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The Lost Chord - a new look at Bode's law

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by Pamela A. Bennett (now Pam Crane)


(A lecture given at the Lodge on 18th November, 1974)

Published in ASTROLOGY (The Lodge Quarterly) Autumn 1975


“Bode's Law. An empirical relationship which gives the approximate proportionate distances of the planets from the Sun. It was put forward by J. E. Bode (1747-1826), but is erroneously called a law. Attempts to find a theoretical basis for it have failed; and, furthermore, it is greatly in error for Uranus and the further planets. To the numbers 0, 0.3, 0.6, 1.2, 2.4, etc., the number 0.4 is added, giving the numbers 0.4, 0.7, 1.0, 1.6, 2.8, etc., which are the approximate proportionate mean distances of the planets taken in order from the Sun. The belt of minor planets may be regarded as corresponding to the number 2.8.” (Paul Muller, ‘Collins Concise Encyclopaedia of Astronomy’.)

Nevertheless, this ‘law’ has been the basis for much astronomical theory ever since its mathematical formulation by the German Astronomer and Director of Berlin Observatory, Johann Elert Bode, in 1778 - the idea having originated with his compatriot Wolf in 1741 and been taken up in 1772 by Titius of Wittenburg. It played a significant part in the discovery by Verrier in 1846 of the planet Neptune, even though this planet, once its orbit was established, proved to be the first body to differ drastically in mean distance from the original expectation.
Because the law almost works, give or take a slight discrepancy here and there between the calculated figure and the actual orbital position, there is no sense in dismissing it out of hand on Neptune's account; but a re-investigation is necessary, to see if some crucial factor escaped Bode's notice. A lot of work has been done in this area, though to my knowledge no-one has yet established a new and satisfying formulation (despite some very interesting calculations by Ronald Harvey in England and Charles Roark in the U.S.A.) which combines not only astronomical fact and mathematical elegance, but also a philosophical framework on which to pin the meaning that we as astrologers know to be an essential property of the cosmos.
The theory which follows does not pretend to be the last word on the subject, but does offer a structural concept long-sought, and consistent with other phenomena of cosmic inter-relationships. Interested minds may find ways to refine it further.

The Law of Saturn

Bode's Law, as we have seen, is based on a simple mathematical series, beginning with 0 and 3, and then doubling at each step. It is because of this constant doubling that Neptune fails to conform to the pattern. Now, it is not too frequently mentioned that in fact Pluto's mean distance from the Sun does match quite closely the calculated position that follows Uranus' orbit according to the Bode system. Perhaps Pluto once again validates the law, while Neptune, interposed between these two orbits, asks for a further property of the law to be revealed. The mean distance of Neptune from the Sun lies approximately half-way between Uranus and Pluto; an underlying symmetry is suggested. It occurred to me that were the orbits dependent on the mean distance of the planet Saturn, a nice relationship might be acknowledged where Uranus' distance is twice that of Saturn, and Pluto's four times Saturn. The discrepancies here would be relatively small compared to the gross errors in the original scheme. Yet there were still the intra-Saturnian planets to consider; these still conformed closely to Bode's Law. What was the answer?

It seemed reasonable to take Saturn as the fiducial in this way; not only for the above reasons, but on principle, since the essence of Saturn - as we understand it - is definition, the framework, the limiting factor. Taking therefore Saturn's number 96 in the Bode series as our fiducial, a fairly satisfying set of relationships emerged from the following series: (96/32) + 4, (96/16)+ 4, (96/8)+ 4, (96/4)+ 4, (96/2)+ 4, (96)+ 4, (2x96)+ 4, (3x96)+ 4, and (4x96)+ 4 ... each result divided by 10.

But Mercury found no place in this scheme, and the addition of the constant 4 to 96 at Saturn's position merely added a wide discrepancy that did not arise if the 96 was left to stand in its own right. Maybe there was no need to add the constant to the fiducial, for some reason. And perhaps Mercury would obey a further variation of the law as Neptune did.

The variation had to be 96/24, without the addition of the constant, if it were not to be (96/0) + 4, the only logical alternative. Yet 96/24 did not follow naturally in the series as it stood, nor did a divisor of 0 seem an appropriate sequel to a divisor of 32.

The answer to the latter problem lies in the limits of the structure under examination: the fiducial taken is the relationship between Saturn and the Sun - therefore there are two limits: Saturn and not-Saturn, 96/1 and 96/0, the all and the nothing, the defined dominion of matter and time, and the total yielding-up of that dominion to the supremacy of the spiritually creative and timeless fire.

