Table 1. Number values of the ten Sephiroth in the four Worlds.
ORDER OF ANGELS
(Angel of the Presence)
|Chaioth ha Qadesh
(Holy Living Creatures)
|Rashith ha Gilgalim
(Herald of the Deity)
(The Sphere of the Zodiac)
(God in multiplicity)
(Contemplation of God)
(Benevolence of God)
(Severity of God)
(God the Creator)
(Like unto God)
The Solar Light.
(Lord of Hosts)
(Grace of God)
|Tarshishim or Elohim
(God of Hosts)
(Sons of God)
The Stellar Light.
|SHADDAI EL CHAI
(Almighty Living God)
(Strong Man of God)
The Lunar Flame.
(The Lord and King)
(Souls of Fire)
The Breaker of the Foundations.
The Sephiroth exist in the four Worlds of Atziluth, Beriah, Yetzirah and Assiyah. Corresponding to them are the Godnames, Archangels, Order of Angels and Mundane Chakras (their physical manifestation). This table gives their number values obtained by the ancient practice of gematria, wherein a number is assigned to each letter of the alphabet, thereby giving a number value to a word that is the sum of the numbers of its letters.
(All numbers in this table referred to in the article are written in boldface).
1. Fibonacci numbers in the Pythagorean musical scale
Table 2 displays the first 55 notes in the Pythagorean musical scale above the tonic of the first octave (red numbers are Fibonacci numbers):
Table 2. The first 55 notes above the tonic of the Pythagorean scale.
The first seven octaves of the Pythagorean scale ending in the note C8 with tone ratio 27 = 128 comprise 50 notes, of which 21 notes are overtones and 29 are partials. ELOHIM, the Godname of Binah with number value 50, and EHEYH, the Godname of Kether with number value 21, prescribe the first seven octaves. They comprise the tonic C1, the last note C8 and the 48 notes between them. The latter consist of the six octaves C2–C7 and 42 notes made of 14 overtones and 28 partials. The first 55 notes above the tonic ending with note B8 comprise 26 overtones and 29 partials. YAHWEH (Hebrew: YHVH), the Godname of Chokmah with number value 26, prescribes this set of 55 notes. It consists of the seven octaves C2–C8 and 48 notes made up of 19 overtones and 29 partials.
That this is not just a coincidence is indicated by the fact that the letter values of YHVH denote the numbers of overtones in groups of notes whose numbers are Fibonacci numbers. H = 5 (he) denotes the 5 overtones in the first 21 notes, V = 6 (vav) denotes the six overtones in the next 13 of the first 34 notes, H = 5 denotes the 5 overtones in the next 8 notes and Y = 10 (yod) denotes the ten overtones in the last 13 notes. The letter values of YAHWEH divide up the 55 notes above the tonic into sets whose numbers are always members of the Fibonacci series! YAH (Hebrew: YH), the partial version of the full Godname with number value 15 specifies the 15 overtones in the last 21 notes, the two letters Y and H denoting the subsets with, respectively, 13 notes and 8 notes. This division in the Godname occurs at C6, the 5th octave with tone ratio 32. The first 5 overtones span 3 octaves, as do the next 11 overtones and the last 15 overtones. The 55 notes from D1 with tone ratio 9/8 to B8 with tone ratio 243 span an interval of 216. This is the number value of Geburah. The interval between B8 and the next note C9 is 256/243, which is the Pythagorean leimma. The ratio of the tone ratios of the 55th and 21st notes is 243/8, which is that of the 34th note! The 21:34 division in the Fibonacci number 55 creates a 3:5 division of the 8 octaves spanned by the 55 notes. The numbers 3, 5 & 8 are themselves consecutive Fibonacci numbers! Counting from the 55th note, the 21:34 division generates the same 3:5 division. The first 32 notes have ten overtones. As 32 is
not a Fibonacci number, we see that only counting backwards from the 55th note generates the gematria numbers of the letters of YAHWEH in the order in which they are written in Hebrew:
This is remarkable because it demonstrates that the pattern of overtones is prescribed by this Godname not in the left-to-right way that it is written in English but in the right-to-left way that it is written in its natural, Hebrew language!
