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)

The scales I., II., and III. may now be rendered as complete as scale IV., simply by multiplying upwards by 2 from (¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇), and (¹⁄₁₅), thus:—

I.(1)(⁸⁄₉)(⁴⁄₅)(²⁄₃)(⁴⁄₇)(⁸⁄₁₅)(¹⁄₂)
II.(¹⁄₂)(⁴⁄₉)(²⁄₅)(¹⁄₃)(²⁄₇)(⁴⁄₁₅)(¹⁄₄)
III.(¹⁄₄)(²⁄₉)(¹⁄₅)(¹⁄₆)(¹⁄₇)(²⁄₁₅)(¹⁄₈)
IV.(¹⁄₈)(¹⁄₉)(¹⁄₁₀)( )(¹⁄₁₂)( )(¹⁄₁₄)(¹⁄₁₅)(¹⁄₁₆)

We now find between the beginning and the end of scale I. the quantities (⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅).

The three first of these quantities we find to be the remainders of the whole indefinite quantity contained in (1), after subtracting from it the primary harmonic quantities (¹⁄₉), (¹⁄₅), and (¹⁄₃); we, however, find also amongst these harmonic quantities that of (¹⁄₄), which being subtracted from (1) leaves (³⁄₄), a quantity the most suitable whereby to fill up the hiatus between (⁴⁄₅) and (²⁄₃) in scale I., which arises from the omission of (¹⁄₁₁) in scale IV. In like manner we find the two last of these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest of the two parts into which 7 and 15 are susceptible of being divided. Finding the number 5 to be divisible into parts more unequal than (2) to (3) and less unequal than (4) to (7), (³⁄₅) naturally fills up the hiatus between these quantities in scale I., which hiatus arises from the omission of (¹⁄₁₃) in scale IV. Thus:—

I.(1)(⁸⁄₉)(⁴⁄₅)(³⁄₄)(²⁄₃)(³⁄₅)(⁴⁄₇)(⁸⁄₁₅)(¹⁄₂)
II.(¹⁄₂)(⁴⁄₉)(²⁄₅)( )(¹⁄₃)( )(²⁄₇)(⁴⁄₁₅)(¹⁄₄)
III.(¹⁄₄)(²⁄₉)(¹⁄₅)( )(¹⁄₆)( )(¹⁄₇)(²⁄₁₅)(¹⁄₈)
IV.(¹⁄₈)(¹⁄₉)(¹⁄₁₀)( )(¹⁄₁₂)( )(¹⁄₁₄)(¹⁄₁₅)(¹⁄₁₆)

Scale I. being now complete, we have only to divide these latter quantities by (2) downwards in order to complete the other three. Thus:—

I.(1)(⁸⁄₉)(⁴⁄₅)(³⁄₄)(²⁄₃)(³⁄₅)(⁴⁄₇)(⁸⁄₁₅)(¹⁄₂)
II.(¹⁄₂)(⁴⁄₉)(²⁄₅)(³⁄₈)(¹⁄₃)(³⁄₁₀)(²⁄₇)(⁴⁄₁₅)(¹⁄₄)
III.(¹⁄₄)(²⁄₉)(¹⁄₅)(³⁄₁₆)(¹⁄₆)(³⁄₂₀)(¹⁄₇)(²⁄₁₅)(¹⁄₈)
IV.(¹⁄₈)(¹⁄₉)(¹⁄₁₀)(³⁄₃₂)(¹⁄₁₂)(³⁄₄₀)(¹⁄₁₄)(¹⁄₁₅)(¹⁄₁₆)

The harmony existing amongst these numbers or quantities consists of the numerical relations which the parts bear to the whole and to each other; and the more simple these relations are, the more perfect is the harmony. The following are the numerical harmonic ratios which the parts bear to the whole:—