Chapter 2

The Relationship of the Harmonic

by Johannes Kepler Icon

Proportions to the Five Regular Figures.

This relationship is diverse and manifold.

However, there are in the main 4 degrees. For either the token of the relationship is taken solely from the external appearance which the figures have; or along with the actual construction of the edge the same proportions emerge which are also harmonic; or they result from figures which have already been constructed, either separately or in conjunction; or finally they are equal to or close to the proportions of the spheres of the figure.

In the first degree the proportions of which the characteristic or major term is 3 have an affinity with the triangular face of the tetra hedron, octahedron, and icosahedron; those of which the major term is 4, on the other hand, with the square face of the cube; and those of which it is 5 with the five-sided face of the dodecahedron.

This similarity of the face can also be extended to the minor term of the proportion. Thus wherever the three fold is found next to a term in the proportion of continuous doubling,^^ that proportion is considered to have affinity with the three figures first named, such as 1:3 and 2:3 and 4:3 and 8:3, and so on.

Where on the other hand there is the fivefold, that proportion is to be completely appropriated to the dodecahedric marriage, such as 2:5 and 4:5 and 8:5, and similarly 3:5 and 3:10 and 6:5 and 12:5 and 24:5. The relationship will be less plausible if the sum of the terms should express this similarity, as in the case of 2:3 the terms added together make 5, as if therefore, 2:3 would have affinity with the dodecahedron.

Relationship on account of the external appearance of the solid angle is similar. It is trilinear in the primaries, quadrilinear in the octahedron, quinquelinear in the icosahedron.

Thus if one term of the proportion participates in the threefold, the proportion will have affinity with the primary solids; but if it participates in the fourfold, with the octahedron; and lastly if in the fivefold, then with the icosahedron.

But in the case of the feminine ones this relationship appears more attractive because the appearance of the angle is also adopted by the characteristic figure, hidden inside, the square in the octahedron, the pentagon in the ico sah ed ro n .T h u s 3:5 on account of both would belong to the icosahedric sect.

The second degree of relationship, which is based on origin, should be conceived as follows. First, there are some harmonic proportions of numbers which have affinity with one marriage or family, that is to say the individual perfect proportions with the cubic family.

On the other hand there is a proportion which is never expressed by whole numbers, and is only demonstrated in numbers by a long series of them which gradually approach it. This proportion is called “divine,” insofar as it is perfect; and it reigns in different ways through the dodecahedric marriage.

Hence, the following harmonies begin to sketch out this proportion:

  • 1:2 (It is most imperfect here)
  • 2:3
  • 3:5
  • 5:8 (It is most perfect here)

It would be more perfect if onto 5 and 8 added together, making 13, we were to superimpose 8, if that were not already ceasing to be harmonic.

Moreover, to establish the edge of the figure the diameter of the globe must be divided.

The octahedron demands its division into two, the cube and the tetrahedron into three, and the dodecahedric marriage its division into five. Therefore, the proportions are distributed among the figures in accordance with these numbers, which express the proportions.

Also the square of the diameter is divided, or the square of the edge of the figure is formed, from a certain portion of it.

Then the squares of the edges are compared with the square of the diameter, and establish proportions as follows: for those of the cube, 1:3, of the tetrahedron 2:3, of the octahedron 1:2.

Hence for the combined pairs, for those of the cube and tetrahedron 1:2, for those of the cube and octahedron 2:3, for those of the octahedron and tetrahedron 3:4.

The sides of the dodecahedric marriage are inexpressible.

Third, the figures already established give rise to harmonic proportions in various ways.

For either the number of sides of the face is compared with the number of edges of the whole figure, and the following proportions emerge: for the cube 4:12 or 1:3, for the tetrahedron 3:6 or 1:2, for the octahedron 3:12 or 1:4, for the dodecahedron 5:30 or 1:6, for the icosahedron 3:30 or 1:10. O r the number of sides of the face is compared with the number of faces: then the cube gives 4:6 or 2:3, the tetrahedron 3:4, the octahedron 3:8, the dodecahedron 5:12, the icosahedron 3:20. Or the number of edges or angles of the face is compared with the number of solid angles: and the cube gives 4:8 or 1:2, the tetrahedron 3:4, the octahedron 3:6 or 1:2, the dodecahedron with its wife 5-20 and 3‘-12, that is T.4.

Or the number of faces is compared with the number of solid angles; and the cubic marriage gives 6:8 or 3:4, the tetrahedron the proportion of equality, the dodecahedric marriage 12:20 or 3:5. Or the number of all the sides is compared with the number of solid angles; and the cube gives 8:12 or 2:3, the tetrahedron 4:6 or 2:3, the octahedron 6:12 or T2, the dodecahedron 20:30 or 2:3, the icosahedron 12:30 or 2:5.

The solids are also compared with each other. If the tetrahedron is concealed in the cube, and the octahedron in the tetrahedron and cube, by inscribing them geometrically, the tetrahedron is a third of the cube, the octahedron is half the tetrahedron, and a sixth of the cube, so that the octahedron which is inscribed in the globe is also a sixth part of the cube which circumscribes the globe. The volumes of the remaining solids are inexpressible.

The fourth kind or degree of relationship is more appropriate to the present work,^^ as what is sought is the proportion of the pheres inscribed in the figures to the spheres circumscribing them, and what is calculated is the harmonic proportions which come close to them.

For only in the tetrahedron is the diameter of the inscribed sphere expressible, that is as one third of the circumscribed sphere; but in the cubic marriage the proportion, which is unique to that case, is similar to lines which are expressible only in square.

For the diameter of the inscribed sphere is to the diameter of the circumscribing sphere in the semitriple p ro p o rtio n .A n d if you compare the actual proportions with each other, the proportion of the tetrahedric spheres is the square of the proportion of the cubic spheres.

In the dodecahedric marriage, the proportion of the spheres is again unique, but inexpressible, a little greater than 4:5. Therefore, the harmonic proportions which are close to the proportion of the cubic and octahedric spheres are the following: 1:2 as the next greatest, and 3:5 as the next smallest; while the harmonies which are close to the proportion of the dodecahedric spheres are 4:5 and 5:6, the next smallest, and 3:4 and 5:8, the next greatest.

But if for particular reasons 1:2 and 1:3 are to be appropriated to the cube, as the proportion of the cube’s spheres to the proportion of the tetrahedron’s spheres, so the harmonies 1:2 and 1:3, which are allotted to the cube, will be to 1:4 and 1:9 which have to be allotted to the tetrahedron, if indeed it is right to use this analogy; for these proportions are the squares of the harmonies mentioned. And because 1:9 is not harmonic, its place will be taken by the nearest harmonic, 1:8, for the tetrahedron. But to the dodecahedric marriage, using this analogy, will belong approximately 4:5 and 3:4.

For just as the proportion of the cubic spheres is approximately the cube of that of the dodecahedric, so also the cubic harmonies 1:2 and 1:3 are approximately the cubes of the harmonies 4:5 and 3:4.

For 4:5 cubed is 64:125; and 1:2 is 64:128. Similarly 3:4 cubed is 27:64, and 1:3 is 27:81.

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