The Relationship of the Harmonic

Proposition 13: The extreme motions of Jupiter and Mars should have had as their harmonies, for the greater in fact 1:8, about a triple diapason, and as the lesser 5:24, a minor third above the double diapason.

The cube has been allotted 1:2 and 1:3.

• But the proportion of the spheres of the tetrahedron, which is located between Jupiter and Mars, designated as the triple, is the square of the proportion of the spheres of the cube, designated as semitriple, then it was appropriate that proportions for the motions should also be fitted to the tetrahedron which are the square of the cubic proportions.

The proportions which are the squares of 1'2 and 1-3 are in fact 1:4 and 1'9.

But 1:9 is not harmonic, and 1:4 has already been taken up for the octahedron.

Therefore, the neighboring harmonies to these proportions had to be taken, by Axiom 1.

The ones which are neighbors to it are:

• 1:8 as the smaller
• 1:10 as the greater.

Between these, the choice is made by affinity with the tetrahedron, which has nothing in common with the pentagon, since L’lO is of the pentagonic group.

However, the tetrahedron’s greater affinity is with F'8, on many accounts in Chapter 2.

Further, on 1:8, another feature is that just as 1:3 is the greater proportion of the cube, and 1:4 the greater of the octahedron, because they are powers of the proportions between the spheres of the figures, so also L'8 should have been the greater proportion of the tetrahedron.

For as its body is twice that of the octahedron inscribed in it, as stated in Chapter I, so also the term 8 of this tetrahedric proportion is twice the term 4 of the octahedric proportion.

Further, since 1:2, the lesser proportion of the cube, is one diapason, and 1-4 the greater proportion of the octahedron is two diapasons, so in this instance 1:8, the greater proportion of the tetrahedron should have been three diapasons.

It had to have more diapasons than two because since the lesser tetrahedric harmony must necessarily be the greatest of all the smaller harmonies in the other figures (inasmuch as the proportion of the tetrahedric spheres is the greatest of all those of the figures), the greater tetrahedric harmony also should have exceeded the greater harmonies of the others in its number of diapasons.

Last, the threefold nature of the intervals in diapasons has kinship with the triangular type of the tetrahedron. It has a certain perfection in accordance with the universal perfection of the Trinity, since the eightfold also, its term, is the first of the cubic numbers, which are perfect in quantity, that is to say of three dimensions.

II. To 1:4, or 6:24, the neighboring harmonies are 5:24, in fact the greater, and the lesser 6:20 or 3’-10.

Again, however, 3:10 is of the pentagonic group, which has nothing in common with the tetrahedron.

But 5-‘24 on account of the numbers 3 and 4 (of which the numbers 12 and 24 are offspring) has kinship with the tetrahedron.

For we neglect the other lesser terms, that is to say 5 and 3, here, because their degree of affinity with the figures is the lowest, as may be seen in Chapter 2.

In addition, the proportion of the spheres of the tetrahedron is the triple; and the proportion of the converging distances ought also to be the same size, about, by Axiom II.

But according to Chapter 3, the proportion of the converging motions is approximately the inverse of the sesquialterate of that of the distances, whereas the sesquialterate of the triple proportion is about that between 1000 and 193.

Therefore, in units in which the motion of Mars at aphelion is 1000, Jupiter will be a little greater than 193, much less than 333, a third part of 1000. Therefore, not the harmony 10:3, that is 1000:333, but the harmony 24:5, that is 1000:208, holds the place between the converging motions of Jupiter and Mars.

Proposition 14: The extreme motions’ own proportion in the case of Mars should have been greater than a diatessaron, 3:4, and about 18:25

For let precisely the harmonies 5:24 and 1:8, or 3:24, be attributed in this case to Jupiter and Mars in common, by Proposition 13.

Divide the lesser, 5:24, into the greater, 3:24. The quotient is 3:5, the product of both planets’ own proportions. But Jupiter’s own proportion alone was found in fact in Proposition XI above to be 5:6. Divide that, therefore, into the produvt of their own motions, 3:5. That is, divide 25:30 into 18:30. The quotient is Mars’ own pro­ portion, 18:25, which is greater than 18:24 or 3:4. However it will become yet greater, if by the following arguments the greater common proportion 1:8 were to be increased.

