Tag: double square

Cologne Cathedral Facade as Double Square

image: The Gothic cathedral of Cologne by night, by Robert Breuer CC-SA 3.0

On the matter of facades of Gothic cathedrals, I hark back to previous work (February 2018) on Cologne cathedral. This was published in a past website that was destroyed by its RAID backup system!

As we have seen with Chartres, some excellent lithographs with scales can often exist online from which one can interpret their sacred geometrical form and even the possible measures used to build that form. The Gothic norm for a facade seem more closely followed at Cologne facade which has two towers of (nearly) equal height.

We saw at Chartres that an underlying geometry using multiple squares may have been used to define a facade and bend it towards a suitable presentation of astronomical time, in a hidden world view that God’s heaven for the Earth is actually to be found in the sky as a pattern of time. This knowledge emerged with the megaliths and, in the medieval, it appeared again in monumental religious buildings built by masons who had inherited a passed-down but secret tradition.

A Prologue to Cathedral Music

In my book Matrix of Creation I observed that the Lunar Year of 12 months appear to be like Plato’s World Soul, of 6:8::9:12 only raised by a fifth (3/2) to be (9:12::13.5:18). The number 12 is then the 12 lunar months of the lunar year and the 13.5 are the 13.5 lunations which are the synodic period of Jupiter (398.88 days). The synod of Saturn (378 days) is then caught between the 12 and 13.5, near the geometric mean of the octave 9::18, as a location known to tuning theory as the upper Just tritone of 64/45 (= 1.422), a prime example of the diabolicus in music. That is the Moon appears to be a central part and factor of an astronomical instrumentality relating Jupiter and Saturn, the two outer gas giants of the solar system.

Without knowledge of geocentric astronomy, megalithic metrology, sacred geometry, and the study of numbers (the four higher parts or Quadrivium of the Traditional Arts), it is impossible to read such monuments, and the truths placed within them.

The Double Square

The properties of the double square, here proposed as a vertical 2 by 1 rectangle embracing the whole facade, are to be seen in many other posts you might want to reference (this link opens a new search tab). It seemed to me that the key orientation was the crossing of the lower square’s diagonals, a location where Chartres has its Rose Window and in this case, the domed top of a major rectangular window.

Referring to the diagram below, the bottom square is the cosmic octave’s “ballast” of 9 lunar months and the top square the “active portion” of an octave in which 12 lunar months (or lunar year) is the fourth note of the octave uplifted numerically by 3/2. Saturn’s synod of 12.8 months is 12.8/9 = 64/45, musically √2 which is the length of the lower square’s diagonals which cross the arch of the main window. The red arrow thus signifies by its arc the location of Saturn as the tritone (geometric mean) of the octave.

The Façade of Cologne as the double octave of Plato’s World Soul elevated by 3/2.

The left tower is slightly lower that the right, indicating that the Saturn synod (378) is less than the Jupiter synod (399). Musically, Jupiter is 3/2 is the fifth in the octave 9::18, numerically 13.5 lunar months. If one halves the right side of the upper square into two, this is where the fifth belongs and this point is also a whole tone (9/8) above the lunar year, whilst Saturn is 16/15 above the lunar year as 12.8 lunar months.

Plato’s World Soul, transformed

In a single figure, the transformation of Plato’s World Soul of 6:8::9:12, as simplest solution, then masked the hidden doctrine that, in lunar months, the very same is implemented in the relationship of the outer giant planets to the Moon as lunar year but trasformed by a musical fifth. The dominant and subdominant are the lunar year and Jupiter synod, with the Saturn synod providing the “satanic” tritone which acts in denial of the octave “god”. This octave of 19::18 has only survived in the Supplemental Glyphs of the Olmec (additional to long counts), who appear to have received it from the collapsing Bronze Age of the Eastern Mediterranean around 1500 BC (see my Sacred Number and the Lords of Time).

Abandoning the geocentric perspective of the planets for the heliocentric “washed away [this] baby with the bathwater”, that the moon was the intermediary in simple numbers of months of the principle of cosmic harmony in the higher worlds. Holding us back from seeing the old perspective is our fond belief that cosmic design was part of religious fantasies in which God, gods or angels had made the sky of the earth. Whilst we know so much about space, time has been neglected for its astronomical action upon the present moment within which change is the prime phenomena, as the Buddha said “change is the only thing that does not change.”

Starcut Diagram: geometry to define tuning

This is a re-posting of an article thought lost, deriving in part from Malcolm Stewart’s Starcut Diagram. The long awaited 2nd edition Sacred Geometry of the Starcut Diagram has now been published by Inner Traditions. Before this, Ernest McClain had been working on tuning via Gothic master Honnecourt’s Diagram of a Man (fig. 2), which is effectively a double square version of the starcut diagram.

