Titius, Bode and the planetary row
After Kepler’s thorough discussion of planetary order, relatively little progress was made in the theory of the harmony of the spheres until the physicist Johann Titius believed he had made an astonishing discovery in 1766. The Wittenberg-born university professor, who is also said to have built the first lightning rod, pointed out a peculiarity of the planetary arrangement in a footnote to a book. The remark was taken up in 1772 by another German astronomer, Johann Elert Bode, and has since been known as the Titius-Bode series.
One of its versions reads:
A=0.4 + 2n *0.075
n stands for the series of natural numbers, starting with Mercury at 1, A for the mean distance. Venus is given 2 and so on up the series of planets. The distance from the earth to the sun automatically becomes the unit distance.
Distance Sun/Earth = 0.4+8*0.075 = 1The distance between the Sun and Venus is calculated in the same way:
Distance Sun/Venus = 0.4+4*0.075 = 0.7The distance of the sun to Mars:
Distance Sun/Mars = 0.4+16*0.075 = 1.6The discovery of Uranus in 1781 by Herschel, a few years later, seemed to confirm the formula in the most beautiful way. However, a conspicuous gap appeared in the series of planets known until then.
The Titius/Bode numbers of all the planets – expanded so that they become natural numbers – read as follows (the actual distance ratios known today can be found right next to them):
| Mercury |
4 |
3,9 |
| Venus |
7 |
7,2 |
| Earth |
10 |
10 |
| Mars |
16 |
15,2 |
| ? |
28 |
? |
| Jupiter |
52 |
52 |
| Saturn |
100 |
95,5 |
| Uranus |
196 |
192 |
It seemed from the series that there must be another planet hidden between Mars and Jupiter, and everyone who was anyone in astronomy set out in search of one. In September 1800, a historic meeting took place in the private observatory of the Bremen civil servant Johann Schröter. Schröter, his assistant Karl Ludwig Harding, Baron Franz Xaver von Zach – a friend of Herschel’s – and the amateur Heinrich Olbers founded the «Himmlische Polizey» (Heavenly Police) with the aim of systematically searching the sky for the predicted body. Experts throughout Europe were also encouraged to do so by letter.
The request also reached Giuseppe Piazzi at the observatory in distant Palermo. Piazzi had wrested a masterpiece of modern optics from the most famous telescope maker, the Englishman Jesse Ramsden, in a reasonable amount of time. This is quite an achievement when you consider that the University of Dublin, for example, had to wait 27 years for an order from the fanatical perfectionist and the Greenwich Observatory threw in the towel after six years of waiting and canceled the order.
Piazzi began measuring several thousand fixed stars and, on January 1, 1801, discovered a small point of light in the constellation of Taurus that could not be a star and which apparently also moved in the following nights. Piazzi’s observations found their way to Zach, who published them in his Monthly Correspondence». There they attracted the interest of Carl Friedrich Gauss, the Johann Sebastian Bach of mathematics, whose likeness adorned the ten-mark bill that had since been withdrawn from circulation.
Gauss calculates the probable orbit of the celestial body on the basis of Piazzi’s limited data. On December 31, 1801, another sighting of the object is made, which Piazzi has now named «Ceres». Soon afterwards, a second body was discovered in the vicinity of Ceres – it was given the name «Pallas» – and in 1804 a third, which was christened «Juno». By the present day, a further 10,000 are to be added – the so-called asteroid belt is discovered – more or less where the Heavenly Police suspected another planet.
However, the planets Neptune (discovered by Galle in 1846) and Pluto (Tombaugh, 1930), which were discovered later, do not fit into the Titius-Bode series at all. It therefore quickly lost its significance:
| Neptune |
388 |
300 |
| Pluto |
772 |
394 |
At first, the Titius/Bode series appears to be a confirmation of celestial harmony. However, this is only the case at a very superficial level. Remarkably, only a few of the Titius/Bode ratios define useful musical intervals. Between Mercury and Mars there is more or less a ratio of 1:4, which covers two octaves. There is almost an octave ratio between Jupiter and Saturn, but the deviation is around four percent, i.e. very close to a major seventh. The same applies to the ratio of Saturn to Uranus and Neptune. Venus does not define any useful intervals with the rest of the planets because of the 7, and the asteroids also have a 7 in the prime factorization (28 = 22*7).
A few final observations need to be made on the idea of the harmony of the spheres, not least because the idea continues to be mothballed, especially in esoteric circles. Until recently, for example, no one really thought about how far removed the relationship between two planetary distances and simple numerical ratios is from a coincidence.
To get even a vague idea of the accuracy that must be achieved if you want to obtain significant deviations from random values, consider the following: Let’s idealize the planetary orbits and think of them as concentric circles around the sun. Moreover, imagine a planet placed at any location between the sun and another planet. How large is the maximum possible deviation from one of the concentric circles defined by the twelve semitones?
The planetary distances above an ideally selected semitone grid.
If, for example, the Sun-Uranus distance is taken as an octave,
planets placed randomly between them will deviate from the
distance of a semitone by no more than around 4 percent
of the octave distance.
The distance between two semitones is one twelfth if the distance between the sun and the outer planet is a unit distance. So if a second planet is as far away as possible from one of the semitone circles, it must lie exactly in the middle between two of them. This means that its distance is half of one twelfth, i.e. one twenty-fourth. However, this corresponds exactly to the maximum percentage deviation. In other words: the ratio of two randomly selected distances between planets and the sun deviates by a maximum of 4.16 percent from a tone ratio. Maximum! As a rule, it is likely to be smaller, and even if only seven planets are considered, some of them are likely to be in a seemingly astonishingly exact tonal relationship to each other. It is also like the assignment of the number 666 to the names of the Roman emperors in the Book of Revelation: anything can be done with a little imagination.