A further answer to both problems lies in the equivalence of Mercury's mean distance with the constant 4 ... 96/24 = 4.
Perhaps it is simply the Sun-Mercury relationship that is added as the constant through the series, or its near equivalent defined by 96/24.
As the fiducial 96 is so close to the tenth of the mean distance of Saturn, it seemed logical to experiment with a series based on the actual orbit. Working all the way through with the semi-axis major as given for each of the planets, the result is satisfying; and a constant of Saturn/24 proves more effective than taking the actual semi-axis major of Mercury's orbit - pointing even more strongly to the validity of Saturn as the key to the whole conformation of the Solar system as we know it.

However, after Saturn itself, the use of the constant breaks down. Uranus is too close to double Saturn's distance to argue for the addition of any figure; Neptune and Pluto demand four times the value of the constant to bring the result close enough to the real figure. Examination of the interplanetary distances (based again on the semi-axis major of each orbit) suggested that the main cause of the problem here was the excess distance between Uranus and Neptune - since the difference between the semi-axes major of Neptune and Pluto comes very close to the Sun-Saturn fiducial, and the Saturn-Uranus difference. Because of this discrepancy, the semi-major axis of Pluto exceeds twice that of Uranus by 1.02115 AU. One cannot fix compasses at Uranus and include the limits of the known solar system in a perfect circle, passing through the Sun and Pluto.

Uranus and the Ellipse.
But one does not find perfect circles on the Solar system - as Johannes Kepler eventually realised in 1609, when he published his first two laws of planetary motion.
One finds, instead, ellipses.

It is possible to enclose the known limits of the Solar system (taking each planet as at the distance defined by the semi-axis major of its orbit) within an ellipse of eccentricity 0.02588, with Uranus at one focus. This ellipse conforms closely to that formed by the Earth's own orbit, with an eccentricity of 0.0167. (Eccentricity is characteristic of Uranus ...!) Within this framework are contained two almost perfect circles: one with Saturn at its centre and passing through Sun and Uranus, the other with Neptune at its centre, and passing through the ‘twin’ focus (F2) of Pluto and Uranus in the ellipse. In addition, a pair of intersecting ellipses, one with Saturn and F2 at the foci, the other with Neptune and Uranus, pass respectively through Sun and Neptune, and Pluto and Saturn. A triangle drawn to connect either Saturn or Neptune with the midpoint of Uranus/F2 (centre of the Sun/Pluto ellipse) and the point of intersection of the two latter ellipses has its horizontal and hypotenuse in the Golden ratio, focussing the cosmic harmony at the centre of the Sun/Pluto ellipse.

The mathematics of the Law of Saturn now make sense; Saturn remains unaltered, as it is the fiducial. Uranus remains as twice the distance of Saturn. To achieve symmetry after Uranus, the interfocal distance of 1.02115 AU is added to 3 x Saturn for Neptune, and to 4x Saturn for Pluto. Within the bounds of Saturn's orbit, although it is the Sun/Saturn axis that determines the mathematics of the series, the actual symmetry of the planetary inter-relationships begins with the first planet from the Sun, Mercury, which forms the link between the central Sun and the encircling system - Mercury as the messenger of the Gods - hence the necessity of adding his ideal distance of Saturn/24 to all subsequent figures based on successive halving of the intra-Saturnian distances. A circle of radius Jupiter-Saturn actually passes more nearly through Venus than Mercury: this may have a significance of its own.

The symmetries may be appreciated better in the scale diagram here.

The Cosmic Chord.
I have entitled this paper ‘The Lost Chord’ ... but this is a chord that has always been with us; only lost, because not clearly established as a resonance pervading and binding our immediate firmament. Clearly, the successive division of the Saturn fiducial by two implies a series of octaves rising from a ground note - which for convenience we may take as Middle C - 5 octaves in all. Doubling the ground note lowers it one octave out to Uranus, and quadrupling it to Pluto lowers it one octave further. Neptune, dividing the ‘string’ at 3 x Saturn, brings in the perfect 5th at a low G; Mercury, at 1/24 of the ground note blends into the chord at a high F - the perfect 4th - the very interval that spanned the four strings of the first lyre, the creation of Hermes for the Sun-God and spell-binding instrument of Orpheus. The harmony thus engendered perhaps finds its closest human expression in the AUM, though which one may attune the subtle self to higher states of being.

