no beats at all, if there were no upper partial or combinational tones, and hence the consonant intervals discovered in the former chapter would be in no respect distinguished from adjacent intervals; there would in fact be no distinction at all between wide consonant intervals and absolutely dissonant intervals.

Now such wider intervals between simple tones are known to produce beats, although very much weaker than those hitherto considered, so that even for such tones there is a difference between consonances and dissonances, although it is very much more imperfect than for compound tones. And these facts depend, as Scheibler shewed,^[5] on the combinational tones of higher orders.

It is only for the Octave that the first differential tone suffices. If the lower note makes 100 vibrations in a second, while the imperfect Octave makes 201, the first differential tone makes 201 - 100 = 101, and hence nearly coincides with the lower note of 100 vibrations, producing one beat for each 100 vibrations. There is no difficulty in hearing these beats, and hence it is easily possible to distinguish imperfect Octaves from perfect ones, even for simple tones, by the beats produced by the former.^[6]

For the Fifth, the first order of differential tones no longer suffices. Take an imperfect Fifth with the ratio 200 : 301; then the differential tone of the first order is 101, which is too far from either primary to generate beats. But it forms an imperfect Octave with the tone 200, and, as just seen, in such a case beats ensue. Here they are produced by the differential tone 99 arising from the tone 101 and the tone 200, and this tone 99 makes two beats in a second with the tone 101. These beats then serve to distinguish the imperfect from the justly intoned Fifth even in the case of two simple tones. The number of these beats is also exactly as many as if they were beats due to the upper partial tones.^[7] But to observe these beats the two primary tones must be loud, and the ear must not be distracted by any extraneous noise. Under favourable circumstances, however, they are not difficult to hear.^[8]

For an imperfect Fourth, having, say, the vibrational numbers 300 and 401, the first differential tone is 101; this with the tone 300 produces the differential tone 199 of the second order, and this again with the tone 401 the differential tone 202 of the third order, and this makes 3 beats with the differential tone 199 of the second order, that is, precisely as many beats as would have been generated by the upper partial tones 1200 and 1203, if they had existed. These beats of the Fourth are very weak even when the primary tones are powerful. Perfect quiet and great attention are necessary for observing them.^[9]And after all there may be a doubt whether by strong excitement of the primary tones, weak partials may not have arisen, as we already considered on p. 159b, c.^[10]

The beats of an imperfect major Third are scarcely recognisable, even under the most favourable conditions. If we take as the vibrational numbers of the primary tones 400 and 501, we have : —

The tones 202 and 198 produce 4 beats. Scheibler succeeded in counting these beats of the imperfect major Third.^[11] I have myself believed that I heard them under favourable circumstances. But in any case they are so difficult to perceive that they are not of any importance in distinguishing consonance from dissonance. Hence it follows that two simple tones making various intervals adjacent to the major Third and sounded together would produce a uniform uninterrupted mass of sound, without any break in their harmoniousness, provided that they do not approach a Second too closely on the one hand or a Fourth on the other. My own experiments with stopped organ pipes justify me in asserting that however much this conclusion is opposed to musical dogmas, it is borne out by the fact, provided that really simple tones are used for the purpose.^[12] It is the same with intervals near to the major Sixth; these also shew no difference as long as they remain sufficiently far from the Fifth and Octave. Hence although it is not difficult to tune perfect major and minor Thirds on the harmonium or reed pipes or on the violin, by sounding the two tones together and trying to get rid of the beats, it is perfectly impossible to do so on stopped organ pipes or tuning-forks without the aid of other intervals. It will appear hereafter that the use of more than two tones will allow these intervals to be perfectly tuned even for simple tones.

Intermediate between the compound tones possessing many powerful upper partials, such as those of reed pipes and violins, and the entirely simple tones of tuning-forks and stopped organ pipes, lie those compound tones in which only the lowest of the upper partials are audible, such as the tones of wide open organ pipes or the human voice when singing some of the obscurer vowels, as oo in too. For these, the partials would not suffice to distinguish all the consonant intervals, but the addition of the first differential tones renders it possible.

