The Relationship of the Harmonic

Proposition 24

Two planets which change the kind of harmony ought to make the difference between their own proportions of the extremes of their motions a diesis, and one’s own proportion ought to be greater than a diesis; and by their motions at aphelion they should make one of the sixths, by those at perihelion the other.

For since the extreme motions make two harmonies, differing by a single diesis, that can happen in three ways: for either the motion of one planet may remain constant, and that of the other vary over a diesis; or else both may vary over half a diesis and they make 3:5, a major sixth, when the superior planet is at aphelion, and the inferior one at perihelion, but making a deviation from those intervals, mutually going to meet each other, the superior up to perihelion, the inferior down to aphelion, they may make 5:8, a minor sixth; or last, one may vary its motion more than the other from aphelion to perihelion, and there may be a difference of one diesis, and so there may be a major sixth between the two at aphelion, a minor sixth between the two at perihelion.

However, the first way is not legitimate, for one of these planets would have no eccentricity, contrary to Axiom IV The second way would be less beautiful and less convenient: less beautiful, because less harmonic. For the two planets’ motions’ own proportions would have been unmelodic, as anything less than a diesis is unmelodic.

It is better, however, for one planet alone to labor under this unmelodic littleness. Indeed, it could not even have come about, because on this basis the extreme motions would have strayed from the positions in the system, or notes in the musical scale, contrary to Part 22.

Also it would have been less convenient, because the sixths would have occurred only at those moments at which the planets would have been at opposite apsides: there would have been no range over which these sixths, and thus the universal harmonies resulting from them, could have occurred.

Therefore, the universal harmonies, when all the positions of the planets had been brought back to the restrictions of definite and unique points on their orbits, would have been very rare, contrary to Axiom XIX. There remains therefore the third way, in which indeed each of the planets varies its own motion, but one more than the other by one perfect diesis at least.

Proposition 25: Of the planets which change the kind of harmony, the upper ought to have a proportion between its own motions less than a minor tone, 9:10, and the inferior less than a semitone, 15:16.

For either they will make 3:5 with their motions at aphelion, or at peri­ helion, as has already been stated. Not with those at perihelion; for in that case the proportion of the motions at aphelion would be 5:8. Therefore, the inferior planet would have one diesis more in its own proportion than the superior planet, by the same previous statement. However that is contrary to Axiom X. Therefore, they make 3:5 with their motions at aphelion, 5:8 at perihelion, less than in the former case by 24:25. But if the motions at aphelion make a hard sixth, 3:5, then the motion of the superior planet at aphelion will make with the motion of the inferior at perihelion more than a hard sixth, for the inferior will add the whole of its own proportion. In the same way, if the motions at perihelion make a soft sixth, 5’-8, the motion of the superior planet at perihelion and that of the inferior at aphelion will make less than a soft sixth; for the inferior takes away the whole of its own proportion. But if the inferior planet’s own proportion equalled a semitone, 15:16, in that case as well as the sixths a diapente could also occur, because a soft sixth diminished by a semitone becomes a diapente: but that is contrary to Proposition XXIII. Therefore, the inferior planet has less than a semitone in its own interval. And because the superior planet’s own proportion is greater than the inferior planet’s own, by one diesis, whereas a diesis added to a semitone makes a minor tone, 9:10, therefore the superior planet’s own proportion is less than a minor tone, 9:10.

Proposition 26: Of planets which change the kind of harmony, the superior ought to have had a double diesis, 576:625, that is nearly 12:13

For the interval between its extreme motions, or a semitone, 15:16, or something intermediate, separated from either the former or the latter by a comma, 80:81; whereas the inferior planet ought to have either a simple diesis, 24:25, or the difference between a semitone and a diesis, which is 125:128, that is nearly 42:43, or last, in a similar way something inter­ mediate, separated by a comma, 80:81, either from the former or the latter, that is the former having a double diesis, the latter a simple diesis, both intervals diminished by a comma.

For the superior planet’s own proportion ought to be greater than a diesis, by XXV, but less than a tone, 9:10, by the previous Proposition.

But in fact the superior one ought to exceed the inferior by one diesis, by XXIV. And harmonic beauty urges that these planets’ oxvn proportions, if owing to their small size they cannot be harmonic, should at least be among the melodic, if that is possible, by Axiom I.

But the only two melodic intervals smaller than a tone, 9:10, are the semitone and the diesis, and these differ from each other not by a diesis but by some smaller interval, 125:128. Therefore, the superior planet cannot have a semitone and the inferior a diesis, at the same time; but either the superior will have a semitone, 15:16, and the inferior 125:128, that is 42:43, or the inferior will have a diesis, 24:25, but the superior a double diesis, 12:13 nearly.

But since the two planets have equal rights, therefore if the nature of melody had to be violated in their own proportions, it had to be violated equally in both cases, so that the difference between their own intervals could remain exactly a diesis, to differentiate the necessary kinds of harmonies, by XXIV.

The nature of melody was equally violated in both cases if the factor by which the superior planet’s own proportion fell short of a double diesis, or exceeded a semitone, was the factor by which the inferior’s own proportion fell short of a simple diesis, or exceeded the interval 125:128.

Furthermore, this excess or shortfall should have been a comma, 80:81, because again no other interval was demonstrated from the harmonic proportions, and so that the comma should be expressed among the heavenly motions in the same way as it was expressed in the harmonic proportions, that is to say only by the excess and shortfall of the intervals between each other.

For among the harmonic intervals the comma is the distinction between the tones, major and minor, and is not noticed in any other way.

It remains for us to investigate which intervals are to be preferred of those suggested, whether it should be the dieses, simple for the inferior planet, double for the superior, or rather a semitone for the superior, and 125:128for the inferior.

The dieses have the winning arguments. For although the semitone has been expressed in various ways in the musical scale, yet its partner 125:128 has not been expressed. On the other hand, both the diesis has been expressed in various ways, and the double diesis in a way, that is in the resolution of tones into dieses, semitones and limmata; for in that case, as has been stated in Book 3 Chapter 8, two dieses follow next to each other in two positions. Another argument is that in making the proper distinction between the kinds the diesis has rights, the semitone none.

Therefore, greater attention should have been paid to the dieses than to the semitone. The outcome of all this is the fol lowing: the superior planet’s own proportion ought to be 2916:3125, or 14:15, nearly; the inferior’s own proportion 243:250, or 35:36 nearly.

Do you ask whether the highest creative wisdom would have been taken up with searching out these thin little arguments? I answer that it is possible for many arguments to escape me.

But if the nature of harmony has not supplied weightier arguments, that is in the case of proportions which descend below the size of all the melodic intervals, it is not absurd for God to have followed even these, however thin they may appear, since he has ordered nothing without reason.

For it would be far more absurd to declare that God has snatched these quantities, which are in fact below the limit of a minor tone prescribed for them, accidentally.

Nor is it sufficient to say that He adopted that size because that size pleased Him. For in matters of geometry which are subject to freedom of choice it has not pleased God to do anything without some geometrical reason or other, as is apparent in the borders of leaves, in the scales of fishes, in the hides of wild beasts, and in their spots and the ordering of their spots, and the like.