The Torus.
A question one must ask in advancing this whole concept, is this: if it has validity as a structural basis, does the Sun/Pluto ellipse exist in actual fact?
The first thing one must acknowledge is that scarcely ever are these 9 known planets, plus the asteroids (I have been taking the semi-major of Ceres - now in 2010 ranked as a dwarf planet - for the purposes of calculation) likely to line up neatly in fair imitation of this diagram, and I doubt that their orbits are at any point so related that all ten bands of heaven actually reproduce the precise intervals here shown; whether they are or are not, I am so far unaware. But because these relationships do appear to exist, it may be that despite irregularities they do form a real matrix of some kind in which each body has its proper place plus a limited degree of freedom to vary - indeed it is characteristic of life in all its vital forms that the perfect idea, the mathematical norm, underlies each of a myriad manifestations of the idea without more than the veriest few being drawn into total conformity. Such a conformity, the crystallisation of an idea, is the death of evolution.

Bearing this in mind, where might one find the real matrix of the Sun/Pluto ellipse? Not level with the Ecliptic plane, unless one were to postulate an infinity of ellipses invisibly overlapping like the petals of some inconceivable flower; but perpendicular to that plane it would fall naturally into place as the cross-section of a great multiple torus ... “a figure generated by the revolution of a circle or other conic section about a straight line in its own plane ... a coil or transformer of that shape.”

A torus enveloping a star. The obvious candidate is the magnetic, gravitational, or etheric field of the Solar system itself, greater than the sum of its parts perhaps, the mighty field of the governing Sun plus the lesser fields of the planets and their moons - all in motion inter-reacting with flows and stresses that blend and break, confuse and clear; every subtle change the seed of a new idea.

The last seven words of the definition of a torus given above are a reminder that it was Rodney Collin in 1954 ('The Theory of Celestial Influence') who conceived of the Solar System as “the factors and components of an enormous transformer for receiving current at a given tension and stepping it up for delivery at eight different voltages”, and I refer readers to his book for a detailed picture of his version of “the long body of the Solar System” as extended in time. The coils I envisage here are those formed by the lines of magnetic force that sweep in their vibrant arcs from pole to pole of the focal star, each ‘shell’ of the multiple torus being occupied by an orbiting body ... and since these subtly resonant shells are curved, the discrepancies that remain (small as they are proportionally) in the framework as here conceived, may be attributable to the inclination of a given planet's orbital plane to that of the Sun's celestial equator, which would determine how far in toward the Sun it was drawn along the retaining curve. A re-examination of the theory on this basis might cast a clearer light still on the properties of the proposed ellipse.

The form of the torus has been found most effective to date in generating the type of magnetic ‘bottle’ that alone can fleetingly hold plasma - the stuff of the Sun - in an Earth environment; and here we find the Sun itself at the heart and centre of a greater, natural torus. One might conjecture that an even more effective laboratory torus might be designed incorporating the structural relationships of the Solar System, since the function - on a vastly diminished scale - is analogous.

The symmetry in the figure suggests a complementary field-form emanating from the orbit of Pluto; to what extent this might be a complete reflection of the Sun-centred shells is highly debatable, but it is interesting that it is at the orbit of Uranus that the gap lies between the twin circles centred on Saturn and Neptune, for it is the Uranian leap of faith, flash of insight, change in consciousness, shock from complacency, that bridges the sometimes desperate human gap between material and spiritual experience as symbolised by Saturn and Neptune respectively; order is shaken into a realm of chaos, chaos vibrates into fresh patterns of order as the galvanising, activating charge of Uranus at the torus’ core shorts across the gap - the ‘higher octave’ of Mercury forming the second key link in the chain of communication from the central star to the remote neighbour of the abyss. The poles of Uranus, unlike any other planet in the system, are almost aligned with the plane of its orbit, so that its field must be at virtual right-angles to all other planetary fields, including the torus itself. What a perfect configuration for the reception and transmission of energies! What a medium of communication - both as a centre and a means! What frictional charge may be set up through the active inter-section of fields of that focal core!

Ouranus the Begetter

One of the highest endowments of Uranus is the potential for creative genius in the individual. Refer to the etymology of the word ‘genius’ and you find that it stems from the Latin root gignere, genitum, to beget. Greek mythology in recounting the history of the Pantheon tells that the oldest of the gods was Ouranos, who took to wife the Earth-goddess, Ge (the close similarity of form of their respectively generated ellipses - of Earth's orbit and of the Torus' section - is interesting here, plus the fact that the interfocal distance of the ellipse is a close approximation to the distance from Earth to Sun) and sired his ambitious son, Kronos (Saturn), which lies at the centre of the circle passing through Uranus and Sun, the node creating the octave above Sun/Uranus. Kronos - Time - subsequently castrated his venerable father, stripping him of his creative power and usurping his role of divine authority in the eternal sky* ... but the act of origination lay historically with Ouranus, just as the Torus ellipse depends on Uranus’ focal position between Sun and Pluto for its generation.

