Hence both the rapidity of the beats, and the width of the interval, enter into the question of consonance. Helmholtz judges that in the middle and higher regions of the musical scale, when the beats reach 33 per second, the dissonance reaches its maximum. Both slower and quicker beats have a less grating or dissonant effect. When the beats are very slow, they may be of advantage to the music; and, when they reach 132 per second, their roughness is no longer discernible. Thanks to Helmholtz, whose views I have here sought to express in the briefest possible language, we are now in a condition to grapple with the question of musical intervals, and to give the reason why some are consonant and some dissonant to the ear. Circumstanced as we are upon earth, all our feelings and emotions, from the lowest sensation to the highest æsthetic consciousness, have a mechanical cause: though it may be forever denied to us to take the step from cause to effect; or to understand why the agitations of nervous matter can awaken the delights which music imparts. Take, then, the case of a violin. The fundamental tone of every string of this instrument is demonstrably accompanied by a crowd of overtones; so that, when two violins are sounded, we have not only to take into account the consonance or dissonance of the fundamental tones, but also those of the higher tones of both. Supposing two strings sounded whose fundamental tones, and all of whose partial tones, coincide, we have then absolute unison; and this we actually have when the ratio of vibration is 1:1. So also when the ratio of vibration is accurately 1:2, each overtone of the fundamental finds itself in absolute coincidence with either the fundamental tone or some higher tone of the octave. There is no room for beats or dissonance. When we examine the interval of a fifth, with a ratio of 2:3, we find the coincidence of the partial tones of the two so perfect as almost, though not wholly, to exclude every trace of dissonance. Passing on to the other intervals, we find the coincidence of the partial tones less perfect, as the numbers expressing the ratio of the vibrations become more large. Thus, the dissonance of intervals whose rates of vibration can only be expressed by large numbers, is not to be ascribed to any mystic quality of the numbers themselves, but to the fact that the fundamental tones which require such numbers are inexorably accompanied by partial tones whose coalescence produces beats, these producing the grating effect known as dissonance. § 6. _Graphic Representation of Consonance and Dissonance_