The facts of the relations of tones, the elements, that is, of melody and harmony, are as follows. We cannot avoid the observation that certain tones "go together," as the phrase is, while others do not. This peculiar impression of belonging together is known as consonance, or harmony. The intervals of the octave, the fifth, the third, for instance, that is, C-C', C-G, C-E, in the diatonic scale, are harmonious; while the interval of the second, C-D, is said to be dissonant. Consonance, however, is not identical with pleasingness, for different combinations are sometimes pleasing, sometimes displeasing. In the history of music we know that the octave was to the Greeks the most pleasing combination, to medieval musicians the fifth, while to us, the third, which was once a forbidden chord, is perhaps most delightful. Yet we should never doubt that the octave is the most consonant, the fifth and the third the lesser consonant of combinations. We see, thus, that consonance, whatever its nature, is independent of history; and we must seek for its explanation in the nature of the auditory process. Various theories have been proposed. That of Helmholtz has held the field so long that, although weighty objections have been raised to it, it must still be treated with respect. In introducing it a short review of the familiar facts of the physics and physiology of hearing may not be out of place. The vibration rates per second of the vibrating bodies, strings, steel rods, etc., which produce those musical tones which are consonant, are in definite and small mathematical ratios to each other. Thus the rates of C-C' are as 1:2; of C-G, C-E, as 2:3, 4:5. In general, the simpler the fraction, the greater the consonance.