(Et Qua ratione Distantijs Siderum Intervalla Systematis Harmonici debeant accommodari)

Achilles Tatius (in the Isagoge to the Phaenomena of Aratus, ch. 7, “On the order of the spheres”) says [that the ancients treated of the celestial harmony]. Of their harmonic motion there treat — in the Canon, as has been said — Aratus, and Eratosthenes in the Mercury [Hermes]; nor were Hypsicles, Thrasyllus, and Adrastus of Aphrodisias the [first] authors of this doctrine, but the Pythagoreans, who hold that the universe is moved by [number] and order. Boulliau [Bullialdus] too, in the prolegomena to his Astronomia Philolaica, [reports] under the name of François Viète a Harmonicon caeleste, which Pierre Dupuy (Puteanus) gave to Fr. Marin Mersenne to use, and to him indeed …

[The opening lines of Chapter VIII are clipped at the column’s left edge in the original scan; the gist is secure (a roll-call of ancient and modern authorities on celestial harmony), with a few connective words supplied. The text breaks off at “huic ve-” - catchword “rò” (verò); it continues on p. 526 (PDF 561), within Chapter VIII.]

(printed p. 526 — Chapter VIII (fitting the harmonic intervals to the distances of the stars), continued. The First Opinion, of the Pythagoreans, is examined: its prior form, giving the ancient distance-estimates in stadia and Earth-semidiameters, is shown to be far too small against observed parallaxes; its posterior musical form, from Pliny and Censorinus, assigns tone-intervals between the spheres summing to six tones, the octave, tabulated with the resulting absurd parallaxes. Then the Second Opinion, of the Platonists, begins, with Macrobius on the World-Soul woven from music.)

…[which Pierre Dupuy gave to Fr. Marin Mersenne to use, but] which seems to have been filched from him. But since we do not know the opinion of these [men], we shall pass down to the views of others.

[Margin: The 1st Opinion — of the Pythagoreans.]

[I.] The First Opinion was that of the Pythagoreans, which we shall try to fish out from Pliny, Plutarch, and Censorinus. Pliny, then (bk. 2, ch. 21), says: “Many have attempted to track out the intervals of the stars from the earth, and have declared the Sun to be distant from the Moon nineteen times as far as the Moon itself is from the earth. But Pythagoras, a man of shrewd mind, reckoned it to be from the earth to the Moon about 126,000 stadia; from her to the Sun double that; and thence to the twelve signs [of the Zodiac] triple” — in which opinion was also our Gallus Sulpicius. Twin to this passage, in part, is that of Plutarch (On the Opinions of the Philosophers, bk. 2, ch. 31), where he says that Empedocles [held] the Moon to be twice as distant from the Sun as from the earth; the Mathematicians, that the Moon is distant from the Sun nineteen times as much as from the earth itself; and Eratosthenes, that the Moon is distant from the earth 780,000 stadia.

[Margin: The distance of the Moon from the Earth and from the Sun, according to the ancient Mathematicians.]

I should believe these Mathematicians — who determined the distance of the Moon from the Sun to be nineteenfold the distance of the Moon from the earth — to be Aristarchus and Hipparchus and their followers. For Aristarchus, in his book On the Magnitudes and Distances of the three bodies (proposition 7), had demonstrated that the distance at which the Sun is from the earth is greater than eighteenfold the lunar distance from the earth, but less than twentyfold — that is, about nineteenfold. And Hipparchus [put] the Moon’s distance at about 64 terrestrial semidiameters, and the Sun’s at 1200, which to 64 is about as 19 to 1.

It is therefore astonishing that Eratosthenes assigned to the lunar distance only 780,000 stadia, that is, 97,500 Italian miles. For (as said in bk. 2, ch. 7) Eratosthenes set the diameter of the Earth at 80,181 stadia, and its semidiameter at 40,090½; and 780,000 stadia divided by 40,090½ yield, for the lunar distance, a full 19 terrestrial semidiameters. But much more shamefully must Pythagoras have blundered, who allowed this distance no more than 126,000 stadia, which divided by 8 make 15,750 Italian miles — that is, scarcely 4 terrestrial semidiameters — though the lunar parallax protests, from which we already derive a distance from the earth of more than 51 semidiameters (bk. 4, ch. 14). And that distance doubled would make the Sun’s distance 8 terrestrial semidiameters, and tripled, the distance of the Fixed stars 12 semidiameters of the Earth; so that, if we followed the ancient Pythagorean opinion, the Sun and the Fixed stars would lie under a parallax of several degrees, against the evidence of observations. These rudiments being set down, let us pass to the other opinion of his.

[Margin: The later opinion of Pythagoras.]

[II.] Having finished ch. 21, then, Pliny (bk. 2, ch. 22) subjoins, saying: “But Pythagoras sometimes, by a musical reckoning, calls a tone the distance by which the Moon is from the earth; from her to Mercury half of that; and from him to Venus nearly the same; from her to the Sun a sesquiple [a tone and a half]. From the Sun to Mars a tone — that is, as much as from the earth to the Moon; from him to Jupiter half; and from him to Saturn half; and thence a sesquiple to the Zodiac: so that seven tones are made, which they call the Diapason harmony — that is, the universality of concord. In it Saturn is moved in the Dorian, Mercury in the phthongus, Jupiter in the Phrygian,” and the like in the rest — with a subtlety more pleasant than necessary.

