(An et Qua Ratione Motus Planetarum ex Proportionibus Harmonicis Determinati fuerint a Deo)

[I.] Here, indeed, Johannes Kepler triumphs (Epitome of Copernican Astronomy bk. 6, pp. 477, 501, and 900; Mysterium Cosmographicum ch. 14, 20, 21; but chiefly in his book of Harmonics) — in whose preface he relates that he began to speculate on this matter 22 years before, and spent the best part of his life in Astronomy to this end: namely, to show that the whole nature of Harmony, however great it is — with all its parts expounded in book 3 — is to be found among the celestial motions. From Kepler a few things were selected by Pierre Hérigone (Cursus Mathematicus vol. 5, from p. 573) and by Mersenne (on Genesis 4, p. 1558, etc.); but more fully by Athanasius Kircher (Musurgia bk. 10, p. 376).

[Margin: Ptolemy’s doctrine on the harmony of the celestial motions.]

But before Kepler, Ptolemy (Harmonics bk. 3, from ch. 9) took up this very argument for himself to treat — whom, therefore, it is fitting to hear first.

[II.] Ptolemy, then (Harmonics bk. 3), [teaches as follows]:

• (ch. 9 — the motion in longitude [the diurnal motion]): he likens the Rising and Setting of the stars to the gravest voices, and the Meridian ascents to the acutest — both because, just as those who train the voice begin from the gravest, and when they have ascended to the acutest, descend until they end in the gravest (and so are moved from silence, through the acutest voice, to silence), so the stars pass from rising, through the meridian, to setting, from one privation of aspect to another; and because the lower places (to which the points of rising and setting are likened) render the gravest sounds, but the higher, like the meridian, the acutest.
• (ch. 10 — the motion in altitude [distance]): in the motion at the maximum distance from the earth the Diatonic genus reigns (because in it two intervals are not less than the rest); in the minimum distance, the Enharmonic (because in it both [outer] intervals are less than the rest); and in the mean distance, the Chromatic (because in it the Lichanos, the index-string, holds the middle of the tetrachord, standing midway between the Diatonic and the Enharmonic).
• (ch. 11 — the motion in latitude): he compares it to the changes that occur among the modes (the transgression of the genera): the Mixolydian and Hypodorian (the extreme modes) answer to the maximum latitude toward North and South; the [Dorian], the middle, to when the planets lack latitude and are on the Ecliptic (or in declination, or on the Equator).
• (ch. 12 — the aspects to the Sun): the remaining order of the tetrachords he likens to the remaining aspects — the tetrachords of the disjunct strings (diezeugmenon) to the intervals from occultation to the appearances of the stars and to the oppositions with the Sun (acronychal risings and full moons); the conjunct tetrachords (synemmenon) to the quadratures; and so of the rest.
• (ch. 13 — the 360-degree circle): the whole circle divided into 360 parts, he assigns to the Trine aspect 120 parts, to the Quadrate 90, to the Sextile 60, and to the Opposition 180; the Proslambanomenos being taken at 180 parts… [he gives, of the middle strings, 20; to the last diezeugmenon 90; to the last hyperbolaeon 60].
• (ch. 14): he inquires into the consonances Diapente, Diatessaron, and some others, through the division of the circle into 360 parts — as certain aspects are of the whole circle, or of a portion of it due to another aspect — yielding a sesquitertian, sesquialteran proportion, and the like.
• (ch. 15): finally he says that the sound of Saturn and Mars (the malefics) makes a Diatessaron consonance with the sound of the benefics — namely, that the last [string] of the hyperbolaeon (the “excellent”) of Saturn consonates with the last of the diezeugmenon (the “disjunct”) of Jupiter; the last of the synemmenon (the “conjunct”) of Mars with the Mese, or middle [string], of Venus; and that the sound of Saturn belongs rather to the Solar sect, [and that of Mars …]

[The catchword “tis” (Mar·tis) points to p. 532 (PDF 567), continuing Ptolemy’s doctrine and then Kepler’s, within Chapter X.]

(printed p. 532 — Chapter X continued, the heart of Kepler’s celestial harmony. Ptolemy’s doctrine ends with Riccioli’s verdict that he partly plays the poet; then Kepler’s own doctrine follows: the Sun as “Choragus” dividing the circle into 720 harmonic parts, and the Third (Harmonic) Law — periodic times as the 3/2-power of mean distances, confirmed by Wendelinus for Jupiter’s satellites. Kepler’s tables of apparent diurnal motions then show that the convergent motions of adjacent planets form near-perfect consonances.)