2. Fibonacci numbers in the Tree of Life & 1-tree
The geometrical compositions of the Tree of Life and 1-tree are shown below:
Points Lines Triangles Total
Tree of Life: 10 22 16 48
1-tree: 11 25 19 55
The 48 geometrical elements comprising the Tree of Life correspond to the 48 corners of the seven separate polygons and to the 48 notes up to B8 other than octaves. The extra seven geometrical elements (one point, 3 lines & 3 triangles) needed to convert the Tree of Life into the 1-tree correspond in the inner Tree of Life to the centres of its seven regular polygons and in the Pythagorean scale to the seven octaves up to the note B8.
Outer Tree of Life
As discussed in more detail in Article 50,2 the Lower Face of the Tree of Life is the red kite-shape in Figure 1 whose corners are Tiphareth, Netzach, Hod, Yesod and Malkuth. It comprises 21 geometrical elements, where 21 is the eighth Fibonacci number. This is the number of elements in the similarly-shaped Upper Face of the 1-tree whose corners are Kether, Chokmah, Binah, Chesed, Geburah and Tiphareth. As the two Faces share a corner, 20 of the elements in the Upper Face are part of the 34
Figure 1. The 55 geometrical elements of the 1-tree comprise the 21 elements of its Lower Face (red) and the 34 elements in its remainder (black).
geometrical elements that make up the 1-tree other than its Lower Face, leaving (34–20=14) elements outside both Faces. The 21:34 division of the 55 elements in the 1-tree has its counterpart in the tetractys array of the first ten integers adding to 55 (notice that 55 is also the tenth Fibonacci number):
The sum of the integers at the corners and centre of the array is 21, and the sum of the integers at the corners of the hexagon is 34. 21 is the sum of the first six integers 1–6 and 34 is the sum of the last four.
Inner Tree of Life
The 48 sectors of the seven separate regular polygons have 55 corners (Fig. 2). They comprise the seven centres of the polygons and their 48 corners. Their musical counterparts in the first seven octaves are the seven octaves C2–C8 and the 48 notes comprising 19 overtones and 29 partials. The extra seven
Figure 2. The tenth Fibonacci number 55 is the number of corners of the 48 sectors of the seven regular polygons making up the inner form of the Tree of Life.
elements needed to convert the Tree of Life with 48 elements into the 1-tree are their counterpart.
If we add together corners of sectors of polygons, there are only two possible combinations of polygons with 21 corners of their sectors:
4 + 6 + 11 = 21, leaving 5 + 7 + 9 + 13 = 34
5 + 7 + 9 = 21, leaving 4 + 6 + 11 + 13 = 34.
But neither case allows the Fibonacci number 21 to split into the Fibonacci numbers 8 and 13. Hence, combinations of corners of sectors do not form Fibonacci numbers that are the sum of the two previous Fibonacci numbers. Either the centres of a subset of the seven polygons are counted with the corners of other polygons (this case was analysed in Article 503 or all seven centres are added to the corners of a subset of the seven polygons. In view of the correspondences between the 55 corners of the sectors of these polygons and the 55 geometrical elements of the 1-tree and between the seven centres and the seven elements added to the Upper Face when the Tree of Life with 48 elements becomes the 1-tree, it follows that the seven centres should belong to the 34 corners of sectors, for this number is the number of elements in the Upper Face containing the seven extra elements. The possible combinations are:
3 + 6 + 12 = 21 and (4 + 5 + 8 + 10) + 7 = 34,
4 + 5 + 12 = 21 and (3 + 6 + 8 + 10) + 7 = 34,
5 + 6 + 10 = 21 and (3 + 4 + 8 + 12) + 7 = 34,
3 + 8 + 10 = 21 and (4 + 5 + 6 + 12) + 7 = 34,
3 + 4 + 6 + 8 = 21 and (5 + 10 + 12) + 7 = 34.
(’7’ denotes the seven centres). Only the last two combinations shown above allow the split 21 = 8 + 13. This leaves the two possible compositions for the remaining 34 corners of sectors:
(5 + 10 + 12) + 7 = 34,
(4 + 5 + 6 + 12) + 7 = 34.
