by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://www.smphillips.mysite.com
Abstract
As there are 27 intervals below the octave between the
eight notes of a musical scale, the eight Church musical modes have (8×27=216)
intervals below the octave. Of these, (8×6=48) are notes above the tonic,
leaving (21648=168) intervals that are not notes. They comprise 78 major
seconds, thirds & fourths, and 90 other intervals. This reproduces the
gematria number values of the Hebrew words ‘Cholem’ and ‘Yesodeth’ in the
Kabbalistic name of the Mundane Chakra of Malkuth. The Church modes comprise
seven types of notes and their seven complements, i.e., 12 intervals below the
octave. The 216 intervals below the octave comprise (21612=204) repetitions of
these basic types of intervals. 134 of them have the tone ratios of notes of
the Pythagorean musical scale and 70 are nonPythagorean. Formally including
the repeated unit interval and octave, there is a minimum number of 206
repetitions of the 14 basic intervals. Articles 32 and 33 showed that the human
skeleton conforms to the pattern of the Tree of Life blueprint. Its 206 bones
are the counterpart of these 206 repeated intervals. The latter include 137
intervals and 69 complements (136 Pythagorean, 70 nonPythagorean). This is how
the number 137 defining the finestructure constant in physics distinguishes
repeated intervals from their complements. 168 repeated intervals are not notes
or complements. Three overlapping Trees of Life represent the three dimensions
of the human body. Their inner form consists of 42 polygons with 206 corners
symbolizing its bones. The 80 corners of the hexagons and dodecagons denote the
80 bones of the axial skeleton. The 126 corners of the 30 other polygons denote
the 126 bones of the appendicular skeleton. The 206 bones/intervals are also
symbolized by the 206 yods associated with the last four enfolded polygons of a
single Tree of Life (137 yods in the hexagon, octagon & decagon, 69 yods in
the dodecagon).They are symbolized also by the 206 vertices of the sectors of
the triangles making up the seven pyramids whose bases are the seven regular
polygons of the inner Tree of Life. The first four pyramids have 80 vertices
symbolizing the 80 bones in the axial skeleton. The last three pyramids have
126 vertices symbolizing the 126 bones of the appendicular skeleton. The 206
bones/intervals are also symbolized by the 206 yods making up the eight types
of polygons formed by vertices of the disdyakis triacontahedron. The 34 corners
of the sectors of the seven polygons above the central one denote the 34 single
bones of the axial skeleton. Outside the shared edge of the last four polygons
are 496 yods other than corners of sectors. They symbolize the 496 particles
transmitting the forces between superstrings. 168 yods shape the boundaries of
their 36 sectors. Their counterparts in the disdyakis triacontahedron are its
168 edges above and below the 12 vertices in its equator. They represent the
168 repetitions other than notes of the 12 basic types of intervals below the
octave between the notes in the eight Church musical modes.

1
1. Interval composition of the eight Church musical
modes Article 16^{1} analysed the notes and intervals of the eight musical modes used
in plainsong by the Roman Catholic Church for hundreds of years. They consist of four
‘authentic modes’: Dorian (D scale), Phrygian (E scale), Lydian (F scale) and Mixolydian (G
scale), and four ‘plagal modes’: Hypodorian (A scale), Hypophrygian (B scale), Hypolydian (C
scale) and Hypomixolydian (D scale):
Although the same D scale as the Dorian mode, the Hypomixolydian mode has a
different dominant, or reciting note, and a different finalis, or ending
note. The C scale (Hypolydian mode) is the Pythagorean musical scale. The tone ratios of its
notes are:
1 9/8 81/64
4/3 3/2
27/16 243/128
2
Table 1 shows the tone ratios of the notes in the eight Church musical
modes:
Table 1.
Musical scale

