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:— I. (1:1) (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 following are the principal numerical relations which the parts in each scale bear to one another:—