…[and that the sound of Saturn belongs rather to the Solar sect, but that] of Mars to the Lunar sect; wherefore all the configurations of Saturn to Jupiter are beneficent, but of the aspects of Saturn to the Sun only the Trines are beneficent (as more consonant than the rest); so too only the trine configurations of Mars to Venus and the Moon are beneficent; further, the configurations of Saturn to the Moon and Venus are evil, and those of Mars to the Sun all dangerous. Let it suffice us to have indicated these, so that it may appear that Ptolemy in these matters partly rather plays the Poet (as seemed also to Kepler, in the appendix to his Harmonics), and partly, out of the Astrological faculty or vanity, hunts for a harmony in the heaven.

[Margin: The proportion between the Periods and the Distances of the Planets.]

[III.] But Kepler (Epitome of Astronomy bk. 4, p. 477; and Harmonics bk. 3, ch. 6) teaches that the least number suitable for determining all the parts of the Monochord — for constituting the system of the double Diapason, that is, of the soft and hard song [minor and major] — is 720; and since, from the ancient observations of Aristarchus and from more recent ones, the apparent diameter of the Sun at apogee subtends 30 minutes (he says), which is 1/720 of the whole circle, this diameter was determined of such a quantity that the first body — that is, the Sun, the Choragus [choirmaster] of the celestial music — should divide the circle, for the earth-dwellers and the contemplating creature, according to the Harmonic laws: that is, into 720 parts. Which number can be divided into very many aliquot parts — by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, and 48 — that is, in order, into the parts 360, 240, 180, 144, 90, 80, 72, 60, 48, 45, 40, 36, 30, 15. (The consonances he determines from the sides of the figures inscribable and demonstrable in the circle we have already taught — ch. 4, Scholia 1 and 2; and Harmonics bk. 4, ch. 5 and 6; and Epitome bk. 6, p. 901.) He teaches that sublunary nature is so often sensibly stirred and stimulated to act, as often as the planets are configured harmonically among themselves — that is, when their aspects and radiations occur at such a distance from one another under the Zodiac as the harmonic proportions require (of which Aspects we shall speak in the following chapter; for now we treat only of the motions themselves).

[Margin: The proportion between the periods and the distances of the planets (Kepler’s Third Law).]

Of these, Kepler treats (Epitome bk. 6, from p. 90, and in the whole of Harmonics bk. 5, especially ch. 3), where he affirms, as a most certain thing, that the proportion between the periodic times of any two Planets is precisely the sesquialteran [3/2-power] proportion of their mean distances from the Sun — provided the mean be the arithmetic mean between the two diameters of the elliptic orbit, which is a little less than the longer diameter. So that, for example, if from the period of the Earth (which is 1 year) and from the period of Saturn (which is 30 years) you take the third part of the proportion (that is, the cube roots), and then make the double of this proportion (by squaring the roots), there will come forth the most exact proportion of the distances of Saturn and the Earth from the Sun. For the cube root of 1 year is 1, and its square 1; but the cube root of 30 years is a little more than 3, and its square a little more than 9: therefore Saturn’s mean distance from the Sun is a little more than ninefold the mean distance of the Earth. This Keplerian proposition Hérigone also accepts (Cursus Mathematicus vol. 5, p. 573); and the same proportion between the periods and the distances of the Satellites of Jupiter was noted by Wendelinus [Govaert Wendelen] in a most learned letter written to me by himself. Kepler proceeds, and (ch. 4) confesses that, in the periodic times of the planets compared among themselves, there are no harmonic proportions — to which Mersenne (on Genesis 4, p. 1558) and Kircher (Musurgia 10, p. 377) readily subscribe. Now the periodic motions of the planets around the Sun are gathered, from all the motions through all the degrees of the whole circuit (long, mean, and small), by Kepler, as in the following table.

