The Causes of Consonances

Proposition 4: A string which is in consonance with either one of two multiples in the proportion of continuous doubling is also in consonance with the 0remaining one; and if it is in dissonance with one, it is also in dissonance with the other.

For by Proposition 3 sounds which are in the proportion of continuous doubling are identical with one another. However, what is in consonance with one of two identical strings is also in consonance luith the other; and the rest follows, by Axiom 6.

Axiom VI was assumed for the sake of this proposition; and this proposition is now of service in examining the parts and remainders of circles. Let know-alls beware of abridging the Propositions and Axioms; for there is no tautology: every­ thing is necessary. Anyone who wants to get through the matter too quickly will get himself into a tangle

Proposition 5: Although the additional sides of stars are constructible, on account of their constructibility they determine the consonant parts of the whole in a circle on the same footing as their fundamental figures do, as

in Axiom I;

However those which cut off a part of a circle which consists of the appropriate number (of the parts which the fundamental figure made) for some inconstructible figures are excepted, when the numbers of the part and of the whole have no common factors.

The first part of this proposition is an axiom. So that it should not be made too general, it had to be restricted by the second part of the proposition.

Now the proof is as follows. For let there be a circle divided by a constructible figure, for example by an icosigon. Now let there be an icosigonal star, the side of which subtends nine of the twentieths made by the icosigon, in the proportion of 9 and 20 which have no common factors. Then since the part has been cut off from the circle, it will certainly be smaller than the whole.

Yet it may be larger than half of the whole, or a quarter, or an eighth, and so on by dividing it repeatedly until some part of the whole in the ratio of continuous halving is less than half of the part with which we are concerned. Thus in our example if the whole is 20, the part with which we are concerned is 9.

Take half of the whole, 10, and half of that again, 5, and a third time. 2i, an eighth of the whole.

That is now smaller than half of nine. Then our part, 9, is to an eighth of the whole circle, 2|, as a circle divided by an unconstructible figure to some part produced by its own division, that is as 18 to 5.

Now the Corollary of Axiom 3 declared that five eighteenths are in dissonance with the whole 18.

Therefore by Axiom V the part produced by our division, 9, will be in dissonance with an eighth of the circle (2| parts by our division).

Therefore by Proposition 4 our part, 9, will also be in dissonance with the whole circle, 20, although its chord is constructible, but at the most remote degree; and its star is among the incongruent figures

Proposition 6: The remainders of circles or strings, after parts in consonance with the whole have been cut off, if they are in the proportion of continuous doubling with their consonant part, are in consonance both with the part cut off and with the whole circle or string.

With the part cut off by Proposition I, with the whole by Proposition IV.

Proposition 7: If such a remainder is in the same proportion to the half or quarter of a circle or string as the whole circle is to some other part of itself which is consonant, it will also be in consonance with the whole circle;

if it is in the same proportion as a dissonant part, it will be in dissonance.

For the whole circle, and its half, and its quarter, are in the proportion of continuous doubling. Hence (by Proposition IV) those remainders which are consonant with such a part of the circle are also consonant with the whole; and those which are in dissonance with the former will also be in dissonance with the latter.

But those remainders are consonant with such a part which are in the same proportion to it as the whole to any consonant part; and those remainders are dissonant with such a part which are in the same proportion to it as the circle to any dissonant part. That is by Axiom V.

Therefore such remainders are also in consonance with the whole circle; and those of the opposite kind are in dissonance with the whole circle. This Proposition is for the sake of the following Proposition VIII.

Propositiion 8

However if a remainder is in the same proportion to a cut off part as the whole circle to any consonant part, it is also in consonance with the cut off part, just as by the previous proposition it was in consonance with the whole. If it is in the same proportion to it as the whole to some dissonant part, it will be in dissonance both with the cut off part and with the whole.

The first branch depends on Axiom V, as does one portion of the second branch also, that the remainder is in dissonance with the cut off part.

However the proof that such a remainder is also in dissonance with the whole is as follows.

For it occupies, in the stated proportion, a position in the whole circle divided by an inconstructible figure. Hence although such a remainder is less than the whole circle, of which it is the remainder, yet it is greater than its semicircle, by the definition of a remainder.

But if it is greater than its semicircle, then a quarter of its circle, that is half of the semicircle, is less than half of this remainder. Hence as the remainder is to a quarter of its circle, so will any circle divided by an inconstructiblefigure be to any part produced by its division.

But such a whole circle is in dissonance with such a part of itself, by Axiom 3.

Therefore the remainder mentioned will also be dissonant with the quarter of its own circle, by Axiom V Therefore it will also be dissonant with the whole of its own circle, by Proposition 7.

of C o ns onanc es 157 Corollary to these Propositions-’^ Therefore there are Consonant Parts 1 . 1 . 1 . 1 2 1 . 1 3 1 3 Consonant Remainders Dissonant Parts 1 …………… 2 …………… 3 3 5 5 Dissonant Remainders ………………………. In respect of the Whole 2 ……………………………. ……………………………. ……………………………. ……………. 4 . . . . ……………. ……………. ………………………. 7 . 7 9 7 11 3 4 5 6 7 ……………………. 8 7 9 ……………………. 10 7 1 1 ……………………. 12 9 111315 . . . 16 1113 17 19 . . . 20 13 17 19 23 . . . 24 And so on.