by
Timaeus, Plato's
treatise on Pythagorean cosmology, the central character, Timaeus of1 Locri (possibly a real person), describes how the Demiurge divided the World Soul into harmonic intervals. Having blended the three ingredients of the World Soul — Sameness, Difference and Existence — into a kind of malleable stuff, the Demiurge took a strip of it and divided its length into portions measured by the numbers forming two geometrical series of four terms each: 1, 2, 4, 8 and 1, 3, 9, 27, generated by multiplying 1 by 2 and 3 (Fig. 1). This became known as Plato’s Lambda because of its resemblance to Λ, the Greek letter lambda. Then, according to Timaeus: “he went on to fill up both the double and the triple intervals, cutting off yet more parts from the original mixture and placing them between the terms, so that within each interval there were two means, the one (harmonic) exceeding the one extreme and being exceeded by the other by the same fraction of the extremes, the other (arithmetic) exceeding the one extreme by the same number whereby it was exceeded by the other.* These links gave rise to intervals of 3/2 and 4/3 and 9/8 within the original intervals. And he went on to fill up all the intervals of 4/3 (i.e. fourths) with the interval of 9/8 (the tone), leaving over in each a fraction. This remaining interval of the fraction had its terms in the numerical proportion of 256 to 243 (semitone). By this time, the mixture from which he was cutting off these portions was all used up” (1). The Table of Lambda, handed down to us by Nichomachus of Geras, makes this obscure statement clearer: The horizontal rows of integers in Figure 2 are geometric progressions by 2 and the rows descending diagonally are geometric progressions by 3. Consecutive vertical numbers are to each other in the ratio 3/2, defining the perfect fifth, whilst consecutive, diagonally paired numbers (shown by arrows) are in the ratio 4/3, defining the musical fourth. The *H is the harmonic mean of two positive numbers a and b (b›a) if (b-H)/(H-a) = b/a. A is their arithmetic mean if (b-A)/(A-a) = 1. 2 tetractys of integers at the beginning of the Table of Lambda:
(it will be called the "Lambda Tetractys") serves as the basis of musical harmony. Using the octave of 6:12, their arithmetic mean is 9, which, in relation to 6, is in the ratio 3:2. This is the perfect fifth. The harmonic mean of 6 and 12 is 8. This proportion, 8:6 or 4:3 is the perfect fourth. But 12:9 is also 4:3, that is, a perfect fourth, whilst 12:8 is also 3:2, which is a perfect fifth. The ratio 9:8 defines a whole tone. The central integer 6, which is absent from the series of integers: 1, 2, 3, 4, 8, 9, 27 in the
Lambda, is pivotal to discussion of the mathematical proportions of the notes belonging to
the musical octave for the following reason. If the integer n is chosen as the fundamental
frequency, then the first overtone, or octave, has a frequency of 2n. The harmonic mean of
these integers is (4/3)n, which is the perfect fourth. The arithmetic mean of n and 2n is
3n/2; this is the perfect fifth. The smallest value of n defining integer values of
Because
of the pivotal importance of the number 6 at the centre of the above tetractys, one can
speculate that Plato concealed his knowledge of and indebtedness to Pythagorean teachings by
revealing in his book
3 Figure 3. The 90 circles symbolise the 90 Trees of Life above the lowest one (denoted by the tetractys) and the sum (90) of the ten integers in Plato’s Lambda tetractys. 4 This unfolding is not in linear time and so the stages of the manifestation represented by the ten points of the tetractys must be considered as simultaneous. The ten Sephiroth of the Tree of Life, which these points symbolise, denote co-existing modes of being, not processes occurring sequentially in time, which characterises only the last Sephirah, Malkuth, the physical universe. All objective Creation manifests as the Divine Qualities of the seven Sephiroth of Construction. They are denoted by the seven points within the three corners of the tetractys symbolising the Supernal Triad, or triple Godhead. Imagine a series of nine
circles converging towards each point of a tetractys (Fig. 3). Each circle symbolises a
Sephirothic hue in the ‘spectrum of consciousness.’ The outermost, three white circles
formally correspond to Kether, Chokmah and Binah and the six coloured circles represent the
Sephiroth of Construction above Malkuth, which is denoted by the red dot at the centre of
each set of concentric circles. The The sum of the ten numbers of Plato’s complete Lambda Tetractys:
is 90, which numbers the
gradations of consciousness beyond the S(n) = 6n + 4, so that S(n+1) – S(n) = 6(n+1) + 4 – (6n + 4) = 6, i.e., there are six such levels between similar Sephiroth of adjacent Trees, every seventh Sephirah being of the same kind. The emanation of the 90 Trees represented by the 90 circles therefore generates (90×6=540) Sephirothic levels. The last of these levels is the
5 commencement
(Kether) of the
This
is the real reason why the
is the number of
Sephirothic levels in 49 overlapping Trees of Life down to the top of their lowest one. The
sum 504 of the nine integers of this tetractys surrounding the central integer 36 is the
number of Sephirothic levels in CTOL down to the top of the seventh tree, whilst the sum 540
of all ten integers is the number of levels in CTOL down to the top of the lowest tree.
