The Relationship of the Harmonic

The Origin of the Eccentricities of the Individual Planets in the Arranging of the Harmonies Between their Motions.

The universal harmonies of all the six planets cannot be the result of accident, above all in the extremities of their motions, which as we have seen all coincided in universal harmonies, except for two, which coincided with the nearest harmonies to the universal ones.

Also it is much less possible for it to come about accidentally that all the positions in the system of an octave established in Book 3 by the harmonic sections should be represented by extremes of the motions of the planets; and least likely of all that the very subtle business of the distinction of the heavenly harmonies into the two kinds, hard and soft, should happen by chance, without the care of a unique Craftsman.

It therefore follows that the Creator, the fountain of all wisdom, the constant advocate of order, the eternal and transcendental wellspring of geometry and harmony, that He, I say, the very Craftsman of that which is in the heavens, has linked the harmonic proportions, which arose from the regular plane figures, to the 5 regular solid figures, and shaped from both classes the one most perfect archetype of the heavens.

In it, just as through the five solid figures there shone forth the ideas of the spheres on which the six stars travel, so it was also through the offspring of the plane figures, the harmonies (deduced from them in Book 3), that the measures of the eccentricities of the individual orbits, for proportioning the motions of the bodies, were given their terms. From these two things a single harmonization came about, and the greater proportions of the orbits yielded nothing to the lesser proportions of the eccentricities, which were necessary for the arranging of the harmonies; and correspondingly, of the harmonic proportions, those which had the greater affinity with each solid figure were chiefly fitted to the planets.

Thus that could come about through the harmonies; and by that logic eventually both the proportions of the orbits, and their individual eccentricities, issued simultaneously from the archetype, whereas the individual peri­ odic times resulted from the breadth of the orbits and the bulk of the bodies.^2’

While I strive to bring forth this line of argument into the light of human understanding by the conventional procedure of geometry, may the author of the heavens himself, the father of understanding, the bestower of mortal senses. Himself immortal and blessed above all, look favorably upon us, and prevent the darkness of our mind from putting forth anything concerning this His work which is unworthy of His majesty, and bring about that we as imitators of God may emulate the perfection of His works, by sanctity of life, for which He has chosen his Church in the lands, and cleansed it of sins by the blood of His son, with the help of the Holy Spirit, and may keep far from us all the dissonances of enmity, all contention, rivalry, anger, quarrels, dissension, sectarianism, envy, provocation, irritating facetiousness, and other works of the flesh. All who have the spirit of Christ will not only share my wish for these things, but will also strive to express them in deeds and to affirm their vocation, spurning all vi­ cious practices of all factions though cloaked and painted over with an outward show of zeal, or of love of truth, or of singular erudition, or of deference to contentious teachers, or any other specious pretext. Holy Father, keep us in the concord of mutual love, so that we may be one, as You are one with Your Son, our Lord, and the Holy Spirit, and as You have made all Your works one by the delightful bonds of consonances; and so that from the restored concord of Your people the body of Your church may be built on this Earth just as You have constructed the heaven itself from harmonies.

I. Axiom

It is fitting that in any place whatever where it could be so, between the extreme motions both of individual planets and of pairs, harmonies ought to have been established of all kinds, so that such variety should adorn the world.

II. Axiom

The five intervals between the six spheres should have corresponded in size to a certain extent with the proportion of the geometrical spheres which are inscribed in and circumscribed about the five regular solid figures, and that in the same order which is natural to the figures themselves.

On this see Chapter I, and the Mysterium Cosmographicum (The Secret of the Universe) and the Epitome of Astronomy, Book IV

III. Proposition

The distances between the Earth and Mars, and the same and Venus, should have been the smallest in proportion to their spheres, and very nearly equal; those between Saturn and Jupiter and between Venus and Mercury, intermediate and again nearly equal; and that between Jupiter and Mars the greatest.^®"

For by II, planets which correspond in position with figures which produce the smallest proportion in their geometrical spheres must similarly make the smallest proportion; those corresponding with figures of intermediate propor­ tion produce an intermediate proportion; those corresponding with a figure of the greatest proportion produce the greatest proportion. But the order which applies between the figures of the dodecahedron and icosahedron also applies between the pairs of planets, one of Mars and the Earth, and the other of the Earth and Venus; and the order which applies to the cube and octahedron also applies to the pair of Saturn and Jupiter and the pair of Venus and Mercury; and last, the order which applies to the Tetrahedron also applies to the pair of Jupiter and Mars —see Chapter III. Therefore, the smallest proportion will be between the planetary spheres first named; but between Saturn and Jupiter a proportion almost equal to that between Venus and Mercury; and last, the greatest between the spheres of Jupiter and Mars.

IV. Axiom

All the planets ought to have their eccentricities, no less than motion in latitude, and also distances from the Sun, the fount of motion, vary­ ing according to the eccentricities. Just as the essence of motion consists not in BEING, but in BECOMING, so also the appearance or shape of the region which a given planet would pass through in its motion does not BECOME solid straight away at the beginning, but by the passage of time in the end acquires not only its length but also its breadth and depth, making a complete threefold of dimensions; and thus it comes about gradually by the linking and accumulation of a great many revolutions that a kind of concave sphere is displayed, having the same center as the Sun, just as by a great many circles of a silken thread, linked with each other and wound together, the dwelling of a silkworm is made

5. Proposition: To each pair of neighboring planets two different harmonies had to

be attributed

For by IV each planet has a longest separation from the Sun, and a shortest. Hence by Chapter III of this Book it will also have a slowest and afastest motion. Therefore, there are two primary comparisons of extreme motions, one of the diverging motions of the two planets, and one of the converging. Now they must necessarily be different from each other, because the proportion of the diverging motions will be greater, that of the converging motions smaller. But there also had to be different harmonies between different pairs of planets, so that this variety might assist the adornment of the world, by Axiom I and also because there are different proportions between the distances, by III. But to each pro­ portion of the spheres there correspond particular harmonic proportions, by their quantitative relationship, as was shown in Chapter V of this Book.

VI. Proposition

The two smallest harmonies, 4:5 and 5:6 have no place among the pairs of planets.

For 5:4 is as 1000:800, and 6:5 as 1000:833 + . But the spheres circum­ scribed round the dodecahedron and icosahedron have a greater proportion to those inscribed, that is to say a proportion of 1000:795, and so on, and these two proportions mark the distances between the planetary spheres which are closest to each other, or the smallest intervals: for in the other regular figures the spheres are further away from each other. However, in this case the pro­ portion of the motions is still greater than the proportions of the distances, unless the proportion of the eccentricities to the spheres were vast, by Number 13 of Chapter /// .‘^’ Therefore, the smallest proportion of the motions is greater than 4:5 and 5:6. Therefore, these harmonies are barred in fact by the regular figures and are granted no place among the planets.