The apparent diurnal motions of the planets, and the “song” of each

(Motus Apparentes Diurni — each planet’s daily angular motion at aphelion and at perihelion, in arc-minutes ′ and seconds ″; and the consonance traced by its own motion-range)

(This is the famous result of Kepler’s Harmonices Mundi: each planet’s own range of angular speed, from slowest at aphelion to fastest at perihelion, spans a musical interval — Saturn a major third, Jupiter a minor third, Mars a fifth, the Earth a semitone, Venus a barely-perceptible diesis, Mercury an octave-and-a-third. Riccioli distinguishes these apparent diurnal motions from the mean diurnal motions, which differ by a few seconds and from which the consonances are exactly computed: e.g. Saturn 1′48″:2′15″ = 108:135 = 4:5; Earth 57′28″:61′18″ = 15:16; Mercury 164′:394′ = 5:12. All six check.)

Harmony of Pairs

(Harmonia Binorum — the harmonic ratio between the extreme diurnal motions of each adjacent pair of planets: Divergent = aphelion-motion of the upper to perihelion-motion of the lower; Convergent = perihelion of the upper to aphelion of the lower)

[Margin: Explanation of the preceding table.]

In the preceding table, then, the first column [“Harmony of Pairs”] indicates the harmonic proportions between the diurnal motions of two planets around the Sun — whether divergent (comparing the aphelion of the upper with the perihelion of the lower) or convergent (the perihelion of the upper with the aphelion of the lower); to mark which, the alphabetic letters are added, which, sought in the second column [the motion-table], signify the aphelion or perihelion motion of the planet. For example, in the first column under the title “divergent” you see a and d, and opposite to them this fraction 1/3; now a, in the second column, signifies Saturn’s aphelion motion, and d, Jupiter’s perihelion; therefore between the diurnal motion of Saturn’s aphelion (which is 1′46″) and Jupiter’s perihelion (which is 5′30″) there is a proportion as of 1 to 3 — for, the motions being resolved into seconds, they are, in diurnal motions, Saturn’s 106″, Jupiter’s 330″, between which is the proportion as 1 to 3. But between b and c — that is, between the convergent diurnal motions of Saturn’s perihelion and Jupiter’s aphelion — the proportion indicated by the fraction 1/2 (namely double, or as 2 to 1); for Saturn’s perihelion motion is 135″ and Jupiter’s aphelion 270″, between which is a most perfect Diapason [octave].

But in the others — except for Jupiter with Mars — the proportions of the motions are so near to the harmonic that, if strings were so tuned, the ears could not easily discern the imperfection of the consonance. So Kepler concludes that there are perfect harmonies: between Saturn’s perihelion and Jupiter’s aphelion, a Diapason [octave]; between Jupiter’s perihelion and Mars’s aphelion, nearly a Diapason-with-soft-third [octave + minor third]; between Mars’s perihelion and the Earth’s aphelion, a Diapente [fifth]; between the perihelia of the same [Mars and Earth], a soft [minor] Sixth; between the aphelia of the Earth and Venus, a hard [major] Sixth; between the perihelia of the same [Earth and Venus] …

[The catchword “Peri” (Peri·helios) points to p. 533 (PDF 568), continuing Kepler’s pairwise harmonies, within Chapter X.]

(printed p. 533 — Chapter X concludes and Chapter XI opens. Kepler’s harmony finishes: the motion-proportions yield musical modes and four voice-parts (Bass to Saturn and Jupiter, Soprano to Mercury), with the Sun as the “Regia” of nature perceiving these harmonies — whence Kepler confirms heliocentrism. Riccioli’s verdict: more ingenuity than solid doctrine, resting on rejected heliocentrism; the celestial harmony of Scripture is only analogical and metaphorical. Chapter XI then opens on the force of astrological Aspects, with Kepler deriving 13 efficacious aspects from inscribed polygons.)

…[between the] perihelia of the same [Earth and Venus], a soft [minor] Sixth; and between Venus’s aphelion and Mercury’s perihelion, or even between their perihelia, a Disdiapason [double octave]. From these and other considerations — but not without many cautions — (ch. 5) he tries to drag the proportions of the planetary motions to the places of the System, or to the keys of the Musical Scale, in the genus of hard and soft song [major and minor]; and (ch. 6) that, in the extremes of those same motions, there are expressed by God, in some way, Musical Tones or Modes; and (ch. 7) that there are given universal Harmonies of all six planets, as it were common Counterpoints, in four forms; and (ch. 8) — although he confesses that in the heaven there are neither sounds, nor any motions (in which he considers Harmonies) that are real, but only apparent from the Sun, and that there is no solid natural cause for comparing the apparent motions of the planets with human voices — yet, by I-know-not-what enticement of congruence and analogy, he assigns the Bass to Saturn and Jupiter, the Tenor to Mars, the Alto to the Earth and Venus, the Descant [Soprano] to Mercury.

