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148 FOURIER'S THEOREM. [VL § 73.
4, &c. He has further shown that each individual wave-form admits of being thus compounded in only one way, and has provided the means of calculating, in any given case, how many and which members of the series will appear, their relative amplitudes and their differences of phase.
When translated from the language of Mechanics into that of Acoustics, the theorem of Fourier asserts, that every regular musical sound is resolvable into a definite number of simple tones whose relative pitch follows the law of the partial-tone series. It thus supplies a theoretical basis for the analysis and synthesis of composite sounds which have been experimentally effected in chapters iv. and v.
When we are listening to a sustained clang, the air at any one point within the orifice of the ear can have only one definite mode of particle-vibration at any one moment. How does the ear behave towards any such given vibration ? It proceeds as follows. If the vibration is simple, it leaves it alone. If composite, it analyses it into a series of simple vibrations whose rates are once, twice, three times &c. that of the given vibration, in accordance with Fourier's theorem. In the former event the ear perceives only a simple tone. In the latter, it is able to recognise, by suitably directed and assisted efforts, partial-tones corresponding to the rate of each constituent into