Ad Quadratum is a convenient and profound technique in which continuous scaling of size can be given to square shapes, either from a centre or periphery. The differences in scale are multiples of the square root of two [sqrt(2)] between two types of square: cardinal (flat) and diamond (pointed).
When a square’s diagonal touches opposite sides sides of a larger square, the two squares differ by sqrt(2) and a square becomes a diamond or visa versa. By repeating this effect a continuum can be generated to form the large patterns found in sacred buildings such as the Vedic Angkor Wat, Christian Gothic cathedrals, Islamic Dome of the Rock and even Stonehenge. The technique is therefore very ancient yet still used. Subjectively such buildings express a high degree of visual order by employing what is called a geometrical progression. Objectively, such patterns appear to convey cosmic principles known to ancient builders but largely forgotten recently.
Here I start by publishing an important diagram that shows how the earliest known references to musical tuning (early 4th millennium BC) on clay cuneiform tablets, using “regular numbers” whose factors are products of only the numbers 2, 3 and 5, led to the cosmological vision of their gods, the primary god, Anu, being a balanced mix of all three numbers as 60 but also called ONE. This is the source of their Sexagesimal or base-60, still employed in measuring angles and time called minutes and seconds. All comes from ONE.
Recently an “early Ptolomaic” tomb was discovered similar in themes to the famous Egyptian Books of the Dead (Middle Kingdom). Normally written on papyrus, they feature multiple tableau of Osiris judging the dead and other scenes. Osiris is a long lasting and perhaps supreme god whose cult was present throughout 3000 years of Dynastic history. I have previously interpreted his throne through drawings but, in the new tomb, he is painted on the walls at least twice and the design of his throne looks like layers of “eggs”. Below is one of the press pictures taken from the Guardian, and the headline is Mummified mice found in ‘beautiful, colourful’ Egyptian tomb.
Osiris could have been seen as a/the god of Harmony and below I explain why harmony may have been thought technically significant at the dawn of our earliest texts, then found in Sumeria 900 miles to the East. The reason I believe musical ratios were significant at the dawn of history because they had naturally emerged from measuring the lunar and solar year and comparing these with the time between loops of the outer planets Jupiter and Saturn.
Sacred Number at the Heart of the World (an article in New Dawn 168)
After the ice receded, late
Stone Age people developed the farming crucial to the
development of cities in the Ancient Near East (ANE). On the Atlantic coast of Europe,
they also developed a now-unfamiliar science involving horizon astronomy. Megalithic
monuments were the tools they used for this, some still seen in the coastal regions
of present day Spain, France, Britain and Ireland. Megalithic astronomy was an exact
science and this conflicts with our main myth about our science: that ours is the
only true science, founded through many historical prerequisites such as arithmetic
and writing in the ancient near east (3000- 1200 BC) and theory-based reasoning
in Classical Greece (circa 400-250 BC), to name but two. Unbeknownst to us, the
first “historical period” in the near east was seeded by the exact sciences of
the megalithic, such as the accurate measurement of counted lengths of time, developed
by the prehistoric astronomers. With the megalithic methods came knowledge and discoveries,
and one discovery was of the harmonic ratios between the planets and the Moon.
The idea that the planets were gods had been
born before the ancient world, through the data of megalithic astronomy and this
megalithic idea was the basis for the religious ideas of the East. Megalithic
astronomy and Near Eastern religious and harmonic ideas have both been written
out of our history of civilization, leaving us with enigmatic monuments and ill-defined
religious mysteries. How this slighting of our real history happened is perhaps
less important than our discovering again the purpose of the megalithic
monuments and of those religious ideas that sprang from the discovery that the
planets were harmonically related to life on Earth.
Gurdjieff first presented his ideas to groups in pre-revolutionary Russia. Amongst his carefully chosen students it was the habit to reconstruct talks and diagrams as much as possible, an endeavour that gave us a textbook of Gurdjieff’s ideas called In Search of the Miraculous (P.D. Ouspensky, 1950). This early form of the teaching wasradically revised and extended by Gurdjieff, now as an author, during the 1920s, producing All and Everything whose part one was Beelzebub’sTales to his Grandson (G.I. Gurdjieff, 1950). Prior to drawing this diagram just after February 1917, Gurdjieff had been presenting ideas about transformation of energies, human and cosmic, using the musical theory surrounding the octave of eight notes. The Diagram of Everything Living was “still another system of classification… in an altogether different ratio of octaves… [that] leads us beyond the limits of what we call ‘living beings’ both higher [and lower] than living beings. It deals not with individuals but with classes in a very wide sense.”
The three henges appear to align to the three notable manifestations to the north west of the northerly moon setting at maximum standstill. The distance between northern and southern henge entrances could count 3400 days, each 5/8th of a foot (7.5 inches), enabling a “there and back again” counting of the 6800 days (18.618 solar years/ 19.618 eclipse years) between lunar maximum standstills.
In a previous article, the 7,500 foot-long Erdevan alignments were seen to have been a long count of the Saros period of 19 eclipse years versus the distance to Mane Groh dolmen of 19 solar years, this probably conceptualized as an 18-19-6 near-Pythagorean triangle, whose inner angle is the bearing from east of Mané Groh. However, the path directly east caused the actual alignments, counting the Saros, to veer south to miss the hill of Mané Bras.