Which opinion Censorinus (On the Birthday, ch. 11) hands down indeed more explicitly, but in the distance of Saturn from the Fixed stars he differs from Pliny (unless the text be corrected); for he says: “Pythagoras taught that this whole world was made by a musical reckoning, and that the seven stars wandering between heaven and earth — which govern the births of mortals — have a rhythmic motion (ἔρρυθμον), and intervals agreeing with the musical diastemata, and render various sounds according to their several heights, so concordant that they sing a most sweet melody, yet one inaudible to us because of the magnitude of the sound, which the narrowness of our ears could not take in.” And a little after: “Pythagoras judged how many stadia there were between the earth and the several stars. (By a stadium, in this measure of the world, is chiefly to be understood what they call the Italian, of 625 feet — that is 125 paces — which make the eighth part of an Italian mile.)” Then he proceeds, saying: “Therefore from the earth to the Moon he thought there were about 126,000 stadia, and that this is the interval of a tone; from the Moon to the star of Mercury (which is called Stilbon, στίλβων), half of that, as it were a hemitone (ἡμιτόνιον); thence to Phosphoros (φωσφόρον), which is the star of Venus, nearly the same — that is, another hemitone; from there onward

to the Sun, three times as much, as it were a tone and a half: so that the star of the Sun is distant from the earth three tones and a half, which is called a Diapente [a fifth]; but from the Moon, two and a half, which is called a Diatessaron [a fourth]. From the Sun to the star of Mars (whose name is Pyroeis, πυρόεις) there is as much interval as from the earth to the Moon, and this makes a tone (τόνον); thence to the star of Jupiter (which is called Phaethon, φαέθων), half of it, which makes a hemitone; and as much again from Jupiter to the star of Saturn (whose name is Phainon, φαίνων), that is another hemitone; thence to the highest heaven, where the Signs are, [another] hemitone. So that from the highest heaven to the Sun the interval is a Diatessaron — that is of two tones and a half — but from the heaven’s summit to the earth there are six tones, in which is the Diapason (διὰ πασῶν) symphony.”

The numbers and proportions of Censorinus Zarlino too described (Harmonic Institutions, part 1, ch. 6), setting from the earth to the supreme heaven — not from his own, but from Pythagoras’s opinion — six tones. For he himself denies that from the earth to the Moon there is a tone, since the earth, being immobile, is unfit to represent any sound, and so to make music with the Moon. Georgius Valla (Music, bk. 1, ch. 2) refers that “sesquiple” in Pliny not to a tone but to a semitone, so that from Saturn to the Fixed stars there are three-quarters of a tone. But Glareanus (Dodecachordon bk. 2, ch. 3) affirms that all the old codices of Pliny have not [the sesquiple] but six tones, and so that for “sesquiple” one should read “semitone”; and he denies that the Diapason can arise, by any musicians’ opinion (much less by Aristoxenus’s), out of seven tones.

[Margin: Pliny must be corrected.]

Certainly, if we consult the scale of Guido taken from Boethius (but augmented and corrected), which we set out in ch. 6, num. 3, we shall see that from the first string to the octave (which make the consonance Diapason) there are indeed seven intervals, but only six tones — namely five whole [tones] and two hemitones. Finally, the most diligent author Censorinus cannot otherwise be reconciled with Pliny. Since he affirms that the Pythagorean stadia are equal to the Italian, and so puts from the Moon to the Earth only 126,000 stadia — that is 15,750 Italian miles, which are about 4 terrestrial semidiameters — we shall easily construct the following little table, with the parallaxes agreeing with the Pythagorean intervals.

Pythagorean Intervals

(Pythagorica Intervalla, ex Plinio correcto & Censorino — “from corrected Pliny and from Censorinus”)

(The original sets three columns: “Distances of the planets among themselves” (the interval-name and its value in Earth-semidiameters — Tone = 4, Semitone = 2, Sesquitone = 6), “And from the Earth” (the cumulative distance in semidiameters), and “Horizontal Parallax” (in degrees and minutes). The intervals sum to 6 tones = the octave; the distances run 4-6-8-14-18-20-22-24 Earth-semidiameters. The parallaxes are printed only for the Moon, Sun, and Fixed stars; each checks as the arcsine of 1/distance — arcsin(1/4) ~ 14°28′, arcsin(1/14) ~ 4°6′, arcsin(1/24) ~ 2°23′ — and their enormity (whole degrees for the Sun and Fixed stars) is exactly Riccioli’s point that the Pythagorean distances are far too small.)

[Margin: The 2nd Opinion — of the Platonists.]

[III.] The Second Opinion was rather of the Platonists than of Plato himself, who rejected the planet-distances determined by Archimedes for no other cause than that they did not keep the proportions due to Music — as Macrobius relates (bk. 2 on the Dream of Scipio, ch. 3), where he has it thus: “Rightly, therefore, is everything that lives captivated by music; for the celestial soul, by which the universe is animated, took its origin from music. This [soul], while it impels the world’s body to its spherical motion, makes a sound — distinguished by unequal intervals, yet marked off by a proportionate reckoning, just as [the soul] was from the beginning woven together.” But these intervals — which in the soul, as incorporeal, are reckoned by reason alone, not by sense — must be sought …

[The text breaks off; the catchword “Vtrum” (“Whether…”) points to p. 527 (PDF 562), continuing ¶III on the Platonists’ opinion, within Chapter VIII.]

(printed p. 527 — Chapter VIII continued. The Second Opinion (the Platonists) concludes: the Timaeus multipliers yield a table of “Platonic Intervals” for the planetary distances, which Riccioli refutes as wildly at odds with demonstrated solar and Saturnian distances — “farewell to this Pythagorean and Platonic Harmony.” The Third Opinion, of Fr. Mario Bettini, then begins: the world-radius as a monochord of 3435 semidiameters, with the Sun at 1145 sounding “Sol” and the planets placed at octave consonances of the whole chord; the page ends with the head of Bettini’s monochord table.)

