HARMONIC ANALYSIS.) The theorem not only deals with the synthesis of a complex form of vibration, but also indicates a method of analysis of such a vibration into its component simple vibrations or harmonics. Thus if x is the resultant displacement at a time t, in a complex vibration of frequency n/ 2r, By suitable choice of the amplitude-values A1, etc., etc., it is possible to analyse or synthesise any form of single valued periodic vibration. Thus the displacement curve repre sented by fig. I (a), may be obtained by adding a sufficient number of odd terms of a sine series. In fig. 1 (b) the first, third and fifth terms are added, in fig. I (c) the first fifteen odd sine terms are added. It will be seen that (c) approximates closely to (a), the more terms taken the more nearly does the syn thesised curve approach the ideal. The mathematical analysis or synthesis of complex wave-forms may become very laborious. In order to simplify the process various mechanical "harmonic analysers" have been constructed (see Millar's " Science of Musical Sounds") which perform the necessary mathematical integrations, by a direct mechanical process. Fourier's theorem and harmonic analysis have a wide field of application not only in the study of sound-vibrations, but in astronomy, meteorology, tide prediction, mechanical and electrical engineering.
(i) Loudness and Intensity.—These two terms refer to the sub jective and physical aspects respectively. The intensity of a sound refers to a definite physical quantity which determines the rate of supply of vibrational energy (proportional to [ampli tude]). Loudness corresponds to the degree of sensation, being necessary to produce a just perceptible increase of sensation is proportional to the pre-existing stimulus." This law indicates a rapid diminution of sensitiveness of the ear with increase of total intensity of the sound.
(ii) Pitch and Frequency.—The frequency of a regular or periodic vibration is the number of vibrations performed per second.
Musical sounds arrange themselves in a natural order according to pitch. The latter depends solely on the predominant fre quency of the vibrations—the greater this frequency the higher the pitch—and on the number of these vibrations reaching the ear per second. The latter stipulation is made to include sounds received from sources of sound in motion. This relation between pitch and frequency is simply verified by means of a revolving toothed wheel striking the edge of a card which produces a sound whose predominant frequency is proportional to the product of speed (revolutions per second) and the number of teeth on the wheel. For a given note the predominant frequency is the same whatever the source of the note, and the ratio of the frequencies of two notes forming a given musical interval is the same in whatever part of the musical range the two notes are situated. The more important consonant intervals with their frequency ratios are Unison I :I, Major Third 5 :4, Fifth 3:2, Major Sixth 8:5, Minor Third 6:5, Fourth 4:3, Minor Sixth 5:3, Octave 2 : i . Notes whose frequencies are multiples of that of a given one, the fundamental, are called harmonics. The frequency ratios defining each note of the diatonic musical scale are The same series of ratios applies to any octave which may be chosen.
(iii) Quality and Wave-Form.—Sounds of the same pitch and loudness, but produced by different means, are distinguished by their quality. Thus the same note produced by a voice, a piano, or a violin, would have distinct characteristics which are at once recognisable by the ear. Very few sounds can be regarded as "pure," that is, free from overtones. The presence or absence of these overtones decides the quality of the sound. A tuning fork emits almost a pure tone whereas a violin note is rich in over tones characteristic of the instrument. Quality depends there fore on wave-form. Fourier's analysis of the wave-form of the sound emitted by a particular instrument tells us which har monics are present and their relative impoi tance.