Harmonies and Solids
Chapters 3 and 4
Chapter 5
Harmony of the World
Chapter 9
A Dissonant Harmony

©2006 LaRouche Youth L.L.P.
A Dissonant Harmony

A Dissonant Harmony—The Asteroid Belt

This is what the younger people, the young adults, in the LYM are studying. They are working progressively, they worked through, to a large degree, worked through the study of the ancient Greek roots of modern science, among the Pythagoreans and Plato. They've gone directly into reliving, page by page, chapter by chapter, the process of the discovery of modern astrophysics by Kepler! They're reliving it. They're not saying, "I learned this." They are reliving the moments of tension, in the work of Kepler, where they get to a chapter, a page, and an unresolved question is posed! Now! What's the answer? Well—they've got to apply their minds to thinking what the answer is, and find the answer. They go to the next chapter. 'Huh—we still don't understand it, but it's a big problem.'

Lyndon LaRouche, January 11, 2007 Webcast


When a young mind sets out to discover the causes for things, to find out what universe he's living in, what is his vehicle? Certainly he must have the means of getting outside of what is familiar to him, and out into the yet-unknown. He has to know what is really “out there” if he's going to discover the reason for its existence.

When the curious mind asks, for example, what sort of creatures inhabit the land across some vast expanse of water, he may certainly travel to that place in order to begin his investigation. However, when the young Greek in 400 B.C., the young Johannes Kepler, or even the young person of today wonders: “What lies in between Mars and Jupiter? It's such large space!”, making travel arrangements becomes a bit more complicated, so a different approach may be needed. Besides, even given the ability to “go there”, as some of man's sensory extensions such as the probe “NEAR-Shoemaker” have (more info on NEAR-Shoemaker here), does simply having a photograph of some rock whirling around in space tell you what principle put it there a long, long time before you were here to ask the question? And more importantly, why is it there at all?

NEAR-Shoemaker and Eros

The asteroid Eros, one of the Near-Earth asteroids between Jupiter and Mars, is seen here in the video on the left (taken by the NASA craft "NEAR-Shoemaker") making a complete rotation. In the images on the right are the asteroid up close (top left), from a distance of about 2300 miles (top right), NEAR-Shoemaker ready for take-off (bottom left), and an artist's depiction of NEAR-Shoemaker in flight (bottom right). (Courtesy of NASA)

To ask the question a different way: how does a discovery enter the mind? If you don't know at one moment, but you do at some later point, the discovery had to get in somehow. Is the answer to our particular query contained within one of those large, strangely shaped rocks we now call asteroids, waiting to somehow transfer into your mind as soon as you get close enough? (We call this the Newton-Apple approach, but it's a bit more hazardous with our particular subject). “Well,” you may say to me, “man has collected mountains of data on the asteroids, which we can study. Perhaps if we compile all the data, the pathway to discovery will become obvious.” It may be true that we have mountains of information, but does knowing, for example, that the asteroid Ceres has a rocky core, is maybe ¼ ice, and has an orbital period of 4.6 years tell me why the space between Jupiter and Mars is filled by thousands of asteroids and comets, rather than one single, “normal” planet?

So, in our quest for discovery of not just data, information, but of cause, the question still remains: what lies within our grasp to explore, which can bring knowledge of this principle into the bounds of our mind?


What's Going on Between Jupiter and Mars?

Kepler, a playful mind, was bothered by the anomalous space between Jupiter and Mars. It's quite large when compared to the intervals just preceding it (like Earth-Mars), which in fact led Kepler to hypothesize in his 1596 Mysterium Cosmigraphicum the existence of one or possibly two yet-undiscovered planets in that space. In the midst of describing his wrestlings with the cause for the number of planets, he says: "I tried an approach by another way, 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. For I thought that in this way I should produce some agreement between the ratios, as the ratios between the pairs would be respectively reduced in the direction of the Sun and increased in the direction of the fixed stars, as the Earth is nearer to Venus, relative to the size of the Earth's circle, than Mars is to the Earth, relative to the size of the circle of Mars. Yet the interposition of a single planet was not sufficient for the huge gap between Jupiter and Mars; for the ratio of Jupiter to the new planet remained greater than that of Saturn to Jupiter; and on this basis whatever ratio I obtained, in whatever way, yet there would be no end to the calculation, no definite tally of the moving circles, either in the direction of the fixed stars, until they themselves were encountered, or at all in the direction of the Sun, because the division of the space remaining after Mercury in this ratio would continue to infinity." (Mysterium Cosmigraphicum, Introduction)