[Margin: The absurdity of the Platonic intervals.]

…[these intervals must be sought by reason, not by sense.] And Archimedes indeed believed he had grasped the number of stadia by which the Moon is distant from the earth’s surface; the Moon from Mercury; Mercury from Venus; the Sun from Venus; Mars from the Sun; Jupiter from Mars; Saturn from Jupiter; and from Saturn’s orb up to the starry heaven itself, he thought he had measured out the whole space by reason. Yet this Archimedean measurement was rejected by the Platonists, as not keeping the double and triple intervals. And they laid it down that this must be believed: that, as great as is [the distance] from the earth to the Moon, double of that is from the earth to the Sun; and as great as is from earth to the Sun, triple of that is from earth to Venus; and as great as from earth to Venus, fourfold is from earth to the star of Mercury; and as great as is from earth to Mercury, ninefold is from earth to Mars; and as great as is from earth to Mars, eightfold is from earth [to Jupiter; and as great as from earth to Jupiter,] twenty-sevenfold is from earth to the orb of Saturn.

This Platonic persuasion Porphyry inserted in his books, by which he infused not a little light into the obscurities of the Timaeus; and he says that they believed the intervals in the body of the world to be filled, after the image of the soul’s weaving, by Epitrites [4 : 3], Hemiolia [3 : 2], Epogdoa [9 : 8], and Hemitones and Limmas, and that thus the concord arises. Which proportions Marsilio Ficino also reports (in his compendium on Plato’s Timaeus, ch. 34), and asserts that they seem to him more probable, and according to Plato’s mind in the Timaeus and in Republic 8 and 10. That we may weigh these intervals, then, we shall assume the lunar distance from the Earth to be nearest to 60 terrestrial semidiameters (per what was said in bk. 4, ch. 4); which posited, the Platonic Intervals will be as may be seen in the following table.

Platonic Intervals

(Intervalla Platonica)

(Verified: each “distance from Earth” is the running product of the proportions — 1, 1·2, 2·3, 6·4, 24·9, 216·8, 1728·27 = 1, 2, 6, 24, 216, 1728, 46656; times 60 = the last column. The multipliers 1, 2, 3, 4, 9, 8, 27 are Plato’s Timaeus soul-numbers.)

There would be, therefore, a distance of the Sun from the Earth of only 120 terrestrial semidiameters — which conflicts with what we demonstrated (bk. 3, ch. 7); but a distance of Saturn from the Earth of about 2,799,360 terrestrial semidiameters — that is, to the solar distance as 23,328 to 1, whereas it ought to be only about 10 to 1, or nearly tenfold, by what was said (bk. 7, sect. 6, ch. 1 and 2). Farewell, then, to this Pythagorean and Platonic Harmony, which does not reconcile reason with the experiments of the senses and with observations — though it ought to have been reconciled, as Ptolemy rightly decides (Harmonics bk. 1, ch. 1 and 2). For it is not enough that these proportions agree with reason, even if they cannot be approved by sense — as Zarlino seems to have admitted (part 1, ch. 6).

[Margin: The 3rd Opinion — Bettini’s.]

[IV.] The Third Opinion is that of our Fr. Mario Bettini (Apiarium 10, Progymnasma 1, propositions 1 and 3, to be read with their Scholia). He supposes, first, that the Sun is distant from the center of the Earth 1145 terrestrial semidiameters, and is in the middle of the Planetary System; and that its distance, taken with the remaining distance up to the Empyrean, makes a Diapente [a fifth]; and therefore that the Sun’s distance from the Earth must be tripled, so that the semidiameter of the whole World may be held as one entire Monochord — namely 3435 terrestrial semidiameters. Whence it follows that, ascending from the earth to the Sun, the Sun occupies the place of the fifth string, and to it agrees the voice which in Guido’s musical Scale is called Sol — fitting to the name of the Sun (Sol).

Then he determines the distances of the planets — indeed, of the heavens above the planets too — according to the consonances of the more recent Musicians, admitted in the common use of the Octochord; but an Octochord purely Dorian, and holding the Diatonic gravity. Further, that he may serve both the ease of Arithmetical operation and the order of the planets found by the more recent Astronomers, and that at the same time the harmonic consonances may disagree as little as possible with observations, he takes some planets at their mean distance, some at their Apogee, some at their Perigee. These being supposed, he determines the intervals of the planets and heavens from the earth, in terrestrial semidiameters, as you see in the following table — in which, as I said, the distance of the Empyrean from the earth’s center, that is, the whole chord, is 3435 terrestrial semidiameters; and:

• the Moon at Apogee — so that by the ancients it be a major Tone, which is the ninth part of the whole chord — is 381 6/9 semidiameters;
• the Venus at Perigee makes the consonance of a Ditone [major third], namely the fifth part of the whole chord, which is 687 semidiameters (so that between the Moon at apogee and Venus at perigee he sets a minor tone);
• the Mercury at Perigee makes the consonance of a Diatessaron [fourth], which is the fourth part of the whole chord, 858 3/4 semidiameters (so that between Mercury and Venus, both at perigee, he sets a major semitone, or Apotome);
• the Sun’s mean distance, as said, 1145 semidiameters, with the residue of the whole chord (which is 2290) makes a Diapente [fifth], and with the whole chord a Diapason-diapente [twelfth]; whence between Mercury and the Sun falls a major Tone, as he supposes;
• the Mars mean distance occupies the terminus of the major Hexachord [major sixth], that is two-fifths of the whole chord, and so 1374 semidiameters;
• to hunt out Jupiter’s distance, he subtracts the major hexachord (1374) from the whole chord (3435), and there remain 2061; of this he takes the ninth part as a major tone, namely 229, and adds it to Mars’s distance, and Jupiter’s distance from the earth becomes 1603 semidiameters;
• Saturn’s distance he makes half of the whole mundane semidiameter, or of the whole chord, that is 1717 1/2 semidiameters, that it may acquire the consonance of a Diapason [octave] with the whole chord;
• the Fixed stars’ distance he makes two-thirds of the whole chord [2290], that they too may sound a Diapente;
• the Crystalline heaven’s distance he determines by the consonance Diatessaron, but so that from earth to the Crystalline there be three-quarters of the whole chord, that is 2574 3/4 semidiameters;
• whence it comes about that the distance of the Empyrean and Crystalline to the whole chord is as 1 to 4, which is the Disdiapason [double octave].

From these we have composed the following table, drawn from his own express words, yet with many consonances added that arise from his intervals, which he passed over in silence.

Harmonic Intervals — from Fr. Mario Bettini

(Harmonica Intervalla, indicated on the line AB as the entire String of the World-Monochord)

(This table only begins here — the line AB is the whole world-monochord, 3435 semidiameters, A being the Earth and B the Empyrean; for each body the proportion of the whole string AB to the outer segment B–[point] gives a consonance. It breaks off after Venus, with the catchword “RESI-”; the remaining bodies — Mercury, Sun, Mars, Jupiter, Saturn, the Fixed stars, the Crystalline, and the Empyrean — follow on p. 528. Note: 3/4 of 3435 is 2576 1/4, so the printed Crystalline value 2574 3/4 appears to be a small slip or rounding; all the other figures check exactly against the 3435 chord.)

[The catchword “RESI-” (Residuum…) points to p. 528 (PDF 563), which continues Bettini’s monochord table, within Chapter VIII.]

(printed p. 528 — Chapter VIII continued. Bettini’s monochord table is completed, then Riccioli critiques his system: though more concinnous than the Pythagorean and Platonic schemes, it contains six errors — impossible lunar and solar distances, incompatibility with lunar and solar eclipses, Jupiter and Saturn placed far too near, and fixed stars so close as to imply a false parallax — all sprung from Bettini’s wish that the Sun sound “Sol” in the fifth place. The Fourth Opinion, Kepler’s harmonic intervals from Harmonics book 5, then begins with the head of his table of planetary aphelion and perihelion distances.)

Remainder of the preceding Table

(Residuum Tabulae Praecedentis — continuing Bettini’s world-monochord table from p. 527; the line AB = 3435 Earth-semidiameters, A the Earth, B the Empyrean)

Further consonances arising from the same intervals (which Bettini “passed over in silence”), measuring the whole string AB against each inner segment A–[point], and segment against segment:

(All verified against the 3435 chord, with B-point = 3435 - [distance]: e.g. BF = 2576¼, so AB:BF = 4:3; BG = 1717½, so AB:BG = 2:1; BK = 1145, so AB:BK = 3:1; BL ~ 858¾, so AB:BL = 4:1. The last added line, “AH to AF, as 2 to 1, Diapason,” is printed thus but does not check: AH:AF = 1374 : 858¾ = 8:5, a minor sixth — the 2:1 octave actually holds for AH:AE = 1374 : 687, Mars-mean to Venus-perigee; the printed “AF” appears to be a slip for “AE.”)

[Margin: Bettini’s. — 1st error.]

[V.] Although the aforesaid symmetry [of Bettini] is much more concinnous than either the Pythagorean or the Platonic, yet it includes many repugnancies and fallacies, and induces a most absonant discord between Astronomy and Harmonics. For, first, it attributes to the Moon so great a distance as no astronomer ever assigned, or can assign, saving the parallaxes. For no one ever found, or will find, in it a horizontal parallax greater than 1°43′, or less than 51½′ — of which the former imports a distance from the earth of about 33½ terrestrial semidiameters, the latter of about 67 — as is certain from Geometry and from what was said (bk. 4, ch. 14). And this very manner of determining the distances of the stars by parallaxes Bettini himself approves (Apiarium 8, Progymn. 3, prop. 9; Progymn. 4; and Aerarium Philosophiae vol. 2, p. 73). [His lunar distance of 381, then, is impossible.]

[Margin: 2nd error.]

Secondly: from the lunar distance of 381 6/9 semidiameters which he sets, it would follow that the Moon could never be eclipsed by the Earth’s shadow — which is against the experience of every age, and against all who are even slightly versed in Astronomy; nay, against what the same Father most learnedly teaches concerning the lunar eclipses made by the Earth’s shadow (Apiarium 8, progymn. 2 and 3, prop. 11). For no one ever raised the height of the terrestrial shadow beyond 282 semidiameters of the Earth (as is clear from bk. 3, ch. 11, Probl. 8). Nor may you say that hence only the eclipse of the Moon at apogee is removed, but not below the apogee; for, the 282 semidiameters of the shadow being subtracted (at the most favorable), from the lunar distance of 381 there remain 99 terrestrial semidiameters — a difference between the Moon’s perigee and apogee that no one assigns; and yet we know that the Moon, even at apogee, has sometimes fallen into a total eclipse, and so far below the apex of the Earth’s shadow.