Saturn, octave offspring of Sun-Uranus, is however -  as we have seen - the chief determinant of the subsequent harmonic structure, just as Kronos became the sire of the next generation in the Pantheon, notably Zeus (Jupiter), Poseidon (Neptune), Hades (Pluto), Hera (Juno), Demeter (Ceres) and Hestia (Vesta), who in turn deposed their autocratic parent and divided the rulership between them. At this point the reference to the 6 planetary octaves beyond Earth/Moon is complete, for Zeus fathered Mars and formed Pallas without jeopardising further the seat of power. As for the two innermost planets, the role of Hermes/Mercury has already been discussed; Aphrodite/Venus is not the direct offspring of the divine hierarchy but arose perfectly formed from the sea. The relationship of this myth to the implications of the Torus is by no means clear ... unless there is a connection with Venus' proximity to the Earth/Moon system at inferior conjunction, and the fact that the Moon is the nearest of all orbs to Venus when such a conjunction coincides with the period of New Moon - a symbol of birth and beginning, of union between Man and Woman, of relationship, of Love. Perhaps she was also called into being by the desire of Mars, since her planet lies roughly half-way between Mars and the Sun. The Law of Saturn, however, supports her kinship with Earth/Moon, as her place is based on Saturn/32, the octave of the ground-note above Sun/Earth which is also Sun/Moon. She completes a hierarchy of seven heavens, that is 7 full octaves of the ground note, within which Mercury and Neptune create the natural harmonies.

It is ironic that Ouranus was the very progenitor of the powers that ultimately robbed him of his supreme authority; but it points clearly to the Uranian principle of social reform - the reformer himself has to have the creative originality, insight, daring, evolutionary drive and persuasive magnetism to transform worlds, systems and peoples. But if his often self-appointed task is fully to succeed, his living creation must be invested with an equal share of that vitalising power in order to perpetuate the new structures, the altered attitudes, and to continue the evolutionary process. Revolution is almost invariably countered by Reaction - as the first fruit of Ouranus was the insecure and dictatorial Kronos - but rarely fails eventually to precipitate change, and it was the third generation, the grandchildren of the old innovator, who put into practice the principle of power-sharing that is cousin to democracy; humanity too, for Kronos was not the victim of savage retaliation for his abuses of parent and children, but removed to a safe place where he was allowed to retain his guardianship of Time in seclusion and advanced age. Against such a traumatic background, it is no surprise that the Uranian impulse is directed against control and the imposition of external rules; the rules to be followed must be initially of his making, and Ouranus was tyrannical in his creative power. Let me quote from Alan Oken’s account of the Ouranus myth:
“The Hindus knew him as the great god Varuna, 'The Universal Encompasser, the all-embracer ... Space, the maker of Heaven and Earth, since both manifested out of his seed'. In this and in the more 'modern' mythology of the ancient Greeks, we see that Uranus partakes of the attributes of the Solar Force, for he is an active creator, a power from whom other gods are formed ... he did not allow his children to see the light of day, but buried them deep within the Earth ... The children of the gods (that’s us) are buried in darkness until the passage of time (evolution) can bring their consciousness out into the light. . . . The light of consciousness is electrical in nature, and if you put 200 volts of energy into a fuse only capable of handling 100 volts, you blow it. So Father Ouranus put his children into the darkness of the Earth (material form) so that through Time (Saturn), they could learn how to cope with increased voltages of truth. Thus they would not only remain alive, but also be filled with added life".
The imagery is echoed in the form of the Torus (see Fig. 2, here).

Uranus, partner of the central life-giving and light-giving Sun, generates with the Sun the torus that channels their double creativity of Fire and Air into the evolution of life and consciousness in the Solar System. The outermost shell of the torus, inhabited by Pluto, intimate of the alien darkness, is also - paradoxically - its closest surface to the sun. Enclosed by this ultimate shell (if it is the ultimate) and therefore shielded from the insupportable glory of that galactic son, the evolution of form and mind continues in harmony with the planetary rhythms. Planetary consciousness, focussed in mankind on Earth, expands in each growing spirit octave by octave, till, conquering the realms of matter, mortality and time within the shell of Saturn, it receives the first shock of illumination at the Uranian core. See how in the symmetry of the ellipse the space between the intersections of the twin ellipses takes the form of the section of a lens; if only by analogy, it highlights the role of Uranus as the Breaker - the cosmic focus and refractor of white light into the seven spectral colours, of unity into diversity of being, the splitter of atoms, loosing the power of the particle. Vibrating to the vertiginous low note of the 6th octave, the mind-spirit is only half-way to the outer shell, and at his deepest into the heart of the torus; yet one leap of faith and imagination across the interfocus, one octave lower, and that cosmic awareness has spanned the gulf to confront the mind-dizzying darknesses of the realm of Pluto, on the outer curve of the torus generated by “the Universal Encompasser ... the all-embracer". He has gone through the transformation from the personal to the collective, from the ordinary to the transcendent realm of consciousness as the transformer acted on his vital potential stage by stage, and as the environment of the intra-Uranian tori of circular section transformed to the out-reaching of the elliptical envelope. But he is still at the pole of that immense axis along which he has travelled ...

