January 2006
Part 2
6. Formative power of sound
7. Planets and geometry
8. Planetary distances
9. Solar system harmonies
10. Intelligent habits
11. Sources
In the late 18th century, German physicist Ernst Chladni demonstrated the organizing power of sound and vibration in a visually striking manner. He showed that when sand is scattered on metal plates, and a violin bow is drawn across them, the resulting vibrations cause the particles to move to the places where the plate is almost motionless, producing a variety of beautiful, regular, intricate patterns.*
*Joscelyn Godwin makes an interesting comment on this phenomenon: ‘Once, passing by a crowded dance hall where rock was being played, I could not help perceiving the floor of the hall in terms of a Chladni plate, and the dancers appeared for all the world like the jumping, helplessly manipulated grains of sand’ (1995, p. 246).
Fig. 6.1 Chladni figures.
A century after Chladni, Margaret Watts-Hughes created images by
placing a powder or liquid on a disk then letting it vibrate to the sound of a sustained musical note.
She experimented with several musical instruments but had most success using her voice. The
particles arranged themselves into geometric shapes, flower patterns (such as pansies, primroses,
geraniums, and roses), or the shape of a fern or a tree. The higher the pitch, the more complex the
patterns produced; a powerful sustained note produced an imprint of a head of wheat.
Fig. 6.2 Figures generated by the voice of Margaret Watts-Hughes.
In the 1950s the study of wave phenomena was continued by Swiss
scientist and anthroposophist Hans Jenny (1904-1972), who named the field
‘cymatics’. Using crystal oscillators (which allow precise frequencies and
amplitudes to be used), he vibrated various powders, pastes, and liquids, and succeeded in making
visible the three-dimensional effects of sound. He produced an astonishing variety of awe-inspiring
geometrical and harmonic shapes, including life-like flowing patterns, which he documented in
photographs and films.
Jenny, too, found that higher frequencies produced more complex shapes. A low frequency produced a simple central circle surrounded by rings, while a higher frequency increased the number of concentric rings. Even higher frequencies created shapes resembling petals, butterflies, or crustaceans, zebra patterns, mandala-like patterns, and images of the five platonic solids. As the frequency rises, the dissolution of one pattern may be followed by a short chaotic phase before a new, more intricate, stable structure emerges. If the amplitude is increased, the motions become all the more rapid and turbulent, sometimes producing small eruptions. Under certain conditions Jenny was able to make the shapes change continuously, despite altering neither frequency nor amplitude.
Fig. 6.3 Hexagonal pattern produced by light refracting through a small sample
of
water (about 1.5 cm in diameter) under the influence of vibration. The figure is in
constant
dynamic motion. (Jenny, 2001, p. 112; courtesy of Jeff Volk)
Fig. 6.4 A round heap of lycopodium powder (4 cm in diameter) is made
to
circulate by vibration. At the same time two centres of eruption rotate at
diametrically opposed
points. (Jenny, 2001, p. 108; courtesy of Jeff Volk)
Fig. 6.5 Jenny built a tonoscope to translate the human voice into visual patterns in sand.
Left to right: ‘oh’ sound, ‘ah’ sound, ‘oo’ sound. (Jenny, 2001, p. 65; courtesy of Jeff Volk)
Peter Guy Manners found that an acoustic recording
of the Crab Nebula caused sand to form a pattern strikingly like the nebula itself. As the
near-infrasonic sound was played, the two swirling arms pulled in, forming a tight ball. Towards the
end of the audiotape, the sand became very highly compacted and then suddenly exploded,
throwing sand off the table.
John’s Gospel begins: ‘In the beginning was the Word, and the Word was with God, and the Word was God.’ The Egyptian Book of the Dead, contains a parallel passage: ‘I am the Eternal, I am Ra ... I am that which created the Word ... I am the Word ...’ The Hindu tradition teaches that ‘Nada Brahma’ (the world is sound). The underlying idea is that everything we see is a divine word – or vibration – that has solidified and become manifest, the vibrations originating in the inner, more ethereal realms. All of nature is essentially rhythmic vibration. Everything from subatomic particles to the most intricate lifeforms, from planets to galaxies, comprises resonating fields of pulsating energy in constant interaction. In the poetic words of Cathie Guzzetta:
The forms of snowflakes and faces of flowers may take on their shape because they are responding to some sounds in nature. Likewise, it is possible that crystals, plants, and human beings may be, in some way, music that has taken on visible form. (D. Campbell, ed., Music: Physician for times to come, Quest, 1991, p. 149)
If we draw a circle representing the earth – which has a mean radius (in round numbers) of 3960 miles – and then draw a square around it, the square will have a perimeter equivalent to 31,680 miles. If we then draw a second circle with a circumference equal to the perimeter of the square, its radius will be 5040 miles (using 22/7 as a good approximation to pi (π), as the ancients often did) – or 1080 miles more than the smaller circle. Just as 3960 miles is the radius of the earth, 1080 miles is the radius of the moon. In other words, the relative dimensions of the earth and moon square the circle!
