# Melodic intervals

##### 2 minutes • 401 words

Melodic intervals are defined as being all the differences between consonances which are smaller than the double interval.

The natural faculty of hearing does not admit any other intervals as melodic but those which arise from this subtraction.

Thus the consonant intervals take their origin from geometry and the constructible figures, but the melodic intervals from the actual consonances; and the melodic intervals stand in relation to the consonances just as in geometry the Apotomae (inexpressible lines) stand to those expressible in square, for the former are also defined by subtraction of an expressible line from an expressible line.* *

Furthermore there is one method of comparison or abstraction which is general or arithmetic, and another which is special, and proper to harmony.

But it is by arithmetical means that consonances which are less than double are selected, so that one of them is not a part of another, in the sense of being indicated by some harmonic mean, as in the previous chapter.

There are between These Melodic- concords intervals In notes, by anticipation: 1^! ’ - i l —

- J - ———♦ . - - J - ’ - f c —
- I H I 6 The harmonic comparison of consonant intervals re fers to their origin, and to the degree of height which is assigned to each of them on account of its origin.

For every greater term in the comparison of propor . . . X A tions is represented by one and the same whole circle, 2 5 and the complete string which is analogous to it, as common to all harmonic divisions.

Therefore we have to find for all the numbers representing the greater terms in the seven harmonic divi sions, that is to say 2, 3, 4, 5, 6, 5, 8, a lowest common multiple, 120; and the whole string must be divided into the same number of equal parts, so that the sound of the whole string is established as the com mon greater term of all the consonances made by the divisions, and the smaller terms must be fitted in such a way that when set along side each other they set up the melodic intervals, which are investigated in this Chapter.

However the results are the same as before, arithmetically. This, therefore, is the origin of the dissonant melodic intervals, to which we shall give their names a little later on.