[Margin: 3rd error. — 4th error.]

Thirdly: a total eclipse of the Sun could indeed happen if the Moon were distant from the earth 381 and the Sun 1145 semidiameters, as he sets; for, the apparent diameters of the luminaries being preserved, such an eclipse would be impossible (as is clear from bk. 5, ch. 9); yet Bettini himself admits a total solar eclipse (Apiarium 8, Progymn. 2, prop. 10), and is bound to admit it from the history of eclipses, of which we [treat] (bk. 5, ch. 20). Fourthly: the lunar distance of 381 semidiameters cannot cohere with the solar distance of 1145 which he posits, as is proved by Aristarchus’s problem founded on the phase of the lunar dichotomy [half-moon] — which Bettini himself praises (Aerarium vol. 1, p. 629); for, the lunar distance being put at 381 6/9, or nearly 382, it follows that the Sun is distant from the earth at least 7299 terrestrial semidiameters — indeed in truth much more, the [dichotomy] angle being posited, which we observed at the time of the dichotomy, as is clear from what was demonstrated (bk. 3, ch. 7, probl. 3).

[Margin: 5th error. — 6th error.]

Fifthly: he places Jupiter and Saturn far nearer to the Sun and the Earth than the prosthaphaereses [equations of the orbs] and the commensurations of the orbs (demonstrated by Copernicus and others) require; for these require that the distance of Jupiter be nearly fivefold the distance of the Sun from the earth, and that of Saturn nearly tenfold (as we taught, bk. 7, sect. 1, ch. 1 and 2); whereas by his [scheme] Jupiter’s distance to the Sun’s distance is less than 14 to 11, and Saturn’s less than 16 to 11. Sixthly: he gives the [Fixed] stars a distance from the earth of 3435 — so small that no astronomer could ever assign it, since from it would follow a horizontal parallax of a minute and a half, which all astronomers know to be false.

[Margin: Other incongruities.]

Now all these errors arose from this: that he wished the Sun’s distance both to be in the fifth place and to utter the voice “Sol,” and to make a Diapente from the supreme heaven with the whole semidiameter of the World; and [that he wished] the Moon to make a tone, that is, to contain one-ninth of that whole interval in its distance, and so forth. I pass over that he assigns to Jupiter — not the least of the planets — no worthy consonance; that he makes a certain consonance answer to four planets outside their perigee position, to the Moon outside apogee, to the Sun and Mars outside their mean distance — as though indeed they did not usually keep the harmony; which is laid down too inconsistently by one who has taken upon himself the demonstration of the Harmonic proportions observed by God in the intervals of the planets. Finally, that the Sun may hold the place of the fifth string, the Earth is assumed by him as the first string — though both [the Earth] and the Empyrean, by their immobility, are unfit to represent a sound.

[Margin: The 4th Opinion.]

[VI.] The Fourth Opinion is that of Johannes Kepler (Harmonics bk. 5, ch. 4), where he sets out the extreme intervals from Tycho’s observations — that is, the distances of the planets from the Sun — in which, at the aphelia and perihelia, the harmonies are found, except for Mars and Mercury, as appears from the Table which he exhibits, as here below. For the understanding of which it must be supposed (from ch. 3 of the same book) that the extreme Convergence of two planets is when their apsides are nearest — namely at the Perihelion of the superior and the Aphelion of the inferior planet; but the extreme Divergence is the opposite apsides of the two planets — namely the Aphelion of the superior and the Perihelion of the inferior. Let the Table now stand:

Intervals compared with the Harmonics

(Intervalla Comparata cum Harmonicis — of which the radius of the Annual Orb [the Earth’s orbit] is 1000)

(This table only begins here — it sets each planet’s distance from the Sun at aphelion and perihelion, the Earth’s orbital radius being 1000, and compares each planet’s own aphelion-to-perihelion interval with a musical consonance. Saturn’s interval, 10052 : 8968 = 1.121, lies between a minor tone (10:9) and a major tone (9:8); Jupiter’s, 5451 : 4949 ~ 11:10, fits no clean consonance; Mars’s, 1665 : 1382 ~ 1.205, is near 6:5. It breaks off after Mars, with the catchword “RESI-”; the remaining bodies — Earth, Venus, Mercury — and the comparison columns follow on p. 529.)

[The catchword “RESI-” (Residuum…) points to p. 529 (PDF 564), which continues Kepler’s table, within Chapter VIII.]

(printed p. 529 — Chapter VIII continued. Kepler’s table of intervals is completed, and its use explained: single planets’ own aphelion-perihelion intervals yield no clean harmony, but the convergent and divergent intervals between adjacent planets do — yet Kepler himself denies that harmonic proportions are to be sought among mere lengths, harmony’s true subject being the motions. The Fifth and Riccioli’s own Opinion then begins: the distances of planets and fixed stars are not determined by harmonic proportions, supported by Aristotle, Pliny, Macrobius, Zarlino, Glareanus, Mersenne, and Kircher’s doctrine of a harmony of disposition rather than sound.)

Remainder of the preceding Table

(Residuum Tabulae Praecedentis — completing Kepler’s “Intervals compared with the Harmonics” from p. 528; annual-orb radius = 1000)

(The full index-key for both halves of Kepler’s table — a = Saturn aphelion (10052), b = Saturn perihelion (8968); c = Jupiter aph. (5451), d = Jup. per. (4949); e = Mars aph. (1665), f = Mars per. (1382); g = Earth aph. (1018), h = Earth per. (982); i = Venus aph. (729), k = Venus per. (719); l = Mercury aph. (470), m = Mercury per. (307).)