The square is the simplest of two dimensional structures to draw, giving access to many fundamental values; for example the unit square has the diagonal length equal to the square root of two which, compared to the unit side length, forms the perfect tritone of 1.414 in our decimal fractional notation (figure 1 left). If the diagonal is brought down to overlay a side then one has the beginning of an ancient series of root derivations usually viewed within the context of a double square, a context often found in Egyptian sacred art where “the stretching of the rope” was used to layout temples and square grids were used to express complex relationships, a technique Schwaller de Lubitz termed Canevas (1998). Harmonically the double square expresses octave doubling (figure 1 right).

Figure 1 left: The doubling of the square side equal 360 units and right: The double square as naturally expressing the ordinal square roots of early integers.

Musical strings have whole number lengths, in ratio to one another, to form intervals between strings and this gives geometry a closer affinity to tuning theory than the use of arithmetic to calculate the ratios within a given octave range. The musicology inferred for the ancient world by Ernest G. McClain in his Myth of Invariance (1976) was calculational rather than geometrical, but in later work McClain (Bibal 2012-13) was very interested in whatever could work (such as folding paper) but was especially interested in the rare surviving notebook of 13th century artist Villard de Honnecourt, whose sketches employed rectilinear frameworks within which cathedrals, their detailing, human and other figures could be drawn.

“I believe we have overlooked Honnecourt as a prime example of what Neugebauer meant in claiming Mesopotamian geometry to approach Renaissance levels illustrated in Descartes. If Honnecourt is 13th c. then he seems more likely to be preserving the ancient picture, not anticipating the new one.”

This draws one into significant earlier traditions of sacred art in Egypt (Canevas) and in Indian temple and statue design, and to Renaissance paintings (see end quote) in which composition was based upon geometrical ideas such as symmetry, divisions into squares and alignments to diagonals. Figure 2 shows one of Honnecourt’s highly stylised sketches of a man, using a technique still in use by a 20th century heraldic artist.

Ernest McClain, Bibal Group: 18/03/2012

Figure 2 The Honnecourt Man employing a geometrical canon.

The six units, to the shoulders of the man, can be divided to form a double square, the lower square for the legs and the upper one for the torso. The upper square is then a region of octave doubling. McClain had apparently seen a rare and more explicit version of this arrangement and, from memory, attempted a reconstruction from first principles (figure 3), which he shared with his Bibal colleagues, including myself.

Figure 3 McClain’s final picture of the Honnecourt Man, its implied Monochord of intervals and their reciprocals.

To achieve a tuning framework, the central crossing point had been moved downwards by half a unit, in a double square of side length three. On the right this is ½ of a string length when the rectangle is taken to define the body of a monochord. McClain was a master of the monochord since his days studying Pythagorean tuning. Perhaps his greatest insight was the fact that the diagonal lines, in crossing, were inadvertently performing calculations and providing the ratios between string lengths forming musical intervals.

Since the active region for octave studies is the region of doubling, the top square is of primary interest. At the time I was also interested in multiple squares and the Egyptian Canevas (de Lubitz. 1998. Chapter 8) since these have special properties and were evidently known as early as the fifth millennium BC (see Heath 2014, chapter two) by the megalith builders of Carnac. In my own redrawing of McClain’s diagram (figure 4) multiple squares are to be seen within the top square. This revealed that projective geometry was to be found as these radiant lines, of the sort seen in the perspective of three dimensions when drawn in two dimensions.

Figure 4 Redrawing McClain to show multiple squares, and how a numerical octave limit of 360 is seen creating lengths similar to those found in his harmonic mountains.

Returning to this matter, a recently developed technique of populating a single square provides a mechanism for studying what happens within such a square when “starcut”.

Figure 5 left: Malcolm Stewart’s 2nd edition book cover introducing right: the Starcut Diagram, applicable to the top square of Honnecourt’s octave model .

Malcolm Stewart’s diagram is a powerful way of using a single square to achieve many geometrical results and, in our case, it is a minimalist version that could have more lines emanating from the corners and more intermediate points dividing the squares sides, to which the radiant lines can then travel. Adding more divisions along the sides of the starcut is like multiplying the limiting number of a musical matrix, for example twice as many raises by an octave.

A computer program was developed within the Processing framework to increase the divisions of the sides and draw the resulting radiants. A limit of 720 was used since this defines Just intonation of scales and 720 has been identified in many ancient texts as having been a significant limiting number in antiquity. Since McClain was finding elements of octave tuning within a two-square geometry, my aim was to see if the crossing points between radiants of a single square (starcut) defined tones in the just scales possible to 360:720. This appears to be the case (figure 6) though most of the required tone numbers appear along the central vertical division and it is only at the locations nearest to D that eb to f and C to c# that only appear “off axis”. The pattern of the tones then forms an interesting invariant pattern.

Figure 6 Computer generated radiants for a starcut diagram with sides divided into six.