Only the last one is consistent with the division 34 = 21 + 13 (13 = 7 + 6 and 21 = 4 + 5 + 12). Hence, this division is between the 21 corners of the triangle, octagon & decagon and the 34 corners of sectors that are either centres of the seven polygons or corners of the square, pentagon, hexagon & dodecagon.
Figure 3. The corners and centres of the five Platonic solids form consecutive Fibonacci numbers.
3. Fibonacci numbers in the Platonic solids
As discussed in Article 50,4 the five Platonic solids have 55 vertices & centres (Fig. 3). They comprise the 21 vertices & centres of the tetrahedron, octahedron & cube and the 34 vertices & centres of the icosahedron & dodecahedron. Notice that this 21:34 division splits the complete set of 5 regular polyhedra into two subsets: one of 2 solids and one of 3 solids, and that the numbers 2, 3 & 5 are consecutive Fibonacci numbers. The 55 vertices & centres correspond to the 55 corners, edges & triangles making up the 1-tree. The 21 vertices & centres of the first 3 Platonic solids correspond to the traditional Upper Face of the Tree of Life rather than to its Lower Face because, intuitively speaking, it is natural to associate the simplest solids with the upper part of the Tree of Life, whose downward progression represents the development of more complex forms from simpler ones. Indeed, its trunk (Fig. 4) consists of the sequence of the first four regular simplexes*: point→line segment→triangle→tetrahedron. They are composed of 26 geometrical elements.5 YAHWEH (YHVH or יהוה) prescribes the trunk, the value 10 of the first letter yod (Y or י) denoting the 10 vertices, the value 5 of the first letter he (H or ה) denoting the 5 edges & triangles in the line segment and triangle, the value 6 of vav (V or ו) denoting the 6 edges of the tetrahedron and the value 5 of the second he (H or ה) denoting the 5 triangles & tetrahedra in the tetrahedron.
Figure 4. Geometrical composition of the trunk of the Tree of Life.
The 1-tree contains two tetrahedra (see Fig. 1) and 55 vertices, lines & triangles. These 57 geometrical elements correspond to the 57 points that are either corners of sectors of the seven separate regular polygons of the inner Tree of Life or the endpoints of the root edge that becomes their shared edge when they become enfolded in one another. Their musical counterparts are the 57 notes in the first eight octaves. The 1-tree consists of the 26 elements of its trunk prescribed by YAHWEH, the Godname of Chokmah, and the 31 elements of its ‘branches’ prescribed by EL, which is the Godname of Chesed, the Sephirah below Chokmah on the Pillar of Mercy of the Tree of Life. One of these elements is the eleventh vertex located at Daath and another is the tetrahedron in the Upper Face. The two elements correspond to the tonic and to the eighth octave — the 57th note. This leaves 29 geometrical elements (15 edges, 14 triangles) in its branches that correspond to the 29 partials in the first 55 notes of the Pythagorean scale above the tonic. The 26 elements in the trunk correspond to the 26 notes that are overtones. This means that the 55 vertices, lines & triangles making up the 1-tree do not correspond to these 55 notes, for they do not include the tetrahedron that corresponds to one of the 26 overtones. The two endpoints of the root edge correspond to the vertex located at Daath and to the tetrahedron in the Upper Face. The 55 corners of sectors correspond to the remaining 55 vertices, lines, triangles & tetrahedra in the 1-tree. The 48 corners of sectors correspond to the 48 vertices, edges & triangles of the Tree of Life, whilst the seven centres correspond to the 3 edges, 3 triangles & one tetrahedron added when the Tree of Life becomes the 1-tree.
Figure 3 indicates that the numbers of points, lines & triangles in the sets of 21 and 34 are:
21 = 5t + 16b,
34 = 20t + 14b,
where the suffixes ‘t’ and ‘b’ refer to the trunk and to the branches of the 1-tree. The tetrahedron has 5 vertices & centres and the octahedron & cube has 16 vertices & centres. The icosahedron has 13 vertices & centres and the dodecahedron has 21 vertices & centres. Hence, the regular polyhedral counterparts of the 5 triangles and 20 vertices & edges of the trunk of the Tree of Life is the 5 vertices & centre of the tetrahedron and the 20 vertices of the dodecahedron. Their musical counterparts are the 25 overtones above the first octave in the first 55 notes. The polyhedral counterparts of the branches of the 1-tree composed of 16 vertices & edges and 14 triangles are the 16 points that are either vertices of the icosahedron or centres of the octahedron, cube, icosahedron & dodecahedron and the 14 vertices of the octahedron and cube.