The Cry of the Bull

... and he is far from the sublime brilliance with which he would be united. In 'A Mithriac (sic) Ritual', presented with a commentary by G. R. S. Mead under the imprint of the Theosophical Society in 1907, the Candidate arrives at a stage after his fifth utterance of worship and supplication to God: "For that a man ... of this (ichor), which at Thy Hands today hath undergone the transmutation of re-birth . . where 'Upon this utterance He shall come to the Pole, and thou shall see Him moving round as on a path. Then gaze intently, and send forth a prolonged ‘bellowing’, like to a horn-note...”
In the commentary to this passage the author states that ‘The "bellowing" is in Greek the utterance of the sound “moo” - and this is in some sort an inversion of the sacred syllable om.’

Two correspondences suggest themselves from these remarks: one is that to reverse the sound ‘om’ might symbolise the desire to return to the Divine Source of the chord that has vibrated outward with the expanding consciousness; the other is that the ‘moo’, the ‘bellowing’ must be connected with the ‘power of the bull’, or creative power, the use of which for spiritual operations constituted the greatest secret of Mithraic ritual. The bull is Taurus. Is it pure coincidence that the common pronunciation of Taurus is identical with that of Torus? Were there significance in this, one might further speculate that with the concept of the torus, and the linking role played by its outer envelope between the kingdoms of Light and Darkness, we may be coming close to an encounter with the child-bride of Pluto - Proserpine/ Persephone - who descends for six months of the year into the Underworld, to return in joy to the embraces of her mother, Ceres/Demeter and the heavenly family for the seasons of Spring and Summer, between the months of shoot and seed, coming forth in Taurus, rejoining her dread husband in Scorpio. Perhaps she has her planet on a further vibrant ethereal shell beyond Pluto; perhaps she is the lovely invisible ringing ring of the torus itself? It is not impossible ; her name, Persephone, contains the root of sound; Pluto was a name also in ancient times of the Sun-god, and is thus linked with the Sun. In the identity of Lucifer again his origin is among the heavenly host, close to God, brightest of angels - until pride and the yearning for power brought him to his terrible banishment. No soul fell lower or further than he who became the ‘Prince of Darkness’ when love of Power usurped the power of Love.

Or it may be that Persephone’s orb is a Moon of Pluto, sharing with her shadowy lord the outer shell that leads from the death of Self back to the resurrection in the heart of Light, that makes the ultimate union between Centre and Periphery, resolving the paradox, of the Manifold and the One - enabling the awakened eye to penetrate and finally emerge from an illusion, the solar game of shades and forms played with the prism. The final, paradoxical act of Pluto is to draw the life-contents out of the imprisoning darkness and into the light, un-burying the children of Ouranus. Only the warder of the prison can unlock its gates; and first the imprisoned must win back their right to freedom. The spirit must demand its rights from Uranus, before its long-sought deliverance through the dark gate of Pluto. Having reached that point, were he even to cry out in despair at its desolation and jump blindly into the void, he can only fall into his earned communion with the Love of the Sun. Only those who will not go through that forbidding door are lost.

Orpheus held the secret of the journey in his lyre, the gift of the Light-god Apollo and creation of Mercury (known to the Hindu as the Wind God’s son, to whom all places are equally accessible) and was enabled to pass into and out of the realm of Pluto and his Queen in his quest for dead Euridyce with the help of the lyre’s charmed chord. Each of us is on the same perilous adventure in search of the lost parts of our souls. Armed with all we possess of love, integrity, intelligence and faith, we may learn sufficient of our inner music to sound our own unfathomable depths, till all their levels, octave beyond octave, are echoing back to us the true name of the awakening whole.
The chord is lost—it hums in the dormant hills, awaiting musicians.

* * * * * * * * * * * *

(* Aquarius, fixed Air - Co-rulers Saturn and Uranus; distribution of the Solar Light of Leo as the circulation distributes the life-giving blood from the heart.)