A. Compound Tones consisting of the prime and its Octave. These cannot delimit Fifths and Fourths by beats of the partials, but are able to do so by those of the first differential tones.

a. Fifth. Let the vibrational numbers of the prime tones be 200 and 301, which are accompanied by their Octaves 400 and 602; all four tones are then too far apart to beat. But the differential tones

give two beats. The number of these beats again is precisely the same as if they had been produced by the two next upper partials.^[13]

b. Fourth. Let the vibrational numbers of the primes be 300 and 401, and of the first upper partials 600 and 802; these cannot produce any beats. But the first differential tones give 3 beats, thus^[14]: —

For Thirds it would be necessary to take differential tones of the second order into account.

B. Compound Tones consisting of the prime and Twelfth. Such tones are produced by the narrow stopped pipes on the organ (Quintaten, p. 33d, note). These are related in the same way as those which have only the Octave.

a. Fifth. Primes 200 and 301, upper partials 600 and 903. First differential tone

b. Fourth. Primes 300 and 401, upper partials 900 and 1203. First differential tone

Even in this case the beats of the Third cannot be perceived without the help of the weak second differential tones.

C. Compound Tones having both Octave and Twelfth as audible partials. Such tones are produced by the wide (wooden) open pipes of the organ (Principal, p. 93d', note). The beats of the upper partials here suffice to delimit the Fifths, but not the Fourths. The Thirds can now be distinguished by means of the first differential tones.

a. Major Third. Primes 400 and 501, with the Octaves 800 and 1002, and Twelfths 1200 and 1503. First differential tones^[15]

b. Minor Third. Primes 500 and 601, Octaves 1000 1202, Twelfths 1500 and 1803. Differential tones^[16]

c. Major Sixth. Primes 300 and 501, Octaves 600 and 1002, Twelfths 900 and 1503. Differential tones

In fact not only the beats of imperfect Fifths and Fourths, but also those of imperfect major and minor Thirds are easily heard on open organ pipes, and can be immediately used for the purposes of tuning.

Thus, where upper partials, owing to the quality of tone, do not suffice, the combinational tones step in to make every imperfection in the consonant intervals of the Octave, Fifth, Fourth, major Sixth, major and minor Third immediately sensible by means of beats and roughness in the combined sound, and thus to distinguish these intervals from all those adjacent to them. It is only perfectly simple tones that so far make default in determining the Thirds; and for them also the beats which disturb the harmoniousness of imperfect Fifths and Fourths, are relatively too weak to affect the ear sensibly, because they depend on differential tones of higher orders. In reality, as I have already mentioned, two stopped pipes, giving tones which lie between a major and a minor Third apart, will give just as good a consonance as if the interval were exactly either a major or a minor Third. This does not mean that a practised musical ear would not find such an interval strange and unusual, and hence would perhaps call it false, but the immediate impression on the ear, the simple perception of harmoniousness, considered independently of any musical habits, is in no respect worse than for one of the perfect intervals.^[17]

Matters are very different when more than two simple tones are sounded together. We have seen that Octaves are precisely limited even for simple tones by the beats of the first differential tone with the lower primary. Now suppose that an Octave has been tuned perfectly, and that then a third tone is interposed to act as a Fifth. Then if the Fifth is not perfect, beats will ensue from the first differential tone.

Let the tones forming the perfect Octave have the pitch numbers 200 and 400, and let that of the imperfect Fifth be 301. The differential tones are

These beats of the Fifth which lies between two Octaves are much more audible than those of the Fifth alone without its Octave. The latter depend on the weak differential tones of the second order, the former on those of the first order. Hence Scheibler some time ago laid down the rule for tuning tuning-forks, first to tune two of them as a perfect Octave, and then to sound them both at once with the Fifth, in order to tune the latter.^[18] If Fifth and Octave are both perfect, they also give together the perfect Fourth.

