effect. Thus, we may suppose that after the number of beats becomes greater than 132 per second, the interval of nervous repose grows evanescent, so that the nerve escapes the injury of excessive stimulation, just in the same way as it does under a strictly continuous stimulation. Further, we may reason that in order to produce this effect, the two tones must be near enough to act upon the same fibre of Corti. Hence, in the lower regions of the scale the aërial beats produced by given intervals, though they are not too rapid to be recognised, do not affect sensibility, because the tones producing them are too unlike in the numbers of their vibrations to act upon the same nervous appendage of the ear. In other words, though there may be aërial beats of the required rapidity, they cannot pass into nervous beats.*

From these considerations one may see how it is that two notes of approximately equal height conflict with each other in the harsh effect of dissonance; for they are sufficiently near one another to act on the same nerve-fibre. But it does not so readily appear why

notes far removed from one another in the diatonic scale are also discordant. How is it, for instance, that an intensely discordant sound is afforded by striking simultaneously the G of the treble clef and the higher F sharp? The answer to this question our author finds in his theory of upper-tones. The first and strongest upper-tone in the G clang is that of the octave above; and since this is only a semitone removed from the other note, F sharp, it clashes with this, and produces the discordant impression. Similarly an upper-tone of one note may produce discord with that of another. When, for example, the lower C of the treble clef is

* Helmholtz says that equal intervals do not combine in producing precisely the same degree of harshness at different levels of the scale, the same intervals being less rough towards the upper parts, harsher towards the lower regions. He does not, however, offer a final explanation of this slight deviation from the principle stated in the text.

+ The presence of these beats may occasionally be detected even in a single clang, if it is very rich in partial tones; since it is possible that the higher upper-tones, those beyond the eighth, may approximate so closely in pitch as to affect the same nervous fibre. These effects are most conspicuous in clangs of low pitch; and this accords with the fact, already alluded to in a previous note, that similar intervals are somewhat more harsh in the lower than in the higher regions of the scale. Helmholtz shows that this phenomenon must be taken into account in determining the lowest continuous tones of the scale which the ear can perceive. It is because of this greater liability to dissonance among its upper-tones in a low-pitched, than in a high-pitched clang, that singers always strive after a high compass of voice. The superior value of high clangs is due

struck with the next A flat above, it may be easily seen from the illustration of the order of the upper-tones already given, that the second upper-tone of the C is only a semitone removed from the first upper-tone of the G. Accordingly, we find that this combination of notes, though rendered familiar as an element of the chord of the lower A flat, has in it something strange and half-sad, and forms a characteristic interval of the C minor key.*

In this way Helmholtz establishes an invariable connection between the degree of discordance of two notes and the number and prominence of these disturbing shocks. The clearly pronounced discords involve disturbances among the ground-tones and the lowest and most powerful upper-tones. From these highly disagreeable mental impressions, up to the sweetest harmonies, there is a decreasing scale of painful effect which is determined by the decreasing number and prominence of these disturbing beats. Just as the harshest discord corresponds to the greatest number and prominence of these beats, so the purest harmony answers to their perfect absence. This is the one discoverable physiological ground of the pleasure of musical harmony, and no other is required. Harmony is only known as contrasted with discord, and a finer sense of disturbing elements, whether due to superiority of natural organism or to higher culture, discovers a conjunction of notes which was once felt to be harmonious, to be painfully discordant. To use the author's own words, we may say that "Consonance is a continuous; dissonance, an intermittent sensation of tone."

The great experimental proof that harmony thus rests on a relative and negative ground, is afforded, according to our author, by the case of those instruments, such as stopped organ-pipes, in which the upper-tones, and consequently their mutual disturbances, are almost entirely absent. These instruments not only fail to yield the rich quality of clang, but are further wanting in the peculiar effects of harmony and discord. All their combinations of notes

partly to the less prominent character of their dissonant upper-tones, and partly to the fact that in the higher regions slight discrepancies of pitch occasion a much larger number of beats, which peculiarity serves to render the musical feeling for pitch in these instances much more accurate.

Besides the upper-tones, other partial tones may also co-operate in producing discord. These are the so-called combination-tones, which arise from the simultaneous utterance of particular notes. Their effect, however, in our ordinary instruments is very insignificant, and it becomes appreciable only when, as in the case of stopped organ-pipes, the upper-tones and their effects are almost completely wanting.

seem to the ear indifferent and colourless; and those which on other instruments afford the sweetest harmony, are said to be scarcely preferable to the rest.

Our author seeks in a very interesting manner to trace out the influence of these colliding upper-tones in the actual formation of our modern harmonic system, and in the various historical developments of harmony. Highly instructive as these researches are, our present space does not allow us to follow them in detail. One or two examples of the method of inquiry pursued must suffice.

