CHAPTER III. (Continued)

On Proportions and their Differences.

Although the Musician may achieve his goal through continuous quantity Physical measurement, such as the length of a string on an instrument., by measuring lines, strings, and so on, because that dimension must ultimately be reduced to numbers—namely, by how many parts one line exceeds another in length, and by how much it is shorter—therefore I have decided to discuss in this place only proportion, which usually arises from discrete quantity Individual units or numbers that can be counted, rather than a continuous physical length., or from the comparison of number to number.

It is clear that the definition of proportion brought forward above is general and fits any kind of proportion; however, a specific one must be established that suits no discipline other than Music. It is as follows: Musical Proportion is the condition or relation of a sounding number to a sounding number. This definition is clear in itself from what has been said above and requires no further explanation.

This proportion can be of two kinds: equality and inequality. The proportion of equality arises from the comparison of an equal number to an equal number, such as 1 to 1, 2 to 2, 3 to 3, etc. The proportion of inequality is produced by the comparison of unequal numbers, for example, 1 to 2, 3 to 4, 5 to 8, etc.

It is established that musical intervals—that is, consonances and dissonances—are produced by the diversity of sounds; from this it is gathered that those proportions whose terms or extremes rely on the equality of numbers are neither suitable for producing intervals, nor can the unison be included under the name of "intervals" for that same reason Since an interval requires a "gap" or distance between two different pitches, a unison (1:1 ratio) technically has no distance.. A different judgment must be made about the unison in practice, which I shall carry out when I speak about each consonance and dissonance individually. Furthermore, from the comparison of an unequal number to an unequal number, many classes of proportions can exist, which the Musician is accustomed to restricting to five: the Multiple, Superparticular, Superpartient, Multiple Superparticular, and Multiple Superpartient genera. The following chapters will demonstrate that each of these genera encompasses almost infinite species.

CAPUT IV.

CHAPTER IV.

Concerning the Multiple Genus.

The Multiple genus is a proportion of number to number where the larger includes the smaller twice, thrice, four times, etc. For example, 2:1 (double), 3:1 (triple), or 4:1 (quadruple). If