The case is similar, when two simple tones have been tuned to be a perfect Fifth, and we interpose a new tone between them to act as a major Third. Let the perfect Fifth have the pitch numbers 400 and 600. On intercalating the impure major Third with the pitch number 501 in lieu of 500, the differential tones are

The major Sixth is determined by combining it with the Fourth. Let 300 and 400 be the vibrational numbers of a perfect Fourth, and 501 that of an imperfect major Sixth. The differential tones are

If we tried to intercalate an interval between the tones forming a perfect Fourth, and having the vibrational numbers 300 and 400, it could only be the subminor Third with the vibrational number 350. Taking it imperfect and = 351, we have the differential tones

These intervals 8 : 7 and 7 : 6 are, however, too close to be consonances, and hence they can only be used in weak discords (chord of the dominant Seventh).^[20] The above considerations are also applicable to any single compound tone consisting of several partials. Any two partials of sufficient force will also produce differential tones in the ear. If, then, the partials correspond exactly to the series of harmonic partials, as assigned by the series of smaller whole numbers, all these differentials resulting from partials coincide exactly with the partials themselves, and give no beats. Thus if the prime makes n vibrations in a second, the upper partials make 2n, 3n, 4n, &c., vibrations, and the differences of these numbers are again n, or 2n, or 3n, &c. The pitch numbers of the summational tones fall also into this series.

On the other hand, if the pitch numbers of the upper partials are ever so slightly different from those giving these ratios, then the combinational tones will differ from one another and from the upper partials, and the result will be beats. The tone therefore ceases to make that uniform and quiet impression which a compound tone with harmonic upper partials always makes on the ear. How considerable this influence is, we may hear from any firmly attached harmonious string after we have fastened a small piece of wax on any part of its length. This, as theory and experiment alike shew, produces an inharmonic relation of the upper partials. If the piece of wax is very small, then the alteration of tone is also very small. But the slightest mistuning suffices to do considerable harm to the tunefulness of the sound, and renders the tone dull and rough, like a tin kettle.

Herein we find the reason why tones with harmonic upper partials play such a leading part in the sensation of the ear. They are the only sounds which, even when very intense, can produce sensations that continue in undisturbed repose, without beats, corresponding to the purely periodic motion of the air, which is the objective foundation of these tones. I have already stated as a result of the summary which I give of the composition of musical tones in Chapter V., No. 2, p. 119a, that besides tones with harmonic upper partials, the only others used (and that also generally in a very subordinate manner) are either such as have a section of the series of harmonic upper partials (like those of well tuned bells), or such as have secondary tones (as those in bars) so very weak and so far distant from their primes, that their differentials have but little force and at any rate do not produce any distinct beats.

Collecting the results of our investigations upon beats, we find that when two or more simple tones are sounded at the same time, they cannot go on sounding without mutual disturbance, unless they form with each other certain perfectly definite intervals. Such an undisturbed flow of simultaneous tones is called a consonance. When these intervals do not exist, beats arise, that is, the whole compound tones, or individual partial and combinational tones contained in them or resulting from them, alternately reinforce and enfeeble each other. The tones then do not coexist undisturbed in the ear. They mutually check each other’s uniform flow. This process is called dissonance.^[21]

Combinational tones are the most general cause of beats. They are the sole cause of beats for simple tones which lie as much as, or more than, a minor Third apart.^[22] For two simple tones they suffice to delimit the Fifth, perhaps the Fourth, but certainly not the Thirds and Sixths. These, however, will be strictly delimited when the major Third is added to the Fifth to form the common major chord, and when the Sixth is united with the Fourth to form the chord of the Sixth and Fourth, \(^6_4\).

Thirds, however, are strictly delimited, by means of the beats of imperfect intervals, in a chord of two compound tones, each consisting of a prime and the two next partial tones. The beats of such intervals increase in strength and distinctness, with the increase in number and strength of the upper partial tones in the compounds. By this means the difference between dissonance and consonance, and of perfectly from imperfectly tuned intervals, becomes continually more marked and distinct, increasing the certainty with which the hearer distinguishes the correct intervals, and adding much to the powerful and artistic effect of successions of chords. Finally when the high upper partials are relatively too strong (in piercing and braying qualities of tone), each separate tone will by itself generate intermittent sensations of tone, and any combination of two or more compounds of this description produces a sensible increase of this harshness, while at the same time the large number of partial and combinational tones renders it difficult for the hearer to follow a complicated arrangement of parts in a musical composition.