Amazingly, the three numbers 36, 288 and 540 measure the number of emanations of,
respectively, seven, 49 and 91 Trees of Life — 6 of trees above the lowest tree in, respectively, seven, 49 and 90 trees. We see now why Plato spoke of its numbers being the ‘numbers of the World Soul.’ They signify more than the meaning of the mathematical harmony of the tones of the musical scale, which the World Soul was supposed to embody. Numbers of the Lambda tetractys refer to realms of consciousness beyond the physical universe mapped by the 7-tree, or physical plane, this being to the 49-tree, or ‘cosmic physical plane,’ what the latter is to CTOL itself.
common point in 9-d space. This means that a point on each closed curve is located
by different sets of nine numbers — its spatial coordinates in superstring space-time. All
ten curves constituting the superstring represent the motion of ten points defined by
(10×9=90) numbers, as well as by the time coordinate, whose value is common to them all. As
the microscopic ‘World Soul,’ the indwelling Divine Life in the subatomic world, the
superstring embodies 90 numbers — the sum of the numbers in Plato’s Lambda when considered
as a tetractys. Its 91 hidden, space-time variables (hidden only in the sense that its
currently conceived form as a simple closed string is really a bundle of ten such
loops) correspond to the 91 Tree of Life in CTOL, with the coordinate of time being
analogous to the lowest tree.
7 others generates two new Lambda Tetractyses:
The three tetractys arrays can be regarded as forming three faces of a tetrahedron with the number 1 at a corner, the face opposite it being formed by their bases:
As successive integers on the left slope of the new Lambda are twice the previous one, the missing central integer not shared with the other Lambdas is 2×12 = 24 =1×2×3×4. The generalisation of the Platonic Lambda creates a tetrahedral array of 20 integers (4): Figure 5. The tetrahedral generalisation of the Platonic Lambda. Their sum is 350. This number is the sum of the four integers 1 =
1 1 and the 16 (=4 8 integers on its boundary is therefore (100+200=300). Notice that the proportions of these sums for integers on the edges of the tetrahedron are 100:200:300 = 1:2:3. The four central integers 24, 12, 8 and 6 define, respectively, in terms of the fundamental tone 6, the frequencies 4/1 = 4, 4/2 = 2, 4/3 = 4/3 and 4/4 = 1. They are thus determined by the integers 1, 2, 3 and 4, which are symbolised by the four rows of the tetractys. The 20 integers are arranged in four layers containing 1, 3, 6 & 10 integers. The sum of the ten integers in the highest three layers of the tetrahedron shown in Figure 5 is 65, which is the number value of Adonai, the Godname of Malkuth. The three of these at the centres of tetractyses sum to 26, which is the number value of Yahweh, the Godname of Chokmah, whilst the sum of all four integers at centres of tetractyses is 50, the number value of the Godname Elohim assigned to Binah, the next Sephirah after Chokmah.
