The first cube (above left) is a single cube of side length one. One is its own cornerstone. The first cubic number is two to the power of three, with side length two and volume equal to eight cubes that define the unit corner stone.

In modern thinking, and functional arithmetic, volume increases with side length but the cube itself, as archetype of space, is merely divided by the side length of the unit cornerstone, which is 1/8th the volume and therefore reciprocal to the volume of 8 leaving the cube singular.

This may not seem important but, by dividing a whole cube, one is releasing more and more of the very real behavior that exists between numbers, within the cube. For example the number 8 gives relations between numbers 1 to 8, such as the powers of two {1,2,4,8} and the harmonic ratios {2/1, 3/2, 4/3,5/4,6/5}. These can give an important spine of {4/3, 5/4,6/5} which equals 6/3 = 2 of the yet to (numerically) be octave of eight note classes. Moving to side length 3, the cube of three is twenty seven (27), as seen in figure above, top right. To obtain it, the corner stone must be side length 1/3, and volume 1/27 so that, in these units, the volume of the cube is 27.

If one were to reciprocally double the 1/3 side length, each cornerstone unit would have 8 subunits, so that the volume of 27 would be times 8 which equals Plato’s number of 216. Another view is then that the cornerstone side length has divided the bottom right cube into six units which number, 6, cubed, is 216 a perfect number for Plato.

By accepting the cube of one as the whole, this form of thinking reciprocally divides that whole side length to generate an inner structure within the whole cube of one, equal to the denominator of the reciprocation. The role of the whole is then to be the arithmetic mean between a number and its reciprocal. This procedure maintains balance between what is smaller than the whole (the reciprocal) and what is larger than the whole (in this case the volume).

In ancient tuning theory this was expressed by the two hexchords descending and ascending from the tonic (we might call *do*), expressed by the two hands. The octave of eight and the cube are both wholes to be broken into by numbers greater than one by means of reciprocation.

Ernest G. McClain revealed the scale of such thinking was massive, whilst also but secretly reciprocal, so that a limiting number could express how different wholes will behave due to their inner diversity of numbers at work within them.

*In this example, a musical code for planetary resonance is revealed within the metrology of the Parthenon (above), implied tone set (right) and octave mountain of numbers below 1440 (left bottom). In this case the number 24 has been multiplied by 60 to give a limiting number of 1440. The cornerstone in this case is bottom left of the mountain = 1024, a pure power of 2 ^{10}.*

By simply quoting a limiting number, in passing, ancient texts could, in the hands of an initiate, create an enormous world of tonal and impied religious meaning – through a kind of harmonic allusion.

It is only by using the conceptual approach of the ancients, that their intellectual life can be recovered – just by adding the waters of number and some powers of imagination.