The Relationship of the Harmonic

Proposition 16: The proportions of the motions of Venus and Mercury combined, their own in each case, should have amounted to about 5:12.

For divide the smaller harmonic proportion, 3:5, attributed in common to this pair, by Proposition XV, into the greater of them, 1:4 or 3:12, by Xll.

The quotient, 5:12, is the product of the two’s own proportions. Thus of Mercury alone the extreme motions’ own proportion is less than 5:12 by the amount of Venus’own proportion.

This is to be understood from these primary arguments.

For below by the secondary arguments, with the inclusion of the common harmonies of the two as a kind of yeast, it will turn out that Mercury’s own proportion alone holds 5:12.

Proposition 17: The harmony of the divergent motions of Mars and the Earth could not have been less than 5:12.

For Mars alone in its motions’ own proportion has got more than a diatessaron, and more than 18:25 by XIV. Now their lesser harmony is a diapente, 2:3, by XV. Therefore these two parts combined make 12:25. But the Earth also must have its own proportion, by Axiom 4.

Then since the harmony of the divergent motions consists of the three elements stated, that will be greater than 12:25. But the harmony next greater than 12:25, or 60-125, is 5-12, that is to say 60-144.

Hence if a harmony is needed for this greater proportion of the motions of the two planets, by Axiom I it cannot be less than 60:144 or 5-12.

Up to this point, therefore, all the other pairs of planets have been fitted to their pairs o f harmonies by necessary arguments.

Only the pair of the Earth and Venus so far has been allotted one harmony alone, 5:8, by the Axioms adopted until now. We shall therefore now search further for its other harmony, that is the greater, or that of the diverging motions, making a new start.

A P o sterio ri Arguments.'

Axiom 18

The universal harmonies of the motions must have been established by the combination of six motions, especially through the extreme motions.

This is proved by Axiom I.

Axiom 19

The same universal harmonies must have occurred over a certain range of the motions, that is to say so that they should happen all the more often.

For if they had been confined to particular points in the motions it could have come about that they would never occur, or certainly very rarely.

Axiom 20

As the most natural distinction of the kinds of harmonies is into hard and soft, as has been proved in Book III, so universal harmonies of both kinds must have been arranged between the extreme motions of the planets.

Axiom 21

Different types of harmonies of both kinds must have been organized so that the beauty of the world might be expressed in harmony through all possible forms of variation, and that by the extreme motions, or at least some of them.

By Axiom I

Proposition 22

The extreme motions of the planets must have represented positions or notes in the system of a diapason, or notes in the musical scale.

For the origin and comparison of harmonies which start from one common term generated the musical scale, or division of the diapason into its positions or notes, as proved in Book 3.

Therefore, since different harmonies between the extremes of the motions are required, by Axioms I, XX and XXI, hence in some system, heavenly or in the harmonic scale, a real differentiation between the extremes of the motions is required.

Proposition 23

There must have been one pair of planets between the motions of which no harmonies could exist except the two sixths, the major, 3:5, and the minor, 5:8,

For since there was a necessary distinction between the kinds of harmonies, by Axiom XX, and that by the extremes of the motions at the apsides, by XXII, because only the extremes, that is to say the slowest motion and the fastest, need to be defined so as to arrange and order them, the intermediate tunings emerge of their own accord during the actual passage of the planet from its slowest motion to its fastest, without particular attention.

Therefore, this ordering could not have come about except when a diesis, or 24:25, was marked out by the extremes of the two planetary motions, on account of the fact that the kinds of harmonies are distinguished by a diesis, from what has been explained in Book III.

But a diesis is the difference either between the two thirds, 4:5 and 5:6, or between the two sixths, 3:5 and 5:8, or the same harmonies augmented by one or more intervals of a diapason.

However, the two thirds, 4:5 and 5:6, had noplace between pairs of planets, by Proposition VI; but neither had thirds or sixths augmented by the interval of a diapason been found anywhere, except for 5:12 in the pair of Mars and the Earth, and that only with its comrade 2:3.

Thus the intermediate proportions 5:8 and 3:5 and 1:2 were equally ad­ mitted. Then it remains for the two sixths, 3:5 and 5:8, to be given to one pair of planets. But also the sixths alone had to be conceded to the variation of those motions, in such a way that they neither extended their terms to include the next greater interval of one octave, 1-2, nor contracted them to the narrowness of the next lesser of a diapente, 2:3. For although it is true that if two planets make a diapente with the converging extremes of their motions, and a diapason with the diverging, the same planets can also make sixths, and so can also tra­ verse a diesis, yet that would not smack of the singular providence of the Orderer of the motions. For the diesis, the smallest of the intervals, which is potentially hidden in all the greater harmonies which are included between the extremes of the motions, would then in fact itself be traversed by the intermediate motions, which vary continuously with the tuning. Yet it is not marked out by their ex­ tremes, since the part is always less than the whole, that is the diesis than the greater interval 3:4, which is between 2:3 and 1:2, which would in this case be supposed to be wholly marked out by the extremes of the motions.