grave harmonic, while all the other intervals have, shows that the natural laws are not confined to one style of working.

When the major scale has been generated, with its three chords, the subdominant, tonic, and dominant, by the primary mathematical ratios, it consists of forms and orders which in themselves are adapted to give outgrowth to other forms and orders by the law of duality and other laws. All the elements, orders, combinations, and progressions in music are the products of natural laws. The law of Ratio gives quantities, form, and organic structure. The law of Duality gives symmetry, producing the minor mode in response to the major in all that belongs to it. The laws of Permutations and Combinations give orders and rhythms to the elements. The law of Affinity gives continuity; continuity gives unity; and unity gives the sweetness of harmony. The law of Position gives the notes and chords their specific levities and gravities; and these two tendencies, the one upward and the other downward, constitute the vital principle of music. This is the spiritual constitution of music which the Peter Bell mathematicians have failed to discern:

If the effects of notes and chords had depended entirely on their mathematical ratios, then the effect of the subdominant, tonic, and dominant would have been alike; for these three chords have exactly the same ratios. It is the law of position which gives the tonic chord its importance, and not any special ratios embodied in its structure. The ratio of 2 to 1 has a pure, unmixed, invariable character, always realized in the interval of the octave. The notes produced from 1 by the first, second, and third powers of 3 have different degrees of centrifugal force. The character of the notes produced by the first power of 5 depends on the character of the notes from which they are derived, namely, 1, 3, and 9. The final character of the notes and chords derived by the same ratios is determined by the amount of force which they have acquired from the way in which they have been derived, and from their position in the system; and no matter where these notes may afterwards be