The problem pursued Kepler into his older years, even after he discovered that the very large interval between the two planets could be accounted for by the ratio between the spheres of the tetrahedron. Despite the near perfect correspondence of the Jupiter-Mars distances to the spheres of the tetrahedron, the harmonies which accompany the space do not share the perfection. In fact, the intervals between and involving Mars and Jupiter are some of the least perfect among the planets. In Chapter 4 of Book V, to reinforce his discovery that the converging and diverging (aphelion to perihelion) extreme motions of all of the planets form one harmonic system, he also demonstrates that there are harmonies on the “same side” (from aphelion to aphelion, and perihelion to perihelion)—except in the case of Mars and Jupiter. With this pair, we get “1:7 [as] the proportion between the motions at perihelion: in fact this alone so far is unmelodic,” and “4:9 divided by 29:30, that is 40:87, another unmelodic interval between the motions at aphelion.” This is the only pair of planets which produces such dissonances. Kepler also goes through three slightly different intervals for the proportion of Mars's own extreme motions, before determining it to be 25:36, something slightly flatter in pitch than 2:3 (more on this here). "These, then," Kepler says, "are the harmonies with each other allocated to the planets; and there is none of the direct comparisons (that is to say between convergent and divergent extreme motions) which does not come very close to some harmony, so that if the strings were tuned in that way, the ears would not easily be able to detect the imperfection, except for the excess of the single one between Jupiter and Mars."

Though not putting forward anything definitive, Kepler never ignored the fact that there seems to be something anomalous occurring at this particular area in our harmonic solar system.


My Own Investigation

Quite bothered by the anomaly as well, and also having the benefit of scientific work since Kepler's time, which (beginning with C.F. Gauss in 1801) demonstrated the existence of an asteroid belt between Jupiter and Mars, I decided to apply my understanding of Kepler's method to discovering its possible cause.

Given that Kepler has proven our solar system to be a composed, harmonic system, not just an amalgam of objects that happen to fly around in the same general area, my first hypothesis was that it must be impossible to divide the space between Jupiter and Mars in a harmonic way (more on what a harmonic division is here), which is why you wouldn't have a single planet traveling in a harmonic orbit.

In order to prove this, I took the interval 5:24, which is the converging interval of Mars and Jupiter, and examined it to see if it were possible to divide it harmonically into three intervals (Mars to the New Planet, the New Planet's own interval, and the New Planet to Jupiter). Below you can see the two possibilities.

In the diagram above, the interval 5:24 is first divided into 1:2 and 5:12. The 1:2 remaining whole, 5:12 is again divided into 5:6 and 1:2, or another possible division of 5:8 and 2:3. This makes the two possible divisions of 5:24: 1:2 x 5:6 x 1:2, or 1:2 x 5:8 x 2:3. See if you can divide it into 3 harmonic intervals in any other way.

So for example, if the New Planet made a 1:2 interval with the aphelial motion of Mars, and a 5:8 interval with the perihelial motion of Jupiter, then it would be left 2:3 as its own interval. 5:8 times 2:3 times 1:2 is equal to 5:24, filling up the entire space perfectly. I was incredibly perplexed. There is a possible harmonic division! (In fact, more than one!) Why doesn't there exist a planet with such characteristics in that spot? After all, according to Kepler's “Axiom I” of Book V, “It is fitting that in any place whatever where it could be so, between the extreme motions both of individual planets and of pairs, harmonies ought to have been established of all kinds, so that such variety should adorn the world.” There must be another reason why this harmony couldn't exist.

I then took a step back, and asked: “if the planetary orbits are organized by the harmonies, but also uphold the ordering of the platonic solids for their relative distances, what solid could belong to the orbit that I seek?” At the prompting of a friend, I turned to the truncated tetrahedron, which is not a Platonic solid, but an Archimedean solid, one of secondary perfection (for more on the implications of this, see Book II). The truncated tetrahedron is formed by cutting each of the edges of the tetrahedron into thirds and lopping off the corners.

If you look at the series of diagrams below, you see first the tetrahedron with its circumscribing and inscribing spheres. Next is the truncated tetrahedron inside of that. Third, you see the truncated tetrahedron with it's inscribing sphere, and finally the additional sphere which circumscribes it (the truncated tetrahedron actually has two inscribing spheres—one touching the triangular faces, and one the hexagonal faces, which it shares with the tetrahedron).


According to the distance/speed relationships which Kepler discovered among the extreme motions of two different planets (see pedagogy on “Proposition 13”), the interval of the converging motions will be larger than the interval of their corresponding distances. Using the same method as Kepler does in the Mysterium Cosmographicum for determining the proportion of a solid's inscribing and circumscribing spheres, I found that if the tetrahedron's circumscribing sphere (A) has a radius of 1000 units, then the truncated tetrahedron's circumscribing sphere (B) would be 638.3 units, the first inscribing sphere (C--touching the triangular faces) would be 555.6 units, and the second (D--the one shared by the tetrahedron) would be 333 units.