These relations are of the utmost importance for the use of different instruments in the different kinds of musical composition. The considerations which determine the selection of the proper instrument for an entire composition or for individual phrases in movements written for an orchestra are very multifarious. First in rank stands mobility and power of tone in the different instruments. On this there is no need to dwell. The bowed instruments and pianoforte surpass all others in mobility, and then follow the flutes and oboes. To these are opposed the trumpets and trombones, which commence sluggishly, but surpass all instruments in power. Another essential consideration is expressiveness, which in general depends on the power of producing with certainty any degree of rapid alterations in loudness at the pleasure of the player. In this respect also bowed instruments, and the human voice, are pre-eminent. Artificial reed instruments, both of wood and brass, cannot materially diminish their power without stopping the action of the reeds. Flutes and organ pipes cannot greatly alter the force of their tone without at the same time altering their pitch. On the pianoforte the strength with which a tone commences is determined by the player, but not its duration; so that the rhythm can be marked delicately, but real melodic expression is wanting. All these points in the use of the above instruments are easy to observe and have long been known and allowed for. The influence of quality proper was more difficult to define. Our investigations, however, on the composition of musical tones have given us a means of taking into account the principal differences in the effect of the simultaneous action of different instruments and of shewing how the problem is to be solved, although there is still a large field left for a searching investigation in detail.

Let us begin with the simple tones of wide stopped organ pipes. In themselves they are very soft and mild, dull in the low notes, and very tuneful in the upper. They are quite unsuited, however, for combinations of harmony according to modem musical theory. We have already explained that simple tones of this kind discriminate only the very small interval of a Second by strong beats. Imperfect Octaves, and the dissonant intervals in the neighbourhood of the Octave (the Sevenths and Ninths), beat with the combinational tones, but these beats are weak in comparison with those due to upper partials. The beats of imperfect Fifths and Fourths are entirely inaudible except under the most favourable conditions. Hence in general the impression made on the ear by any dissonant interval, except the Second, differs very little from that made by consonances, and as a consequence the harmony loses its character and the hearer has no certainty in his perception of the difference of intervals.^[23] If polyphonic compositions containing the harshest and most venturesome dissonances are played upon wide stopped organ pipes, the whole is uniformly soft and harmonious, and for that very reason also indefinite, wearisome and weak, without character or energy. Every reader that has an opportunity is requested to try this experiment. There is no better proof of the important part which upper partial tones play in music, than the impression produced by music composed of simple tones, such as we have just described. Hence the wide stopped pipes of the organ are used only to give prominence to the extreme softness and tunefulness of certain phrases in contradistinction to the harsher effect of other stops, or else, in connection with other stops, to strengthen their prime tones. Next to the wide stopped organ pipes as regards quality of tone stand flutes and the flue pipes on organs (open pipes, blown gently). These have the Octave plainly in addition to the prime, and when blown more strongly even produce the Twelfth. In this case the Octaves and Fifths are more distinctly delimited by upper partial tones; but the definition of Thirds and Sixths has to depend upon combinational tones, and hence is much weaker. The musical character of these pipes is therefore not much unlike that of the wide stopped pipes already described. This is well expressed by the old joke that nothing is more dreadful to a musical ear than a flute-concerto, except a concerto for two flutes.^[24] But in combination with other instruments which give effect to the connection of the harmony, the flute, from the perfect softness of its tone and its great mobility, is extraordinarily pleasant and attractive, and cannot be replaced by any other instrument. In ancient music the flute played a much more important part than at present, and this seems to accord with the whole ideal of classical art, which aimed at keeping everything unpleasant from its productions, confining itself to pure beauty, whereas modern art requires more abundant means of expression, and consequently to a certain extent admits into its circle what in itself would be contrary to the gratification of the senses. However this be, the earnest friends of music, even in classical times, contended for the harsher tones of stringed instruments in opposition to the effeminate flute.

