Now there is need Urania, of a Grander Sound, while I ascend by the harmonic stair of the celestial motions to higher things, where the true archetype of the fabric of the world is laid up and preserved. Follow me, modern musicians, and attribute it to your arts, unknown to antiquity: in these last centuries, Nature, always prodigal of herself, has at last brought forth, after an incubation of twice a thousand years, you, the first true offprints of the Universal Whole. By your harmonizing of various voices, and through your ears, she has whispered of herself, as she is in her innermost bosom, to the human mind, most beloved daughter of God the Creator.
johannes kepler, chapter vii, book v
Now that we have investigated the ellipse and established all the planets into both the hard and soft scales, using daily apparent arc-lengths as observed from the sun, we see how Kepler begins to make clear how rarely neighboring planets actually occupy extremes at the same moment in time in order to produce Harmony.
So here we are, Kepler has taken us through a journey of many hypotheses. Since the Mysterium Cosmographicum, Kepler always thought that God created mankind with the idea that man could discover and know how the Universe was ordered. With this idea in Kepler's mind, he knew that somehow, the world was organized harmonically, and that he could discover the existence of harmony in the many ways that the Universe expressed itself. He understood that the world was one single universe that we looked to with our many different senses, and that to seek harmony, which Kepler considered the higher ordering principle among things, you have to look at the ironies that arise in a plurality of subjects, and investigate the similarities that these different fields express in terms of proportion. The Creator, to Kepler, organizes a host of many effects with a single, simple cause.
Let's see what Kepler does once he finds the harmony in the Solar System.
1. The Harmony of the Planets own Motion
We've seen already how to obtain the measurements in which the harmonic proportions occur. If you were on the sun and you viewed the planets from that perspective you would notice two extreme motions in two distinct locations. The motions are taken as arcs traversed in one Earth day. In other words, comparing proportions of daily arcs traveled at perihelion and aphelion.
Here, you have the apparent daily arcs as viewd from the sun converted to seconds. The two extremes, aphelion and perihelion, are compared to produce actual intervals. Kepler then compensates in very small amounts to produce actual harmonic intervals and tones.
Here are the planets own intervals placed on musical staffs. Jupiter's song is transposed in this diagram (the note that was |h| in Saturn's song is now the |g|) Mercury is also transposed down from it's original notes
You can see that planets produce Harmonies between the two extreme motions. Take the case of Saturn's extremes. Between 107" and 135" you produce the proportion of 108:135 which is the same as 4:5 and according to our harmonic proportions it fits nicely with the major third.
Kepler also finds that the Moon produces a harmony. If you take the hourly arcs observed from the Earth (the Moons fount of motion) at apogee in quadrature (26'26" or 1586") and at perigee at the Syzygies (35'12" or 2112" ) you get almost exactly the same ratio of a Diatessaron which is the proportion of 3/4.
Now a planet never occupies its own perihelion and aphelion at the same time, so you can never "hear" this interval played at the same time. Instead, you hear it as a voice moving through a piece of music leaping from note to note.
2. The Harmony of Diverging and Converging Motion
Here is the next step: do the planets only make harmony with themselves, or do they correspond to their neighbors and how and where will this occur and in what time? Not only do they correspond to their neighbors, but do they correspond as a bounded structure?
Now that we have found the extremes of planets motions we can now begin to compare the extremes of one planet to the extremes of another neighboring planet. There are four different types of comparisons of the extremes; diverging, converging, and two types of same side extremes (for an explanation of what diverging and converging are click here). Don't forget that at this point Kepler is comparing motions not distances.
Remember from Kepler's New Astronomy that all of the planets' apsides are not parallel to each other, so don't think of diverging and converging as a distance between planets, but as a relationship of the planets of the entire system to the Sun.
Between the planets diverging and converging motions you have these harmonies:
Here you have the diverging and converging chart, the column on the left represents the planets daily motions at perihelion and aphelion as observed from the body of the Sun, the blue and red boxes represent the two extremes being compared, notice that the diverging motions of Mars and Jupiter are off by an interval larger than a Diesis.
All the planets make harmonies with each others diverging and converging motions. Now these are not "perfect" intervals as they would like you to think in the modern musical conservatories. On the contrary, they express all the harmonic intervals within discrepancies "smaller than a diesis, 24/25" which Kepler says is the largest discrepency tolerated by musicians ( diesis in Greek means "Slacking of the String"). That's pretty good considering the fact that in Kepler's time they didn't have instruments which could measure with the precise detail that we are capable of doing today. The only interval that has a discrepancy larger than the diesis is the diverging motion of Mars and Jupiter which is off by 18/19 (or about a limma), and we will get into that in a later pedagogy.
Kepler then moves on to say that you have multiple planetary harmonies produced between 3 or more planets. From my own investigation I found out just how rare these occurences are. Nevertheless they occur, which remarkably demonstrates that the solar system is not a pair wise interaction of hard bodies floating in space as Sir Isaac Newton would have it, rather it is a interactive dynamic system that is organized around harmonics! Kepler even hypothesizes that Creation was demarcated by a time long ago when all six known planets were in harmony;
But if three planets must combine together to form a single Harmony, many alterations must be awaited. However, there are many harmonies, so that they occur all the more easily, when all the nearest planets catch their neighbors; and triple harmonies between Mars, the Earth and Mercury seem to occur rather often. However, harmonies of four planets now begin to be scattered over centuries, and those of five planets over myriads of years. However, an agreement together of all six is hedged about by very long gaps of ages; and I do not now whether it is altogether impossible for it to occur twice, by a precise rotation, and it rather demonstrates that there was some beginning of time, from which every age of the world has descended. But if there could occur one six fold harmony, or one outstanding among several, that undoubtedly could be taken as characterizing Creation.
With modern technology and a little hard work, we did exactly as Kepler instructed and produced, to the best of our abilities, the Solar Sytem of Kepler's time, and assembled the Music of the Spheres as Kepler would have heard it, and made it accessible through the limitations of the internet, for all who can hear to listen to the music of the Heavens.
If you want to view the data that was used visit the Archives page, under data sets.
Here you see the actual intervals represented by the animation. In truth the solar system ranges over 7 octaves, but because of the limitations of our ears, the sounds that represent the planets have been divided by octaves so that the proportions are maintained. This is in accordance with Kepler's principles by Book III, Chapter I, Proposition I.
Thus it is no longer surprising that Man, imitating his Creator, has at last found a method of singing in harmony which was unknown to the ancients, so that he might play, that is to say, the perpetuity of the whole of cosmic time in some brief fraction of an hour, by the artificial concert of several voices, and taste up to a point the satisfaction of God his Maker in His works by a most delightful sense of pleasure felt in this imitator of God: Music.
Johannes kepler, book v chapter Vii
What kind of a universe is this? You mean that the Solar System is not chaotic and random like I was told? What does music have to do with the solar system?! What about the discrepancy between Mars and Jupiter's diverging motion? What does this tell us about the nature of investigating the infinitesimally small? What does Music have to do with Science?!
I'd like to end this section with a famous quote....