SCIENTIFIC BASIS.] MUSIC 105 Loud. Not loud. 1.2. 3. 4 . 5 . 6. 7 . 8. 9. 10 fff ff f m f m P P PP PPP Magnitude 1 serves for sounds louder than those used in music, 10 for sounds softer than those used in music, microscopic sounds, so to speak. It seems reasonable, without going into detail, to assume that the mechanical ratio of any two consecutive magnitudes is the same. The general expression of this law is, The measure of sensation is the logarithm of the mechanical excitation. It appears probable that the ratio of energy corresponding to one of the above differences of magnitude is somewhere about 2 or 3. The corresponding ratio depending on amplitude or compression would be from 1 4 to 1 *7, but these quan tities are not known with any accuracy. 4. The pitch depending on the period or frequency and the loudness on the amplitude or magnitude of the changes, there remains on the one hand the quality of tone, and on the other the form of the vibration or the manner in which the motion takes place between the prescribed limits. These may be expected to correspond with each other, and in fact they do so. The peculiarities of form of vibration are most easily discussed in the case of a musical string whose vibrations are started or maintained in a given manner. The smoothest and purest quality of tone that can be produced is known as a simple tone. When a string produces a simple tone its motion is such that its shape at any moment is that of a curve of sines, and that every point of the string executes oscillations according to the pendulum law. Simple tones are also produced by any vibrating surface which moves according to the pendulum law. The method for producing simple tones given by Helmholtz, and com monly employed, is to use tuning-forks as the sources of sound, and present their extreme faces to the opening of a resonator or air-chamber arranged so as to vibrate to the same note as the fork. Resonators may be conveniently made from wide- mouthed bottles with flat corks having holes bored in them. The dimensions are usually found by trial, though data exist for their calculation. Simple tones have also been produced by fitting a sort of organ- pipe mouthpiece into the corks of such bottles. The mouths require to be cut up much higher than usual ; the notes produced are of an exceedingly full and pure char acter. Such bottle- notes can be blown from an organ- bellows, and being easily manipulated are very suitable for experiments on the properties of simple tones. The law of Ohm states that the simple tone or pendulum vibration is that to which the sensation of pitch is at tached in its simplest form. If the motion which consti tutes the vibration of a note be of any other type, it is capable of being analysed by the ear into a series of simple tones according to what is called Fourier s Theorem. This is most simply described in connexion with stretched strings, assuming that the notes which are exhibited in the form of the string pass over into the air through the sound-board without essential alteration of quality, which appears to be true in a general way. Fourier s Theorem, as applied to a string, states that the motion of the string is equivalent to the sum of the motions which would result if there were a curve of sines of the whole length, two curves of sines each of half the length, three each of one- third the length, and so on, the amplitudes being deter mined when the total motion to be represented is given. This equivalence is true mechanically ; the law of Ohm says that it is also true for the ear. Hence a great presumption that the ear acts by a receptive mechanism obeying the laws of mechanics. The notes formed by the division of a string into two, three, or more parts are commonly called harmonics. They are also called overtones ; but this word includes such cases as those of bars, &c., where the notes produced by these divisions are not harmonious with the fundamental. Har monics play an important part in the theory of consonant combinations, but the theory of consonance cannot be rested entirely upon the properties of harmonics. CONSONANCE AND DISSONANCE. It was already known in ancient times that lengths of the same stretched string having the ratio of any two small whole numbers form consonant intervals, or perhaps we may more correctly say smooth combinations, since the interval of a fourth (3 : 4) is regarded as a dissonance in technical music, though it is a smooth combination. The question why smooth combinations are associated with small whole numbers is known as the Pythagorean ques tion. The modern knowledge that the length of a given string is inversely as the vibration number refers the ques tion more generally to vibration numbers rather than to lengths of string. This question has been answered by Helmholtz ; we proceed to give a short account of his answer, with some slight modifications. It has been long known that when two notes form an imperfect unison, or nearly form almost any smooth combination, flutterings or beats are heard. These have been already described in the case of imperfect unisons where two notes differ but little from each other in pitch. They exist also, in most cases, where two notes nearly, but not quite, form a smooth combination. According to Helmholtz, beats are the cause of the sensation of disson ance, and to seek further for this cause we must seek the cause of beats. We may note that this must be taken with some limitation, since the fourth is regarded as a dissonance, though it presents no beats. In the case of imperfect unisons there is no difficulty. Such beats have been long explained as arising out of the alternate coin cidences and oppositions of the motions or pressures arising from the two sets of vibrations. In other cases, however, this explanation is not applic able. Explanations similar in principle have been given by Smith, an English writer of the last century; but these only amount to reckoning the recurrence of certain configurations arising from the superposition of the two sets of motions. No hypothesis is made as to the actual nature of the receptive mechanism of the ear, and no attempt is made to determine of what sounds the beats consist, nor how such sounds arise. We have already seen that the ear receives separately notes which are more than one or two semitones apart. They appear to be received on different parts of the aural mechanism. The production of the beats in the case of imperfect fifths, octaves, ifcc., where the impulses fall on different parts of the receptive mechanism, appears therefore to be due to secondary causes rather than to the direct superposition of the impulses. Beats of Harmonics. This class of beats arises from the fact that in compound notes containing harmonics a pair of notes representing two small whole numbers gives rise to the coincidence of a pair of harmonics forming a unison, and, if the interval be mistuned, the harmonics form an imperfect unison. Beats of this description are easily identified by the pitch of the harmonics. The imper fect unison gives rise to alternations of sound and silence, or to variations of intensity, of a note having the pitch in question. With practice these variations can be heard with the unaided ear. But the employment of resonators, tuned to the pitch in question and connected with the ear, causes the beat to be heard with great intensity. Jkats of Combination Tones. When two notes are XVII. 14