Hence, having made a step to the Eccentricities (in the very prolix ch. 9, with many axioms — for the most part feigned — ingeniously coordinated to his purpose), he tries to show that the eccentricities of the orbs had to be determined from Harmonic reasons, so that the extremes of the aphelion and perihelion motions might represent the harmonic proportions [given] above; and that therefore the inscriptions and circumscriptions of the orbs in (or about) the five Regular bodies had to yield to these harmonic reasons, and that the intervals of the planets could not be so exactly constructed from the Regular bodies [alone], lest the Harmonies of the extreme motions about the Sun should perish. Finally (ch. 10), since he seems to himself to have grasped a Harmony among the extreme motions of the planets — not real, but apparent to the Sun, or seen from the Sun — he concludes that the Sun is the Royal seat [Regia] of all nature, and that there is in it some Mind, hidden from us, which can perceive those harmonies (since they arise only from the motions subtending angles at the Sun). Hence he tries to confirm the rest of the Sun and the motion of the Earth — without which a great part of those harmonies perishes. Yet the same [Kepler] taught how to investigate the harmonic proportions in the motions seen both from the Sun and from the Earth (Epitome bk. 6, p. 901) — which doctrine we set forth from him on another occasion (bk. 7, sect. 5, ch. 8, num. 7), so that we need not repeat it here.

[Margin: Our Opinion.]

[IV.] Our Opinion, however — from which Mersenne (on Genesis 4, from p. 1558, and p. 1704) and Kircher (Musurgia 10) are not far — is that the aforesaid endeavors of Kepler contain more ingenuity than solid erudition or true doctrine. For, first, a great part of them rests on the immobility of the Sun at the center of the World, on a certain mental force of the Sun apprehending the harmonies, and on the annual motion of the Earth about the Sun — which hypothesis we have already rejected in the last chapters of the preceding Section. Secondly, the three former reasons (adduced ch. 8, num. 8) militate here. Thirdly, since Harmony properly so called (harmonic proportions) is found neither in the periods of the planets compared among themselves, nor in most of the diurnal motions seen from the Earth (where the contemplating creature of the divine works is, accustomed to sensible harmonies), nor in the motions as to latitude, nor among the extreme motions themselves apparent from the Sun so exactly as was fitting and possible for God (if, in determining the motions, He had had the harmonic reasons set before Him in the Archetype); and finally, since those extreme motions of two planets (e.g. the perihelion of Saturn and the aphelion of Jupiter) very rarely concur, but for the most part the motions proceed without these harmonies — it seems rather to be asserted that the Harmony which Scripture and the Fathers and very many wise men recognize in the heaven is to be understood only analogically and metaphorically, by a certain accommodation and likeness: namely, that just as in sensible Harmony, out of unequal and diverse sounds and voices, there arises a concord pleasant to the ears, so out of the motions of the celestial bodies — though diverse and unequal — there follows nevertheless an admirable order toward the end sought by God, and a conspiration of the means toward the same, most delightful to the Angelic and Human intellect contemplating these more deeply. But the other things, sought out with such ingenuity by Kepler and others to establish some such Harmony in the heavens (so that nothing should be lacking to it but a sensible sound), seem mere symbolisms, poetic or rhetorical rather than philosophical — as Kepler himself, in the appendix to his Harmonics (p. 253), speaks of the symbols of Ptolemy and of Robert Fludd (“of the Tides”); and Mersenne (on Genesis 4, p. 1558) judged the same of Kepler’s analogies, calling them symbolisms, at most Oratorical — such as Orators could use to amplify, by such metaphors, the Divine Providence in celestial things. And [Kepler] adds that the eternal Geometry of God is concerned with harmonic proportions rather than with the temperings of figures, colors, tastes, and odors; and that, just as we cannot — by the image of the divine Geometry impressed on our mind — give a reason why this figure, or this mixture of colors, should rather delight the eyes, or why these tastes or odors should please the palate or nostrils more than others (nor in these must one recur to the image of Geometry), so neither, in giving a reason why these sounds rather delight the hearing; and much less, from what is pleasant to the hearing alone, is a reason to be drawn of the celestial motions and of the intelligible beauties.