Looking at the chart below, you can see that the intervals from sphere to sphere are .638, .87, and .6 (ratios as decimals). The proportions of the corresponding motions must be larger, according to the principles of Chapter III Book V, which 5:8 (.625), 2:3 (.666), and 1:2 (.5), respectively, are. Therefore, there does seem to exist a solid (albeit of secondary perfection) allotting distances to our planetary orbit which would permit the space between Jupiter and Mars to be divided harmonically.

Holding my findings before me for examination, I saw what appeared to be a perfectly sound reason why there should be a nice, harmonic planet filling the space between Jupiter and Mars. Such a planet could be a good addition to the celestial chorus. However, seeing that the experimental data politely disagreed with me, I was forced to put aside my handiwork, and set out on a new pathway in my mind.

A Fresh Approach

Unable, at least for the moment, to find cause for the anomaly as I had hoped, that is by making as few assumptions as possible as to what was actually there, and simply applying the principles which Kepler had demonstrated, I relented and brought in the data. The chart below shows the mean and extreme distances as well as motions (both true and apparent) for three of the largest asteroids (Ceres, Pallas, and Vesta), and for the planets.


Comparing the proportions of the three asteroids' distances to my calculations for the spheres of the truncated tetrahedron, I was as delighted as I was shocked. They corresponded almost perfectly.

It's funny how the mind can become quite stunned upon discovering that what it hypothesized to be possible, something which came from inside of it alone (or in this case, the mind of my friend), happens to be the same thing as reality. It's similar to those times when you and a friend come up with the same hypothesis or solution to a problem without dialogue or sharing of your ideas first, because there is something akin in your thinking—except this time your “friend” is the Creator of the solar system.

I thought of Kepler's relating of his process of discovery of the so-called 'Harmonic Law': "If you want the exact moment in time, it was conceived mentally on the 8th of March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances, that is, of the actual spheres..."

I was urged forward by the fact that the solution to the problem plaguing Kepler for his entire life (the relationships of distance/motions across the solar system) was not derived deductively from what came before, but was a completely unique, living hypothesis, to which the mind alone gave birth. I therefore began examining the harmonies between the motions of the asteroids and Jupiter and Mars, and was again a bit befuddled. Despite the rather exact correspondence of the distances to the solid, the harmonies seemed a bit inconclusive, showing several nearly perfect harmonies mixed in with a few rather imperfect ones. Many of the asteroids' motions form harmonic intervals with Mars and Jupiter, while others form dissonances. There must be something more complex going on here.

Taking a look at the four diagrams below. Though many of the asteroids' motions are not harmonic, making consonances with their neighbors, you can see that all can be accounted for by building the system of the octave (both Hard and Soft) from both Jupiter and Mars. In this way, even the dissonant intervals can be included.

The method of building these scales was taken from Book V, Chapter V, where Kepler first brings all of the motions into the same octave by halving or doubling the values. He then determines what the true values of the tones of the scales would be, and finally looks to see if any of the planetary motions would fit them. As you can see, the asteroids participate in the scales tuned to Jupiter and Mars. They don't, however, participate with such consistency in scales tuned to the other planets.

"Well," I thought, "the solid whose form they uphold is somewhat imperfect as well, which fact may explain the harmonic inconsistency. " However, turning next to the intervals which were left for the asteroids to sing alone after forming relationships with their neighbors, I caught sight of the elusive creature out of the corner of my mind's eye. While Pallas (the asteroid with the wildest orbit) ironically sings in harmony with itself (as if to make up for its imperfection), Ceres and Vesta sing almost the same interval: a nice Lydian (or what the Greeks called the tritone, because it was three whole tones of 8:9 in succession).

The asteroids also display a surprisingly harmonic relationship to one another, for the most part.

This means that while there could have been a single harmonic planet filling up the space (as, for example, Jupiter divides the space between Mars and Saturn harmonically), instead at least three of the thousands of objects out there travel in orbits which are quite unharmonious (except perhaps with each other)! What sort of opus is the Creator composing that He would see fit to substitute dissonance for harmony? If He wasn't going to make it harmonic, why not just leave the space empty?

My mind turned naturally to our musical system. There, the Lydian interval is a funny creature, not having a precise, fixed value. To get at it on one level, think of a simple “G-Major” scale. The note which we would say makes the Lydian interval with G is C-sharp, a tone which lies between the fourth (C) and the fifth (D).

But which C-sharp are we talking about? The tone which is a semi-tone above C is slightly different than that which is a semitone below D, and both of these are different than a “tritone” (8:9^3). To add more ambiguity, the string whose length is the geometric mean between the whole string and half also produces a different C-sharp (the reader should not take my word for it—pull out the monochord and experiment!). Click on the buttons below for the four examples.