* A regular n-simplex is the n-dimensional analogue of an equilateral triangle. The 0-simplex is the point, the 1-simplex is the line segment (or straight line), the 2-simplex is the triangle and the 3-simplex is the tetrahedron.
Figure 5. The lowest 26 Trees of Life map 26-dimensional space-time. 21 SLs lie on the central pillar of the lowest 9 Trees mapping the 9 spatial dimensions of
superstrings. 34 SLs lie on the central pillar of the next higher 16 Trees mapping the 16 higher spatial dimensions. The 26th Tree maps the dimension of time.
4. YAHWEH prescribes the seven musical scales
It is not by accident that the first 55 notes above the tonic of the Pythagorean scale should include 26 overtones. We have seen in previous articles6,7 that YAHWEH prescribes the seven types of musical scales because they constitute another holistic system. Table 3 lists the tone ratios of their notes:
Table 3. The tone ratios of the notes in the seven musical scales.
(White cells denote Pythagorean notes between the tonic and octave in grey cells).
There are 26 Pythagorean notes between the tonic and octave:
YAHWEH (YHVH) prescribes the seven scales, its four letter values being the numbers of Pythagorean notes in four sets of scales. Alternatively, the letter values are the numbers of four sets of notes:
This property of a single octave of the Pythagorean scale is the counterpart of the 26 overtones in the first 55 notes after the tonic of eight octaves of this scale — another holistic set.
As they are both parameters of a holistic system, the numbers 26 and 55 are always found associated with each other in sacred geometrical representations of such systems. This, too, is the case, arithmetically speaking, because the sum of the first ten integers arranged in a tetractys is 55, whilst 26 is the number of the combinations of the integers in each row:
Number of combinations
2 3 3
55 = 4 5 6 7
7 8 9 10 15
Total = 26
Another example of this association is the fact that there are 55 SLs on the central Pillar of Equilibrium of the lowest 26 Trees of Life in CTOL (Fig. 5). The Malkuths of these Trees of Life can be regarded as corresponding to the 26 overtones and the remaining 29 SLs on this pillar can be thought of as corresponding to the 29 partials. The correspondence demonstrates that the notes of the Pythagorean scale conform to an underlying Tree of Life pattern. 21 SLs span the central pillar of the lowest nine Trees of Life. As a map of the 26-dimensional space-time predicted by quantum mechanics for spinless strings, each Tree of Life in this context denotes a dimension. The 9-tree represents the nine spatial dimensions predicted by superstring theory. Hence, the 21:34 division of 55 found earlier for the first 3 of the 8 octaves spanned by 55 notes has its counterpart in string theory as the distinction between the nine superstring and 16 bosonic dimensions of space. Fibonacci numbers define the demarcation between the two types of dimensions! The Fibonacci number 5 determines the 1-tree with 5 SLs on its central pillar and the number 13 determines the 5-tree with 13 SLs on its central pillar. The counterpart of the 21:34 division of SLs on the central pillar of the 26-tree is the transition from the Upper Face of the 1-tree with 34 geometrical elements to its Lower Face with 21 elements. Its counterpart in the 5 Platonic solids is the transition from the first four solids with 34 vertices & centres to the final solid — the dodecahedron — with 21 vertices & centre (see Fig. 3). In Article 50 (Part 1), it was pointed out that the faces of the five
Platonic solids have 550 (=55×10) corners, edges and sectors. The faces of the first four Platonic solids have 210 (=21×10) polyhedral corners, edges & sectors, leaving 340(=34×10) corners, edges & sectors.8 Once again, the 21:34 division differentiates the dodecahedron from the first four regular polyhedra. The ancient Greeks believed that the latter were the shapes of the particles of the Elements of Earth, Water, Air and Fire making up the physical universe. Through the shared Fibonacci number 21, they are associated with superstring space-time with nine spatial dimensions, whilst the fifth Platonic solid representing the fifth Element Aether is associated through the Fibonacci number 34 with the purely bosonic string dimensions of space-time.