350 = 250 + 100 between the sum of the integers at the apices of the tetrahedral array of integers and the remaining 16 integers has a remarkable counterpart in the inner form of the Tree of Life (Fig. 6). Its seven regular polygons have 36 corners, ten of which belong to the last
polygon, the dodecagon. This means that the 7n regular polygons enfolded in n overlapping
Trees of Life have (35n+1) corners, where ‘1’ denotes the uppermost corner of the hexagon
enfolded in the nth tree. The latter coincides with the lowest corner of the hexagon
enfolded in the (n+1)th tree. The 7n polygons therefore have 35n corners that are
9 10 are 70 polygons (Fig. 7) with (35×10=350) such
corners, of which (10×10=100) belong to the dodecagon, the tenth of the regular polygons. We
see that generalisation of the Platonic Lambda Tetractys leads to a tetrahedral array of 20
integers that sum to the number of unshared corners of the 70 polygons enfolded in
In view of the fact that the separate 1 + 2 + 3 + 4 + … + 26 = 351, as the 26th triangular number. It is the total number of corners of the 70 polygons enfolded in ten, overlapping Trees of Life. 350 is, therefore, the sum of the 25 integers after 1. These integers have the following remarkable arithmetic connection with the 11 number 248, the number of gauge bosons predicted by
superstring theory to transmit the unified, E In other words, 248 is the arithmetic mean of the squares
of the 25 integers after 1. For any positive integer n, n 2 + 3 + 4 + … + 26 = 350, which is the sum of the 20 integers in the tetrahedral array of generalised Lambda Tetractyses. 350 is prescribed also by the Godname Elohim in the following way. As 35 = 6 i.e., 35 is the sum of the first five odd integers after 1, 350 (=10×35) is the sum of (10×5=50) odd integers, where 50 is the number value of Elohim, the Godname of Binah. The number 350 has a unique connection to the ‘trunk’ of the Tree of Life, comprising the point, line, triangle and tetrahedron (Fig. 9): Its 26 geometrical elements are prescribed by Yahweh, whose number value is 26. When the five triangles of the trunk are turned into tetractyses, with its ten lines each made up of four yods, it comprises 35 yods. They are 12 13 made up of the ten Sephirothic points and 25 other yods. If the number 10 is assigned to each of its 35 yods, the trunk embodies the number 350, the Sephirothic points generating the number 100 (Fig. 10). This division: 350 = 250 + 100 was found earlier for the tetrahedral array of integers.
The four integers 1 350 = 50 + 100 + 200, found earlier for the four integers at centres of tetractyses summing to 50, the four integers at the apices that sum to 100 and the 12 other integers on the edges of the tetrahedron summing to 200, has its correspondence in the trunk of the Tree of Life:
Notice that corresponding yods in the tetrahedron and
trunk are of the 14 tetrahedron whose 16 integers sum to 300 are the
counterpart of the 30 yods with 10 assigned to them that delineate the geometrical
As the fourth triangular number, the tetractys is the fourth stage of differentiation of the Pythagorean Monad: A tetrahedron whose faces are tetractyses symbolises this differentiation in 3-dimensional space because it consists of four layers, each being a triangular number corresponding to one of these stages. The sums of the integers in successive layers of the Lambda tetrahedron shown in Figure 10 are listed above. ‘1’ denotes the mathematical point, the Monad ‘10’ (the sum of the four integers in the highest two layers) signifies its 10-fold differentiation into the 10 yods of the tetractys or the 10 Sephiroth of the Tree of Life. ‘65’
(the sum of the 10 integers in the highest three layers) signifies the 10-fold differentiation of the Tree of Life into the 10-tree, which has 65 SLs. The sum (350) of all 20 integers in the tetrahedron is the number of corners intrinsic to the 70 polygons enfolded in the 10-tree and representing its inner form. The tetrahedral generalisation of the Platonic Lambda therefore signifies in geometric terms the four stages of differentiation of Divine Unity (Monad) into the perfect number 10 (tetractys) expressing the 10-fold nature of God, whilst it expresses arithmetically the four stages: Kether* → Tree of Life → 10-tree (outer form) → 10-tree (inner form) culminating in the inner form of the 10 lowest Trees of Life in CTOL. As was shown in *One of the Kabbalistic titles of Kether is “The Primordial Point.” 15 earlier articles, these encode the group-theoretical forces and the oscillatory structure of superstrings. The sum of the 10 integers in Plato’s Lambda Tetractys is 90 and the sum of the 10 integers in the tetrahedron outside this face is (350–90=260). This is the number of yods in the seven enfolded polygons outside their shared root edge. Just as these polygons 'grow' out of the root edge, so three tetractys faces are generated in succession from the Lambda Tetractys, which is the counterpart of the root edge. Once again, the mathematical archetype embodied in the Godname Yahweh prescribes an archetypal form defining the
mathematics of both the Pythagorean musical scale and superstrings.
