Beats arise in the same way. The rhythmical shocks of two sounding bodies, of unequal pitch, sometimes coincide, sometimes interfere, whereby they alternately augment and enfeeble each other's effects. Hence the shock-like, unpleasant swelling of the tone.
Now that we have made ourselves acquainted with overtones and beats, we may proceed to the answer of our main question, Why do certain relations of pitch produce pleasant sounds, consonances, others unpleasant sounds, dissonances? It will be readily seen that all the unpleasant effects of simultaneous sound-combinations are the result of beats produced by those combinations. Beats are the only sin, the sole evil of music. Consonance is the coalescence of sounds without appreciable beats.
Fig. 12.
To make this perfectly clear to you I have constructed the model which you see in Fig. 12. It represents a claviatur. At its top a movable strip of wood aa with the marks 1, 2 ... 6 is placed. By setting this strip in any position, for example, in that where the mark 1 is over the note c of the claviatur, the marks 2, 3 ... 6, as you see, stand over the overtones of c. The same happens when the strip is placed in any other position. A second, exactly similar strip, bb, possesses the same properties. Thus, together, the two strips, in any two positions, point out by their marks all the tones brought into play upon the simultaneous sounding of the notes indicated by the marks 1.
The two strips, placed over the same fundamental note, show that also all the overtones of those notes coincide. The first note is simply intensified by the other. The single overtones of a sound lie too far apart to permit appreciable beats. The second sound supplies nothing new, consequently, also, no new beats. Unison is the most perfect consonance.
Moving one of the two strips along the other is equivalent to a departure from unison. All the overtones of the one sound now fall alongside those of the other; beats are at once produced; the combination of the tones becomes unpleasant: we obtain a dissonance. If we move the strip further and further along, we shall find that as a general rule the overtones always fall alongside each other, that is, always produce beats and dissonances. Only in a few quite definite positions do the overtones partially coincide. Such positions, therefore, signify higher degrees of euphony—they point out the consonant intervals.
These consonant intervals can be readily found experimentally by cutting Fig. 12 out of paper and moving bb lengthwise along aa. The most perfect consonances are the octave and the twelfth, since in these two cases the overtones of the one sound coincide absolutely with those of the other. In the octave, for example, 1b falls on 2a, 2b on 4a, 3b on 6a. Consonances, therefore, are simultaneous sound-combinations not accompanied by disagreeable beats. This, by the way, is, expressed in English, what Euclid said in Greek.
Only such sounds are consonant as possess in common some portion of their partial tones. Plainly we must recognise between such sounds, also when struck one after another, a certain affinity. For the second sound, by virtue of the common overtones, will produce partly the same sensation as the first. The octave is the most striking exemplification of this. When we reach the octave in the ascent of the scale we actually fancy we hear the fundamental tone repeated. The foundations of harmony, therefore, are the foundations of melody.
Consonance is the coalescence of sounds without appreciable beats! This principle is competent to introduce wonderful order and logic into the doctrines of the fundamental bass. The compendiums of the theory of harmony which (Heaven be witness!) have stood hitherto little behind the cook-books in subtlety of logic, are rendered extraordinarily clear and simple. And what is more, all that the great masters, such as Palestrina, Mozart, Beethoven, unconsciously got right, and of which heretofore no text-book could render just account, receives from the preceding principle its perfect verification.