Corollariesby Johannes Kepler
The harmonic divisions of a single string are seven in number, not more.
The expansion of the numbers which are characteristic of divisions occurs in the following manner. To begin with, the whole is expressed in the form of a fraction, that is to say with unity above as numerator, and unity below for denominator. Then each number separately is put as a numerator, and the sum of the two is put as denominator in each case.
Hence from any given fraction two branches arise, until from the sum occurs the number which indicates an unconstructible figure.
I. Causes of this
I found these seven divisions of the string first sevenfold with hearing as guide, in other words the same number found number as there are harmonies not greater than successively.
a single diapason. After that I dug out the causes both of the individual divisions and of the num ber of the total, not without toil, from the deep est fountains of geometry. Let the diligent reader 13 read what I wrote about these divisions 22 years 2Same ago in The Secret of the Universe, Chapter XII,’’
and ponder how in that passage I was under a delusion about the causes of the divisions and the harmonies, mistakenly striving to deduce their number and the reasons from the number of the five regular solid bodies; whereas the truth is rather that both the five solid figures and the musical harmonies and divisions of the string have a common origin in the regular plane figures. Also by the generosity ofjohannes Georg Heerward, Chancellor of Bavaria, I have obtained the Harmony of Ptolemy, together with the commentary of Porphyry, which I referred to in the passage mentioned; and from the third book of it I have trans ferred the more important part to the Appendix to Books IV and V of this work. Yet I did not find the true causes of the harmonies in them, and consequently no mention occurs even of these divisions and of their sevenfold number. Although I remarked at a fairly early stage that the causes must be sought in the plane figures, and you see the seeds^^ of the matter already scattered in the Chapter referred to, XII, of The Secret, yet they racked me exceedingly for a long time, before all my mind’s doubts were satisfied. For first the constructible figures had to be separated from the inconstructible. Next I had to find the reason why although these divisions came from the figures, the divisions were restricted to seven but the figures extended to infinity. Thirdly, I had to establish the difference between the pentekaedecagon and the other construct ible figures, because I saw that that figure was excluded from the be getting of harmonies, on the evidence of hearing.®^ Also the individ ual chapters had their own more limited hazards, each one of which kept me occupied for a long time. Take for example Proposition V, which I saw had to be added last of all, when I was already writing out the work, which I had not realized until then. For if that were not among the basic assumptions, and if, for instance, seven twentieths had on that account been suitable for setting up a harmony, because they are constructible through three twentieths (in combination with which they make a semicircle); and in that case both seven tenths and five sevenths, and so both two sevenths and one seventh, would be ad judged to make harmonies, which is in all respects rejected both by the ears and by our Axioms. Therefore even by reference to the sole evidence of my book The Secret of the Universe the hearing is sufficiently fortified against the de traction of the sophists, and those who dare to disparage the trust worthiness of the ears on very minute divisions, and their very subtle discrimination of consonance —especially since the reader sees that I followed the evidence of my ears at a time when, in establishing the number of the divisions, I was still struggling over their causes, and did not do the same as the ancients did. They advanced to a certain point by the judgement of their ears, but soon abandoning their leader ship completed the rest of the journey by following erroneous Reason, so to speak dragging their ears astray by force, and ordering them out right to turn deaf. Indeed I have taken extra pains below in Chapter VIII of this Book to ensure that anybody may have a ready opportun ity to consult his hearing under his own colors on these and other divisions of the string, and of weighing up their evidence, so that he can be sure that we are struggling over the causes of what rests on the dependable test of the senses, and are not improvised fictions of my own (a charge of which the Pythagoreans stand accused) and in truded in the place of truth