Tone ratio

D scale

1

9/8

32/27

4/3

3/2

27/16

16/9

2

C scale

1

9/8

81/64

4/3

3/2

27/16

243/128

2

B scale

1

256/243

32/27

4/3

1024/729

128/81

16/9

2

A scale

1

9/8

32/27

4/3

3/2

128/81

16/9

2

G scale

1

9/8

81/64

4/3

3/2

27/16

16/9

2

F scale

1

9/8

81/64

729/512

3/2

27/16

243/128

2

E scale

1

256/243

32/27

4/3

3/2

128/81

16/9

2

D scale

1

9/8

32/27

4/3

3/2

27/16

16/9

2

(White cells denote Pythagorean notes; red cells denote nonPythagorean
notes.)
The seven different types of musical scale have 42 notes between
the tonic and octave. They comprise 26 Pythagorean notes and 16
nonPythagorean notes, showing how the Godname YAHWEH^{2} with number value 26 prescribes the Pythagorean
character of the seven octave species. As 42 is the 21st even integer,
EHYEH, the Godname of Kether, prescribes the seven types of scales. EL ChAI, the Godname of
Yesod with number value 49, prescribes the seven scales because there are
49 intervals between their notes. The eight modes have 48
notes between the tonic and octave comprising 30 Pythagorean notes and 18 nonPythagorean
notes. Including tonics and octaves, the 64 notes are made up of (30+16=46)
Pythagorean notes and 18 nonPythagorean notes. 48 is the number value of
Kokab, Mundane Chakra of Hod. 64 is the number value of
Nogah, Mundane Chakra of Netzach, the Sephirah preceding Hod in the Tree of
Life.
Table 2 lists the numbers of each kind of note/interval found in the eight
Church modes (Pythagorean intervals are those whose magnitudes equal the tone ratios of
notes in the Pythagorean musical scale).
2
Table 2. Numbers of notes/intervals in the eight Church musical modes.
Tone ratios written in red denote nonPythagorean notes or intervals. Tone
ratios of Pythagorean and nonPythagorean notes are listed separately in order of increasing
size. A musical scale has eight notes with 36 intervals (eight ‘1’s — the unit
intervals between each note and itself — and 28 intervals between its notes). The eight modes
have (8×36=288) intervals, where 288 = 1^{1} + 2^{2} +
3^{3} + 4^{4}. This demonstrates how the Tetrad Principle^{3} determines the total number of intervals between the
(8×8=64=4^{3}) notes in the eight scales. They comprise
64 ‘1’s and (8×28=224) intervals between notes. Eight of them are octaves.
(224–8=216) intervals are not octaves. 216 is the number
value of Geburah. The number value 36 of ELOHA, Godname of this Sephirah,
is the number of intervals between the eight notes of a musical mode, whilst its own number
value is the number of intervals other than octaves in all eight modes The
216 intervals comprise 140 Pythagorean intervals, where
140 is the number value of Masloth, the Mundane Chakra of Chokmah,
called ‘the Sphere of the Zodiac.’ They also include 76 nonPythagorean
intervals. YAHWEH ELOHIM, Godname of Tiphareth with number value 76,
prescribes the nonPythagorean tone intervals. Notice that there are 50
nonPythagorean intervals of value 729/512 (×6), 256/243 (×16) and 32/27 (×28) and
26 nonPythagorean intervals of size 128/81 (×10), 16/9 (×12) and 1024/729
(×4). This 50:26 division is reflected in the number values of ELOHIM
(50) and YAHWEH (26).
Including the eight octaves, the eight modes have
(140+8=148) Pythagorean intervals, where 148
is the number value of Netzach, the eighth Sephirah. Of the 288 intervals, eight are octaves,
leaving 280 intervals below the octave, where 280 is the
number value of Sandalphon, the Archangel of Malkuth. There are 64
‘1’s, where 64 is the number value of Nogah, the Mundane Chakra of
Netzach. There are (64+8=72) ‘1’s and ‘2’s, where
72 is the number value of Chesed, the fourth Sephirah. The number value
72 of the first Sephirah of Construction measures the number of
intervals in the eight Church modes that are ‘1’s or ‘2’s, whilst the number value
216 of the second Sephirah of Construction measures the number of intervals
that are not ‘1’s or ‘2’s!
The 14 types of notes in the eight Church modes are, in order of increasing
tone ratios:
1 256/243 9/8 32/27 81/64 4/3 1024/729 
729/512 3/2 128/81 27/16
16/9 243/128
2.
These notes make up the seven different musical scales and belong not just
to the eight modes but to any set of eight scales in which one scale is repeated. They
are the basic musical ‘atoms,’ so to speak, that make up the chain molecules called the Church
musical modes. The 14 different notes of the seven musical scales consist of seven pairs of
notes and their socalled ‘complements,’ the note whose tone ratio is the
3
interval between its partner and the octave:

