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.”

Ernest McClain, Bibal: 18/03/2012:

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 used by heraldic artists.

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[1].

Figure 3 McClain’s final picture of the Honnecourt Man
and the  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
throwing up 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 Malcolm Stewart 2008 book cover (left) introducing the Starcut Diagram (right)
applicable to the top square’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.

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 has been identified in ancient texts as a significant harmonic number. Since McClain was finding elements of octave tuning within a two-square geometry, my aim was to see if the crossing points between radiants 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# can only appear “off axis”. The pattern of the tones then forms an interesting invariant.

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). 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.

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

### Bibliography

Heath, Richard. 2014. Sacred Number and the Lords of Time. Rochester, VT: 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. 2008. Patterns of Eternity: Sacred Geometry and the Starcut Diagram. Edinburgh, Floris