Riemann for Anti-Dummies: Part 14 : The Dissonance that Smiled

The Dissonance that Smiled

By all accounts, Descartes, Newton, Euler, and Kant all shared one common trait: they were grouchy old farts. As such, these poor souls fled from the dissonance and tension by which the universe presents its development to the mind of man. Like their Venetian brethren, who only desired forms of music devoid of Lydian intervals, these minds would not conceive that God would present to them a challenge, by which their own cognitive capacity would be improved. Never could they know that mixture of woefulness and joyfulness that Schiller associates with the sublime. Yet, there is no need to distinguish whether they were grouchy because they hated dissonance, or whether that hatred came because they were grouchy. The lesson to be learned is the same: Grouchy old farts can’t know the minds of Kepler, Leibniz, Kaestner, Gauss, and Riemann, and people who can’t comprehend these great thinkers, become grouchy old farts.

It is in this spirit, that we turn our pedagogical attention to that dissonance in the solar system that today we recognize as the asteroid belt, and the corresponding cognitive transformations that its discovery and investigation produced.

That our solar system would contain an orbit with the characteristics of the asteroid belt, was already affecting human cognition, even before the first asteroid presented itself to human eyes. In his earliest work, the Mysterium Cosmographicum, Kepler had already noticed an anomaly in the organization of the planets in the solar system, with respect to the distances of the known planets from the Sun. While Kepler found that the orbits of the planets, in first approximation, were consistent with the five regular Platonic solids, this ordering produced an anomaly between Mars and Jupiter. This anomaly had impinged on Kepler’s thinking, even before he discovered his polyhedral hypothesis, and, according to his own description, provoked him to arrive at that discovery.

In trying to determine, “why things were such and not otherwise: [namely] the number, size, and the motion of the circles [of the planets],” Kepler first looked for some ratio of numbers that corresponded to the observed distances between the planetary orbits. When this failed, “I tried an approach of remarkable boldness. Between Jupiter and Mars, I placed a new planet and also another between Venus and Mercury, which were to be invisible perhaps on account of their tiny size, and I assigned periodic times to them…. Yet the interposition of a single planet was not sufficient for the huge gap between Jupiter and Mars.” Failing to find a numerical ratio that corresponded to the distances, Kepler tried to find a sequence of inscribed and circumscribed polygons that would correspond to the observed distances. This, too, failed, in the interval between Jupiter and Mars, provoking him to discover the correspondence between the size, number, and motion of the planetary orbits, with the five Platonic solids.

The anomaly between Jupiter and Mars was still indicated under the polyhedral hypothesis, by the placement of the tetrahedron in this gap, as the tetrahedron is the one solid which is its own dual.

This anomaly provoked Kepler to further investigate, and, upon closer examination of the orbit of Mars, he discovered the non-uniform nature of the planets’ orbits. Now, he had a further dissonance. Circumscribing and inscribing spheres around the Platonic solids gave the distances between the circles of the orbits, but the orbits were not circular. They were eccentric. The question posed by this dissonance was: what governed the eccentricities, or, in other words, why was each planet’s eccentricity “this way and not other”?

As he wrote in the Harmonies of the World, “As far as the proportion of the planetary orbits is concerned, between pairs of neighboring orbits, indeed it is always such as to make it readily apparent that in each case, the proportion is close to the unique proportion of the spheres of one of the solid figures; that is to say the proportion of the circumscribed sphere of the figures to the inscribed sphere. However, it is not definitely equal, as I once dared to promise for eventually perfected astronomy….

“From that fact it is evident that the actual proportions of the planetary distances from the sun have not been taken from the regular figures alone; for the Creator, the actual fount of geometry who, as Plato wrote, practices eternal geometry, does not stray from his own archetype. And that could certainly be inferred from the very fact that all the planets change their intervals over definite periods of time, in such a way that each one of them has two distinctive distances from the Sun, its greatest and its least; and comparison of distances from the Sun between pairs of planets is possible in four ways, either of greatest distances or of the least or of the distances on opposite sides when they are furthest from each other, or when they are closest. Thus, the comparisons between pair and pair of neighboring planets are twenty in number, whereas on the other hand there are only five solid figures. However, it is fitting that the Creator, if He paid attention to the proportion of the orbits in general, also paid attention to the proportion between the varying distances of the individual orbits in particular, and that that attention should be the same in each case, and that one should be linked with another. On careful consideration, we shall plainly reach the following conclusion, that for establishing both the diameters and the eccentricities of the orbits in conjunction more basic principles are needed in addition to the five regular solids.”

The “more basic principles” that Kepler discovered concerned the harmonic relationship among the extreme speeds of neighboring planets. The planet’s speed at any moment is a function of its distance from the Sun at that moment; the slowest speed of the planet is at its maximum distance from the Sun (aphelion) and its fastest speed is at its minimum distance from the Sun (perihelion). These extremes are themselves a reflection of the planet’s eccentricity. The solar system chose, so to speak, those eccentricities for the planets that produced the speeds, according to a “more basic principle”. That principle was reflected in the harmonic relationships among those speeds.

Kepler measured those speeds by the arc the planet traversed at perihelion and aphelion, as seen from the Sun, during one Earth day. The results were:

Saturn  at aphelion-  1'30"; at perihelion-  2'15";
Jupiter at aphelion-  4'30"; at perihelion-  5'30";
Mars    at aphelion- 26'14"; at perihelion- 38' 1";
Earth   at aphelion- 57' 3"; at perihelion- 61'18";
Venus   at aphelion- 94'50"; at perihelion- 97'37";
Mercury at aphelion-147' 1"; at perihelion-384' 0";

When these speeds are compared between neighboring planets, their ratios correspond to harmonic musical intervals. Each pair of planets makes two intervals: a converging interval between the perihelion speed of the outer planet with the aphelion speed of the inner one; a diverging interval between the aphelion speed of the outer planet with the perihelion speed of the inner one. These intervals according to Kepler are:

Saturn - Jupiter diverging 1/3;  converging 1/2;
Jupiter- Mars    diverging 1/8;  converging 5/24;
Mars   - Earth   diverging 5/12; converging 2/3;
Earth  - Venus   diverging 3/5;  converging 5/8;
Venus  - Mercury diverging 1/4;  converging 3/5;

These intervals correspond to those Kepler derived for musical intervals with one exception. The converging interval between Jupiter and Mars, deviated from Kepler’s musical intervals by a diesis (otherwise called a quarter-tone). The diesis, Kepler said, is “the smallest interval, by which the human voice in figured melody is almost perpetually out of tune. However, in the single case of Jupiter and Mars, the discrepancy is between a diesis and a semitone. It is therefore evident that this mutual concession on all sides hold exceedingly good.”

That small dissonance would later reveal itself to be a reflection of a further “more basic principle”, whose expression Gauss and Riemann would later provide.