Fig. 7.1 Earth and moon square the circle. Note that 5040 (the radius of the outer circle) = 1x2x3x4x5x6x7 (known as ‘factorial 7’, also written: 7!) = 7x8x9x10 (or 10!/6!). A quarter of its circumference (also equal to the diameter of the earth-circle) = 7920 = 8x9x10x11 (or 11!/7!), and the area of each semicircle = 11!.
Exactly the same proportions and digits (expressed in feet rather
than miles) can be found at Stonehenge (see Michell, 1995, 2001). The outer (sarsen) circle has a mean radius of
50.4 ft and a circumference of 316.8 ft. This is equal to the perimeter of a square drawn round the
smaller (bluestone) circle, which has a radius of 39.6 ft. This radius is also equal to the diameter of
the circle defined by the inner U-shaped structure. This is clear evidence that the
‘English’ foot and mile are at least as ancient as Stonehenge and, like many other ancient
systems of measures, are closely related to the dimensions of earth, moon, and sun.
Fig. 7.2 Ground plan of Stonehenge. The lintels on top of the stones of the outer (sarsen) circle were mortised
to the uprights and jointed at their ends, forming what was once a precision-made, perfectly level platform.
Fig. 7.3 Given its slope angle of 51.83°, the Great Pyramid, too, squares the circle. The length of each
base side divided by the height equals π/2. In addition, the apothem divided by half the base side equals φ.
Fig. 7.4
If a tetrahedron is inscribed in a sphere with the apex placed at
either pole, the three corners of the base will touch the sphere at a latitude of 19.47 degrees in the
opposite hemisphere. This latitude marks the approximate location of major vorticular upwellings of
planetary and solar energy. The primary focus of sunspot activity is about 19.5° N and S. On
Venus, there are volcanic regions at 19.5° N and 25.0° S. Mauna Loa and Kilauea
(Hawaii), Earth’s largest volcanoes, are located at 19.5° and 19.4° N respectively.
On the moon there is a mare-like lava extrusion at 19.6° S. On Mars, Olympus Mons, possibly
the largest volcano in the solar system, is located at 19.3° N. The Great Red Spot of Jupiter is
located at 21.0° S. On Saturn there are storm belts at 20.0° N and S. On Uranus, there are
upwellings causing cooler temperatures at 20.0° N and S, and a dark spot at about 22.5°
S. The Great Dark Spot of Neptune, photographed by Voyager 2 in 1989, was located at 20.0°
S, but when the Hubble Space Telescope viewed the planet in 1994, the spot had vanished –
only to be replaced by a dark spot at a similar location in the northern hemisphere.
Fig. 7.5 Jupiter’s Great Red Spot.
Are the planets of our solar system located at random distances from the sun? The Titius-Bode law, discovered 1766, suggests they are not. The law is obtained by writing down first 0, then 3, and then doubling the previous number: 6, 12, 24, etc. If 4 is added to each number and the sum divided by 10, the resulting numbers give the mean distances of the orbits of the planets in astronomical units (1 AU = the earth’s mean distance from the sun). Uranus, discovered in 1781, fitted the law, as did Ceres, the largest asteroid between Mars and Jupiter, discovered in 1801. However, the law breaks down completely for Neptune and Pluto, which were discovered later. Various efforts have been made to modify the Titius-Bode law to make it more accurate.*
*William R. Corliss, The Sun and Solar System Debris, Sourcebook Project, 1986, pp. 34-42.