Proposition 15

Between the converging motions of Mars and the Earth, of the Earth and Venus, and of Venus and Mercury, the harmonies 2-S, the diapente, 5:8, the soft sixth, and 3:5, the hard sixth, had to be shared; and in that order.

For the dodecahedron and icosahedron, thefigures interposed between Mars, the Earth and Venus, have the smallest proportion between their spheres, circumscribed and inscribed. Then they ought to have the smallest of the possible harmonies, being akin on that account, and so that Axiom II may have its place.

But the smallest harmonies of all, that is 5:6 and 4:5, are not possible, by VI.

Then the figures stated ought to have the harmonies next greater than those, that is either 3:4 or 2'3 or 5:8 or 3:5.

Again the figure interposed between Venus and Mercury, that is to say the octahedron, has the same proportion in its spheres as the cube.

But to the cube as its lesser harmony, which is between its converging motions, belongs the di­ apason, by VIII.

Then by analogy the octahedron ought to have had a proportion of the same size, that is 1:2, as its smaller one, if no diversity is included.

However, diversity is included, to the extent that in the case of the cubic planets in fact, that is Saturn and Jupiter, their individual motions’ own proportions combined produced a total not greater than 2:3. In this case, however, of the octahedric planets, Venus and Mercury, their individual motions’own proportions combined will make a total greater than 2:3, which is easily apparent in the following way.

For suppose that what was required was the proportion between the cube and the octahedron, if it were the only one: let, I mean, the lesser octahedric proportion be greater than those which have been prescribed here, and let it be absolutely as great as was the cubic, that is to say 1:2, whereas the greater was 1:4 by XII.

Therefore, if this is divided by the lesser proportion which we have just assumed, 1:2, there still remains 1:2 as the product of Venus and Mercury’s own proportions. But 1:2 is more than the product of Saturn and Jupiter’s own proportions, 2:3.

The consequence of this greater product is a greater eccentricity, by Chapter III; while the consequence of this greater eccentricity is a lesser proportion between the converging motions, by the same Chapter III.

Hence it comes about from the multiplication of this greater eccentricity by the proportion between the cube and octahedron that a lesser proportion than 1:2 is also required between the converging motions of Venus and Mercury.

It was also appropriate for Axiom I that as the harmony of the diapason was taken up for the cubics, another which was very close should be adapted to the octahedrics, and by the previous proof, one less than 1:2. Now the proportion next smaller than that is 3:5, which, as it is the greater of the three, the figure with the greater proportion between its spheres ought to have had, that is to say the octahedron.

Therefore, the lesser harmonies 5:8 and 2-3 and 3:4 were left for the icosahedron and dodecahedron, figures with a lesser proportion between their spheres.

Now these remaining harmonies were distributed among the two remaining figures. For just as of the figures, although they have equal proportions between their spheres, the harmony 1:2 has in fact been allotted to the cube, but the smaller 3:5 to the octahedron, for the reason that the product of Venus’ and Mercury’s own proportions would exceed the product of Saturn’s and Jupiter’s own proportions, so also in this case the dodecahedron, even though it makes the same proportion between its spheres as the icosahedron, should have had a lesser harmony than the icosahedron, but the closest, on account of a similar reason, that is because the latter figure is between the Earth and Mars, the eccentricity of which had been made large among the superior planets.

Whereas the eccentricities of Venus and the Earth, as we shall hear in what follows, are the smallest. And since the octahedron has 3:5, the icosahedron, of which the spheres have a smaller proportion, has the next, a little smaller than 3:5, that is 5:8.

Therefore, there was left for the dodecahedron either 2-3, which remained, or 3:4; but preferably the former, inasmuch as it is closer to the icosahedric 5:8, as their figures are also similar. But even 3:4 was not possible.

For although among the superior planets Mars’ extreme motions’ own proportion was great enough, yet the Earth, as has already been said, and will be clear in what follows, contributed as its own a proportion too small for the product of the two to exceed a diapente.

Therefore, 3:4 could not have the position, by VII; and all the more so because as will follow in Proposition XLVII the proportion of the converging distances should have been greater than 1000:795.