TO THE KEPLERIAN APPENDIX, 7

...were to be, and so in the rest? We know that the natural motion of Saturn, according to the opinion of the ancients, is completed in a space of 30 years; that of Jupiter completes its journey in nearly twelve years; Mars runs through the Zodiac in the space of nearly two years and five months; the Sun traverses the ecliptic in 365 days and nearly six hours; Venus passes through the Zodiac at nearly the same motion as the Sun, as does Mercury; and we know the Moon passes through all the degrees of the Zodiac in its natural motion in about 28 days. Furthermore, by the logic of these preceding motions, we know the mean motion of Saturn—which was called by the ancients its deferent motion The "deferent" is the large circular path in ancient astronomy that carries the center of a smaller circle, the epicycle—moves upon its own axis, intersecting the axis of the Zodiac or ecliptic, following the succession of the signs at nearly two minutes per day.

But truly, they maintain that the motion of the same planet, caused by the spheres carrying the apogee original: "augem"; the point in an orbit furthest from the center of Saturn—which is proportional to the axis and poles of the ecliptic of the eighth sphere—occurs at a rate of one degree and about twenty minutes every two hundred years. Thus they say the motion of Jupiter is almost the same with respect to the spheres carrying its apogee as that of Saturn, namely 1 degree and 28 minutes. But its deferent motion upon its own axis and poles is far swifter than the mean motion of Saturn; for by this, Jupiter naturally moves nearly 5 minutes daily: by which proportion it must necessarily run through the Zodiac in the predicted time, namely in the space of 12 years. Mars also moves at the same time as Saturn and Jupiter relative to the motion of the spheres carrying the apogee; but by its own motion, it completes about 30 minutes daily, by which proportion it is said to complete the circuit of the Zodiac in nearly two years and 5 months. They say the spheres carrying the Sun’s apogee have the same motion as the other superior planets, namely with Saturn, Jupiter, and Mars. But its deferent, upon its own poles and axis, completes nearly 59 minutes and eight seconds following the succession of the signs; and thus the Sun completes the whole Zodiac in 365 days and nearly 6 hours. And it is the same for the other inferior planets, namely Venus, Mercury, and the Moon.

But let it be admitted that our Author here—along with Copernicus, Schöner, Brahe original: "Brachio", and others of the same sort—entirely denies and seeks to remove the stationary and retrograde motion of the epicycle (for such seems to be the Author's mind in Chapter 4, Book 5), and instead requires peculiar orbits for the spheres carrying the apogee, by which he wishes all Planets to be made eccentric—that is, to change their distances from the Sun. Let this very thing, I say, be granted; yet it will nonetheless be necessary, insofar as the aforementioned authors differ little or nothing in effect from the ancients regarding the natural motion of the Planets, that the Harmony found in his Planetary System be composed of such motions, since the Author has placed his hope upon these same foundations.

Since this is so, it must be asked (and do not think this is said foolishly): why has he rejected the variety of times in their length and shortness, when the progression of these motions does not occur except in physical time? Certainly, a single and simple species of time does not suffice for this work, but rather a manifold one, insofar as the motion of the Planets is manifold and diverse. From this it follows without a doubt that the motion of the Moon relates to the higher planets just as, in musical times, the shortest durations relate to the medium and longer ones. For there is, beyond any controversy, a most exact ratio in the diversity of the motions of the Planets; since the progression of the Moon has a relation to an excellent Cantus the highest voice part, or Soprano, the motions of Mercury, Venus, and the Sun relate to the Altus and Mezzo the middle voice parts, Mars possesses the place of the Contratenor, Jupiter the Tenor, and Saturn claims for itself the Bass in the greatest or longest sounds because of its slowness in motion and its gravity meaning both physical weight and low musical pitch.

What, therefore, is the cause that our Author here, out of an excessive search for intervals, has forgotten himself and so neglected the secrets of time? He is not only content to use only semibreves whole notes; the basic unit of time in the music of this period in his Harmony—perhaps due to a manifold ignorance of the triple differences of times (since in that proportion of temporal numbers, his triangular bodies ought to sound alongside the planes of other species)—but he even delights to fall into a more blatant error by asserting that the durations of sounds are arbitrary and do not require an inquiry into their causes.

They do require, I say, a massive inquiry into their causes, as will most easily appear from a diligent inspection and contemplation of my Temporal Triangles in Book 4 of my Artificial Music. For it is a matter of no small importance to know which temporal notes are entirely perfect, which are perfect once, which twice, thrice, or four times, and which are ultimately completely perfect. Similarly, to investigate the infinite unequal or disparate proportions of temporal notes to one another will require great inquiry; namely, which is simple, which is multiple, and then among the simple ones, which is a multiple double, triple, or quadruple; which is superparticular a ratio of (n+1)/n, like 3:2 or 4:3, superbipartient a ratio of (n+2)/n involving thirds, fifths, sevenths, or supertripartient a ratio of (n+3)/n involving fourths, fifths, sevenths, or superquadrupartient, etc.

Similarly among the multiples: which is a double or triple superparticular, and from those, which are sesquialtera a 3:2 ratio, sesquitertia a 4:3 ratio, sesquiquarta a 5:4 ratio; and in the quadruple, which are sesquialtera or sesquiquarta. And then after that, which is a double multiple superbipartient, superbipartient of thirds or fifths, or supertripartient of fourths or fifths; which is a triple superbipartient of thirds or fifths, or supertripartient of fourths or fifths; which is a quadruple superbipartient of thirds or fifths, and supertripartient of fourths or fifths. Finally, from our Temporal Triangle, it will be possible to distinguish from the notes according to their proportions which notes, looking toward one another, are doubles, which are sesquitertias, which are sesquialteras, and which are finally triples. Furthermore, even the Pauses musical rests, which are silent or understood notes, require the greatest inquiry because of the variation of the predicted proportions; as do the signs indicating the speed or slowness, and the duplicity or triplicity of the notes. Concerning all these things, although I have spoken briefly in my Book 4, nevertheless...