The open organ pipes afford a favourable means of meeting the harmonic requirements of polyphonic music, and consequently form the principal stops. They make the lower partials distinctly audible, the wide pipes up to the third, the narrow ones (geigen principal^[25]) up to the sixth partial tone. The wider pipes have more power of tone than the narrower; to give them more brightness the 8-foot stops, which contain the 'principal work,' are connected with the 4-foot stops, which add the Octave to each note, or the principal is connected with the geigen principal, so that the first gives power and the second brightness. By this means qualities of tone are produced which contain the first six partial tones in moderate force, decreasing as the pitch ascends. These give a very distinct feeling for the purity of the consonant intervals, enabling us to distinguish clearly between consonance and dissonance, and preventing the unavoidable but weak dissonances that result from the higher upper partials in the imperfect consonances, from becoming too marked, but at the same time not allowing the hearer’s appreciation of the progression of the parts to be disturbed by a multitude of loud accessory tones. In this respect the organ has an advantage over all other instruments, as the player is able to mix and alter the qualities of tone at pleasure, and make them suitable to the character of the piece he has to perform.

The narrow stopped pipes (Quintaten),^[26] for which the prime tone is accompanied by the Twelfth, the reed-flute (Rohrflöte)^[27] where the third and fifth partials are both present, the conical open pipes, as the goat-horn (Gemshorn),^[28] which reinforce certain higher partials^[29] more than the lower, and so forth, serve only to give distinctive qualities of tone for particular parts, and thus to separate them from the rest. They are not well adapted for forming the chief mass of the harmony.

Very piercing qualities of tone are produced by the reed pipes and compound stops^[30] on the organ. The latter, as already explained, are artificial imitations of the natural composition of all musical tones, each key bringing a series of pipes into action, which correspond to the first three or first six partial tones of the corresponding note. They can be used only to accompany congregational singing. When employed alone they produce insupportable noise and horrible confusion. But when the singing of the congregation gives overpowering force to the prime tones in the notes of the melody, the proper relation of quality of tone is restored, and the result is a powerful, well-proportioned mass of sound. Without the assistance of these compound stops it would be impossible to control a vast body of sound produced by unpractised voices, such as we hear in [German] churches.

The human voice is on the whole not unlike the organ in quality, so far as harmony is concerned. The brighter vowels, of course, generate isolated high partial tones, but these are so unconnected with the rest that they can have no universal and essential effect on the sound of the chords. For this we must look to the lower partials, which are tolerably uniform for all vowels. But of course in particular consonances the characteristic tone of the vowels may play an important part. If, for example, two human voices sing the major Third \(b\flat\) \(d'\) on the vowel a in father, the fourth partial of \(b\flat\) (or \(b''\flat\)), and the third partial of \(d'\) (or \(a''\)), fall among the tones characteristically reinforced by \(A\), and consequently the imperfection of the consonance of a major Third will come out harshly by the dissonance \(a'' b''\flat\), between these upper partials; whereas if the vowel be changed to o in no, the dissonance disappears. On the other hand the Fourth \(b\flat e'\flat\) sounds perfectly well on the vowel a in father, because the higher note \(e'\flat\) has the same upper partial \(b''\flat\) as the deeper \(b\flat\). But if a in father be inclined towards a in fall, or a in fat, the upper partials \(f''\) and \(e''\flat\) or else \(d'''\) and \(e'''\flat\) might interrupt the consonance. This serves to shew, among other things, that the translation of the words of a song from one language into another is not by any means a matter of indifference for its musical effect.^[31]

Disregarding at present these reinforcements of partials due to the characteristic resonance of each vowel, the musical tones of the human voice are on the whole accompanied by the lower partials in moderate strength, and hence are well adapted for combinations of chords, precisely as the tones of the principal stops of the organ. Besides this the human voice has a peculiar advantage over the organ and all other musical instruments in the execution of polyphonic music. The words which are sung connect the notes belonging to each part, and form a clue which readily guides the hearer to discover and pursue the related parts of the whole body of sound. Hence polyphonic music and the whole modern system of harmony were first developed on the human voice. Indeed, nothing can exceed the musical effect of well harmonised part music perfectly executed in just intonation by practised voices. For the complete harmoniousness of such music it is indispensably necessary that the several musical intervals should be justly intoned, and our present singers^[32] unfortunately seldom learn to take just intervals, because they are accustomed from the first to sing to the accompaniment of instruments which are tuned in equal temperament, and hence with imperfect consonances. It is only such singers as have a delicate musical feeling of their own who find out the correct result, which is no longer taught them.