What the Titius-Bode law essentially means is that planetary orbits become progressively greater by a ratio of approximately 2:1 (the ratio of the octave) with increasing distance from sun. This is brought out in columns 3 and 4 of the table below, in which half the distance between Mercury and Earth is taken as the unit of measurement. Uranus and Pluto have mean orbits close to the exact distances necessary to complete two further octaves. Neptune is located almost exactly half-way between Uranus and Pluto, as though to fill in the half-octave position. This may indicate that it was not an original member of the solar system (theosophy says it was captured from outside our solar system). Likewise, the Titius-Bode law works for Pluto if we ignore Neptune.
| Mean distance of the planets from the sun | ||||
| Planet | Calculated |
Observed |
Perfect octaves in units of distance from Mercury |
Actual units of distance from Mercury |
| Mercury | 0.4 |
0.387 |
0 |
0 |
| Venus | 0.7 |
0.723 |
1 |
1.1 |
| Earth | 1.0 |
1.000 |
2 |
2 |
| Mars | 1.6 |
1.524 |
4 |
3.7 |
| Asteroid belt (Ceres) |
2.8 |
2.767 |
8 |
7.8 |
| Jupiter | 5.2 |
5.203 |
16 |
15.7 |
| Saturn | 10.0 |
9.539 |
32 |
29.9 |
| Uranus | 19.6 |
19.191 |
64 |
61.4 |
| Neptune | 38.8 |
30.061 |
(64x1.5=96) |
(96.8) |
| Pluto | 77.2 |
39.529 |
128 |
127.7 |
Due to its ad-hoc nature, the Titius-Bode law is usually dismissed as a numerical coincidence that has no physical basis. However, the fact that the planets’ distances from the sun follow a pattern can easily be demonstrated by plotting the logarithm of the mean distance of the planets (including the asteroid belt) against their sequential number (1 to 10). The fact that all the points lie very nearly on a straight line proves that gravitation is ‘quantized’. However, there is currently no detailed mainstream theory that explains how gravity works and why it should be quantized. The orbits of satellites around moons show the same quantized spacing, as do the orbits of electrons around an atomic nucleus (in the Bohr model of the atom).
Fig. 8.1
Figs. 8.1 to 8.6 courtesy of John Martineau
(A Little Book of Coincidence, Wooden Books, 2001)
Curiously, the mean orbital radii of the four inner and four outer planets reflect about the asteroid belt. For instance, if we multiply together the orbital radii of Venus, Mars, Jupiter, and Uranus, we get virtually the same value as multiplying the orbital radii of Mercury, Earth, Saturn, and Neptune (5.51 x 1034 km as against 5.56 x 1034 km).
The spacing of the planets shows many geometrical regularities. For example:
Fig. 8.2 Three circles touching: if Mercury’s mean orbit passes through the centres
of the three circles, Venus’ orbit encloses the figure (99.86% accuracy).
Fig. 8.3 Left: The mean orbits of Mars and Jupiter can be drawn from four touching circles
or a square (99.995%). Right: A related pattern spaces Earth’s and Mars’ orbits (99.8%).
Fig. 8.4 In this diagram, the smaller and larger circles represent not only the relative sizes
but also the orbits of Mercury and Earth; they are related by a pentagram (99.1%).
Fig. 8.5 This diagram shows Earth’s and Saturn’s relative sizes
and orbits; they are related by a 15-pointed star (99.3%).
Fig. 8.6 Earth’s and Jupiter’s mean orbits can be created by spherically nesting
three cubes, or three octahedra, or any threefold combination of them (99.89%).
The 17th-century astronomer Johannes Kepler discovered a remarkable relationship between a planet’s mean distance from the sun and the time it takes to orbit the sun: the ratio of the square of a planet’s period of revolution (T) to the cube of its mean distance (r) from the sun is always the same number (T²/r³ = constant). For instance, measuring T in earth-years and r in astronomical units, we get:
| Venus: | 0.61521²/0.7233³ |
= 1.0002 |
| Earth: | 1.0000²/1.0000³ |
= 1.0000 |
| Mars: | 1.88089²/1.5237³ |
= 1.0000 |
| Jupiter: | 11.8623²/5.2028³ |
= 0.9991 |
Orthodox science has no real explanation for this, or for the many ‘resonances’ in solar system dynamics. For instance, the periods of Jupiter and Saturn show a 2:5 ratio. The periods of Uranus, Neptune, and Pluto stand in a 1:2:3 ratio. Mars and Jupiter are locked into a 1:12 resonance, Saturn and Uranus are in a 3:1 resonance, and there is a 2:3 resonance between Mercury’s rotational and orbital periods.
What is the nature of such ‘resonances’? Abstract mathematical concepts such as ‘curved spacetime’ shed no light on the matter. A concrete explanation must be sought in the behaviour of the dynamic ether filling space, whose vorticular motions cause the planets and stars to rotate and carry them along in their respective orbits. According to the ether-science model known as aetherometry, T²/r³ is a constant for all the planets because it refers to the constant flux of energy that the solar system as a whole extracts by its primary gravitational interaction with the ether, entailing a nearly constant energy supply for each of its members.
