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The Melodic Relations of the Sounds of the Common Scale.
In describing the form of Syren devised by Helmholtz, it waa mentioned, tbat tbe lower revolving plate was pierced with four circles of 8, 10, 12, and 18 holes, and the upper with four circles of 9, 12, 15, and 16. If only the "8-hole circle" on the lower and the " 16-hole " circle on the upper be opened, while the Syren is working, two sounds are produced, the interval between which, the musician at once recognises as the Octave. When the speed of rotation is increased,both sounds rise in pitch, but they always remain an Octave apart. The same interval is heard, if the circles of 9 and 18 holes be opened together. It follows from these experiments, that when two sounds are at the interval of an Octave, the vibrational number of the higher one is exactly twice that of the lower. An Octave, therefore, may be acoustically denned as the interval between two sounds, the vibration number of the higher of which is twice that of the lower. Musically, it may be distinguished from all other intervals by the fact, that, if any particular sound be taken, another sound an octave above this, another an octave above this last, and so on, and all these be simultaneously produced, there is nothing in the resulting sound unpleasant to the ear.
Since the ratio of the vibration numbers of two sounds at the interval of an octave is as 2 : 1, it is easy to divide the whole range of musical sound into octaves. Taking the lowest sound to be produced by 16 vibrations per second, we have
Thus all the sounds used in music are comprised within the compass of about eight octaves.