2/1 is the frequency of the octave, 3/2 is the frequency of the perfect fifth (arithmetic mean of 2 and 1)) and 4/3 is the frequency of the perfect fourth (harmonic mean of 1 and 2). The complete tetractys version of Plato’s Lambda: 16 generates these ratios because adjacent pairs in the same
row are always in the same ratio 3/2 and corresponding pairs in adjacent rows linked by the
diagonal arrows are in the same ratio 4/3, whilst integers linked by diagonal lines are in the
ratio 2/1. Adjacent members of the same row in the two new Lambda Tetractyses are in the ratios
2/1 and 4/3. This means that the base of the tetrahedron formed by the three Lambda Tetractyses
exhibits Only in this fourth tetractys, which is opposite the apex
of the tetrahedron with 1 Quite apart from their summing to the number of corners
of the 70 polygons enfolded in ten overlapping Trees of Life, with those at the vertices
summing to the number of corners of the ten dodecagons outside the edges they share with other
polygons, the tetrahedral array of integers has the additional merit of generating a Lambda
Tetractys of integers whose natural, nearest neighbour pairings exhibit 17 is appropriate that the number at the apex of this
tetractys determining the tonal ratios of the eight notes of the musical scale should be 8! At
the centre of the fourth tetractys is the note with a musical proportion of 24 (notice that 24
= 1×2×3×4). It has a frequency of 4 relative to the fundamental tone with a musical proportion
of 6 and frequency of 1. It is the
had the power of generating and expressing
the mathematical nature of the physical and spiritual cosmos, that is, of revealing
through sacred geometry the numbers that God has chosen to design His Creation. It is no
accident that connections should exist between the numerical properties of the Lambda
Tetractys and the spiritual map belonging to another mystical tradition, namely Kabbalah,
as this article reveals. Mystical truth is universal and eternal, and the Pythagoreans
realised that one could learn to contemplate it actively through the mathematical vehicle
of the tetractys, which went far beyond the superficial symbolism into which academic
scholarship, lacking mystical insight, downgraded its sublime status. For the
neo-Pythagorean Plato, who was more interested in abstract principles (pure dialectic)
than the world to which they applied, mathematics*The nth Mersenne number is: M 18 was merely a means to an end — a way of purifying the
soul in preparation for contemplating the Absolute on its own transcendental level. For the
Pythagoreans, however, study of their sacred mathematics led the soul into an awareness of God
For many people, the fact that the Lambda Tetractys
generates the tone ratios of music will be merely a mathematical curiosity without even
fundamental significance, let alone, spiritual meaning. This is because their denial of the
reality of a spiritual, or archetypal, basis for logical thought (what the ancient Greeks
called ‘dianoia’) and their rejection of what they regard as a magical way of thinking in terms
of correspondence and analogy makes them unable to recognise as such universal patterns that
link the particulars. For anyone who is a Pythagorean, the connection between music, the Lambda
Tetractys and its tetrahedral generalisation has a more profound significance. It demonstrates
the
3. ‘Levels of consciousness’ are not synonymous with ‘states of consciousness.’ There are a finite, discrete number (91) of the former but an unlimited number of the latter as variations of the continuum of consciousness defined within each of these primary modalities, of which there are seven. 19
4. 5. Particle physics provides the
following counterpart to the tetrahedral generalisation of the Lambda: when the
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