While the difference may be somewhat negligible to the hearing in this sort of example, in living music the difference is something quite precise. Listen to the example below of what precision a dissonance can have in bel canto choral music.

As demonstrated in the musical example, the composer who is a true artist takes the harmonic system as his material, but rather than being bound by its laws, he makes it subject to his idea. In that case, the musical tones are redefined from being the product of a non-living geometry, to being a vehicle for a complex, living idea—one which can be replicated across many centuries.

I almost laughed when I realized that I could no longer proceed as if the solar system were some ancient relic which was made a long time ago, and has been the same ever since. It is instead a developing composition whose Composer must have a specific intention, a specific developmental direction for his creation. “That's quite an assertion you're making!” someone might accuse me. It may be, but what is the purpose of dissonance in human classical composition, if not to play a necessary part in the development of a certain beautiful musical idea?

Kepler states this in his own way at the end of Book III: " ...A considerably better reminder comes from those things which in the government of the world have the ratio of discords: faults in the soul; monsters among animals; eclipses in the heaven; inexpressibles in geometry (which, however, as they arise from the necessity of quantitative matter, in the following books will be far more correctly compared with the variety of the celestial motions); in the works of providence examples of divine anger and vengeance; among rational beings the devil. All of which are arranged by God the supreme Regent for a good end, and the most complete harmony of all things."

As Lyndon LaRouche explains in the clip below, experimental data from the 1980's seems to support the idea of a directed, developing solar system.

The question was now redefined for me. Since the asteroid belt is part of a whole composition, the whole Solar System, the harmonic tensions can not have come from its relationships with Jupiter and Mars alone, as an isolated harmonic space. This means that the dissonance's existing there have to be an expression of a characteristic coming from "behind" the particular harmonies, having more to do with where the composition is going than with Mars and Jupiter, its immediate neighbors. Something similar was demonstrated in Book I in the case of the heptagon. In that case, we had something which appeared to be a part of one domain, but when confronted with our inability to construct it, we were forced to look to the higher domain from which it was casting its shadow.

The same thing is found in musical composition, such as the "Ihr aber seid nicht fleischlich" section of J.S. Bach's "Jesu, meine Freude" motet, where the dissonance between "Fleischlich" (of the flesh) and "Geistlich" (of the spirit) which occurs in the opening section of the piece, is only such from your current realm of thought. As the piece develops to a resolution, Bach demonstrates the necessity of the dissonance in the context of the idea he is driving for, which is in neither of the two elements, but only in something which is beyond both (click here for more on this piece by Bach in a class given at the Boston LYM office). The next question to be taken up, then, is: "To what mode, or what idea, does the asteroid belt belong?" If the harmonies are all part of one harmonic system, then all should be accessible to one another. The only question is by what pathway (or mode) is each accessible to all in a harmonic, lawful way, where the dissonance is no longer dissonant? (Kepler began an investigation of this in Book III Chapter XIV, on the modes, where he found a home for the impure consonances of Chapter XII.) Perhaps the asteroid belt is tuned to a particular harmonic space which is outside the bounds of what I am currently considering.

This is the road down which my current investigation has led me, and so I'm poised at the beginning of a new, slightly different path, charts of data in hand (or linked here for you), prepared to take on another branch of hypothesis.

I urge any truth-hungry mind to continue the investigation begun here, because there is much more work to be done to understand the Creator's composition, which is being performed by the great conductor, our Sun. Perhaps along the way, you too will encounter this wonderful frustration: though you first set out to discover more about the organization of the cosmos, something "out there", quite far from home, instead your findings keep stubbornly leading you back to uncover more and more about the organization of your own mind (as if the answer were in there the whole time), and to reflect on how man organizes things here on earth, such as his musical system. The correspondence of the two can be no coincidence; it is as if the Creator, like Ben Franklin proudly displaying his electricity experiments in a friend's parlor, is pausing to watch his amazed friends excitedly investigate his work, taking joy in their determination to rediscover the power he had harnessed.

The "frustrating" phenomenon of being led in circles, always finding yourself returning to explore the hypotheses which come from nowhere but your own mind, is certainly nothing to be avoided. In fact, it seems to be the sense of the kinship of man to his Creator which was clearly the main guidepost in Kepler's own thinking, which has led to one of the greatest revolutions in modern science to date.

No eerie hunch is wrong.  For man is an image of God, and it is quite possible that he thinks the same way as God in matters which concern the adornment of the world.   For the world partakes of quantity and the mind of man grasps nothing better than quantities for the recognition of which he was obviously created.

Kepler, Life and Letters