Figure 7 http://HarmonicExplorer.org showing the tone circle and harmonic mountain (matrix) for limit 720, the “calendar constant” of 360 days and nights.

Each of the radiant crossing points represents the diagonal of an M by N rectangle and so the rational “calculation” of a given tone, through the crossing of radiants, is a result of the differences from D (equal to either 360 or 720) to the tone number concerned (figure 8).

Figure 8 How the tone numbers are calculated via geometrical coincidence of cartesian radiants which are rational in their shorter side length at the value of a Just tone number

It is therefore no miracle that the tone numbers for Just intonation can be found at some crossing points and, once these are located on this diagram, those locations could have been remembered as a system for working out Just tone numbers.

Bibliography

Heath, Richard.

  • 2014. Sacred Number and the Lords of Time. Rochester, VT: Inner Traditions.
  • 2018. Harmonic Origins of the World: Sacred Number at the Source of Creation. Inner Traditions.
  • 2021. Sacred Geometry: Language of the Angels. Inner Traditions.

Lubitz, R.A. Schwaller de.

  • 1998. The Temple of Man: Apet of the South at Luxor. Vermont: Inner Traditions.

McClain, Ernest G. 

  • 1976. The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas Hays.

Stewart, Malcolm.

  • 2022. Sacred Geometry of the Starcut Diagram: The Genesis of Number, Proportion, and Cosmology. Inner Traditions.

Double squares: Venus and the Golden Mean

The humble square, with side length equal to one unit, is like the number one. It’s area is one square unit and, when we add another identical square to one side, the double square appears. Above right the Egyptian Djed column is shown within a double square. The Djed is the rotating earth which the gods and demons have a tug of war over. This is also a key story in the Indian tradition, called The Churning of the Oceans, where the churning creates both the food of the gods (soma) and every wonderful thing that emerges upon the Earth. In this, the double square symbolized the northern and southern hemispheres of the Earth. The anthropomorphic form Djed shown above has elbows indicative of the Double square.

Figure 1 The churning of the ocean (Samudra Manthan in Sanskrit)

The Djed appears to be the general principle of rotation of, and apparent motion around, the earth.

The god Isis is (as a planet) Venus and is shown (fig.2) offering up the sun disk: another Djed is below, with her Ankh symbol of Life atop the Djed, now having female arms . This sun most probably points to the practical year as 365 days which is 5/8 of the Venus synod of 584 days. (This ratio of 1.6 is the sixth note of the octave 1 to 2.)

In figure 2, two female attendants provide the duality which one might take to be her two famous manifestations of (firstly) the brightest Evening Star, as the sun goes down, and then (after that) the brightest Morning Star before the sun rises. Above there is duality again with three baboons either side of the sun, perhaps representing the six visible planets: Moon, Venus and Mercury: Jupiter, Saturn, Mars and their “tug of war”.

Figure 2 The creation of Horus-Ra from out of an ankh with female arms atop a djed. from Budge 1899, also fig. 7.8 of Richard Heath, The Harmonic Origins of the World.

The numbers 5 and 8 are Fibonacci approximations {1 2 3 5 8 13 21 34 …} to the golden mean, a transcendent number {1.618034…} which rational numbers can only approximate. Venus and the Earth have clearly settled into orbits around the sun resonant with Fibonacci ratios since the Venus orbital period (224.701 days) is 8/13 of the solar year. And it is this fact that eventuates in what we see on Earth, namely the manifestations of Venus every 8/5 of a practical year. of 365 days.

Figure 3 The double square, its in-circle and out-circle manifesting golden rectangles around itself.

In this post, I developed a result sent to me, that a square drawn within the upper hemisphere of a circle must define a golden mean rectangle either side from its height of 1 and the remaining radius of 0.618034… and so it can be seen that the divine principle of the Golden Mean emanates from the double square, either side of each square, when the double square is embraced by a circle drawn from its center. Obviously, on Earth and between orbits (of Venus and Earth), the Golden Mean (also called Phi) has to be approximated by whole number ratios but the principle is present within the geometry and its out-circle. Schwaller de Lubicz thought the dynastic Egyptians held the Golden Mean to be “the fundamental scisson” (literally “scissor cut”) in the range one to two and, its reciprocal can be seen to share the portion over 1 (figure 3).

One can see that geometry and the early numbers would have been seen as two aspects of what we call space and time, in which “things” are separate from each other in Existence but somehow conjoined within Eternity. What we call order is in fact an achievement of harmony made possible by the arranging and fitting of parts to form a coherent whole. It is this insight which gave meaning to their study of geometry and numbers from the prehistoric onwards, into the recorded history of early civilizations. The meaning for Life on Earth became encoded within ancient and prehistoric symbols, whose geometrical and numerical language of expression went to the heart of phenomena.