Sacred Number

The name of the Ka'aba means "cube" whilst the building itself is evidently not a cube (with identical side lengths). In Sacred Number and the Origins of Civilization much was made of the fact that the square root of three appears between opposite corners of a cube with unit length (the root of two arising between the opposite corners of its square sides).

Root three appears as a very important proportion in the Gothic style of building whilst many sacred cubes are noted in sacred building in general, the Temple of Solomon and Leto [Egypt], Hindu temples, the Delphic altar to Apollo, and the cube of the New Jerusalem.

In his third book, Meditations on the Koran, Earnest McClain suggests that the original prototype of the cube is the cube of 60, which was the base number of the Sumerian and then Babylonian civilisations. From this base, still in use today (within time and angular measure) McClain shows the ability of 60³ to generate a tone map that defines a hexagonal array of musical notes, all within a single octave but held between 216,000 [=60³] and 432,000, a very significant number in, for example, the Vedic tradition as being the length of the Kali Yuga or dark age.

McClain Yantra for 432,000. Only this yantra can yield a hexagonal region where tones and their reciprocals co-incide. The hexagon has 37 tones because 37 is a hexagonal number, viz chapter 10 and the cube of Metatron.
[from p 90 of  Meditations on the Koran by Earnest McClain, Nicolas-Hays, Ney York, 1981]

We need to remember that a hexagon is a cube seen from one corner (and projected upon a flat surface) and that, in McClain's world, all ancient texts were written with numerical tuning theory in mind as a key plot generator. This last point has not been acknowledged yet by the worlds of scholarship or religion because the implications are too profound, subversive to the traditional ideas behind religious texts as communications from God or gods (rather than intimations of the divine world).

Returning to the Ka'aba, McClain gives a splendid account of its possible origins, the background of Mohammed's re-purposing of the shrine and the meaning of its non-cubic design.

Its width, length and height were probably intended to be 10 by 12 by 16 and McClain points to these proportions as being similar to the temple of Poseidon in Plato's myth of Atlantis, namely "6:3 plethora (full)" meaning expressive of 6:5:4:3 in that 5:6 lies between width and length and 3:4 between length and height. These ratios are minor third and perfect fourth.

One should note that the proportion between the width and the height is 6/5 times 4/3 which equals 8/5 the synodic period of Venus [584 days] in terms of the "practical" year of 365 days. The variation of width, length and height can lead to a greater tonal achievement that may be significant.

Why then should the polar radius of the Earth be, according to the ancient model, 3456 royal miles (as I pointed out in Sacred Number)? 3456 is 12 cubed [1728] times 2 and "the doubling of a [cubic] altar" was considered a challenge for the ancients, proved theoretically impossible in modern times. New Jerusalem was declared in Revelations to be a cube of 12,000 per side and as I found, one royal mile [8/7 miles] equals 6000 Greek feet of 176/175 English feet. This makes 1728 royal miles equal to the volume of a cube side length 2 royal miles of 12,000 such feet. [This I found in potentio within the layout of Washington DC].

The volume of a relatively small cube of 12 royal miles, projected to form a hexagon upon the surface of the Earth, generates an allusional volume one half of the polar radius. This relates to both tuning theory and the doubling of the altar. Such a cube has only powers of 3 and 5 plus 2 that figure within 10 = 2 times 5, and 12 = 4 times 3. However, 2 royal miles is 1 over 1728 of the polar radius, that is 1 over 12³ of its length.

Taking up the "doubling of volume" for a 12 side cube, the result is a cube 12 times the cube root of two in side length. The latter is very nearly 1.26 [actually 1.25992105, one part in 16,000 different] and this raises some interesting "ancient metrological" considerations, for John Neal's Standard Canonical transform is 126/125 or 1.008 of a foot. We can then see that 1.25 is 5/4 [the major third] and that this multiplied by 126/125 yields a highly accurate ability to create a cube that has doubled in volume. 5/4 times 126/125 is 1.26 which then times a side length of 12 gives 15.12 for the double volume cube's sides.

It appears then that this metrological ratio happens to be a practical means to the "doubling of the cubic altar" problem.

In Sacred Number I suggest that prehistoric people learnt about the natural structure of number through counting the periods of celestial events, most simply using the day as the counting period. The problem with counts is that they are periods and not frequencies; we are more familiar with musical tones and not the wavelengths of the sounds that we hear.

When looking at Venus and the Earth, the practical year is the fundamental frequency and is higher at 365 days than that of Venus at 584 days because the reciprocal of 584 is smaller than the reciprocal of 365. We should remember that

• 365 is 5 x 73 days and
  
• 584 is 8 x 73 days and also,
  
• as in Matrix of Creation, 73 days is
• 4 times 18.25 days which is
  
• 0.618 of a lunar month, i.e. related to the Golden Mean, and
  
• 1/20th of the practical year.

Venus, in terms of frequency, is held between the frequency of one year and that of two years, the latter forming low do for an octave with the year itself as the high do, as below.

The numbers in the blobs are times 73 days period lengths,
the lower frequencies having a larger number 

Chapter 2 explains how these sorts of octave come into existence due to relatively small numbers, for example this one with this sequence of intervals (the arrows such as 9/8, the whole tone and 16/15 the half tone) emerges out of the numbers between 24 and 48.  As a scale this would be the Ionian Mode - two tones then a semi tone - three tone then a semi tone - to make up the octave.

