The Causes of Consonances

Part 1: Dissonance and Consonance

“Dissonant” means “asymphonic”

“Consonant” means “symphonic.” This is divided into:

• “identical” which we shall adopt instead of “homophonic”. This has 2 kinds:
  • single-sounding
  • identical by opposition
• “non-identical” which we shall adopt instead of “diaphonic.”

Part 2: Part and Parts

In geometry the terms “part” and “parts” are different.

“Part” is used for that of which the whole is a multiple in a certain proportion, such as double, triple, quadruple

“Parts” is used when not a single and unique whole but a quantity of wholes is a multiple of them.

Thus, 1 7th is called a part because the whole circle is 7 times the part. But 3 sevenths are called not a part but parts, because a total of three circles is seven times the arc.

Here however we shall not use that distinction; but we shall call a part one of the fractions mentioned as much as the other.

That is, every fraction which is expressible^^ we shall call a part, though with the restriction, provided it is not greater than a semicircle.

The term “remainder” on the other hand will be used for what is left, being not less than a semicircle, when an expressible portion [in length] is subtracted from the whole.

The distinction between a remainder and a part is extremely necessary, because a part can be a consonance, and its remainder a dissonance, as we shall see.

Part 3: A string [chord] is any length which can emit a sound

It is not the line subtended by an arc of a circle, as in geometry.

This sound is elicited by motion.

“String” is abstract in reference to the length of any motion whatever, or to any other length whatever, even if it is conceived in the mind.

Axiom 1: The diameter of a circle, and the sides of the fundamental figures expounded in Book I, which have a proper construction, mark off a part of the circle which is consonant with the whole circle.

How the circle can be stretched out, so that it emits sounds, and how it must he fastened to a hollow body, so that resonance occurs, either at one mark so that the whole sounds, or at two, so that the parts sound, it would be a lengthy business to expound here.

However it was necessary to start in this way because it is not only a question of melody, which is harmony realized in sounds, but the underlying reference to an interval in abstraction from sound must be understood.

As far as music is concerned, it is sufficient that a string stretched out straight can be divided in the same way as when it is bent round into a circle it is divided by the side of the inscribed figure.

Corollary: The consonances are infinite, because the constructible figures are infinite.

However it is not yet time to speak of the identification of consonances, which does not make itself very obvious. On this point the Pythagoreans sought in their numbers, as causes, the bounds of the size of consonant intervals, which only the human hearing fixes for them, which is not of infinite power.

The restriction of the number of consonances by the abstract harmonic intervals is therefore only accidental, and not causal. Even the musicians of today themselves overstep the Pythagorean bounds, to say nothing here about celestial harmonies.

Axiom 2

To the same extent as the construction of a side is remote from the first degree, the consonance of a part of a circle, cut off by the side, with the whole circle, deviates from the most perfect consonance of unison; or, the allotted place of the figure of which it is the side among other figures is the same as the place of that consonance among the others.

This subordinate axiom will be adoptedfor the identification of consonances, with respect to their giving pleasure.

Axiom 3

The sides of the regular and star figures which are not constructible mark off a part of the circle which is dissonant from the whole circle. The same applies to the side of a figure which is in fact constructible but not in its own right, nor by a proper construction.

Or in place of the lack of a proper construction, consider lack of congruence, as in Book II. By both methods the fifteen sided figure is excluded.

This axiom will round off and complete the cause of consonance which I am substituting for the Pythagorean abstract numbers which have been repudiated.

Corollary

Then these parts are dissonant , 2 , 3

• 4 1 , 2 , 1 , 2 , 3, 4, 1 , 2 , 3, 4, 3, 1 , -
• 4, 2 , 1 , I, 2 , 3, 4,

1 , - I, 2 , 3, 4, 1

5 5, 5,

5, 5, 5, .. . . . . . . . . . . 6

. . .

• 7 . . . . . . 8 ………… 6 , 7,
• 7, - ………… . . . . . . … 6 , 7, 8 , 9 from the whole 7 9 II 13 14 15 173« 18 19 and so on to infinity.

Axiom 4

Figures which have kindred constructions for their sides, also give rise to kindred harmonies. Through this axiom the origin and cause of the harmonic proportions will be proved superabundantly.

Axiom 5

Strings or arcs of a circle, of equal tensions, having to each other, with respect to their length, the same proportion as the Part or residue of a circle has to the whole circle, also have the same consonance or dissonance, although it occurs between different limits or sounds.

Let it thus be understood in the abstract that a circle stands in certain harmonic proportions to its part; and that within whatever various limits, whether sounds, or soundless motions they are found, they are always harmonic.

This axiom is added, because not all harmonic proportions arise immediately from the circle itself, directly from its division by means of a regular figure, but some accrue which are generated from the prior ones themselves, up to a certain limit, as we shall see in the propositions.

The application of this axiom is in Propositions VII and VIII.