Note

Complement


1. 
1

2

1×2 = 2

2. 
256/243

243/128

256/243×243/128 = 2

3. 
9/8

16/9

9/8×16/9= 2

4. 
32/27

27/16

32/27×27/16 = 2

5. 
81/64

128/81

81/64×128/81 = 2

6. 
4/3

3/2

4/3×3/2 = 2

7. 
1024/729

729/512

1024/729×729/512 = 2

The vertical line drawn in the row of 14 tone ratios separates the first
seven notes from their complements. Between the tonic and octave are six Pythagorean notes and
six nonPythagorean notes, i.e., the 14 notes comprises eight Pythagorean notes and six
nonPythagorean notes. As noted earlier, the 216 intervals between notes that
are not ‘1’s or ‘2’s comprise 140 Pythagorean intervals and
76 nonPythagorean intervals. There are (216–12=204)
repetitions of these basic note intervals comprising (140–6=134)
Pythagorean intervals and (76–6=70) nonPythagorean intervals.
The 48 notes between the tonic and octave of the eight
musical modes can be represented by points on a sphere (Fig. 1). Its North Pole denotes the octave, the South Pole denotes the tonic
and the 36 intermediate notes of the six scales other than the two D scales
are spread along three Great Circles spaced 120° apart, the 12 notes of the two D scales
being distributed along their shared, vertical diameter. The Godname ELOHIM with number value 50 prescribes the 50 notes on the
sphere and the Godname ELOHA with number value 36 prescribes the
36
notes that surround its axis, which is made up of 14 notes. The numbers of
notes with a given tone ratio are shown below:
The tone ratio 729/512 appears only once as a
note, according to Table 1, but six times
4
as an interval, according to Table 2, i.e., five times as an interval between notes above the tonic.
Similarly, the tone ratio 1024/729
appears only once as a note but four times as an interval, i.e., three times as an interval
between notes above the tonic.
The spherical distribution of 50 notes representing the
eight Church musical modes constitutes a holist system with a Tree of Life pattern shown by the
50 corners of the two sets of the first six enfolded polygons making up the
inner form of the Tree of Life (Fig. 2). The ends of the ‘root edge’ shared by each set of polygons symbolize
the tonic at the South Pole and the octave at the North Pole. The 24 corners outside the
root edge of one set denote the 24 notes. Their mirror image counterparts in the other set
denote the 24 complements of these notes. The
six external corners of an octagon denote the six notes above the tonic up
to the perfect fourths of the Dorian and Hypomixolydian modes arranged along the vertical axis
of the sphere. Their six counterparts in the other octagon denote the six complementary notes.
As shown in Article 40,^{4} this 24:24 division is a characteristic feature of holistic
systems.
Table 3 gives the numbers of intervals, notes and intervals other than notes,
i.e., intervals between notes above the tonic.
Table 3. Interval composition of the eight Church musical modes.
5
Each different tone ratio appears more than once. There are
168 intervals other than notes. This is remarkable because
168 is the number value of Cholem Yesodeth, the Mundane Chakra of
Malkuth, which has been shown in previous articles to appear in many different contexts because
it characterises all holistic systems (for example, it is the structural parameter of
superstrings). This is excellent evidence of the holistic nature of the eight Church musical
modes. More remarkable still, Table 3 shows that the number value 78 of Cholem (Fig. 3) is the number of intervals other than notes that are major seconds,
thirds or perfect fourths and that the number
value 90 of Yesodeth is the number of the nine remaining types of
intervals.
We found earlier than there are 204 repetitions of the basic set of 12
intervals between the tonic and octave of the eight Church musical modes. As they have
216 intervals between their tonics and octaves and as 12 notes of the two D
scales lie on the axis of the three Great Circles along which their notes are arranged, there
are also 204 intervals other than notes of the Dorian mode and its plagal counterpart, the
Hypomixolydian mode with the same set of notes:
1 9/8
32/27
4/3 3/2
27/16 16/9 2
Instead of arranging their notes along the axis, the 12 basic notes
between the tonic and octave may be arranged along it, the 48 notes (30
Pythagorean, 18 nonPythagorean) between the tonic and octave of the eight modes being assigned
to points along four Great Circles. One halfcircle represents an authentic mode and the other
halfcircle represents its plagal counterpart. Including the tonic and octave, there are now
not 38 notes but 50 notes (32 Pythagorean, 18 nonPythagorean) lying on
6
Great Circles. Including the 12 basic types of notes (6 Pythagorean, 6
nonPythagorean) between the tonic and octave that make up the eight musical modes, we now have
62 notes arranged on the sphere and its axis. They comprise (32+6=38)
Pythagorean notes and (18+6=24) nonPythagorean notes. (38–2=36) Pythagorean
notes and 24 nonPythagorean notes are distributed around the tonic and octave at its poles.
(36–6=30) Pythagorean notes and (24–6=18) nonPythagorean notes surround its
axis, on which are the tonic, the octave, six Pythagorean notes and six nonPythagorean
notes.
Compare this pattern of 62 notes with the
62 vertices of the disdyakis triacontahedron (Fig. 4). The Catalan solid with the most edges, it was proved in previous
articles to be the polyhedral version of the inner form of the Tree of Life. It has the 12
(B) vertices of an icosahedron, the 20 (C) vertices of a dodecahedron and 30 (A) vertices
that are apices of pyramids attached to the 30 Golden Rhombic faces of the rhombic
triacontahedron whose 32 vertices belong to these Platonic solids. With a straight line
passing through two diametrically opposite C vertices, this vertical axis is surrounded by
30 A vertices, (20–2=18) C vertices and 12 B vertices. The following correspondence exists
between the 60 vertices between the apex and nadir and the 60 notes either on Great Circles
or on the central axis between the tonic and the octave at the Poles:
30 A vertices
30 Pythagorean notes;
12 B vertices
12 basic notes between tonic and
octave;
(2+18) C vertices 2 Pythagorean notes
(tonic/octave) + 18 nonPythagorean notes.
The two opposite C vertices denote the tonic and octave, the remaining 18 C
vertices denote the 18 nonPythagorean notes, the 30 A vertices denote the 30 Pythagorean notes
and the 12 B vertices denote the 12 basic notes between the tonic and octave of the eight
musical modes. This correlation between the sacred geometry of the disdyakis triacontahedron
and the spherical representation of the Church modes demonstrates the holistic nature of each
system. It also gives a defining role to the icosahedron.
The spherical distribution with notes of the two D scales arranged along the
axis comprise the tonic, octave, eight Pythagorean notes and four nonPythagorean notes on the
axis and 24 Pythagorean notes and 12 nonPythagorean notes surrounding it. This does not
correlate with the disdyakis triacontahedron in a natural way because both Pythagorean and
nonPythagorean notes have to be assigned to vertices of a given type, apart from the fact that
there are no notes for the 12 B vertices. Even if the 12 basic types of notes between the tonic
and octave were assigned to the 12 B vertices and the 50 notes to the 30 A and
20 C vertices, their 32 Pythagorean notes other than the tonic and octave and their 16
nonPythagorean notes cannot be assigned to either the 28 A & 20 C vertices (axis passing
through two opposite A vertices) or the 30A and 18 C vertices (axis through two C opposite
vertices) without both types of notes being assigned to each type of vertex. In the case where
the central axis was made up of the 12 basic notes, their six Pythagorean notes and six
nonPythagorean notes were assigned to the 12 B vertices. So there is mixing here as well, but
only for B vertices, whereas, when the notes of the two D scales lie on the axis, each type of
note has to be assigned to both A and C vertices.
An alternative way of correlating notes with vertices is to consider the
axis of the polyhedron passing through two A vertices. There are 12 vertices in the equator, 24
vertices between it and the apex and 24 vertices between it and the lowest vertex. This
(24+12=36):24 pattern correlates with the 36 Pythagorean
notes and 24 nonPythagorean notes in the basic set of 12 notes and in the 48
notes of the eight modes.
Table 4 shows the number of repetitions of intervals in the eight Church
modes.
7
The 204 repetitions of the basic set of 12 notes between the tonic and
octave comprise 136 intervals (91 Pythagorean, 45 nonPythagorean) and 68 complements (43
Pythagorean, 25 nonPythagorean). There are 134 repeated, Pythagorean intervals and 70 repeated
nonPythagorean intervals. There are 36 repeated notes/complements (24
Pythagorean, 12 nonPythagorean) and (204–36=168) repeated
intervals that are not notes/complements (78 Pythagorean intervals with tone ratios up to that
of the perfect fourth, 90 Pythagorean & nonPythagorean intervals other than these).
Just as there are 168 intervals between notes above the
tonic in the eight Church modes (78 intervals that are seconds, thirds and perfect fourths, 90
other intervals), so there are 168 intervals other than notes that are
repetitions of the 12 types of notes between the tonic and octave (78 Pythagorean intervals
that are seconds, thirds and perfect fourths, 90 others). Once again, two sets of intervals
define the number values of the Hebrew words ‘Cholem’ and ‘Yesodeth’ in the name of the Mundane
Chakra of Malkuth. This cannot be coincidental because the two numbers refer to
similar sets of notes. The 168 intervals consist of 110 Pythagorean
and 58 nonPythagorean ones.
The Tetrad Principle defines the 136 repeated intervals up to 1024/729
because
136 = 
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

