The Causes of Consonances

Proposition 1: The consonance of a half with the whole, apart from unison, is the only one which is in the first degree simple, perfect and identical, that is identical by opposition.

For that which is in the nature of a figure is made up of diverse elements, and is therefore not simple or identical. For a figure has area, and parts in respect of its area, and angles which dijfer in position.

On the other hand that which is not in the nature of a figure, because of course it is without breadth of area, and in that respect also without parts, and angles, being merely a straight line, and of a measure equal to that proposed, is therefore itself both simple, and the same in its measure, that is identical.

The regular figures are in fact of the former kind, when they are inscribed in a circle; the diameter of a circle is of the latter kind.

For:

1. all the sides offigures diverge equally from the center: the diameter passes through the center itself.

2. A chord,^ which divides a circle from a point with the side of a figure as measure, when it has proceeded to do that a number of times, eventually returns with the other end of the side to the first point: the diameter on the other hand, passing through the center itself, returns at once at thefirst repetition to the initial point.

3. The rest of the figures possess both length of sides and area of the surface which they surround: the diameter, which neither surrounds nor encloses any part of a plane, in continual repetitions wholly coincides with itself every second time.

4. The other figures when they divide a circle make many parts: the diameter makes the smallest of all numbers of parts, that is two; for if it is to divide the whole, it absolutely could not make fewer parts than two.

5. And since the diameter is the measure with which the sides of the figure are to be compared, for purposes both of conception and of construction, the sides of the remaining figures are more laborious to draw, and are brought to achievement of knowledge with a more imperfect degree of construction; but the diameter of a circle is drawn according to the simplest law, so that it passes through the center, from one point on the circle to the one opposite, and is equal to itself, and the measure of itself.

6. Also the sides of the figures in a single division of the circle, or in the cutting off of a part, make unequal portions, and a part which is smaller than the remainder: the diameter leaves a part cut off which is equal to the remainder.

Now this proportion of equality is pure and simple and perfect, because parts which are equal among themselves, are as far as mensuration is concerned the same thing.

7. Lastly, the other figures do indeed divide the circumference of the circle into a number of equal parts, but the area of the circle into a number of unequal parts, because one—that is, the area of the figure—is left in the middle which is larger than any one of the segments: the diameter divides not only the circumference but at the same time the area also into two equal parts.

But Axiom 2 says that the character of the side or line which divides the circle consonantly passes over to the consonance itself.

Therefore the consonance of the part which the diameter cuts offfrom the circle, that is of the semicircle with the whole circle, is simple, perfect, and identical.

Also by Axiom V all other lengths which are to each other as the whole circle is to half of itself, make the same, that is identical, perfect, and simple consonance. Further, in the case of numbers (not certainly of abstract and counting numbers, but of lengths which are counted numbersthe double proportion, that is between I and 2 and also between equal multiples of them, gives rise to identical consonance.

Note here how the diameter through all its simplicities and perfections is nevertheless not as simple as a point, but remains a line bounded by two points of the circle, cutting the circle in opposite positions, and establishing two parts. Just as those parts, although they are equal to each other, are individually less than their own whole, so also an identical consonance is nevertheless not a unison; and of notes although they are in identical consonance yet one is smaller, the other larger.

That is, the former is high, the latter low, corresponding with the former, so to speak, from the opposite side; so this is called an identical consonance by opposition.

You have, then, from the diameter of the circle the true cause through which the sound of a whole string with the sound of half the string, though they are different from each other, is yet taken by the hearing as in a way the same in comparison with the other consonances.

Others seek for the cause of this identity of sound in the number of the 8 notes, vainly, since this identity of sound is by nature prior to the division of this interval into the seven melodic parts by which the eight sounds are designated. However it is not yet time to give a name to that consonance, nor to the rest; for that must be deferred to Chapter V.

Yet notice also the fact that other parts also are identical consonances although they are not established by the diameter, but not in the first degree, nor through the figures, but through their propagation, which is the subject of the following propositions.

Proposition :

If of two parts of a circle the smaller is to the larger as the larger is to the whole circle, in some other proportion than successive doubling, then if the larger is in consonance with the whole circle, the smaller part will be in dissonance with it.

For after the double comes the triple. Now successive tripling puts in third place the ninth part of the whole circle, successive multiplication by 5 the twenty-fifth part; and successive multiplication by six implies the ninth part, successive multiplication by ten the twenty-fifth part, because six times six is 36 which is four times nine, and ten times ten is 100, which is four times 23.

Likewise for the rest. But a ninth, and a twenty-fifth, and similar parts are dissonant from the whole, by Axiom III. See Proposition XLVII in thefirst Book.

Proposition 3

Strings in the proportion of successive doubling are in identical consonance with each other, but those in more distant proportion are in consonance at a more remote degree.

For the three nearest are to each other as the whole circle is to the half, and to the quarter respectively. But both the half and the quarter are in consonance with the whole circle, by Axiom I.

Also the quarter is in consonance with the half, by Axiom V. Therefore all the three nearest proportions are in consonance with each other.

Further, the consonance of the quarter with the whole circle is also identical. For a whole and its half are in identical consonance, by Proposition I.

So also is the quarter with the half, by the same Proposition: hence by Axiom VII the quarter is also in identical consonance with the whole circle; and by Axiom V any fourfold is with the single.

The ratio which is between the first, second, and third proportionals will be the same as that between the second, third, and fourth, and so on continuously between the three which are nearest to each other. Therefore all proportionals which are in the proportion of successive doubling are in identical consonance with each other.

Notice therefore in such cases the distinction between consonance as a genus and identical consonance as a species. Fourth, eighth, sixteenth, and similar parts also are in consonance by Axiom I and the figures. Tetragon, Octagon, and so on: but they are in identical consonance, on account of the progressive generation of this class of figures from the bisection of the circle.

For if it had not had this derivation their consonances would not have been identical. For as all figures make either many parts of the circle, if they are equal, or unequal parts, if they make only two of them, since they enclose an area, they do not divide the area of the circle equally, nor do their sides pass through its center, nor do they return to the same point, nor are they equal to its diameter:

Furthermore consonances derived from figures of the tetragonal class would in some way have amplified themselves to the hearing, and stretched the mind by the manifest variety and diversity of their notes, as do the conso­ nances which come from the other figures, which consist of a number of sides which is not a product of successive doubling, by Proposition I.

However not all power has been removedfrom this class offigures of varying the consonances and diverting them from the purity of identical consonance (just as they themselves have regressed from the simplicity of the diameter).

For first, although the consonance of the part of the circle cut off by the figure is converted into pure identical consonance (on account of the said derivation of the parts of the circle, from the original bisection), yet the degrees of identical consonance become more remote, for the smaller which is in identical consonance by opposition with the one next larger than itself becomes continually higher in pitch as the points of opposition are multiplied.

Thus the intervals of the notes continually increase.

Secondly, identical consonance does indeed remain in the part (as in division by the diameter), but not at all in the remainder; for this remainder in the later figures becomes continually inferior as far as its harmonic nature is concerned. But there follow particular propositions about such remainders.

On the other hand, it is not only the tetragonal class which generates identical consonances; but also the other classes, to the same extent as they partake of bisection, also make identical consonances.

For the part of a circle cut off by the side of a subsequent figure is always in identical consonance with the part cut off by the side of an antecedent figure, as the remaining propositions relate.

Thus the analogy holds good in all its branches. The application of this proposition is in the following one.