The elements of the Pythagorean system of harmonic number, so far as can be gathered from the quotations I have given above, seem to be simply the indivisible monad (1); the duad (2), arising from the union of one monad with another; the triad (3), arising from the union of the monad with the duad; and the tetrad (4), arising from the union of one duad with another, which tetrad is considered a perfect number. From the union of these four elements arises the decad (10), the number, which, agreeably to the Pythagorean system, comprehends all arithmetical and harmonic proportions. If, therefore, we take these elements and unite them progressively in the following order, we shall find the series of harmonic numbers (2), (3), (5), and (7), which, with their multiples, are the complete numerical elements of all harmony, thus:— 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 4 = 7 In order to render an extended series of harmonic numbers useful, it must be divided into scales; and it is a rule in the formation of these scales, that the first must begin with the monad (1) and end with the duad (2), the second begin with the duad (2) and end with the tetrad (4), and that the beginning and end of all other scales must be continued in the same arithmetical progression. These primary elements will then form the foundation of a series of such scales. I. (1) (2) II. (2) (3) (4) III. (4) (5) (6) (7) (8) IV. (8) (9) (10) ( ) (12) ( ) (14) (15) (16) The first of these scales has in (1) and (2) a beginning and an end; but the second has in (2), (3), and (4) the essential requisites demanded by Aristotle in every composition, viz., “a beginning, a middle, and an end;” while the third has not only these essential requisites, but two intermediate parts (5) and (7), by which the beginning, the middle, and the end are united. In the fourth scale, however, the arithmetical progression is interrupted by the omission of numbers 11 and 13, which, not being multiples of either (2), (3), (5), or (7), are inadmissible. Such is the nature of the harmonic law which governs the progressive scales of numbers by the simple multiplication of the monad. I shall now use these numbers as divisors in the formation of a series of four such scales of parts, which has for its primary element, instead of the indivisible monad, a quantity which may be indefinitely divided, but which cannot be added to or multiplied. Like the monad, however, this quantity is represented by (1). The following is this series of four scales of harmonic parts:— I. (1) (¹⁄₂) II. (¹⁄₂) (¹⁄₃) (¹⁄₄) III. (¹⁄₄) (¹⁄₅) (¹⁄₆) (¹⁄₇) (¹⁄₈) IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)