In Kepler’s time, probability calculation did not yet exist and it would not have occurred to the astronomer to check exactly how far his constructions could be from a purely random result. Interestingly, no one later made such general considerations. It was not until 2001 that the German author Hartmut Warm tackled the problem of a comprehensive probabilistic investigation of harmonic systems[1] – with devastating results. He comes to the conclusion that “only Kepler’s assignment of the values for the planets visible from Earth on Saturn at aphelion and perihelion shows a certain deviation from a random distribution.”
Warm also mentions the attempts to make the harmony of the spheres audible in Kepler’s sense:
So we are not dealing with music in the human sense, but with a collection of howling buoys squawking away at different heights and at different speeds. In order to avoid such an idea, Kepler primarily relied on the fixed values of a planet at perihelion or aphelion. Without the tuning of the tones, however, pure melodies or sublime songs are not possible.
As is usual with probability calculations, the mathematical details are complex. Anyone interested in this should refer to Warm’s book. Warm concludes that “Kepler’s ideas about the planetary harmonies of angular velocities must be laid to rest”. However, he offers a replacement based on planetary velocities. Depending on the minor semi-axis of the planetary orbit ellipse, he finds three points on the path of the planets with a striking property: they define numerical ratios corresponding to interval ratios that are far removed from all randomness. However, he embellishes his ingenious and clever statistical-astronomical considerations with almost touching fantasies in the manner of an ancient world view:
The fact that the velocities have turned out to be physically tangible carriers of the harmony of the spheres is basically only too logical. A musical instrument, if you imagine the structure of the variable stars as such, must in some way produce vibrations that are in resonant relationships with each other. This can only be thought of in terms of orbits or movements at different speeds. However, as has now been discovered, orbital periods are highly unsuitable for reasons of planetary system stability, as resonances of small whole numbers could have devastating consequences. Incidentally, it is also known today that the space between the planets is not an absolute vacuum, but is filled by the solar wind. Even if this is a little speculative, it can certainly be imagined as a propagation of the vibrations generated in this way. Mankind’s age-old idea of a cosmos filled with music and Johannes Kepler’s attempt to scientifically prove this, even if it is only audible in the mind, with the help of the planetary laws he discovered, have for the first time found real confirmation – I would not presume to call it proof.
Hartmut Warms’ book is worth reading, even if you don’t want to follow his metaphysical and theological speculations. After all, analogies or regularities in cosmic phenomena have their own appeal. However, preoccupation with the logic of simple numerical relationships and geometric symmetries today only contributes insignificantly to the understanding of musical questions and distracts from the crucial problems. Questions such as: What does music mean? What forces play a role in the development of a harmonic language? Why does the time axis structure itself into a complex web of differently weighted impulses? and many more similar questions are completely beyond the reach of a theory of spherical harmony, symmetry or resonance. Trying to bring them into such a system anyway is like trying to explain the expressive diversity of architecture using only the laws of physical statics.
The system of «harmony of the spheres» with its static perspective has become a barren, dead metaphor for modern man. The proponents of the idea have not succeeded in presenting the aforementioned framework theory that would make the coupling of musical phenomena with cosmological conditions plausible and predictable. On the contrary: over the centuries, the idea has become more and more absurd and nonsensical – and ultimately the constructions of interval relationships with cosmological facts have also become more and more contradictory.
This does not mean that the idea in and of itself has always been a mistake. Quite the opposite is the case. In the intellectual history of mankind, from Europe to China, it has helped to overcome one of the most important methodological questions, namely that of what inspiration one should be guided by in the search for the fundamental laws of nature. The idea of the harmony of the spheres is an expression of an ancient paradigm in cosmology – which still plays a role in fundamental thinking today – but the construction of the basic musical material also bears the seeds of its overcoming: the belief that the laws that define nature at its very foundation are simple and that the universe expresses itself in clear and elegant symmetries always leads to a dead end sooner or later. The compromises already inherent in the mathematical structure, which have to be made if you want to construct a “beautiful” scale, show that the fundamental laws of nature do not exhibit the elegant simplicity that ancient Greek and medieval Christian man would have liked to see in nature.
As the discussion of information theory in the next chapter will show, simplicity also means a lack of expression. In other words, elegance and depth are to a certain extent mutually exclusive. And precisely because mathematical precision is inherent in the musical building material, it helped to overcome intellectual prejudices. In this respect, Kepler, who is sometimes ridiculed today, was probably a more radical, scientific and modern person than Isaac Newton, for example, who uncritically indulged in astrological and esoteric ideas throughout his life. Kepler wholeheartedly resisted giving up the idea that the planets revolved in concentric circles in the sky, but then allowed himself to be convinced by his material; he resisted the idea that the harmony of the spheres, as Pythagoras might have imagined it, was untenable and discovered the third planetary law in the process. In the best sense of the word, he is what the modern science theorist Karl Popper had in mind as an ideal: a seeker who is suspicious of eternal truths, who therefore subjects his theories to the toughest tests and is prepared to abandon them if the empirical material speaks against them. Only a few have shown such greatness in the course of the history of science. As a rule, scientists hold on to opinions once they have formed them, even if they themselves know that they are already outdated.