The tenth Fibonacci number 55 determines the nine superstring dimensions in another way. The 55th SL from the bottom of CTOL is Chesed of the ninth Tree of Life mapping the ninth of the spatial dimensions of superstrings. This SL is the 496th SL from its top. The tetractys representation of the 550 SLs of CTOL:
550 = 55 55 55
55 55 55 55
implies that the central number 55, which occupies the position of the point symbolizing Malkuth in the tetractys counterpart of the Tree of Life, characterizes the physical universe in a fundamental way. Remarkably, the gematria number value of Malkuth is 496 (see Table 1). Even more remarkably, this is the number of spin-1 gauge bosons that mediate the unified interactions between superstrings with nine spatial dimensions mapped by the nine lowest Trees in CTOL. We saw earlier that the 21:34 division of SLs on the central pillar of CTOL demarcates the nine superstring dimensions from the 16 bosonic string dimensions. The 21:34 division of all SLs up to the first Sephirah of Construction of the ninth Tree of Life distinguishes the 3 large-scale dimensions of space from the 6 compactified dimensions of superstrings because, counting from the bottom of CTOL, there are 21 SLs up to Binah of the third Tree of Life mapping the third spatial dimension and 34 SLs in the next 6 Trees of Life above it up to Chesed of the ninth Tree of Life. Below are listed analogous examples of the fundamental bifurcation of holistic systems created by these Fibonacci numbers:
1st 3 octaves 1st 5 octaves
last 15 overtones 1st 11 overtones
Lower Face of 1-tree 1-tree outside its Lower Face
1st 3 Platonic solids or dodecahedron icosahedron & dodecahedron or 1st four Platonic solids
9 superstring dimensions 16 bosonic string dimensions
3 large-scale dimensions 6 compactified dimensions
A musical chord is a set of 3 or more notes that sound simultaneously. There are some combinations of 3 notes that, when played together, produce a sensation of harmony, rest and stability. They are the so-called ‘consonant chords’ and there are only two types of them: major and minor chords. The rest of the chords are dissonant. An example of a consonant chord well-known to musicians is the C major chord composed of the notes C, E & G. They are the first, third & fifth notes of the eight-note Pythagorean scale. The numbers 1, 3, 5 & 8 are all Fibonacci numbers. The following consideration indicates that this is not coincidence but indicative of how the human ear is tuned to a harmony determined by the Fibonacci numbers, just as the connection of the latter to the Golden Ratio has been recognised by artists and architects as creating forms with proportions that please the eye: the first 21 notes after the tonic span 3 octaves, the next 34 notes span 5 octaves and 55 notes are between the tonic and 8th octave. The same Fibonacci numbers re-appear! Just as the octave is a musical whole of 8 notes, so these 55 notes solely belonging to 8 octaves constitute another analogous whole in which each note is replaced by an octave. Just as there are 27 intervals less than an octave between the 8 notes of a musical scale, so there are 27 harmonics (notes with integer tone ratios) below the 8th octave. The 21:34 division separates the first 3 octaves from the last 5. Its counterpart in the single octave is
Figure 6. The Sri Yantra.
E, the major second, and its inversion (the interval between it and the first octave). The 34:21 division separates the first 5 octaves from the last 3 octaves. Its counterpart is G, the perfect fifth. In terms of the letter values of YHVH, the division is set by that between VH and YH. The perfect fifth represents the fundamental division of the octave interval. It is set by this ancient division of Tetragrammaton, the sacred Name of God, because YAH (Hebrew: YH) was the form used particularly by the prophet Jacob and his people.9
5. The 21:34 division in the 2-dimensional Sri Yantra
Revered by Hindus as the most powerful and sacred of the Yantras, or meditative images, the Sri Yantra (Fig. 6) consists of nine interlocking triangles. Five triangles point downwards and four triangles point upwards. This generates 42 triangles arranged in four sets of 8, 10, 10 & 14. They surround a downward-
pointing triangle at the centre of which is a point, or bindu. Article 3510 analysed the 2-dimensional Sri Yantra, showing its equivalence to the Tree of Life because the former consists of 70 points and the 43 triangles of the latter comprise 70 points when its 16 triangles are tetractyses (Fig. 7).