It defines the 78 Pythagorean intervals up to the perfect fourth because
78 = 
1

2

3

4

12



5

11



6

10

9

8

7

It defines the 168 intervals other than notes that are
repetitions of the basic set of 12 notes between the tonic and octave because
168 =

3

5

7

9

25



11

23



13

21

19

17

15

8
The Tetrad Principle defines the 45 repeated, nonPythagorean intervals in
this set because 45 is the fourth hexagonal number after 1.^{5} It defines the 70 repeated, non Pythagorean intervals in the set
of 204 repetitions of the basic set of 12 notes because 70 is the fourth member after 1 of
the class of fourdimensional, tetrahedral numbers.^{6} It defines the 36 repeated notes and their
complements because 36 is the sum of the first four even integers
and the first four odd integers. It defines the 64 notes of the
eight modes because 64 = 4^{3}. It defines the 16 tonics and
octaves because 16 = 4^{2}. It defines the 48 notes between the
tonic and octave because 48 = 4^{3} – 4^{2}. The Tetrad
Principle always defines global properties of holistic systems, e.g., the four authentic and
the four plagal Church modes, and the four notes in a mode and their four complements.
Table 5 displays the number of pairs of repeated intervals and their
complements and the numbers of intervals and their complements that are unpaired.
Table 5. Numbers of paired and unpaired repeated intervals.
Interval

