Kepler’s Postscript on Bartolus, September 1622

Postscript.

I thank Abraham Bartolus for the gift he sent. Since I received it through you, and know nothing more of the man’s position or circumstances, I thought I should take up his undertaking with him through you as go-between, and so as arbiter and judge as well. I think I am bound to do so, and publicly, because he is the first, so far as I know, to admit harmonies among the configurations of the planets in public practice, and to credit them with an effect on the weather. Since I believe I was his guide in this, both in my miscellaneous writings and above all in book IV of the Harmony of the World, I think it my business to see how skilfully anyone follows in my footsteps. For if the public good does not demand it (since there will be those who, thinking the matter disputed, will judge it no concern of theirs), my own need at least seems to require it: that readers should not hunt for the sense of my philosophy in those who seem to follow me rather than in my own words.

So this is what I wish to learn from that famous man, in return for the goodwill he has shown me: for what reasons does he credit with an effect those intervals of the good planets in the zodiac which are to the whole zodiac as the difference in length of two strings of the same tension (which give melodic sounds) is to the longer string? For either he was led to this by pure experience, comparing the single intervals with the weather; or by pure natural reason; or by both, by turns and mixed; or, finally, he follows me as his authority. If he pleads pure experience, I ask him to say so in plain words, and also, if possible, to send me some particular evidence of the examination he has made. If he follows pure reason, what is it? For I should like that explained to me too. Since I too have given a whole book to this search for reasons, I am all the more spurred to see others’ thoughts and compare them with mine. For I went by a mixed method. I took the primary aspects on the trust of the observations of the ancients; I let myself be led to their causes by Ptolemy and Cardano. From the causes, carefully worked out, derived from the first source of the highest philosophy and so enlarged, I first came to suspect that more aspects were effective. I tested these further aspects on the weather. By testing I found still more of them effective, the semisextile and the quincunx;Aspects of 30° and 150°, two of the new aspects Kepler added to Ptolemy's five. and not I alone, but others too, at Basel and in the Netherlands. So I saw that in my search for causes I had so far stumbled somewhat over the rules of κατ᾽ αὐτό and κατ᾽ ἄλλο. Going back, then, to the true constitution of the causes, I took in with them those that the clearest experience had revealed, and in turn excluded those (the sesquadrates) that my first reasoning from causes had established and experience had left in doubt. Among those I took in, I established an order of rank, again agreeing with experience, so that the decile and the tridecile came after the semisextiles and quincunxes. That was my method. Now I want to learn Bartolus’s method too, if he declares himself the author of his intervals. But if, which was the fourth case, he appeals to my example, then my zeal owes him a reminder that he has not understood me fully, either in my first reasons, which I set out in the Mysterium and in De stella nova, or in my later, finished ones, which are in the prolegomena to the Ephemerides and in the Harmony. For as far as appears, he compares the whole circle of the zodiac, cut at the place of one planet and stretched out lengthwise in his mind, with a monochord; and however many places there are along the whole length of the monochord that determine melodic sounds, so many places he makes in which another planet, standing there, forms an effective aspect with the one said above to hold the beginning. I did indeed follow the same comparison in the Mysterium, of the zodiac stretched out lengthwise in the mind with a monochord. But I did not turn all the places that give melodic sounds into aspects, only those that determine consonant ones. So these intervals came out for me then:

But later I found that I had been wrong to consider the zodiac as a straight line; for there is one and the same cause, common both to the aspects and to the consonances of the monochord, which is found in the circle, in so far as it is still a continuous circle, and on that account a true character of the soul. That cause first gives rise to the effective aspects within the circle itself; later, through the comparison of the straight monochord with the circle, it goes out, as it were, from the circle, takes on others arising from straightness, and gives rise separately to the harmonies of the monochord, and, from the harmonies compared with one another, to other melodic intervals, as secondary ones. So these melodic intervals (which are dissonant) cannot be the mothers of the aspects; they are granddaughters and great-granddaughters of the harmonies, the aspects’ sisters. What, then, is that first cause? The division of the circle by inscribed figures, I say, such that the arc of the circle is an expressible part of the whole, and the side of the figure is comparable with the diameter in a certain degree of nearness to the ratio of equality. For in this way only the noble figures, those that have a knowable or mental essence, supply aspects; those that are mental non-entities, that is, that cannot be known from their own nature, are excluded outright; and those whose mental essences have moved further, by more degrees of knowledge, from the ratio of equality are excluded to that extent. So the effective aspects define the intervals of the zodiac that are determined by the side of the dodecagon and the chord subtending five of its sides; the side of the decagon and the chord subtending three of its sides; the side of the hexagon; of the pentagon and the chord subtending two of its sides; the side of the square; and of the triangle.

Of these, 1/10 and 3/10 stand in the second line.Literally “behind the principes”, the second line of a Roman battle order. The side of the octagon and the chord subtending three of its sides do come in too, but they stand one degree further from the ratio of equality than 1/10 and 3/10 themselves. So these are the aspects I am given:

In this way the account of the aspects is separated from the account of the harmonies; and my comparing the harmonies with the aspects in the Mysterium was incidental to the aspects themselves. If we liken the aspects to the House of Austria, and the harmonies to a family of barons in Austria, the House of Austria is not noble because it is distantly related to those barons; on the contrary, those barons are the nobler for being related to the House of Austria.

If Mr Bartolus has not so far understood this account, I ask him to follow it in future, if he wants to fortify himself with my authority; but if, as I said at the start, he has reasons and experience of his own, then let him come down into the arena and try his strength with me, in a wrestling match of reasons or with the boxing glove of experience.

I have not touched here on two other heads: first, the division of the monochord; second, the beginning made from E. For I drew my numbers from the judgement of the ears. When the monochord is divided into 720 equal parts, perfect triple harmonies arise at these divisions: 600. 576. 540. 480. 450. 432. 360. Here, since between 480 and 600, by the judgement of the same ears, there is the same harmony as between 576 and 720, and the ratio between the numbers is the same in both, it follows that the ratio between 576 and 720 should also be divided harmonically in the same way as the ratio between 480 and 600 is divided naturally, namely by the number 540. So I too, with nature as my guide, divided the interval between 576 and 720 likewise, by the number 640; and for the same reason the interval 360. 450 too, by the number 405. And so I get the fixed notes in both kinds of melody, 360. 405. 480. 540. 640. 720, to which two movable ones are added, that is, notes that differ from one kind of melody to the other: in the soft, 450. 600; in the hard, 432. 576. If Bartolus follows another way of dividing the monochord, let him plead either the judgement of the ears or reasons. For I helped the judgement of my ears by tripling the concord, and by comparing the interval not yet divided with the one divided. Without this help, the ears would judge what is melodic in voices near the top and near the bottom less surely and with more doubt.

That Bartolus has a semitone right at the bottom is of little moment. He will surely not say that the monochord stopped at the place of the semitone gives three concords, as I say of the other natural divisions. So he is dealing not with the natural beginning but with the customary one; and that he is free to do. As for my making G the root of my naturally divided monochord, I have given the reasons from the very practice of musicians: let Dassius tell of those; and then let him refute my reasons in book III of the Harmony.