a more comprehensive view of the manner in which they follow the law of numerical ratio just explained than any more lengthened exposition. It is here requisite to mention, that in the construction of these scales, I have not only adopted the old German or literal mode of indicating the notes, but have included, as the Germans do, the note termed by us B flat as B natural, and the note we term B natural as H. Now, although this arrangement differs from that followed in the construction of our modern Diatonic scale, yet as the ratio of 4:7 is more closely related to that of 1:2 than that of 8:15, and as it is offered by nature in the spontaneous division of the monochord, I considered it quite admissible. The figures give the parts of the monochord which would produce the notes. I. { (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)* { C D E F G A B H _c_ II. { (¹⁄₂)* (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃)* (³⁄₁₀) (²⁄₇) (²⁄₁₅) (¹⁄₄)* { _c_ _d_ _e_ _f_ _g_ _a_ _b_ _h_ _c′_ III. { (¹⁄₄)* (²⁄₉) (¹⁄₅)* (³⁄₁₆) (¹⁄₆)* (³⁄₂₀) (¹⁄₇)* (²⁄₁₅) (¹⁄₈)* { _c′_ _d′_ _e′_ _f′_ _g′_ _a′_ _b′_ _h′_ _c′′_ IV. { (¹⁄₈)* (¹⁄₉)* (¹⁄₁₀)* (³⁄₃₂) (¹⁄₁₂)* (³⁄₄₀) (¹⁄₁₄)* (¹⁄₁₅)* (¹⁄₁₆)* { _c′′_ _d′′_ _e′′_ _f′′_ _g′′_ _a′′_ _b′′_ _h′′_ _c′′′_ The notes marked (*) are the harmonics which naturally arise from the division of the string by 2, 3, 5, and 7, and the multiples of these primes. Thus every musical sound is composed of a certain number of parts called pulsations, and these parts must in every scale relate harmonically to some fundamental number. When these parts are multiples of the fundamental number by 2, 4, 8, &c., like the pulsations of the sounds indicated by _c_, _c′_, _c′′_, _c′′′_, they are called tonic notes, being the most consonant; when the pulsations are similar multiples by 3, 6, 12, &c., like those of the sounds indicated by _g_, _g′_, _g′′_, they are called dominant notes, being the next most consonant; and multiples by 5, 10, &c., like those of the sounds indicated by _e_, _e′_, _e′′_, they are called mediant notes, from a similar cause. In harmonic combinations of musical sounds, the æsthetic feeling produced by their agreement depends upon the relations they bear to each other with reference to the number of pulsations produced in a given time by the fundamental note of the scale to which they belong; and it will be observed, that the more simple the numerical ratios are amongst the pulsations of any number of notes simultaneously produced, the more perfect their agreement. Hence the origin of the common chord or fundamental concord in the united sounds of the tonic, the dominant, and the mediant notes, the ratios and coincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:— [Illustration]