The Tetraktys as plan of planetary harmony and the four Elements

Figure 1 The elimination of 5 as a factor in the harmonic mountain for 36 lunar years, resolved using matrix units of one tenth of a month and the limit 360 units.

In a previous post 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.

Relations to the Tetrakrys

One can also see Plato’s World Soul in the Pythagorean Tetraktys (see Keith Critchlow’s Foreword and Robin Waterfield’s Introduction to The Theology of Arithmetic,  Phanes, 1988), as a development of the Lambda diagram as per figure 2.

Figure 2 Filling in the Tetraktys using its invariant directional vectors reveals a bottom register of numbers separated by fifths, like the register of figure one.

Keith Crichlow demonstrates that the Lambda diagram is none other than the Tetraktys, the latter probably a schematic missing in Plato’s Timaeus  35b&c, cryptically referred to by Plato since the uninitiated lacked it. The Tetaktys gives a beautiful organisation of the Pythagorean numbers involving only 2 and 3 as factors perfectly represented by an equilateral triangle growing organically out of ONE.

The Tetraktys of figure 2 differs from the mountain of figure 1 in that

  1. the vertical sense is not due to increasing multiplication by five, but rather decreasing multiplication by 2 or 3, and
  2. the elements were not normalised to a single octave.

The role of 36 within the Tetraktys of figure 2 corresponds to the number of lunar months chosen for the limit in figure 1, and so one can choose to normalise it, through doubling numbers where required, into the octave range 18:36.

Figure 3 Normalising the Tetraktys to the limit of 36

On the bottom register, 27 already fits the octave whilst 12 and 18 must be doubled to 24 and 36, and 8 quadrupled to 32. At that point all that is above shares the numerocity of what lies in the bottom register, being dependent only on their powers of three.

The numbers 32:24:36:27 are identical to the white register of figure 1 and this may well indicate that the secret knowledge of the Pythagoreans, said to be summarised in the Tetraktys in particular, could well have been the astronomical harmonic relations to be found between the double and triple lunar year periods, the double synod of Jupiter and the 32 lunar month period which is also both 945 solar days and 2.5 Saturn synods long. This would have been a very simple means for the Demiurge to have employed, close to the harmonic origin story from one, two and then three.

Number Values of the Four Elements

If the right hand three elements are 12:18:27, and this is reminicent of the numbers generally used to designate three of the four Elements (water, air and fire) by the medieval period, and the four Elements were often shown then(after Plato), as is the case here, as being separated one from the other by fifths – exactly as Plato describes it in Timaeus (ref). However, most schemata differ from the allocation found in the decorated crypt of the Pope’s summer palace, the Anagni Cathedral (figure 4, right).

Figure 4 In the Crypt of Anagni Cathedral is a schematic in which customary allocation of numbers to elements is reversed viz 27 and 8. Yet the form of it then corresponds perfectly to the Tetraktys when the three means of the Lambda diagram are filled-in to make it.

In the planetary matrix of synodic periods, Saturn is distinguished as the cornerstone having no other factors than the powers of two with respect to the lunar year. Hence it is harmonically commensurate (5/2) with the 32 lunar month period which is 8 merely doubled twice to fit the limit of 36. So it would seem that Saturn represents the Earth element. It is quite obvious that the lunar year of 12 lunar months is Water and that half the triple lunar year, of 18 months, is the element Air. The element of Fire is then the double synod of Jupiter (27).

8: Since 32 lunar months is 945 days, then the earth which continually rotates towards the east is the cycle of barrenness, according to Plato, is like the number 2 of octave doubling which provides a container for intervals as a womb enables children to arise. Saturn is also considered “plumbous” or “weighty” and so is naturally represents the solid state of objects and, traditionally, the giver of boundaries being the outermost visible planet.

12: The Moon’s links to water come directly through the tides as first of many traditions such as that it fills with water until “full” and thereafter starts to “empty”.

18: The lunar year also collaborates with the near anniversary of three solar years equalling 37.1 lunar months. This periodicity was studied by the megalithic astronomers of Carnac as a right-angled triangle whose counted base was 36 lunar months symbolised by 36 stones in the kerb marking the base ay Le Manio’s Quadrilateral.

27: After two loops of Jupiter, the middle of the retrograde loop will be punctuated by a full moon (at maximum retrograde) since the Sun is then in line with the Earth and Jupiter. This fact enables the loops of outer planets to be counted in lunar months, as required. The association of Jupiter with fire seems natural since his primary weapon is lightning, the cause of natural fires.  


  1. The numerical Tetraktys,
  2. the account in Timaeus of a harmonic Creation and,
  3. the harmonic realities of Saturn and Jupiter relative to the lunar year,

are so closely related as to suggest the first two were a plan and description of the latter. The numbers referred to are the natural ones, in lunar months and this affects the history of ideas in sourcing Plato’s cosmology (influential in the Arabic Golden Age (800-1100) and subsequent western Quadrivium of numerical arts) on concrete astronomical facts ascertainable from simple observation and counting of time periods, inherited by the Pythagoreans.

The Pythagoreans considered themselves privy to a secret doctrine, and central to this was the school of “theoretical” harmonists in Greco-Roman times, recognizable today under the rubric “Harmony of the Spheres”. Whilst many cunning versions of such a thing have been proposed, the core of the matter rests in the simplicity of the numbers in figures 1 and 2, of the synodic periods of the outer planets in months relative to the lunar year. To think otherwise requires an explanation as to why this is not obviously a simpler and more probable fact. It is easiest to extrapolate megalithic astronomy as having observed and counted the synodic periods of the outer planets and, on comparing these with the lunar year length, discovering the planetary harmonic ratios, hence leaving a legacy of harmonic numbers referring to the heavenly world ancient literature and crafts.

The above is a further development of the theme of my recent book
The Harmonic Origins of the World and is now published in chapter seven of Sacred Geometry: Language of the Angels about English sacred pavements.