Richer in upper partials, and consequently brighter in tone than the human voice and the principal stops on the organ, are the bowed instruments, which consequently fill such an important place in music. Their extraordinary mobility and expressiveness give them the first place in instrumental music, and the moderate acuteness of their quality of tone assigns them an intermediate position between the softer flutes and the braying brass instruments. There is a slight difference between the different instruments of this class; the tenor and double-bass have a somewhat acuter and thinner quality than the violin and violoncello, that is, they have relatively stronger upper partials. The audible partials reach to the sixth or eighth, according as the bow is brought nearer the finger-board for piano, or nearer the bridge for forte, and they decrease regularly in force as they ascend in pitch. Hence on bowed instruments the difference between consonance and dissonance is clearly and distinctly marked, and the feeling for the justness of the intervals very certain; indeed it is notorious that practised violin and violoncello players have a very delicate ear for distinguishing differences of pitch. On the other hand the piercing character of the tones is so marked, that soft song-like melodies are not well suited for bowed instruments, and are better given to flutes and clarinets in the orchestra. Full chords are also relatively too rough, since those upper partials which form dissonant intervals in every consonance, are sufficiently strong to make the dissonance obtrusive, especially for Thirds and Sixths. Moreover, the imperfect Thirds and Sixths of the tempered musical scale are on bowed instruments very perceptibly different in effect from the justly intoned Thirds and Sixths when the player does not know how to substitute the pure intervals for them, as the ear requires. Hence in compositions for bowed instruments, slow and flowing progressions of chords are introduced by way of exception only, because they are not sufficiently harmonious; quick movements and figures, and arpeggio chords are preferred, for which these instruments are extremely well adapted, and in which the acute and piercing character of their combined sounds cannot be so distinctly perceived.

The beats have a peculiar character in the case of bowed instruments. Regular, slow, numerable beats seldom occur. This is owing to the minute irregularities in the action of the bow on the string, already described, to which is due the well-known scraping effect so often heard. Observations on the vibrational figure shew that every little scrape of the bow causes the vibrational curve to jump suddenly backwards or forwards, or in physical terms, causes a sudden alteration in the phase of vibration. Now since it depends solely on the difference of phase whether two tones which are sounded at the same time mutually reinforce or enfeeble each other, every minutest catching or scraping of the bow will also affect the flow of the beats, and when two tones of the same pitch are played, every jump in the phase will suffice to produce a change in the loudness, just as if irregular beats were occurring at unexpected moments. Hence the best instruments and the best players are necessary to produce slow beats or a uniform flow of sustained consonant chords. Probably this is one of the reasons why quartettes for bowed instruments, when executed by players who can play solo pieces pleasantly enough, sometimes sound so intolerably rough and harsh that the effect bears no proper ratio to the slight roughness which each individual player produces on his own instrument.^[33] When I was making observations on vibrational figures, I found it difficult to avoid the occurrence of one or two jumps in the figure every second. Now in solo-playing the tone of the string is thus interrupted for almost inappreciably minute instants, which the hearer scarcely perceives, but in a quartette when a chord is played for which all the notes have a common upper partial tone, there would be from four to eight sudden and irregular alterations of loudness in this common tone every second, and this could not pass unobserved. Hence for good combined performance, a much greater evenness of tone is required than for solo-playing.^[34]