The Venus ratio of 8/5 becomes, in frequency, Venus has a frequency of 5 units relative to the Earth's year "frequency" of 8 units, i.e. it is lower, more bass. The Earth is 8/5 above Venus in frequency and this equals a number of familiar interval sequences shown: 8/5 = 3/2 (the fifth) times 16/15.  A fifth is itself 9/8 times 10/9 times 9/8 times 16/15. And so on, there are many intervals that seamlessly form the octave in general but in this case it can show the position of Venus as a relative frequency.

The essence of Venus is reciprocal that of the Earth, in many ways. If the Venus synod is set up within an astronomy program to trace the lines between where the earth and Venus are, the following picture emerges:

 3D view of the inner solar system, the sun central,
tracing between successive Earth and Venus positions
with a time period of 584 days (cross eyes to view in stereoscopic)

 The two planets experience the other describing a pentagon within the Zodiac or ecliptic. I hope everyone is enjoying the Evening Star as Venus rides high above the sunset, as she does every 8/5 or 1.6 years - which is not that often! As the planet swings around the Sun, on an inner orbit, she will shoot past the Sun to become a Morning Star.

Venus is locked into a Fibonacci orbital relation to Earth, with a synodic period of 8/5 and an orbital period of 13/8 solar years (0.615).

Sounds from the Deep

The continuity of subterranean spaces such as caves into burial mounds, dolmen chambers, and the spaces within temples, churches, cathedrals and crypts, involves more than a structural similarity. All these spaces have the effect of creating sound spaces, an acoustic experience.

Acoustic archaeology is a recent development, popularised by Paul Deveraux a British artiquarian and “ley line hunter”. His book, Stone Age Soundtracks, follows the continuity of acoustics within ancient sacred spaces. This implies that sound as well as proportion should be registered when intepreting such spaces.

Going back to the Stone Age, spanning a period as broad as 30000 to 8000 BC, Iegor Reznikoff with others have found a correlation between resonant acoustics within cave systems and cave paintings at those specific points. Similar findings were made beneath paintings on cliff faces, and there is a surviving tradition in Australia of aboriginal communities performing music and reciting stories below aboriginal art in a multi-media tradition. This indicates that stone paintings were intended in some cases to be supplemented with interpretive elements and were incomplete without the traditional knowledge that accompanied them. That a tradition was in existence is clear from the fact that painting styles often remained constant over thousands of years, as was the case with Egyptian sacred painting.

Work by others within dolmen has identified resonances within a limited range, corresponding with the male voice, that results from dimensions that encouraged a given length of standing wave. Such resonance amplifies a given frequency and can also give the impression of sound arising from out of the space as if independent of the original source.

The acoustic of Thoronet (Chapter 8 of Sacred Number) will have a very deep resonance spread over a range of infra-sound and hence over the higher frequencies octaves above it. Rather than amplication this spread spectrum of receptivity probably does provide high fidelity of the full range of notes as Larcher asserts. The exciting acoustics of the echo chamber is contained therefore as is found in the effects developed in older recording studios.

In terms of acoustic science Thoronet resembles the equal tempered scale in which all notes are given equality of separation, whilst the monks, using the human voice, were employing pure numerical intervals within their strict Gregorian form now called Cistercian chant. In “Earth to the Unknown Power” David Hykes sent the live sound of the concert, digitally encoded, to Le Thoronet Abbey over ISDN telephone lines, played back live through a sound system in the extraordinary Abbey acoustics, then re-encoded and sent back to The Kitchen in New York, placing the audience within the ‘Virtual Abbey.’

The Golden Mean ratio of 1.618 includes all the notes up to 8/5 of 1.6, a note that is just a semitone of 16/15 above the fifth of 3/2. 8/5 is the characteristic ratio between the Earth year of 365 days and the Venus synodic period. The creation of a space such a Thoronet represents the continuity of what was prehistoric experimentation, initially noting sensational effects but maturing with the numerical sciences into a space modelled on cosmic proportions in order to model the temple to the planets and the Sun. This idea bridges gap between theory and practice perhaps?

The style of Gregorian chanting was probably similar to how Norse epics were told. The roots of traditions need to be sought since the pagan-christian divide occurred over hundreds of years, years in which both traditions were borrowing each others clothes so that the Norse became Christians as the Romans had, influencing the new amalgum and injecting new components. It is only church history that need clear cut absolutes, real cultures being far more permissive if not promiscous.

Another enigma might be solved once acoustics is entertained as a core numerical science within the traditional art called “music”. In chapter 3 (now in Metrology Appendix 2), I showed that the ancient measures implicitly embody a network of musical intervals. The use for such a network is only weakly explained until the use of measures to create acoustic resonances is considered.

The speed of sound in air means that there is an inverse relationship between frequency and the wavelength of that frequency. For this reason organ pipes are of different lengths according to frequency and stone age flutes have been found in which the location of note-holes would have corresponded with lengths along the pipe made of, for example, bone. The size of a resonant cavity therefore determines the resonant frequency and it is ancient measures that would have been used to construct acoustic spaces with a resonant character.

If a chamber build in English feet resonates in B, then rebuilding it using the same number of Assyrian feet would mean it resonated at C since Assyrian feet are 9/10 feet long and the chamber would be smaller with a higher resonance.

There is an interesting type of structure that was built identically using different types of foot – The Scottish Brochs.