The speeds of the planets in their orbits represent their pitch-frequencies. Cosmic ‘chords’ are produced when planets come into conjunction (or ‘kiss’), i.e. stand in a straight line with the earth and sun. A certain number of regular planetary conjunctions occur over particular periods of time, and the ratios between these numbers reflect with considerable accuracy the length ratios necessary to produce the diatonic notes of an octave, i.e. the seven notes of a musical scale.
Fig. 9.1 Planetary conjunctions as ‘chords’ (Tame, 1984, p. 239). The line in this diagram represents an octave, divided into seven intervals by eight notes. The line could represent the string of a one-stringed musical instrument. The numbers above the line are the numbers of conjunctions of each planet with the sun and earth, and those below the line are the numbers of years involved.
The planets’ orbits are ellipses with the sun at one of their
foci. Their speeds are therefore variable: they speed up as they approach perihelion (the point in
their orbit nearest the sun) and slow down as they approach aphelion (their greatest distance from
the sun). Following Kepler, Francis Warrain expressed the ratio between the minimum and
maximum speeds of one planet and those of different planets as a musical interval. The results (only
those for the inner planets are given below) rule out chance altogether, and constitute ‘a
powerful argument for the harmonic arrangement of the solar system’ (Godwin, 1995, pp.
132-6). Of 74 tones, as many as 58 belong to the major triad CEG.
| Harmonies of the planets’ angular velocities, as seen from the sun | ||||||||||
| Harmonic number | ||||||||||
1 |
9 |
5 |
3 |
25 |
27 |
15 |
2 |
|||
Mars |
aphelion (l) |
l:m = 2:3 |
C |
|
|
G |
||||
Earth |
aphelion (n) |
n:o = 15:16 |
|
|
B |
C'
|
||||
Venus |
aphelion (p) |
p:q = 24:25 |
|
|
|
G |
G# |
|
||
Mercury |
aphelion (r) |
q:r = 16:27 |
C |
|
|
A |
||||
The time taken by Venus to seemingly orbit the Earth (i.e. a Venus synod) is currently 584 days, so that 5 Venus synods are equivalent to 8 ‘practical’ earth-years (of 365 days). Venus has a sidereal orbital period of 225 days, and 13 of these periods equal 8 practical earth-years. In both cases, the numbers composing these ratios are consecutive Fibonacci numbers, and therefore give approximations to the golden section: 8/5 = 1.6, and 13/8 = 1.625. Venus rotates extremely slowly on its axis: its day lasts 243 earth-days, or 2/3 of an earth-year (the same ratio as a musical fifth). Every time Venus and earth ‘kiss’, Venus does so with the same face looking at earth. Over the 8 years of the 5 kisses, Venus will have spun on its own axis 12 times in 13 of its years.
Thus, in 8 years Venus has 5 inferior conjunctions (when it lies between earth and sun) and 5 superior conjunctions (when it lies on the opposite side of the sun). Plotting either of these sets of 5 conjunctions in relation to the zodiac produces a five-pointed star or pentagram, the segments of the constituent lines being related according to the golden section. There is a slight irregularity, for the pentagram is not completely closed, there being a difference of two days at the top. This irregularity generates a further cycle, as it means that the pentagram will rotate through the whole zodiac in a period of about 1200 years. It is interesting to note that the pentagram was associated with the Babylonian goddess Ishtar-Venus, and that depictions of Venus as a five-pointed star have also been found at Teotihuacan in Mexico. In theosophy, Venus is said to be closely connected with our higher mind (manas), the fifth principle of the septenary human constitution.
Fig. 9.2 Teotihuacan: stellar symbol of Venus dispensing
its influence
downwards towards the earth.
Fig. 9.3 The Venus pentagram.
According to theosophy, the key numbers to the solar system lie in a
combination of the year of Saturn and the year of Jupiter, expressed in earth-years (Purucker, 1973, pp. 3-15).
About 12 earth-years (11.86) make 1 year of Jupiter, and about 30 earth-years (29.46) make 1 year
of Saturn: 12 x 30 = 360, the number of degrees in a circle and the number of days in an ideal
earth-year. (Theosophy says that an earth-year oscillates above and below 360 days over very long
periods of time.)