Number of paired intervals

Number of unpaired intervals

1

0

0

256/243

3

12

9/8

11

28

32/27

13

14

81/64

9

10

4/3

27

6

1024/729

3

0

729/512

3

2

3/2

27

0

128/81

9

0

27/16

13

0

16/9

11

0

243/128

3

0

2

0

0

The 204 repeated intervals include 66 pairs of intervals and their complements (90 Pythagorean,
42 nonPythagorean). ADONAI, the Godname of Malkuth, prescribes the number of repeated pairs
because 66 is the 65th integer after 1. There are 72 unpaired
intervals (44 Pythagorean, 28 nonPythagorean). This is the number value of Chesed, the fourth
Sephirah. Altogether, there are 216 intervals other than octaves. This is the
number value of Geburah, the Sephirah following Chesed. Its Godname ELOHA with number value
36 prescribes the 36 notes between the tonic and octave of
the six modes arranged on the three Great Circles on the sphere. The 204 repeated intervals
comprise 134 Pythagorean intervals and 70 nonPythagorean intervals. As 134 is the
67th even integer, it shows how the number value 67 of Binah
determines the number of repeated, Pythagorean intervals. The Godname EHYEH of Kether with
number value 21 prescribes the 42 nonPythagorean intervals in the 66 pairs of
repeated intervals because 42 is the 21st even integer.
Let us return to the spherical distribution of the 50
notes, with the tonic and octave assigned to the South and North poles, the 12 notes (8
Pythagorean, 4 nonPythagorean) of the two D scales between them arranged along the axis and
the 36 notes of the six other modes between the tonic and octave arranged
along the six semicircles of the three vertical Great Circles inclined at 120° to one another.
The 204 repetitions of the 12 basic types of intervals between notes of the eight Church modes
comprise 134 Pythagorean intervals and 70 nonPythagorean intervals. (134–8=126)
9
Pythagorean intervals are therefore not notes of the D scales. Including the
tonic and octave, there are 10 Pythagorean notes arranged along the central axis, 70 non
Pythagorean intervals and 126 Pythagorean intervals, that is, 206 intervals, where
206 = 126 + 10 + 70,
126 = 37(9/8) + 19(81/64) + 31(4/3) + 25(3/2) +
11(27/16) + 3(243/128),
10 = 1(1) + 2(9/8) + 2(4/3) + 2(3/2) + 2(27/16) + 1(2),
and
70 = 5(729/512) +
15(256/243) + 27(32/27) + 9(128/81) + 11(16/9) +
3(1024/729).
The 206 intervals comprise the tonic, octave, and 204 repetitions of the 12
basic types of notes making up the eight Church modes. They consist of 136 Pythagorean
intervals and 70 nonPythagorean intervals.
2. The Tree of Life nature of the eight Church musical
modes Consider a set of overlapping Trees of Life. The lowest tree
(the ‘1tree’) consists of 19 triangles with 11 vertices. Its ‘trunk’ consists of the
geometrical sequence of a point, line, triangle and tetrahedron defined, respectively, by
Kether, the path connecting Chokmah and Binah, the triangle whose vertices are the
location of Chesed, Geburah and Tiphareth and the tetrahedron formed by Netzach, Hod,
Yesod and Malkuth. The remainder of the 1tree constitutes its ‘branches.’ Now suppose
that the triangles are each divided into their three sectors and that each
sector is then turned into a tetractys (Fig. 5). The 1tree is composed of (19×3=57) tetractyses with 251 yods. Its
trunk is composed of five triangles that transform into 15 tetractyses.
This shows how YAH, the
10
Godname of Chokmah with number value 15, prescribes the
trunk of the 1tree. The trunk comprises 80 yods. There are
(57–15=42) tetractyses in the branches of the 1tree with
(251–80=171) yods. Creating 14 triangles, they comprise 42 centres of
tetractyses. As there are six hexagonal yods inside each triangle on the edges of tetractyses,
there are (14×6=84) such yods in the branches. There are (42+84=126) hexagonal yods inside the
triangles making up the branches of the 1tree.
It was explained in Article 32^{7} that the 80 yods of the trunk of the 1tree symbolize
the 80 bones of the human axial skeleton and that the 126 hexagonal yods in
its branches created by the transformation of each triangle into three tetractyses symbolize
the 126 bones of the appendicular skeleton (Fig. 6). It was also shown that the 206 bones of the human skeleton are
encoded in the inner form of three overlapping Trees of Life representing the three
spatial dimensions of the physical body of Adam Kadmon — the human prototype. As Fig. 5 shows, the three sets of enfolded polygons have 206 corners, the
hexagons and dodecagons having 80 corners that symbolize the
80 bones of the axial skeleton and the remaining polygons having 126
corners that symbolize the 126 bones of the appendicular skeleton. The 42 centres (blueyods)
of the tetractyses in the branches correspond to the 42 corners of the six octagons. The 84
hexagonal (red) yods on edges of tetractyses in the branches correspond to the 84 corners of
the six triangles, the six squares, the six pentagons and the six decagons.
In the case of the 1tree, the 206 yods comprise Kether, Malkuth, 78 yods in
its trunk and 126 yods in its branches. In the case of the polygons enfolded in three trees,
the 206 corners consist of the two topmost corners of the pair of hexagons enfolded in the
third tree, 78 corners of hexagons and dodecagons and 126 corners of the remaining polygons. As
we shall shortly see, this 2:78:126 division has a musical meaning in the context of the
spherical representation of the notes of the Church musical modes. As stated earlier, the
composition of the 126 repeated, Pythagorean intervals is:
126 = 37(9/8) + 19(81/64) + 31(4/3) + 25(3/2) +
11(27/16) + 3(243/128).
In the 1tree and in the inner form of three overlapping trees, the number
126 splits into 42 and 84 (see Fig. 5). There is only one combination of sets with 42 intervals — the
31 perfect fourths and the 11 sixths. The composition of the 10 Pythagorean
notes in the two D scales arranged on the axis is:
10 = 1(1) + 2(9/8) + 2(4/3) + 2(3/2) + 2(27/16) + 1(2).
They correspond to the 10 yods located at the positions of the 10 Sephiroth
of the 1tree and to the 10 external corners of the three hexagons belonging to the set of
21 polygons enfolded on one side of the three overlapping trees. The
composition of the 70 nonPythagorean intervals is:
11
70 = 5(729/512) + 15(256/243) + 27(32/27) + 9(128/81) + 11(16/9) + 3(1024/729).
They correspond to the 70 yods in the 1tree that are not located at
Sephiroth and to the 70 corners of the six dodecagons and the three hexagons in the set of
21 polygons enfolded on the other side.
As the starting point of a musical scale, the tonic should be assigned to
Kether, the starting point of the 1tree. As the completion of a scale, the octave should be
assigned to Malkuth, its lowest point. The eight remaining Pythagorean notes of two D scales
are assigned to the eight Sephiroth between these extremities and to the lowest and highest
corners of the three hexagons that are joined by their corners. Adding them to the 70
nonPythagorean intervals, the 206 intervals consist of the tonic, the octave, 78 intervals (8
Pythagorean, 70 nonPythagorean) and 126 Pythagorean intervals — the same pattern as that found
in the 1tree and in the inner form of three overlapping trees.
This demonstrates the Tree of Life nature of the interval composition of the
eight Church musical modes. The 10 Pythagorean notes on the axis of their spherical
representation act as the ten Sephiroth, the South and North poles corresponding to Kether and
Malkuth. The representation has notes increasing in pitch as their location move northwards
towards the octave at the North Pole. This choice was made in order to reflect the natural
association of height with pitch. However, this convention is arbitrary and the tonic could be
assigned, instead, to the North Pole in order to maintain spatial correspondence with the
1tree.
3. The eight Church musical modes as polyhedral
geometry The semiregular polyhedra consist of the 13 Archimedean
solids and their duals, the 13 Catalan solids. The simplest Catalan solid is the triakis