[Margin: Use of the table.]

The extreme intervals of no single planet, then, allude to Harmonies — except those of Mars and Mercury. For in Mars, if its perihelion 1382 were 1388, it would be to its own aphelion 1665 as 5 to 6, and so would be the consonance of a semiditone (minor third) — from which, however, it is little distant. But in Mercury, although to Kepler it seems to allude to a Diapente, or to the harmony 8 : 5 (the minor Hexachord), to me its extreme intervals seem rather to allude to a Diatessaron (which is between 4 and 3) — or else this interval is to be despised [as negligible].

But if you compare the extreme intervals of different planets among themselves — says Kepler — some light of Harmony shines forth, as is clear to one who contemplates the first column of the preceding table and its index-characters of the extremes:

But Kepler subjoins that these intervals, in so far as they are lengths without motion, are not aptly examined for Harmonies — whose subject is rather motion itself, as to its swiftness and slowness; and therefore, if we seek harmonies, they are not to be sought in the intervals in so far as they are semidiameters of the orbs, but in so far as they are the measure of the motions — that is, rather in the motions themselves; especially since, for the semidiameters of the orbs, nothing can be taken but the mean distances from the Sun, among which the Harmony shines forth less than in the extremes of the aphelia and perihelia. Wherefore Kepler absolutely denies that harmonic proportions are to be sought among the intervals of the planets as such, and apart from motion.

The Fifth and Our Opinion, and the Authorities for it

(Quinta et Nostra Opinio, et Authoritates pro illa)

[Margin: Aristotle’s. — Macrobius’s. — Zarlino’s, Glareanus’s. — Mersenne’s. — Fr. Athanasius’s praise and opinion in matters of Music.]

[VII.] The Fifth — and Our — Opinion, not unlike the preceding and truer, is that the distances of the Planets and Fixed stars — whether from the Earth, or from the Sun, or among themselves — are not to be determined by Harmonic proportions or Musical intervals. Of this opinion, without doubt, was Aristotle (On the Heaven 2, text 52), where, of the Pythagoreans who sought a harmony in the motion and intervals of the heavens, he said that “this is indeed said wittily and elegantly, yet the truth is not so.” And Pliny (bk. 2, ch. 22), where, after relating Pythagoras’s opinion of the intervals of the stars by Tones, Semitones, and Sesquitones, he concludes that these and the like said by him were “with a subtlety pleasant more than necessary.” Of the same opinion too was Macrobius (bk. 2 on the Dream of Scipio, ch. 4), thinking this inquiry belongs to one “showing off, not teaching” — whose opinions we have so often inculcated, as often as occasion returns to us.

To Aristotle and Pliny subscribed in this Zarlino (Institutions part 2, ch. 29) and Glareanus (Dodecachordon bk. 1, ch. 13, and bk. 2, ch. 13), where he says: “But whether the intervals of the orbs, in the heaven itself, stand by the same reckoning as the phthongi [notes] in the Diapason, does not seem to me probable, by whatever genus of mode we may finally constitute them.” And, some things being inserted, [he adds that] Boethius, the true judge of this business — since he saw these things wonderfully varied among the ancients, and Pliny not afraid of a subtlety pleasant more than necessary — so tempered [the matter] as nevertheless to set both opinions before the eyes; and finally, declaring his own opinion, [says] that “not without reason does this notion seem to Aristotle pleasant to say rather than likely to be true.”

Nor is there doubt that Mersenne (on Genesis 4, verse 21, p. 1558) was of the same opinion, when he says: “I know it is not necessary that the mutual distance of the Planets among themselves and with the Earth be exactly [proportioned] to their sounds and magnitudes, so that their Music may represent their economy [order] — though would that it could be done to the line, that we might have some commerce with [their] voices and instruments.” And (p. 1559) he makes light of those very allusions and consonances which we have reported from Kepler in the preceding number, because the opinions of Astronomers about the distances of the stars are diverse — which [of them] the Musician should choose [he cannot tell]. Again (p. 1703), where he reports the opinion of certain [astrologers] saying that the Diapason and Disdiapason are governed by the Sun, the Diapente by Venus, the Diatessaron by [Mars], and the Diapason-diapente by Jupiter, he says (on the following page) that all these are to be repudiated, and adds: “In vain, therefore, will there be harmonists, if [they hold it to be] from the earth to the Moon a tone; from the Moon to Mercury a semitone, and from him to Venus as much; from the earth to the Sun a Diapente; from the Moon to the Sun a Diatessaron; from the Sun to Mars a tone, thence to Jupiter a semitone, as much from Jupiter to Saturn, and from Saturn to the Firmament; and consequently from the Sun to the Firmament a Diatessaron, and from the earth to the Firmament a Diapason.” For it is sufficiently clear, by Geometric and Astronomical experience, that those spaces are wrongly constituted (as appears from article 3, and from many other places, concerning the distance of the heavens from one another); but the conclusion must be drawn from the proportion of the motions alone.

Most recently, Fr. Athanasius Kircher treats best of all of the Music reigning in the heaven and in all Nature — especially in Musurgia bk. 10, which he entitles the Decachord of Nature (Register 2, paragraph 2); inquiring what the harmony of the celestial bodies is, and in what it consists, he concludes that their Harmonic consent consists “not in the periodic numbers of the motions, nor in any sensible collision of the celestial bodies, but in nothing else than in their wonderful disposition, and a certain ineffable proportion, conspiring into unity — by which the mundane bodies so correspond to one another that, one being removed or changed, the harmony of the whole would deservedly perish,” and so on. Which harmony, as we have said, consists in the wonderful disposition and most proportionate [arrangement] of one body …

[The catchword “ris” (corpo·ris) points to p. 530 (PDF 565), which continues Kircher’s view and Riccioli’s conclusion, within Chapter VIII.]

