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).
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.
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.
Though megalithic astronomers could look at the
sky, their measurement methods were only accurate using horizon events. Horizon
observations of solstice sunrise/set each year, lunar extreme moonrises or settings
(over 18.6 years) allowed them to establish the geometrical ratios between
these and other time periods, including the eclipse cycles. In contrast, the
synod of Jupiter is measured between its loops in the sky, upon the backdrop of
stars, in which Jupiter heads backwards each year as the earth passes between
itself and the Sun. That is, Jupiter goes retrograde relative to
general planetary direction towards the east. Since such retrograde movement
occurs over 120 days, Jupiter will set 120 times whilst moving retrograde. This
allowed megalithic astronomy to study the retrograde Jupiter, but only when the
moon is conjunct with Jupiter in the night sky and hence will set with Jupiter
at its own setting.
The diatonic or natural scale, consisting of
five whole tones and two opposed
semitones, is most familiar today in the white notes of the piano [Apel. see Diatonic]. On the piano this would be
called C-major, which imposes the sequence of tones (T) and semitones (S) as
T-T-S-T-T-T-S in which the initial and final tetrachords are identically T-T-S,
leaving a tone between F and G, the two fixed tones of the Greek tetrachordal
The diatonic scale is … an abstractum; for all we have is five tones and two semitones a fifth apart [until] we fix the place of the semitones within the scale, thereby determining a definite succession …, [and] we create a mode. [Levarie. 213].
Musical Morphology,. Sigmund Levarie and Ernst Levy. Ohio:Kent State 1983. 213.
One can see that the tones are split by the major diatonic into one group of two (T-T) and one group of three (T-T-T), so the semitones are opposed (B-F) towards the tonic C as in figure 1.
Letters such as C are called note classes so as to label the tones of a diatonic scale which, shown on the tone circle, can be rotated into any key signature of twelve keys including flattened or sharpened notes, shown in black in figure 1. We will first show how these black notes came about naturally, due to two aspects of common usage.
The note classes arose from the need of choral
music to notate music so that it could be stored and distributed. When we “read
music” today, the tablature consists of notes placed within a set of five lines
with four gaps, and two extendable areas above and below in which only seven
note classes can be placed, seven being the number of note classes in the modal
diatonic and the number of white keys on the keyboard, which is the other
aspect of usage.