tetrahedron (Fig. 7), a polyhedron with eight vertices, 18 edges and 12 faces formed by
attaching tetrahedra to the four faces of a tetrahedron. Each edge is the side of an
internal triangle with the centre of the polyhedron at its corner. It was shown in Article
29^{8} that the triakis tetrahedron is composed of 137 vertices,
edges & triangles. In other words, it embodies the number that is central to modern
physics in that its reciprocal is the socalled “finestructure constant,” which measures
the strength of the coupling of electric charges to the electromagnetic field. An axis
passing through the centre and joining two opposite vertices consists of five geometrical
elements (three vertices and two edges of internal triangles). (137–5=132) geometrical
elements surround the axis and 136 elements surround its centre. This is the geometrical
counterpart of the 136 repeated intervals in the eight Church modes up to the note with tone
ratio 1024/729, their highest note
12
below the six complementary notes below the octave. The distinction between
the tonic and these 136 repeated intervals corresponds to the geometrical distinction between
the centre of the triakis tetrahedron and the 136 geometrical elements surrounding it. The six
vertices surrounding its axis symbolize the six notes above the tonic up to that with tone
ratio 1024/729. Alternatively, the eight vertices of the polyhedron correspond to the eight
Pythagorean notes in the Dorian and Hypomixolydian modes lying on the axis of the spherical
distribution of the 50 notes, whilst the other 126 geometrical elements
surrounding the axis correspond to the 126 repetitions of the Pythagorean intervals in the
eight modes. The triakis tetrahedron is therefore the polyhedral counterpart of the spherical
arrangement of repeated Pythagorean notes and intervals. It is prescribed by ADONAI, the
Godname of Malkuth, because the minimum number
of geometrical elements needed to create it is
65,^{9} which is the number value of ADONAI.
We found in Article 27^{10} that, when the sectors of the faces and internal triangles of
the triakis tetrahedron are each divided into three sectors and the latter then converted
into tetractyses, it has 361 yods in 90 tetractyses. This is the number of acupoints in the
14 meridians of the human body, as established in Chinese acupuncture (Fig. 8). The 54 internal tetractyses and the
36 tetractyses in its faces conform to the 36:54 division
in the Lambda tetractys,^{11} whose 10 numbers determine the properties of holistic
systems:
According to Table 1 of Article 27,^{12} the triakis tetrahedron has 168
geometrical elements surrounding its axis when its internal triangles are each divided into
three sectors. They correspond to the 168 intervals other than notes in the
eight Church modes, i.e., the 168 intervals between their 56 notes above
the tonic (78 seconds, thirds and perfect fourths, 90 other intervals — see Table 3). They also correspond to the 168 repetitions of the
basic set of 12 notes between the tonic and the octave (78 seconds, thirds & perfect
fourths, 90 other intervals — see Table 4)). Once again, this shows that
13
the geometrical composition of the triakis tetrahedron mirrors the
composition of the intervals between musical notes. The 168 elements consist
of 24 corners, 78 edges and 66 triangles. The number value 78 of Cholem is the number
of edges and the number value 90 of Yesodeth is the number of vertices and triangles.
The same 78:90 division appears in the triakis tetrahedron and in the two compositions
of intervals! This is not coincidental, because both this polyhedron and the set of eight
Church musical modes are manifestations of the universal
paradigm governing holistic systems. As such, they must embody the gematria
number value 168 of the Mundane Chakra of Malkuth.
4. The last four
enfolded polygons map the human skeleton
We saw in Section 3 that the inner form of three overlapping Trees of Life consists of 42
polygons with 206 corners. These 206 geometrical degrees of freedom symbolize the 206 bones in
the human body whose three dimensions are represented by these three trees. This information is
embodied also in the last four of the seven polygons making up the inner form of a single Tree
of Life (Fig. 9). These polygons have 36 sectors prescribed by ELOHA,
the Godname of Geburah with number value 36. They have
14
15
136 vertices, edges and triangles outside the root edge shared with the
other set of polygons. This is the number of repeated intervals in the eight Church modes up to
1024/729. 206 yods are associated with the last set of four regular polygons. That this is not
due to coincidence is demonstrated by the fact that 34 yods can be associated with the hexagon
that symbolize the 34 single bones of the axial skeleton, 46 yods associated with the octagon
symbolize the 46 bones of the axial skeleton forming pairs and 126 yods in the decagon and
dodecagon symbolize the 126 bones of the appendicular skeleton. That such a detailed
correlation could arise by chance is highly implausible. Moreover, the 80 yods
associated with these four polygons that symbolize the 80 bones of the axial
skeleton consist of two yods on the root edge and 78 yods outside it. This 2:78:126 pattern was
found in Section 2 in the 206 yods of the 1tree, which comprise Kether, Malkuth, 78 yods of
the trunk and 126 yods in its branches, as well in the three sets of polygons enfolded in three
trees, whose 206 corners consist of the two topmost corners of the pair of hexagons enfolded in
the third tree, 78 corners of hexagons and dodecagons and 126 corners of the remaining
polygons. It demonstrates that the 2:78:126 division is intrinsic to a holistic
system. It manifests in the human skeleton as the distinction between the axial skeleton, which
protects the vital organs, and the appendicular skeleton, which provides mobility and
dexterity.
The same pattern was found in Article 41 in the set of seven pyramids whose
bases are the seven regular polygons. The seven polygons divided into their 48
sectors are the projections of these pyramids. When their internal triangles and faces are
divided into three sectors, the 432 sectors of the seven pyramids have 206 vertices^{13} (Fig. 10). They comprise 80 vertices in the pyramids whose
bases are the first four polygons and 126 vertices in the last three pyramids. Their musical
counterpart is the spherical distribution of 50 notes belonging to the
eight Church modes, with the 12 basic notes between the tonic and octave arranged along its
axis as the generative musical elements. The seven pairs of basic notes and their
complements correspond to the seven pairs of apices of pyramids and centres of their
bases.
5. The disdyakis triacontahedron embodies numbers 137, 206 &
168 There are 15 sheets of vertices
perpendicular to an axis joining two diametrically opposite C vertices of the disdyakis
triacontahedron. The central layer is a hexagon with six A vertices. On either side,
sandwiched between the two opposite C vertices, are 6sided polygons with six A vertices
and six C vertices, and five triangles with three A, three B, three C, three A &
three B vertices. When the sectors of the polygons are turned into tetractyses, the eight
sheets in onehalf of the polyhedron contain 206 yods (Fig. 11).^{14} They are made up of 34 corners and centres of the
uppermost seven polygons, 46 yods that are either centres of these polygons or on the edges
of the central hexagon and 126 other yods. They symbolize, respectively, the 34 single bones
of the axial skeleton, the 46 bones in it that are grouped in pairs and the 126 bones of the
appendicular skeleton. The bone composition of the human body is defined by the eight
different sheets of polygons in the disdyakis triacontahedron.
When the 60 sectors of the 15 polygons are each divided into three
triangles, the 180 triangles have 135 vertices,^{15} that is, 134 vertices surround the centre of the
polyhedron. They correspond to the 134 repeated Pythagorean intervals in the eight Church
modes. Including the two C vertices not belonging to polygons, the disdyakis triacontahedron