It has been remarked that the form of the northern alignments of Edeven were similar
to those starting at Le Menec’s egg-shaped stone circle 4.25 miles away, at a
bearing 45 degrees southeast. Whilst huge gaps have been caused in those of
Edeven by agriculture, the iconic Le Menec alignments seem to have fared better
than the alignments of Kermario, Kerlescan and Petit Menec which follow it
east, these being known as the Carnac Alignments above the town of that name.
One similarity between alignments is the idea of starting and terminating them with ancillary structures such as cromlechs (stone kerb monuments), such as the Le Menec egg and, despite road incursion, a3-4-5 structure similar to Crucuno, aligned to the midsummer sunset by a length 235 feet long. This is the number of lunar months in the 19 year Metonic period and is factored 5 times 47. Another similarity may be seen in Cambray’s 1805 drawing of these Kerzerho alignments, at the head of ten stone rows marching east (figure 1).
In "Planetary Resonances with the Moon" I explored the astronomical matrix presented in The Harmonic Origins of the World with a view to reducing the harmonic between outer planets and the lunar year to a single harmonic register of Pythagorean fifths. This became possible when the 32 lunar month period was realized to be exactly 945 days but then that this, by the nature of Ernest McClain’s harmonic mountains (figure 1) must be 5/4 of two Saturn synods.
Using the lowest limit of 18 lunar months, the
commensurability of the lunar year (12) with Saturn (12.8) and Jupiter (13.5)
was “cleared” using tenths of a month, revealing Plato’s World Soul register of
6:8::9:12 but shifted just a fifth to 9:12::13.5:18, perhaps revealing why the
Olmec and later Maya employed an 18 month “supplementary” calendar after some
of their long counts.
By doubling the limit from 18 to three lunar
years (36) the 13.5 is cleared to the 27 lunar months of two Jupiter synods,
the lunar year must be doubled (24) and the 32 lunar month period is naturally
within the register of figure 1 whilst 5/2 Saturn synods (2.5) must also
complete in that period of 32 lunar months.
A recent plan of the Kaaba indicates that its walls, by their odd-number proportionality, symbolised the numerical origins of musical harmony through the first six numbers, 1, 2, 3, 4, 5 & 6, sometimes called the Senarius meaning “existence out of six”.
According to Plato, the world was created using the rules of musical harmony in a scheme involving perfect fifths of 3/2, fourths of 4/3 and tones of 9/8, leaving “leftover” semitones of 256/243: a rudimentary musical scale. This only used prime numbers 2 and 3 and multiplications with themselves and each other; a system called Pythagorean tuning. The Kaaba incorporates another prime number 5, called the human number, this enabling two more large intervals called thirds, the major third of 5/4 and minor third of 6/5. Using 5 enables more and better scales to be formed and fills in the gap between 4 and 6 to show all the large intervals in the first six numbers, 1:2:3:4:5:6. This fuller tuning system has been found in the ancient Near East as long ago as the Sumerians, in their tuning texts on cuneiform (c. 3000 BC onwards).
Our modern globes are based upon political boundaries and geographical topography yet they have geometrical predecessors, which described the world as an image, diagram or schemata. The original idea for the form of the world was summarised within a simple two dimensional geometry, like an eastern mandala or yantra.
Such a diagram was built into the Cosmati pavement at Westminster Abbey, built by Henry III and dedicated to the Saxon King and Saint Edward the Confessor. This exotic pavement became the focus for the Coronations of subsequent English then British monarchs.
My crucial entré to planetary harmony came when I noticed musical ratios in the synodic time periods of Jupiter and Saturn relative to the lunar year. This approach differs from the norms for “harmonies of the spheres” (a.k.a. Musica Universalis) which are geometrical and spatial, rather than temporally harmonic.
The planetary harmony I found within synodic periods became the subject of my new book The Harmonic Origins of the World (pub. 2018). These synodic ratios have been parts of my work from c. 2000, then expressed as “matrix diagrams” (Matrix of Creation, figure 2 below). In my new book, I show how ancient tuning theory seems to have presented the same information, in a different type of matrix (see figure 4).
Below I connect the outer planets using two additional (and useful) kinds of diagram, the right-angled triangle (figure 1) and the Pentad (figure 5), the latter developed in the 20th century within a discipline called Systematics.
It is said that we are transiting from the Age of Pisces to the Age of Aquarius in a backward precession through the 12 zodiacal signs. Examining the numbers that define these Ages is one of the core themes of this book. The basic premise is that stories – some of them handed down orally since Neolithic times – enable us to identify the inner spiritual aspects within our material world and participate in the evolution of human consciousness foretold by ancient myths. The author is greatly influenced by G.I. Gurdjieff and his Law of Seven, albeit with revisions of his own.
Readers such as myself, for whom mathematics is not their strong suit, need not be daunted by the many sets of figures presented in this book. They are important as supporting evidence for the theories presented, and their comprehension is made easier by the use of diagrams. Moreover, the fractions and ratios are often related to musical octaves and the Do-Re-Mi music-reading system.