The pianoforte takes the first place among stringed instruments for which the strings are struck. The previous analysis of its quality of tone shews that its deeper octaves are rich, but its higher octaves relatively poor, in upper partial tones. In the lower octaves, the second or third partial tone is often as loud as the prime, nay, the second partial is often louder than the prime. The consequence is that the dissonances near the Octave (the Sevenths and Ninths) are almost as harsh as the Seconds, and that diminished and augmented Twelfths and Fifths are rather rough. The 4th, 5th, and 6th partial tones, on the other hand, on which the Thirds depend, decrease rapidly in force, so that the Thirds are relatively much less distinctly delimited than the Octaves, Fifths, and Fourths. This last circumstance is important, because it makes the sharp Thirds of the equal temperament much more endurable upon the piano than upon other instruments with a more piercing quality of tone, whereas the Octaves, Fifths, and Fourths are delimited with great distinctness and certainty. Notwithstanding the relatively large number of upper partial tones on the pianoforte, the impression produced by dissonances is far from being so penetrating as on instruments of long-sustained tones. On the piano the note is powerful only at the moment when it is struck, and rapidly decreases in strength, so that the beats which characterise the dissonances have not time to become sensible during the strong commencement of the tone; they do not even begin until the tone is greatly diminished in intensity. Hence in the modern music written for the pianoforte, since the time that Beethoven shewed how the characteristic peculiarities of the instrument were to be utilised in compositions, we find an accumulation and reduplication of dissonant intervals which would be perfectly insupportable on other instruments. The great difference becomes very evident when an attempt is made to play recent compositions for the piano on the harmonium or organ.

That instrument-makers, led solely by practised ears, and not by any theory, should have found it most advantageous to arrange the striking place of the hammer so that the 7th partial tone entirely disappears, and the 6th is weak although actually present,^[35] is manifestly connected with the structure of our system of musical tones. The 5th and 6th partial tones serve to delimit the minor Third, and in this way almost all the intervals treated as consonances in modem music are determined on the piano by coincident upper partials; the Octave, Fifth, and Fourth by relatively loud tones; the major Sixth and major Third by weak ones; and the minor Third by the weakest of all. If the 7th partial tone were also present, the subminor^[36] Seventh 4 : 7, as \(c^{\prime 7}b^{\prime}\flat\), would injure the harmoniousness of the minor Sixth; the subminor Fifth 5 : 7, as \(c^{\prime 7}b^{\prime}\flat\), that of the Fifth and Fourth; and the subminor Third 6 : 7, as \(g^{\prime 7}b^{\prime}\flat\), that of the minor Third; without any gain in the more accurate determination of new intervals suitable for musical purposes.

Mention has already been made of a further peculiarity in the selection of quality of tone on the pianoforte, namely that its upper notes have fewer and weaker upper partial tones than the lower. This difference is much more marked on the piano than on any other instrument, and the musical reason is easily assigned. The high notes are usually played in combination with much lower notes, and the relation between the two groups of notes is given by the high upper partials of the deeper tones. When the interval between the bass and treble amounts to two or three Octaves, the second Octave, higher Third and Fifth of the bass note, are in the close neighbourhood of the treble, and form direct consonances and dissonances with it, without any necessity for using the upper partials of the treble note. Hence the only effect of upper partials on the highest notes of the pianoforte would be to give them shrillness, without any gain in respect to musical definition. In actual practice the construction of the hammers on good instruments causes the notes of the highest Octaves to be only gently accompanied by their second partials. This makes them mild and pleasant, with a flute-like tone. Some instrument-makers, however, prefer to make these notes shrill and piercing, like the piccolo flute, by transferring the striking place to the very end of the highest strings. This contrivance succeeds in increasing the force of the upper partial tones, but gives a quality of tone to these strings which does not suit the character of the others, and hence certainly detracts from their charm.

In many other instruments, where their construction does not admit of such absolute control over the quality of tone as on the pianoforte, attempts have been made to produce similar varieties of quality in the high notes, by other means. In the bowed instruments this purpose is served by the resonance box, the proper tones of which lie within the deepest Octaves of the scale of the instrument. Since the partial tones of the sounding strings are reinforced in proportion to their proximity to the partial tones of the resonance box, this resonance will assist the prime tones of the higher notes, as contrasted with their upper partials, much more than it will do so for the deep notes. On the contrary, the deepest notes of the violin will have not only their prime tones, but also their Octaves and Fifths favoured by the resonance; for the deeper proper tone of the resonance box lies between the prime and 2nd partial, and its higher proper tone between the 2nd and 3rd partials. A similar effect is attained in the compound stops of the organ, by making the series of upper partial tones, which are represented by distinct pipes, less extensive for the higher than for the lower notes in the stop. Thus each digital opens six pipes for the lower octaves, answering to the first six partial tones of its note; but in the two upper octaves, the digital opens only three or even two pipes, which give the Octave and Twelfth, or merely the Octave, in addition to the prime.

