Combination of Musical Sounds—The smaller the Two Numbers which express the Ratio of their Rates of Vibration, the more perfect is the Harmony of Two Sounds—Notions of the Pythagoreans regarding Musical Consonance—Euler’s Theory of Consonance—Theory of Helmholtz—Dissonance due to Beats—Interference of Primary Tones and of Over-tones—Mechanism of Hearing—Schultze’s Bristles—The Otoliths—Corti’s Fibres—Graphic Representation of Consonance and Dissonance—Musical Chords—The Diatonic Scale—Optical Illustration of Musical Intervals—Lissajous’s Figures—Sympathetic Vibrations—Various Modes of illustrating the Composition of Vibrations § 1. _The Facts of Musical Consonance_ The subject of this day’s lecture has two sides, a physical and an æsthetical. We have to-day to study the question of musical consonance—to examine musical sounds in definite combination with each other, and to unfold the reason why some combinations are pleasant and others unpleasant to the ear. Pythagoras made the first step toward the physical explanation of the musical intervals. This great philosopher stretched a string, and then divided it into three equal parts. At one of its points of division he fixed it firmly, thus converting it into two, one of which was twice the length of the other. He sounded the two sections of the string simultaneously, and found the note emitted by the short section to be the higher octave of that emitted by the long one. He then divided his string into two parts, bearing to each other the proportion of 2:3, and found that the notes were separated by an interval of a fifth. Thus, dividing his string at different points, Pythagoras found the so-called consonant intervals in music to correspond with certain lengths of his string; and he made the extremely important discovery that _the simpler the ratio of the two parts into which the string was divided, the more perfect was the harmony of the two sounds_. Pythagoras went no further than this, and it remained for the investigators of a subsequent age to show that the strings act in this way in virtue of the relation of their lengths to the number of their vibrations. Why simplicity should give pleasure remained long an enigma, the only pretence of a solution being that of Euler, which, briefly expressed, is, that the human soul takes a constitutional delight in simple calculations. The double siren (Fig. 163) enables us to obtain a great variety of combinations of musical sounds. And this instrument possesses over all others the advantage that, by simply counting the number of orifices corresponding respectively to any two notes, we obtain immediately the ratio of their rates of vibration. Before proceeding to these combinations I will enter a little more fully into the action of the double siren than has been hitherto deemed necessary or desirable. [Illustration: FIG. 163.]