The whole process of evolution can be summed up as a descent of divine consciousness-centres or monads into matter, and their subsequent re-ascent to spirit, enriched by the experience gained on their aeons-long evolutionary journey. This process can be symbolized by two interlaced triangles, known as Solomon’s seal or the sign of Vishnu, the upward-pointing triangle representing spirit, and the downward-pointing triangle representing matter. Significantly, as Saturn and Jupiter revolve around the sun, they mark out two interlaced triangles around us every 60 years! The upward triangle is formed by their conjunctions and the downward triangle by their oppositions. Once again, there is a slight irregularity: after 60 years the conjunction does not take place at exactly the same point; there is a gap of 8 degrees, so that the interlaced triangles slowly rotate through the entire zodiac in a period of 2640 years. There are 432 of these 60-year Jupiter/Saturn cycles in a precessional cycle of 25,920 years.
Fig. 9.4 Conjunctions and oppositions of Jupiter and Saturn.
The sun is the heart and brain of the solar kingdom and the regular
sunspot cycle is akin to a solar heartbeat. The sunspot cycle has a major impact on earth,
especially terrestrial magnetism and the climate. Over the past 250 years its length has varied
irregularly between 9 and 14 years, averaging 11.05 years. Sunspots peak shortly after Jupiter
passes the point in its orbit closest to the sun. (The ‘ideal’ sunspot cycle is said in
theosophy to be 12 years, so there would be one such cycle for each year of Jupiter.) The sunspot
maximum does not occur exactly in the middle of the sunspot cycle. The ascending part of the cycle
has a mean length of 4.3 years – very close to the figure of 4.22 years that would divide the
11.05-year cycle exactly according to the golden section.*
*Theodor Landscheidt, ‘Solar activity: a dominant factor in climate dynamics’, www.john-daly.com/solar/solar.htm.
To attribute the order, harmonious proportions, and marvellous recurring patterns in nature to pure chance is absurd. The pagan philosopher Cicero wrote:
If anyone cannot feel the power of God when he looks upon the stars, then I doubt whether he is capable of feeling at all. From the enduring wonder of the heavens flows all grace and power. If anyone thinks it is mindless then he himself must be out of his mind. (On the Nature of the Gods, Penguin Classics, 1972, 2.55)
However, the traditional theological picture of ‘God’ as a supreme selfconscious being who thinks, plans, and creates – even managing to make the entire universe out of nothing – is untenable. If ‘he’ is a being, he must be finite and limited, and have a relative beginning and end. Whereas if the divine is infinite, it cannot be a thinking being, separate from the universe, but must be one with it, as taught by pantheism. The divine essence would then be synonymous with boundless space, or infinite consciousness-life-substance. And since nothing can come from nothing, it must always have existed.
Materialistic scientists prefer to attribute the patterned order of the cosmos to ‘laws of nature’, with new ones emerging ‘spontaneously’ as evolution proceeds. But this explains nothing, for the word ‘law’ simply denotes the regular operations of nature – the very regularities that the term is supposed to explain! Patterns in the living world are sometimes attributed to genetic programmes, but this is merely a declaration of faith, since all that genes are known to do is provide the code for making proteins – not for arranging them into complex structures. Moreover, it is hard to swallow the conventional claim that genetic programmes themselves originated by chance.
If some patterns arose through random genetic mutations and natural selection, it would mean, for example, that the reason the golden angle is widely found in leaf arrangements throughout the plant kingdom is because it contributed to their survival. This implies that at one time most plant species did not embody the golden angle – but such a claim can never be tested. What we do know is that life emerged on earth incredibly quickly, that new and fully functional types of organisms have tended to appear on earth incredibly quickly, and that there is no evidence whatsoever for vast periods of trial-and-error experimentation along darwinian lines.
The theosophic tradition, or ancient wisdom, teaches that nature’s patterns and regularities are better seen as habits, an expression of its ingrained instinctual behaviour, of the tendency for natural processes to follow grooves of action carved in countless past cycles of evolution. For it teaches that all worlds, on every conceivable scale, reembody again and again. And behind these habits lies an all-pervading consciousness, the universe being composed of interworking hierarchies of intelligent and semi-intelligent ‘beings’ or energy-forms, from elemental to relatively divine. The order of nature also reflects the essential interconnectedness of all things, and the fact that the same basic patterns and processes recur on widely different scales.
All monads, or units of consciousness, are said to progress through a series of kingdoms towards a state of relative perfection in the system of worlds in which they are then evolving, before passing, after a long rest, into other world systems, on other planes. Humans, in their present stage of rebellious selfconsciousness, often succumb to the temptation to misuse their free will for selfish and shortsighted ends, creating discord and suffering. But it lies within our power to attune ourselves to the fundamental harmony of our inner, spiritual selves – sparks of the universal ‘Self’ – and to become voluntary coworkers with nature in the great cosmic adventure of evolution.