is made up of 137 vertices. This is how it embodies the number 137 defining the
finestructure constant. It is the counterpart of the 137 geometrical elements composing the
triakis tetrahedron (see Section 3).
According to Fig. 9, 206 yods are associated with the last four enfolded polygons. The
16
17
dodecagon has 69 yods outside the root edge. Associated with the hexagon, octagon and
decagon are (20669=137) yods. Both the number of bones and the finestructure constant number
137 are embodied in the last four polygons!
Figure 12 shows the spectacular way in which the most important number in
physics is embodied in the inner form of the Tree of Life. When the 94 sectors of the two
sets of enfolded polygons are each divided into three triangles and the latter then turned
into tetractyses, the resulting 282 tetractyses contain 1370 yods, i.e.,
the yods in 137 tetractyses! 282 is the number value of Aralim,
the Order of Angels assigned to Binah.
The last four polygons also embody the number at the heart of superstring
theory, which predicts that 496 particles transmit the forces between
superstrings. There are 496 yods in the last four enfolded polygons other than
their corners and centres (Fig. 13).^{16} Lest this should be thought a coincidence, it must be pointed out
that the first and fourth of these have 248 yods and the second and third
polygons have 248 yods. 248 is the dimension of the
superstring gauge symmetry group E_{8} and 496 is the dimension of
E_{8}×E_{8}', the gauge symmetry group associated with the
E_{8}×E_{8}' heterotic superstring. It is highly implausible that it could
be mere chance that both the superstring numbers 248 and
496 should be the exact yod populations of certain combinations of
polygons.
18
Of the 180 edges of the disdyakis triacontahedron, 12 edges (four AB, four
BC & four AC) are in the equatorial plane perpendicular to an axis joining two
diametrically opposite A vertices (Fig. 14). There are 84 edges (28 AB, 28 BC & 28 AC) above these and
84 similar edges below them. The number 168 is the number of edges that
generate the 3dimensional form of the disdyakis triacontahedron. That this number
defines its shape can also be seen from the following consideration: the polyhedron has 60
vertices, 180 edges and 120 triangular faces (360 geometrical elements) surrounding any axis
passing through two opposite vertices. When these vertices are A vertices, there are seven
sheets of vertices perpendicular to the axis.
The fourth sheet of vertices in the equatorial plane of the polyhedron
defines a 12sided polygon with 12 vertices and 12 edges, totalling 24 geometrical elements.
(360–24=336) elements lie above and below the equator, 168 on each side.
168 edges therefore shape the polyhedron above and below its equator, each
part being composed of 168 geometrical elements.
As discussed in Article 40,^{17} this beautiful property of the disdyakis triacontahedron is the
geometrical manifestation of the 168 automorphisms of PSL(2,7), the
symmetry group of the Klein quartic equation. This group is isomorphic to the symmetry group
SL(3,2) of the Fano plane representing the algebra of the octonions.^{18}
The 180 edges of the disdyakis triacontahedron are sides of internal
triangles that meet at its centre. Suppose that each of the 180 internal triangles is divided
into their three sectors. This creates (3×180=540) triangles. Including its 120 triangular
faces, the polyhedron is composed of (120+540=660) triangles. There are 62
vertices in its faces, one vertex at its centre and 180 vertices of the 540 triangles in its
interior, totalling 243 vertices. The line joining each polyhedral vertex to the centre is an
edge of a triangle.
19
There are three edges of sectors inside each original, interior triangle.
The number of edges = 180 + 62 + 3×180 = 782. The number of vertices, edges
and triangles = 243 + 782 + 660 = 1685. The central axis is composed of three vertices and two
edges of triangles, i.e., five geometrical elements. Therefore, 1680 geometrical elements
surround the axis of the disdyakis triacontahedron. Apart from the Pythagorean factor of 10 (a
tetractys is composed of 10 yods), the number value 168 of the Mundane Chakra
of Malkuth determines the geometrical composition of the disdyakis triacontahedron! As
discussed in many previous articles,^{19} the number 1680 is the number of circularly polarized oscillations
made in each of the 10 closed curves of the heterotic superstring as it winds five times
around its axis of spin, each revolution comprising 336 oscillations and each
halfrevolution comprising 168 oscillations. As the microscopic
manifestation in spacetime of the Tree of Life, the Mundane Chakra of Malkuth is
the superstring constituent of the protons and neutrons inside atomic nuclei. Its gematria
number value 168 is the number of circular oscillations made during a
halfrevolution of each closed curve, the superstring being created by 16800 such
oscillations.
6. The last four enfolded polygons embody the number
168 The last four enfolded regular polygons have
168 yods outside their shared edge on the boundaries of their
36 tetractyses (Fig. 15). This demonstrates par excellence the
shapedefining character of this number, discussed in many previous articles
as a structural parameter of the heterotic superstring. Moreover, the
168 yods divide into the 84 yods on edges of the 18 sectors of the
octagon and decagon and the 84 yods on the edges of the 18 sectors of the hexagon and
dodecagon — the same two pairs of polygons that define the split of the
superstring parameter 496 into 248 and
248. We encountered a similar division earlier for the disdyakis
triacontahedron, which has 84 edges above its equator and 84 edges below it when its axis
passes through two diametrically opposite A vertices. The 336 yods outside the root edge on
the boundaries of both sets of the last four polygons correspond to the 336 geometrical
elements in the faces of the disdyakis triacontahedron that are above and below its
equator.
The seven pyramids whose bases are the seven regular polygons constituting
the inner form of the Tree of Life have 336 vertices, edges & triangles surrounding their
axes.^{20} The socalled ‘npyramid’ (a pyramid with an nsided regular
polygon as its base) has 7n geometrical elements surrounding the axis joining its apex and
base centre. The 3pyramid, 4pyramid, 5pyramid & 12pyramid have 168
elements surrounding their axes, as do the 6pyramid, 8pyramid & 10pyramid.
The Tetrad Principle defines the numbers 84 and 336 because
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}
and
336 = 2^{2} + 6^{2} + 10^{2} + 14^{2}.
The eight Church musical modes have 216 intervals below the octave, of
which 48 intervals are notes, leaving 168 intervals that are
not notes. There are (216–48=168) rising
intervals and 168 falling intervals, the latter having tone ratios that are
reciprocals of those of the former. This is the musical counterpart of:
1. the (168+168) yods on the edges of the two sets of the
last four enfolded polygons;
2. the (168+168) geometrical elements above and below the
equator of the disdyakis triacontahedron;
3. the (168+168) geometrical elements surrounding the axes of
the seven pyramids whose bases are the seven regular polygons of the inner Tree of Life.
20
All are examples of holistic systems that are characterized by the same set of parameters,
such as 168, 137 and 206.
References
^{1} Phillips, Stephen M. Article 16: “The tone intervals of
the seven octave species and their correspondence with octonion algebra and superstrings,”
(WEB, PDF).
^{2} 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). They have number values obtained by the practice of
gematria, wherein each letter of the Hebrew alphabet has a number assigned to it, giving a
number value to a word that is the sum of the values of its letters. The Hebrew names, their
translations and their number values are shown in the table below:
Table 1. Number values of the ten Sephiroth in the four Worlds.