There is also a somewhat similar relation in the human voice, although it varies much for the different vowels. On comparing the higher and lower notes which are sung to the same vowel, it will be found that the resonance of the cavity of the mouth generally reinforces relatively high upper partials of the deep notes of the bass, whereas for the soprano, where the note sung comes near to the characteristic pitch of the vowel, or even exceeds it, all the upper partials become much weaker. Hence in general, at least for the open vowels, the audible upper partials of the bass are much more numerous than those of the soprano.

We have still to consider the artificial reed instruments, that is the wind instruments of wood and brass. Among the former the clarinet, among the latter the horn are distinguished for the softness of their tones, whereas the bassoon and hautbois in the first class, and the trombone and trumpet in the second represent the most penetrating qualities of tone used in music.

Notwithstanding that the keyed horns used for so-called concerted music have a far less braying quality of tone than trumpets proper, which have no side holes, yet the number and the force of their upper partial tones are far too great for the harmonious effect of the less perfect consonances, and the chords on these instruments are very noisy and harsh, so that they are only endurable in the open air. In artistic orchestral music, therefore, trumpets and trombones, which on account of their penetrative power cannot be dispensed with, are seldom employed for harmonies, except for a few and if possible perfect consonances.

The clarinet is distinguished from all other orchestral wind instruments by having no evenly numbered partial tones.^[37] To this circumstance must be due many remarkable deviations in the effect of its chords from those of other instruments. When two clarinets are playing together all of the consonant intervals will be delimited by combinational tones alone, except the major Sixth 3 : 5, and the Twelfth 1 : 3. But the differential tones of the first order, which are the strongest among all combinational tones, will always suffice to produce the beats of imperfect consonances. Hence it follows that in general the consonances of two clarinets have but little definition, and must be proportionately agreeable. This is really the case, except for the minor Sixth and minor Seventh, which are too near the major Sixth, and for the Eleventh and minor Thirteenth, which are too near the Twelfth. On the other hand, when a clarinet is played in combination with a violin or oboe, the majority of consonances will have a perceptibly different effect according as the clarinet takes the upper or the lower note of the chord. Thus the major Third \(d' \space f'\sharp\) will sound better when the clarinet takes \(d'\) and the oboe \(f'\sharp\), so that the 5th partial of the clarinet coincides with the 4th of the oboe. The 3rd and 4th and the 5th and 6th partials, which are so disturbing in the major Third,^[38] cannot here be heard, because the 4th and 6th partials do not exist on the clarinet. But if the oboe takes \(d'\) and the clarinet \(f'\sharp\), the, coincident 4th partial will be absent, and the disturbing 3rd and 5th present. For the same reason it follows that the Fourth and minor Third will sound better when the clarinet takes the upper tone. I have made experiments of this kind with the clarinet and a bright stop of the harmonium, which possessed the evenly numbered partial tones, and was tuned in just intonation^[39] and not in equal temperament. When \(b\flat\) was played on the clarinet, and \(e'\flat\), \(d'\), \(d'\flat\), in succession on the harmonium, the major Third \(b\flat\) \(d'\) sounded better than the Fourth \(b\flat\) \(e'\flat\), and much better than the minor Third \(b\flat\) \(d'\flat\). If, retaining \(b\flat\) on the clarinet, I played \(f\), \(g\flat\) , \(g\) in succession on the harmonium, the major Third \(g\flat\) \(b\flat\) was rougher, not merely than the Fourth \(f\) \(b\flat\), but even than the minor Third \(g\) \(b\flat\).

This example, to which I was led by purely theoretical considerations that were immediately confirmed by experiment, will serve to shew how the use of exceptional qualities of tone will affect the order of agreeableness of the consonances which was settled for those usually heard.

Enough has been said to shew the readiness with which we can now account for numerous peculiarities in the effects of playing different musical instruments in combination. Further details are rendered impossible by the want of sufficient preliminary investigations, especially into the exact differences of various qualities of tone. But in any case it would lead us too far from our main purpose to pursue a subject which has rather a technical than a general interest.