By conceiving the absence of repugnant elements in the partial tones of clangs to be the one condition of harmony, Helmholtz seeks to define the various degrees of consonance between notes, and to classify them as follows. First of all we have "absolute consonance" when the ground-tone of one note coincides with an upper-tone of the second, as happens in the case of the octave and in that of the twelfth, or fifth above the octave. In these cases the effect of beats from adjacent upper-tones may be looked on as evanescent. Secondly, there are the "perfect consonances" of the fifth and fourth, as C-G, C-F. Of these two, the fifth is the most free from disturbing upper-tones; and this interval has, accordingly, always been recognised as a harmonious combination. The fourth is less completely free from disturbing upper-tones, and hence the repeated disputes as to its harmonic character. Next in order come the "middle consonances," or those of the major sixth and third, C-A, C-E, in which the presence of antagonistic upper-tones is much more apparent. Finally, there remain the "imperfect consonances of the minor sixth and third, C-A flat, C-E flat, in which, as their minor character suggests, there is a distinctly recognisable effect of dissonance. These intervals exhaust all the more common and familiar combinations of the major and minor keys. Others, that are frequently used in musical compositions, as, for instance, those of the dominant seventh (G-F in the key of C), are in reality distinctly discordant, though they are admitted, on other grounds, as transient elements in sequences of harmonies.*

* Helmholtz has a method of determining what upper-tones conflict in the several intervals, by regarding each interval as a discordant deviation from an adjacent consonant interval. Thus the third may be looked on as an imperfect fourth, and vice versa. By adopting this conception he shows that "in every consonant interval those upper-tones conflict which coincide in the adjacent interval." The mathematical reader may demonstrate this for himself, by help of the table already given.

After thus applying the phenomena of upper-tones to the explanation of the effects of harmony, our author proceeds to investigate their bearing on melody. He accepts it as an indisputable historical fact that the feeling for melody may be developed in the absence of harmonic experience; so that he deems it a mistake to derive all melodic gratification from remembered impressions of harmony. He finds a basis of natural melodic affinity in the effects of upper-tones. Thus, for example, it follows from what has been said respecting the order of upper-tones, that when a note is succeeded by its next octave above, we hear" (in the second clang) "a part of that which we have just heard, and hear at the same time nothing new which we have not already heard." Similarly, when a sequence of fifths is played, as C-G-D-A, each new clang carries on, so to speak, a part of the impression of the preceding one. In the case of less perfect melodious intervals, as those of the third and sixth, the common element which links together the sequent clangs is less conspicuous. The author defines two degrees of natural affinity between sequent tones, as determined by this element of a common upper-tone. Clangs are related in the first degree, when they have two equal partial tones, the degree of affinity being, of course, greater when this common upper-tone is in the first and influential region of the series of upper-tones. The second or indirect degree of affinity belongs to clangs which are related directly in the first degree to the same third clang. By help of the first degree of affinity, it is possible "to construct rationally" the principal intervals of the major key, that is to say, transitions from C to E, F, G, A, and C. The less melodious intervals of the second and seventh, C-D, C-B, rest on indirect affinity, the mediating clang being the fifth of the tonic, or G. Similarly, the minor interval, C-A flat, may be regarded as a relation effected through the higher octave of the tonic, C.

Helmholtz seeks to prove the influence of this natural affinity between sequent tones, by means of a very learned and interesting examination of the various known melodic scales, ancient and modern. Into these verifications our present limits do not allow us to follow him. Suffice it to say, that he succeeds in bringing together an impressive array of facts in support of the view, that the human ear has always been guided, even if unconsciously, in the selection of tuneful sequence, by the action of these connecting uppertones. The intervals most generally adopted in all systems, are precisely those in which the natural affinity is greatest, namely, the

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octave, fifth and fourth; this last, moreover, having had an increased value as the fifth inverted (C-G inverted gives G-C). Thus we find that some of the oldest of known scales were simply successions of fifths (C-G-D-A, etc.), and so represented the closest degree of melodic affinity. Similarly, among the Gaelic scales, one is wanting in the third and sixth, another in the third and seventh, that is to say in the notes most distantly related to the key-note. At the same time, as Helmholtz reminds us, since this natural law of melodic affinity is itself of a very elastic character, we must not expect to find perfect uniformity among the several systems. He admits, too, that other requirements, besides those of melodic affinity, have served to determine the various forms of scales. Of these the principal one is the need of a continuous series of equal intervals, which may enable the ear to estimate the distance of all intervals by help of some simple unit of measure. This unit is, in our modern European systems, the semitone interval; and this limit was, according to Helmholtz, recognised in the fixed, as distinguished from the variable notes of the Greek scales. In modern Arabic music, again, the octave is divided into twenty-four quarter-tones. The delicacy of the ear's sensibility in discriminating the pitch of tones appears to have had only a remote influence on the construction of melodic scales, since it is known that the organ can distinguish much finer shades of pitch than either the half-tone or quarter-tone.*

Helmholtz employs the method of difference, too, in verifying his conclusions respecting the dependence of melodic pleasure on the influence of upper-tones. Just as the proper pleasure of harmony is wanting when the combining notes are destitute of upper-tones, so, he thinks, the proper pleasure of melody is wanting when the sequent notes are poor in the quality of timbre. A melody played on stopped organ pipes, or whistled with the mouth, does not, he says, afford a pure melodic enjoyment. If it be a familiar air which

Helmholtz considers that the transitions of sequent chords, like those of single notes, are determined by a certain natural affinity, namely the common possession of some one clang. But this part of his exposition leads us into considerations of a more psychological or æsthetic character, and has but little connection with the physiological theory which we are now examining. Similarly, too, his discussions of the technical laws which control transition from key to key, and other processes of the art, though they frequently recognise the consequences of this theory, involve the consideration of other requirements which belong less to any peculiarities of our auditory organ than to the general laws of our emotional nature.