MUSIC 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 question. The modern knowledge that the length of a given string is inversely as the vibration number refers the question 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 dissonance, 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 coincidences and oppositions of the motions or pressures arising from the two sets of vibrations.

In other cases, however, this explanation is not applicable. 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, Sc., 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 imperfect 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.

Beats of Combination Tones.---When two notes are sounded loudly at the same time they give rise to the appearance of certain other notes within the ear, which are called combination tones. Call the vibration numbers of the two notes p and q. Then the first combination tone (Tartini tone, difference tone) has the frequency p– q. Other combination tones are also formed, whose frequencies are of the forms p – 2q, p – 3q, and so on. Each of these has its region of greatest intensity when its frequency is smallest consistently with its forming an audible sound. Thus the first difference tone (p – q) is most powerful when p and q differ only by one or more semitones, though the note is still recognizable by the beats it produces when p and q are a fifth or an octave apart. The beats of mistuned consonant intervals other than the beats of harmonics are produced by the formation of imperfect unisons between combination tones and primaries, or among the combination tones themselves.

Intervals of the ,fartn h 1. - These comprise the intervals formed between fundamental and harmonics. The beats of mistuned consonances of the form h : 1, other than the beats of harmonics, consist of variations of intensity of the lower note of the pair. This rule has been established experimentally by the employment of the pure notes furnished by bottles blown from an organ-bellows.

Octave. Let the notes be 100 : 201. p–q= 101, which with 100 gives one beat per second.

Twelfth. Let the notes be 100 : 301. p – 2 q=101, which with 100 gives one beat per second.

Double octave. Let the notes be 100 : 401. p –3q=101. One beat per second as before, and so on.

These explanations satisfy the observation that the beats are on the lower notes of the combinations.

Other Consonant Intervals. - The chief remaining consonance which furnishes beats is the fifth. There is no doubt that the beats of the fifth, other than the beats of harmonics, are chiefly on the octave below the lower note. Hence the following explanation : -