SEPHIRAH

GODNAME

ARCHANGEL

ORDER OF ANGELS

MUNDANE
CHAKRA

1 
Kether
(Crown)
620 
EHYEH
(I am)
21 
Metatron
(Angel of the Presence)
314 
Chaioth ha Qadesh
(Holy Living Creatures)
833

Rashith ha Gilgalim
First Swirlings.
(Primum Mobile)
636 
2 
Chokmah
(Wisdom)
73 
YAHWEH, YAH
(The Lord)
26,
15

Raziel
(Herald of the Deity)
248 
Auphanim
(Wheels)
187 
Masloth
(The Sphere of the Zodiac)
140 
3 
Binah
(Understanding)
67 
ELOHIM
(God in multiplicity)
50

Tzaphkiel
(Contemplation of God)
311

Aralim
(Thrones)
282

Shabathai
Rest.
(Saturn)
317 

Daath
(Knowledge)
474 




4 
Chesed
(Mercy)
72 
EL
(God)
31 
Tzadkiel
(Benevolence of God)
62 
Chasmalim
(Shining Ones)
428

Tzadekh
Righteousness.
(Jupiter)
194 
5 
Geburah
(Severity)
216

ELOHA
(The Almighty)
36

Samael
(Severity of God)
131

Seraphim
(Fiery Serpents)
630

Madim
Vehement Strength.
(Mars)
95 
6 
Tiphareth
(Beauty)
1081

YAHWEH ELOHIM
(God the Creator)
76 
Michael
(Like unto God)
101

Malachim
(Kings)
140

Shemesh
The Solar Light.
(Sun)
640 
7 
Netzach
(Victory)
148

YAHWEH SABAOTH
(Lord of Hosts)
129

Haniel
(Grace of God)
97 
Tarshishim or
Elohim
1260

Nogah
Glittering Splendour.
(Venus)
64 
8 
Hod
(Glory)
15

ELOHIM SABAOTH
(God of Hosts)
153

Raphael
(Divine Physician)
311

Beni Elohim
(Sons of God)
112

Kokab
The Stellar Light.
(Mercury)
48 
9 
Yesod
(Foundation)
80

SHADDAI EL CHAI
(Almighty Living God)
49,
363

Gabriel
(Strong Man of
God)
246

Cherubim
(The Strong)
272

Levanah
The Lunar Flame.
(Moon)
87 
10 
Malkuth
(Kingdom)
496

ADONAI MELEKH
(The Lord and King)
65,
155

Sandalphon
(Manifest Messiah)
280 
Ashim
(Souls of Fire)
351

Cholem Yesodeth
The Breaker of the Foundations.
The Elements.
(Earth)
168 
(Numbers from this table appearing in the article will be written in
boldface)
21
^{3} Phillips, Stephen M. Article 1: “The Pythagorean nature
of superstring and bosonic string theories,” (WEB, PDF), p. 5.
^{4} Phillips, Stephen M. Article 40: “The unification of all sacred geometries
and its implications for particle physics,” (WEB, PDF), pp. 2328.
^{5} Wells, David. “The Penguin dictionary of curious and Interesting numbers,”
Penguin Books, 1988.
^{6} Ibid, p. 60.
^{7} Phillips, Stephen M. Article 32: “Derivation of the bone & classical
acupoint compositions of the human body and their relationship to the seven musical scales,”
(WEB, PDF), pp. 9, 10.
^{8} Phillips, Stephen M. Article 29: “The triakis tetrahedron and the disdyakis
triacontahedron embody the finestructure constant and the structural parameter of the
heterotic superstring,” (WEB, PDF), pp. 45.
^{9} Ibid, Table 3, p. 9, p.10.
^{10} Ref. 7, p. 22.
^{11} Phillips, Stephen M. Article 11: “Plato’s Lambda — its meaning,
generalisation and connection to the Tree of Life,” (WEB, PDF).
^{12} Phillips, Stephen M. Article 27: “How the disdyakis triacontahedron
embodies the structural parameter 1680 of the E_{8}×E_{8} heterotic
superstring,” (WEB, PDF), p. 4, Table 1.
^{13} Phillips, Stephen M. Article 41: “The pyramidal form of the inner Tree of
Life, its counterparts and its encoding of the human skeleton,” (WEB, PDF), p. 6.
^{14} Phillips, Stephen M. Article 36: “The Sri Yantralike pattern of the 15
layers of vertices in the disdyakis triacontahedron and its scientific meaning,” (WEB, PDF), p. 9, Table 4.
^{15} Ibid, p. 11.
^{16} Phillips, Stephen M. Article 34: “The seven layers of vertices in the
disdyakis triacontahedron encode the 206 bones of the human skeleton, the superstring symmetry
groups E_{8} & E_{8}×E_{8} and the superstring structural
parameters 168, 336, 840 & 1680,” (WEB, PDF), p. 39.
^{17} Ref. 4.
^{18} Phillips, Stephen M. Article 15: “The mathematical connection between
superstrings and their micropsi description: a pointer towards Mtheory,” (WEB, PDF), p. 12.
^{19} For example, see ref. 12.
^{20} Ref. 13, p. 5
22
