The first two of the three physiological (subjective) characteristics - Strength. height, timbre - are characterized physically by distance and velocity of the oscillations, respectively. How does timbre manifest itself objectively? Answer: In the form of the oscillations. You can cause a sounding body to record its vibrations: By attaching to a sounding tuning fork a pin and letting a plate, covered with soot, glide underneath it, so that the pin leaves a trace; by illuminating a small section of a sounding string (inside an otherwise dark room) and taking a photos while moving the film sidewards to the direction of vibration. The membrane of the Phon-autograph of E.L.Scott 1861 and the phonograph of Edison are generally convenient means for such studies. The curves of Fig. 317 were obtained by means of the Phon-autograph.
The membrane in the Phon-autograph corresponds to the ear drum and a given curve to an ear's sensing: Its impression of a given timbre. What links the characteristic form of the curve and the characteristic timbre? The curves of Fig. 317 are more complicated than those of a simple wave form, however they are related to them in a simple manner. We know from our study of the superposition of wave motions that several causes of oscillations, acting at the same location simultaneously, add or subtract to or from each other, depending on whether they act in the same direction or not, moreover, that a very complicated motion may result from such superposition (Fig. 318).
Fourier's Theorem. Ohm's Law
JFourier 1843 has shown that every arbitrary periodic form of oscillation - for example, each of the two curves shown in Fig. 317 - can be decomposed into a number of simple forms of vibration, and indeed only in a unique manner; the periods of these oscillations are related to each other as the numbers 1, 2, 3, иии.
Just as the curve d in Fig. 318 can be decomposed into the curves a, b, c, you can decompose the curves of Fig. 317. You should envisage that every partial curve represents a tone, the Hertz number its height, the amplitude its strength. Hence the complicated sound records tell you in the sense of Fourier's Theorem: The sources of sound do not emit a single tone, but a mixture of tones, which differ in height and strength, but not otherwise. Moreover, if you realize that the membrane of the Phon-autograph corresponds to the ear drum and its curve to a definite sensing of a sound, Fourier's Theorem can be reformulated, following Helmholtz: "Every vibratory motion of the air in the auditory channel, which corresponds to a musical sound, can always be represented (and every time only in a unique manner) by simple oscillations, which correspond to the partial tones of this sound."
These partial tones are in no way just a hypothesis, they can actually be heard. Ohm 1843 has formulated the theorem, that the ear senses only one pendulum like, that is, in our sense, simple oscillation as an undivided tone and decomposes each not pendulum like oscillation into a sequence of such oscillations, each of which it perceives as a unit. The reason why we, as a rule, do not sense them individually and require special training or special equipment for it is that the partial tones are mostly less strong compared with the base tone*. Hence the ear senses, as a rule, the base tone at its due height and strength, ascribes to the entire sound the height and strength of the base tone and believes that it hears a single homogeneous tone. However, it is mistaken! According to Helmholtz 1863, we hear the various instruments with different timbres due to the fact that they emit beside the base tone overtones - partial tones which have 2-, 3-, 4- иии times as many oscillations as the base tone and different heights and strengths.
* In all natural and musically employable sounds, the partial tones decrease in strength with height, but in some of the best musical timbres the strength of the lower overtones does not differ much from that of the base tone. The base tone is weaker for the sounds of the piano of the medium and lower octaves than the first or even the first two overtones. (Nernst )
If there were no overtones, that is, only the base tone, or if all instruments had the same overtones, all of them would have the same sound. You cannot differentiate a tuning fork and a reed-pipe at the height of c5 (W.Preyer), because their overtones occur in the 7- and 8-' Octave and are inaudibly high.
For these reasons, you must make a distinction in Acoustics between sound and tone: Sound is the general impression of periodic motion of the air, tone the impression of a simple oscillation. You can only ascribe height to a single tone, you must ascribe, strictly speaking, different heights to a sound, corresponding to its different partial tones. We speak of consonance of different instruments, but every sound, which allows to differentiate overtones, is already a consonance of several tones.
Analysis and synthesis of timbre
Helmholtz 1863 has established the correctness of his opinion analytically and synthetically. He has shown that the sounds of individual musical instruments contain individual mixtures of overtones. Employing tuned resonators, he has analyzed sound sources and determined which overtones besides the base tone are present in a given sound. He was then able to reduce the variability of timbre to the fact, that in different sounds there are present different overtones at varying strength. However, most convincingly, he has proved synthetically the correctness of his views concerning the nature of timbre. If the timbre of an instrument really depends on the fact that beside the base tone you hear tones, the periods of which are 2-, 3- иии times as large as that of the base tone, and the strengths of which is in a certain relationship to that of the base tone, it must then be possible to simulate the timbre of an instrument by generating its base tone and those other tones individually, give them the correct proportions of strength and then mix them. This is exactly what Helmholtz has done by generating the simple tones without overtones by means of oscillating columns of air, which he caused to sound by resonance with tuning forks. Employing in this manner several different tuning forks, he has imitated the vowels of the human voice as well as the tones of organ pipes in different registers and the "nasal sound of the clarinet by several uneven overtones and the soft sounds of the horn by the complete choir of all tuning forks". We can only here refer to the possibility of generation of arbitrary timbres by electro-acoustic means. employed by Nernst in his piano.
If you move the phases of the curves c and b of Fig. 318 relative to a, you obtain quite a different curve, although the partial curves - the overtones - are the same. Does there correspond to this new form of oscillation another timbre, that is, does a phase displacement change the timbre? Helmholtz's answer to this question was no!.
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