Figure 7. The equivalence of the Tree of Life and the Sri Yantra.Two of them — the central point called the “bindu”
and the lowest corner of the central, downward-pointing triangle — are unshared with the 42 surrounding triangles.
Hence, 68 corners of the latter surround the central triangle, 34 corners per set of 21 triangles in each half of the Sri Yantra. There are (34+21=55) corners & centres of these 21 triangles (Fig. 8).
Figure 8. The Fibonacci number pattern in each half of the 2-dimensional Sri Yantra.
The primary 21:34 division found in other sacred geometries differentiates between the corners and centres of the triangles in each half of this ancient symbol of divine creation. It also differentiates between the fourth set of 14 triangles, which have 21 corners & centres, and the first three sets of 28 triangles, which have 34 corners & centres. Below are listed the Fibonacci numbers in the 21 triangles in each half of the Sri Yantra:
The 34 corners & centres of the first three sets comprise 13 corners & centres of the second set and (11+10=21) corners & centres of the first and third sets. The 13 corners & centres of the second set comprise 5 centres and (3+5=8) corners, 3 of which touch sides of triangles. The 21 corners & centres of the first and third sets comprise groups of 8 ((1+2)+5 =3+5) and 13 (5+4+4=5+8). Of the 34 corners, 13 corners either touch sides of triangles (1+2+3) or are the seven outer corners of the fourth set of triangles, leaving 21 other corners (7+5+5+4).
Comparing the 55 corners & centres of each half of the Sri Yantra with the 55 corners, edges & triangles of the 1-tree, the 21 centres of triangles in each half correspond to the 21 geometrical elements in the Lower Face and the 34 other corners & centres in each half correspond to 34 geometrical elements in the remainder of the 1-tree.
Figure 9. This sunflower has 55 spirals of florets arranged in 34 clockwise spirals and 21 anticlockwise spirals.
Finally, an example of how the 21:34 division manifests in flowers is the 21 anticlockwise spirals and 34 clockwise spirals of florets in some sunflowers (Fig. 9), although they manifest other pairs of successive Fibonacci numbers as well. Such numbers optimise the packing of florets so that they occupy equal space, maximising the amount of sunlight to which they can be exposed. On many plants, the number of petals is a Fibonacci number. The aster, black-eyed susan and chicory have 21 petals; plantain and pyrethrum both have 34 petals.11 The Lucas numbers, which are related to Fibonacci numbers,12 also appear sometimes in flowers. We now see that the distinction between 21 and 34 is one that manifests not only in the world of plants but also in the notes of the Pythagorean musical scale and in the dimensions of the space-time of the infinitesimally small superstrings, demarcating the Lower Face of the Tree of Life blueprint from its Upper Face, i.e., the personal and transpersonal levels of consciousness, as well as the centres of the triangles in each half of the Sri Yantra and their corners. Such universality is aptly embodied in the hermetic principle “As above, so below.”
1An excellent discussion of Fibonacci numbers and the Golden Section is “The Golden Section,” by Scott Olsen, Wooden Books Ltd, 2006.
2 Phillips, Stephen M. Article 50 (Part 1): “The Golden Ratio, Fibonacci and Lucas numbers in sacred geometries,” (WEB, PDF), pp. 7, 8.
3 Ibid, pp. 17, 18.
4 Ibid, pp. 31, 32.
5 Phillips, Stephen M. Article 33: “The human axial skeleton is the trunk of the Tree of Life,” (WEB, PDF), p. 8.
6 Phillips, Stephen M. Article 14: “Why the ancient Greek musical modes are sacred,” (WEB, PDF), p. 9.
7 Phillips, Stephen M. Article 31: “The musical nature of the polyhedral Tree of Life,” (WEB, PDF), pp. 6, 7.
8 Ref. 2, pp. 33, 34.
9 Schaya, Leo: “The Universal Meaning of the Kabbalah,” Unwin Paperbacks, 1989, p. 153.
10 Phillips, Stephen M. Article 35: “The Tree of Life Nature of the Sri Yantra and some of its scientific meanings,” (WEB, PDF).
12 Ref. 1, p. 16 and ref. 2, p. 9.