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Scan 24
Harvard Dictionary Of Music(1950), 1950
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1ACOUSTICS
3amplitudes, corresponding to the different magnitudes of the initial disturbance. This leads to the first law of acoustics: The intensity of a sound depends upon the amplitude of the vibration [see *Bel]. Therefore a fading sound will show a vibration curve of gradually diminishing elongations [see below].
5Still more important is another element of variety, namely, that which enters if sounds of different pitch are studied. If the photographic experiment described above is repeated with a shorter tongue, a higher tone will be heard and the resulting curve will show vibrations of narrower width (provided that the speed of the moving film remains unaltered) [Ex. 4]. This means that the single vibration of the higher-pitched tongue takes a shorter time than that of the lower-pitched one. In other words, the higher sound makes more vibrations per second, i.e., has a greater frequency, than the lower sound. This is the basis of the second law of acoustics: The pitch of a sound depends only upon the frequency of the vibration. A sound is audible if its frequency is approximately between 16 and 20,000 cycles; the tones of the piano vary from about 30 to 4,000, those of the violin from about 300 to 3,000. The frequency of a middle A (a'), i.e., of concert pitch, is 440 (or 880, if single vibrations are counted).
7In the above law, the word only is of particular importance. It expresses the fact, known to every musician, that the pitch of a vibrating string is not altered by the greater or lesser force with which the string is plucked, or, in other words, that the pitch does not depend upon the amplitude. The piano player obtains a tone of the same pitch regardless of whether he uses a pianissimo or a fortissimo touch. The same principle is borne out by the fact that a sound does not alter its pitch when it gradually decreases in intensity. This means that a curve representing a fading sound [Ex. 5] will always have the form a, not the form b.
9II. Vibrating Strings. If a violin is plucked or bowed, each single point of the string will make an up-and-down vibration comparable to that made by the lowest point of the steel tongue previously described. All these vibrations have the same frequency, but differ in amplitude. For the purpose of our explanations, the vibration of the string can be considered as being represented by that of its point of highest vibration amplitude, i.e., of the point at which the string is plucked. If this is the middle point of the string, the resulting phenomenon can be roughly illustrated by Example 6.
11III. Frequency, Vibrating Length, and Pitch. The pitch produced by a vibrating string depends upon its material (steel, copper, etc.), its diameter, its tension, and its length. For the present purpose it is sufficient to consider only the latter factor, the others being regarded as constant. These conditions are realized in the case of a single string whose vibrating length can be changed by stopping (violin) or by means of a movable fret (*monochord). The following fundamental law results: The frequency is in inverse proportion to the vibrating length. This means that if the whole string (e.g., one yard) gives a sound of the frequency 600, the string of the half length (one-half yard) gives a sound of the double frequency, 1200, while a string of two-thirds of a yard produces the frequency 600 \( \times \frac{3}{2} = 900 \), etc. More important from the musical point of view is the relation between a given vibration and the pitch of the sound it produces. This problem was investigated and solved by Pythagoras, who established the law relating the pitch of a note to the length of the string by which it is obtained. The results have a more general application, however, if they are expressed in frequencies rather than in vibrating lengths. Thus expressed, they remain unchanged regardless of whether the sound is produced by a pipe or by a string, and they do not depend upon additional factors such as the tension, thickness, or material of the string. The fundamental principle is as follows: If the frequency of a tone is n, that of the octave is 2n, that of the fifth, \( \frac{3}{2} n \), and that of the major third, \( \frac{5}{4} n \). From these
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