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planation partly historic and partly æsthetic, but too com. plex for exposition here.
The 'timbre' of a note is due to its wave-form. Waves are either simple ('pendular') or compound. Thus if a tuning-fork (which gives waves nearly simple) vibrate 132 times a second, we shall hear the note c. If simultaneously a fork of 264 vibrations be struck, giving the next higher octave, c', the aerial movement at any time will be the algebraic sum of the movements due to both forks; whenever both drive the air one way they reinforce one another; when on the contrary the recoil of one fork coincides with the forward stroke of another, they detract from each other's effect. The result is a movement which is still periodic, repeating itself at equal intervals of time, but no longer pendular, since it is not alike on the ascending and descending limbs of the curves. We thus get at the fact that non-pendular vibrations may be produced by the fusion of pendular, or, in technical phrase, by their composition.
Suppose several musical instruments, as those of an orchestra, to be sounded together. Each produces its own effect on the air-particles, whose movements, being an algebraical sum, must at any given instant be very complex; yet the ear can pick out at will and follow the tones of any one instrument. Now in most musical instruments it is susceptible of physical proof that with every single note that is sounded many upper octaves and other ‘harmonics' sound simultaneously in fainter form. On the relative strength of this or that one or more of these Helmiholtz has shown that the instrument's peculiar voice depends. The several vowel-sounds in the human voice also depend on the predominance of diverse upper harmonics accompanying the note on which the vowel is sung. When the two tuning-forks of the last paragraph are sounded together the new form of vibration has the same period as the lower-pitched fork; yet the ear can clearly distinguish the resultant sound from that of the lower fork alone, as a note of the same pitch but of different timbre; and within
the compound sound the two components can by a trained ear be severally heard. Now how can one resultant waveform make us hear so many sounds at once?
The analysis of compound wave-forms is supposed (after Helmholtz) to be effected through the different rates of sympathetic resonance of the different parts of the membranous cochlea. The basilar membrane is some twelve times broader at the apex of the cochlea than at the base where it begins, and is largely composed of radiating fibres which may be likened to stretched strings. Now the physical principle of sympathetic resonance says that when stretched strings are near a source of vibration those whose own rate agrees with that of the source also vibrate, the others remaining at rest. On this principle, waves of perilymph running down the scala tympani at a certain rate of frequency ought to set certain particular fibres of the basilar membrane vibrating, and ought to leave others unaffected. If then each vibrating fibre stimulated the hair-cell above it, and no others, and each such hair-cell, sending a current to the auditory brain-centre, awakened therein a specific process to which the sensation of one particular pitch was correlated, the physiological condition of our several pitchsensations would be explained. Suppose now a chord to be struck in which perhaps twenty different physical rates of vibration are found: at least twenty different hair-cells or end-organs will receive the jar; and if the power of mental discrimination be at its maximum, twenty different 'objects' of hearing, in the shape of as many distinct pitches of sound, may appear before the mind.
The rods of Corti are supposed to be dampers of the fibres of the basilar membrane, just as the malleus, incus, and stapes are dampers of the tympanic membrane, as well as transmitters of its oscillations to the inner ear. There must be, in fact, an instantaneous damping of the physiological vibrations, for there are no such positive after-images, and no such blendings of rapidly successive tones, as the retina shows us in the case of light. Helmholtz's theory of
the analysis of sounds is plausible and ingenious. One objection to it is that the keyboard of the cochlea does not seem extensive enough for the number of distinct resonances required. We can discriminate many more degrees of pitch than the 20,000 hair-cells, more or less, will allow for.
The so-called Fusion of Sensations in Hearing.-A very common way of explaining the fact that waves which singly give no feeling of pitch give one when recurrent, is to say that their several sensations fuse into a compound sensation. A preferable explanation is that which follows the analogy of muscular contraction. If electric shocks are sent into a frog's sciatic nerve at slow intervals, the muscle which the nerve supplies will give a series of distinct twitches, one for each shock. But if they follow each other at the rate of as many as thirty a second, no distinct twitches are observed, but a steady state of contraction instead. This steady contraction is known as tetanus. The experiment proves that there is a physiological cumulation or overlapping of processes in the muscular tissue. It takes a twentieth of a second or more for the latter to relax after the twitch due to the first shock. But the second shock comes in before the relaxation can occur, then the third again, and so on; so that continuous tetanus takes the place of discrete twitching. Similarly in the auditory nerve. One shock of air starts in it a current to the auditory brain-centre, and affects the latter, so that a dry stroke of sound is heard. If other shocks follow slowly, the brain-centre recovers its equilibrium after each, to be again upset in the same way by the next, and the result is that for each shock of air a distinct sensation of sound occurs. But if the shock comes in too quick succession, the later ones reach the brain before the effects of the earlier ones on that organ have died away. There is thus an overlapping of processes in the auditory centre, a physiological condition analogous to the muscle's tetanus, to which new condition a new quality of feeling, that of pitch, directly corresponds. This latter
feeling is a new kind of sensation altogether, not a mere appearance' due to many sensations of dry stroke being compounded into one. No sensations of dry stroke can exist under these circumstances, for their physiological conditions have been replaced by others. What 'compounding' there is has already taken place in the braincells before the threshold of sensation was reached. Just so red light and green light beating on the retina in rapid enough alternation, arouse the central process to which the sensation yellow directly corresponds. The sensations of red and of green get no chance, under such conditions, to be born. Just so if the muscle could feel, it would have a certain sort of feeling when it gave a single twitch, but it would undoubtedly have a distinct sort of feeling altogether, when it contracted tetanically; and this feeling of the tetanic contraction would by no means be identical with a multitude of the feelings of twitching.
Harmony and Discord.-When several tones sound together we may get peculiar feelings of pleasure or displeasure designated as consonance and dissonance respectively. A note sounds most consonant with its octave. When with the octave the 'third' and the fifth ' of the note are sounded, for instance c-e-g-c', we get the 'full chord' or maximum of consonance. The ratios of vibration here are as 4:5:6:8, so that one might think simple ratios were the ground of harmony. But the interval cd is discordant, with the comparatively simple ratio 8:9. Helmholtz explains discord by the overtones making 'beats' together. This gives a subtle grating which is unpleasant. Where the overtones make no 'beats', or beats too rapid for their effect to be perceptible, there is consonance, according to Helmholtz, which is thus a negative rather than a positive thing. Wundt explains consonance by the presence of strong iden tical overtones in the notes which harmonize. No one of these explanations of musical harmony can be called quite Batisfactory; and the subject is too intricate to be treated farther in this place.
Discriminative Sensibility of the Ear.-Weber's law holds fairly well for the intensity of sounds. If ivory or metal balls are dropped on an ebony or iron plate, they make a sound which is the louder as they are heavier or dropped from a greater height. Experimenting in this way (after others) Merkel found that the just perceptible increment of loudness required an increase of of the original stimulus everywhere between the intensities marked 20 and 5000 of his arbitrary scale. Below this the fractional increment of stimulus must be larger; above it, no measurements were made.
Discrimination of differences of pitch varies in different parts of the scale. In the neighborhood of 1000 vibrations per second, one fifth of a vibration more or less can make the sound sharp or flat for a good ear. It takes a much greater relative alteration to sound sharp or flat elsewhere on the scale. The chromatic scale itself has been used as an illustration of Weber's law. The notes seem to differ equally from each other, yet their vibration-numbers form a series of which each is a certain multiple of the last. This, however, has nothing to do with intensities or just perceptible differences; so the peculiar parallelism between the sensation series and the outer-stimulus series forms here a case all by itself, rather than an instance under Weber's more general law.