(printed p. 530 — Chapter VIII continued. The Fifth Opinion’s Kircherian thread concludes: the true celestial harmony is God’s providential disposition of distances and magnitudes for the good of sublunary nature, not any audible sound. Five reasons for Riccioli’s opinion follow: the motions make no sensible sound; any consonances would exceed the range of human music; harmonic ratios of magnitude are not true harmony, which lies in the quality of sounds; the harmonics cannot be reliably located among the disputed distances; and harmony’s arbiters are sense and reason together, per Ptolemy.)

…[the harmony consists in the wonderful disposition and most proportionate analogy] of one mundane body to another, by their interval; and also in the most exactly fitted analogy of the quantity, or magnitude, of each for obtaining its own end. But the end intended by God was not the delight, sensible to hearing, of a sound roused from the heavens, but the production, conservation, and advancement of sublunary nature (especially of living things) toward [its] ultimate end, each in its own manner — which Kircher there goes on to expound excellently and at length. And therefore [God] attributed so great a distance to the Planets that they might produce effects agreeing with sublunary natures: for if the Moon or Sun were much nearer than they are, the former would moisten and chill [things] too much, the latter would dry up and overheat them; and unless, between Saturn and Mars — those discordant, pestiferous, and malign planets — He had interposed Jupiter with its four companions [the four moons], wholesome and most temperate, and on this side the beneficent Sun and Venus, great and intolerable harms would have followed.

This, before Kircher, Pliny had noted (bk. 2, ch. 8), where, having said that the star of Saturn is “of a cold and stiffening nature,” and a little after that the third star, of Mars (which some call Hercules’s) is “fiery and burning from the nearness of the Sun, completing [its course] in about two years,” he at once subjoins: “And therefore, interposed between both by the excessive heat of the one and the cold of Saturn, Jupiter is tempered by both, and made wholesome.” And the Ciceronian Paulus intimated the same in the Dream of Scipio, when he says: “of which [orbs] one globe is held by that [star] which on earth they call Saturn’s. Next is that prosperous and wholesome gleam to the human race which is called Jupiter’s. Then the ruddy one, dreadful to the lands, which you call Mars’s.” But what we stammer out concerning these things, toward indicating a certain specimen of divine Providence, must be understood (if it can be understood by us at all) of innumerable other reasons and proportions chosen by God in ordering [all things] to their end.

Reasons for the Fifth and Our Opinion

(Rationes pro Quinta et Nostra Opinione)

[Margin: 1st reason.]

[VIII.] First: the intervals and motions of the celestial bodies either produce no sound at all — especially in the Planetary system, where the planets move in a fluid and most tenuous ether — or at any rate produce no sound sensible to us, as is clear by experience; wherefore it was truly said by Pliny (bk. 2, ch. 3): “To us who dwell within, the world glides on silently, alike by day and night.” They are therefore not ordained by God for sound; and accordingly the harmonic proportions are not to be sought in them — those proportions which, in human voices, are consonant to our ears so as to produce a concinnity pleasant to them; for, the end being taken away, the order and proportion due to that end is taken away too. And much more [is this so] in the intervals themselves, taken apart from motion — since, as Kepler rightly said, the proper subject of harmony is not immobile quantity, but mobile: that is, the very motion of the bodies.

[Margin: 2nd reason.]

Secondly: even if the celestial bodies did, by their motion, produce a sound sensible to us, yet the consonances of their sounds would not fall within those terms and limits within which the consonances of human voices fall — which, on account of our weakness or the necessity of [our] nature, we confine within the bounds of the Boethian system or the Guidonian scale. For since the lung, throat, palate, tongue, teeth, and lips are not ordained to this end only (that, by forming the voice, we may modulate sweetly), but also to many other ends, these organs had to be so framed that they could descend only to a certain limit of gravity, or ascend [only] to [a limit] of acuteness. But in other animals there are those that can form a graver or acuter voice than we; and in the strings and pipes of organs far more and subtler differences can be found — how much more in the motions of the stars, mingled with such great variety. In vain, therefore, would we measure the laws of that [celestial] concord by the over-short and meager norm of our [human] harmony; it would be just as if we wished to cramp the Angelic songs — in the innumerable bodies which [the angels], by God’s permission, could assume — to our [human] measures.

[Margin: 3rd reason.]

