same sound, because the length of the vibrations of each will be the same; and when there is a difference in the tone, it will be found that there is a difference in the length of the vigrations. This fact can be demonstrated by varying the vibrations—by tightening or loosening the strings, and thus shortening or lengthening the vibrations, when it will be perceived that the shorter the vibration the higher will be the pitch or tone. The length of vibration, then, determines the question of harmony. Here appears the great law of harmony in musical sound throughout the universe, which is commensurability. In mathematics, things which will mutually measure each other are said to be commensurable. Now these spheres of atmospheric vibration will always produce concord or harmony of sound. The difference between a third and a fifth is in the difference in the tone, and the difference in tone depends, as already said, upon the length of vibration. The sweetest harmony is the apparent discord, where the vibrations do not chord, but where every fifth coincides; and in this way produces the harmony of the third and fifth. The octave produces it by being repeated twice, so that after all the real octave is as the square of the octave; that is, the octave multiplied into itself; and you arrive mathematically at the law of harmony by following out that principle. The point to which I wish to call your attention is, that what constitutes harmony is simply commensurability in the atmospheric undulations.
Now my nerve-fluid moves by pulsatory movements, as move all other media, and these movements sustain to those of your nerve-fluid commensurable or incom-