Thirdly: If a harmonic proportion is to be sought in the celestial intervals or motions — chiefly because in them some symmetry can be found, or even a double, triple, quadruple, quintuple proportion, etc. (if not most exactly, at least with fractions little vitiating that proportion) — yet, the aforesaid proportions being granted (though not yet conceded), it still does not follow from them, as such, that a true Harmony is found, but [only such as harmonies] are in the qualities of voices and sounds. For so Panaetius said (in his book On the Principles of Geometry and Music): “The faculty of consonances is regarded not in the magnitudes of voices, but in their qualities.” Otherwise, in whatever discrete or continuous quantities are found (reduced by division to discrete ones, in which, besides unity, are the numbers 2, 3, 4, 5, 6, by which all the fourteen consonances reckoned in the first table of ch. 4 are contained), there Harmony would “reign,” taken properly or quasi-properly — and so very many arts would be confounded with one another. Nor must every symmetry be weighed by the Canon of the musical Monochord, or by the norm of harmonic ratios. For who would demand this of the architect in his buildings, or of the apothecary in compounding medicines, or of the general in drawing up a battle-line, or of the shipwright in building ships, or of God Himself and Nature in the structure of the human body, or of animals or plants? It would surely be ridiculous to require that the nerves, intestines, and tendons should have, in the animal body, that measure which the harmonic proportions require; or that the five principal vessels (instruments of the soul) — the brain, heart, liver, kidneys, and spleen — should be so commensurate among themselves, as to quantity or motion, that the brain (for example) should have with another member a Diapason [octave], the heart a Diapente [fifth], the liver a Diatessaron [fourth], the kidneys a Diapason-diapente [twelfth], the spleen a Disdiapason [double octave]. For such a proportion would be unfit for the end to which they are ordained — which is far different from the delight of the ears. The same judgement, therefore, holds of [the proportions] in the celestial intervals and motions, [as] not having that symmetry which is proper to Music — that is, symphonism. Nor is there doubt that many proportions [are] unfit for begetting consonances — such as those arising from the number 7 (and 4) compared with others — and yet are most fit for other effects intended by God and Nature in the heaven and in other bodies.

[Margin: 4th reason.]

Fourthly: Either the harmonic proportions are to be sought in whatsoever intervals of the Planets — and not this, for most intervals do not have them, and so the work of the Divine Tuner [Harmost] would for the most part lack the very harmony which the followers of the Pythagoreans so greatly commend; or [they are to be sought] from the more notable intervals only — namely from the Maximum, Mean, and Minimum distance of the stars among themselves, or from the Sun, or from the Earth, or from [combinations of] all these — and neither can this be obtained: both because the diversity of opinions about these stellar intervals among the more recent Astronomers stands in the way; and because, not even in any single Astronomer’s scheme, are all the maximum, mean, or minimum intervals of the planets so attuned that the commonly received musical consonances are in them; and, finally, because never have all the planets’ Apogees concurred with [other] Apogees, or [their] Perigees with Perigees, in one place under the Fixed stars — or if ever this was [or will be], it does not suffice for a Harmony worthy of God, for that ought to happen always, or for the most part. Just as it does not suffice for the excellence of some chief-musician if the voices and tones are so ordered by him that they consonate twice or thrice but dissonate a thousand times. Not to mention, meanwhile, the variety of the Genera and Modes of Music (indicated in ch. 5), which is so great that it is not certain which Genus is suited to the celestial [bodies].

[Margin: 5th reason.]

Fifthly: Finally, no one is so senseless, or of so stubborn or shameless a brow, as not to grant to Ptolemy (Harmonics bk. 1, ch. 1 and 2) that the arbiters of harmony ought to be Sense and Reason — yet so that Sense, a posteriori, first finds what is near to the truth, while Reason, considering the causes, a priori finds and determines what is exact; or, if Reason first finds what is exact, that finding must nevertheless be received by sense and approved as concinnous — lest either, with the Pythagoreans, we attribute too much to reason, or, with the Aristoxenians, too much to the senses. And that same opinion of his must again be inculcated, from bk. 1, ch. 2, where he says: “It is the harmonist’s aim everywhere to preserve the rational positions of the Canon, in no way repugnant to the senses (by most men’s opinion); just as it is the Astronomer’s aim to preserve the positions of the celestial motions, consonant with the observed revolutions — and these too taken, indeed, from the more evident and more universally apparent [phenomena].” But if we should wish [to determine] the intervals of the Planets from the rules of harmony …

[The catchword “regu-” (regu·lis, “rules”) points to p. 531 (PDF 566), continuing the fifth reason, within Chapter VIII.]

(printed p. 531 — Chapter VIII concludes: harmonic rules for the planetary intervals conflict with observation, so Scripture’s and the Fathers’ “Harmony of the heavens” must be taken metaphorically, not literally. Chapter IX asks whether the magnitude and density of the celestial bodies were determined harmonically, and Riccioli denies it, noting Kepler defined sizes geometrically and Rheita’s proportions contradict observation. Chapter X then opens on whether the planets’ motions were determined by harmonic proportions, beginning with Ptolemy’s treatment in Harmonics 3.9-15.)

(conclusion)

…[But if we should wish to determine the intervals of the planets from the] rules of harmony, they will often be repugnant to the observations evidently made by sense; nor will the commensurations of the orbs, or the prosthaphaereses founded on them, represent the places of the planets such as are detected through Astronomical instruments and accurate observations; and they will so conflict as to be far from that nearness which this business would require, in order that Reason might have a foundation.

[We must say], rather, that God willed to attain other ends through the celestial intervals and motions — yet so that He also proposed this [end]: namely, the beauty of the harmonic ratios, to be wrought for Himself and to be contemplated by the intellectual creature. Since, therefore, observations repeatedly cry out against and dissonate from the harmonic laws, it must be said either that those ends could not be acquired at once through these means, or that God truly did not will it. Accordingly, the assertions of Scripture, of the Fathers, and of the Wise concerning the Harmony and Concord of the heavens are to be taken not in a proper, or quasi-proper, sense — as though nothing else were lacking to [the heaven] in respect of harmonic reckoning except a sound sensible to us — but in a metaphorical sense, and according to a certain analogy and accommodation. In the same way, too, that [text] of Ecclesiasticus 32 must be understood, in political and economic governance: “Have they made thee ruler? Be not lifted up,” etc., “and hinder not the music” — that is, the order and subordination of duties, or the peace and concord of citizens or of a household.