ARTICLE 31
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
The polyhedral Tree of Life is correlated with the eight
Church musical modes. These musical scales, which start and finish with the D
scale, comprise 48 notes between the tonic and octave — a set of 50 notes that
is represented by the 50 corners of the first (6+6) regular polygons of the
inner Tree of Life. They comprise 14 different notes (eight with tone ratios of
the Pythagorean scale, six with nonPythagorean tone ratios). The eight scales
have 168 rising intervals and 168 falling intervals between notes above the
tonic. These intervals are symbolised by the (168+168) yods in the first (6+6)
polygons other than their corners. These are the musical and Tree of Life
counterparts of the 168 automorphisms and 168 antiautomorphisms of the Klein
Quartic. Its symmetry group PSL(2,7) is isomorphic to the symmetry group SL(3,2)
of the Fano Plane representing the multiplication of octonions. Their group of
automorphisms is the exceptional group G_{2}. Its seven pairs of roots
correlate with the seven pairs of notes in the seven musical scales, explaining
why the correspondence exists, viz. Pythagorean music and physics based upon
octonions share the same principles. The 48 vertices of the 144 Polyhedron that
stem from the 48 faces of the underlying disdyakis dodecahedron express the 48
basic degrees of freedom manifested by a holistic system. In the context of the
musical modes, these are notes, which can be grouped into eight sets of six
notes between the tonic and octave. The icosahedron with 12 B vertices in the
disdyakis triacontahedron represents the 12 different notes between the tonic
and octave of these scales. Eight of them consist of fours pairs of notes and
their complements, one of which is Pythagorean and the other nonPythagorean.
The 216 edges of the 144 Polyhedron represent the 216 intervals other than
octaves between the notes of the eight Church musical modes. The 180 edges of
the disdyakis triacontahedron represent the 12 basic notes and the 168 intervals
other than octaves between notes above the tonic. The 13 rising intervals and
the 13 falling intervals between the tonic and the other types of notes in the
Church modes are the musical counterparts of the 13 Catalan solids and their
duals. Just as the disdyakis triacontahedron has 26 sheets of vertices that are
perpendicular to either BB or CC axes and 33 sheets perpendicular to AA, BB
& CC axes, so it is the 26th member of the family of Archimedean and
Catalan solids and the 33rd stage in the development of polyhedra. The disdyakis
triacontahedron contains 1680 vertices, edges & triangles, where 1680 is the
number of yods in the lowest 33 Trees of Life constructed from tetractyses. The
28 polyhedra defined by its 62 vertices have 3360 hexagonal yods in their faces.
This is the number of yods needed to construct the inner Tree of Life from
2ndorder tetractyses. It is also the number of helical turns in one revolution
of the ten whorls of the basic unit of matter described over a century ago by
Annie Besant & C.W. Leadbeater, showing how the disdyakis triacontahedron
embodies geometrically the structural parameter of this object, identified in
earlier work by the author as the E_{8}×E_{8} heterotic
superstring constituent of up and down quarks.

1
1. The ancient musical scales/Church
modes Historically speaking, musical scales were always divided into
eight notes because the ancient Greeks regarded them as composed of two tetrachords (sets
of four notes). If the pitch, or ‘tone ratio,’ of the starting note (‘tonic’) of a scale
is given the value of 1, the eighth note of the scale (‘octave’) has a tone ratio of 2,
that is, it has twice the frequency of the tonic and is the tonic of the next higher set
of eight notes. The arithmetic mean of these two frequencies is (1+2)/2 = 3/2. This is
the tone ratio of the ‘perfect fifth,’ socalled because it is the fifth note in the
ascending scale, counting from the tonic. The musical scale based entirely on octaves and
fifths is called the ‘diatonic scale.’ The tone ratios of the eight notes making up an
octave of this scale are:
1 9/8
(9/8)^{2} 4/3
3/2 27/16
243/128 2
This diatonic scale is also called the ‘Pythagorean scale' because
Pythagoras is generally thought to have discovered its mathematical basis. It comprises five
tone intervals (T) of 9/8 and two intervals (L) of 256/243, called in Greek the
leimma, or ‘left over,’ which corresponds to the modern semitone, although slightly
less than it. The interval pattern of the Pythagorean scale is:
tone–tone–leimma–tone–tone–tone–leimma.
T T
L
T T
T L
In the C scale, the tonic is labelled ‘C’ and subsequent notes in the scale
are labelled D, E, F, G, A & B, the octave being written as C'. Their tone ratios are:
1

9/8

(9/8)^{2}

4/3

3/2

27/16

243/128

2

C

D

E

F

G

A

B

C

As proved in Article 14,^{1} the six notes above
the tonic of the C scale can form only two triplets of notes with the same proportions of their
tone ratios. They are (D, F, A) and (E, G, B), corresponding members of which are separated by
a tone interval (Fig. 1).
The seven notes D–C' above the tonic of the C scale therefore comprise three
pairs of notes (D, E), (F, G) & (A, B) and the octave C' as well as a (3:3:1) pattern. The
two triplets (D, F, A) and (E, G, B) correspond in the Tree of Life (Fig. 2) to the two triads of Sephiroth of Construction:
ChesedGeburahTiphareth and NetzachHodYesod, whilst the last note of the C scale, the
octave C', corresponds to Malkuth, the last Sephirah of Construction, which completes the
emanation of the Tree of Life. The three pairs of notes (D, E), (F, G) & (A, B)
separated by a tone interval correspond to the pairs of Sephiroth of Construction on the
three pillars of the Tree of Life. This parallelism
2
suggests that the musical scales, both collectively and in their
mathematically perfect version — the Pythagorean scale, conform to the Tree of Life, the
Kabbalistic representation of Adam Kadmon, or ‘Heavenly Man.’
This article will add to the evidence presented in Article 14 for this bold
assertion by showing that a remarkable similarity exists between the intervallic composition of
the musical scales and the disdyakis triacontahedron, the most complex of the Catalan solids,
which was proven in Articles 2230 to be one of the two polyhedra that make up the
polyhedral representation of what the author has called the “inner Tree of Life” (to be
discussed later).
Many modern musical scholars hold the view that the ancient Greek musical
modes, such as the Dorian and Phrygian modes, were not different octave species but
different
3
keys of the same scale. Article 16 refuted this belief with two
arguments:
1. the ancient Greeks would not have used two different musical terms —
‘harmonia’ and ‘tonos’— to mean the same thing, namely, key, instead of scale and key;
2. the Pythagorean mathematician Nichomachus stated^{2} (quoting Pythagoras and Plato) that the ancient Egyptians
ascribed 28 sounds to the universe, indicating that they were aware of the seven octave
species, which have 28 notes with tone ratios belonging to the Pythagorean scale. As it is
known^{3} that many musicians of ancient Greece, as well as learned men
like Pythagoras and Plato, studied music with the Egyptians, it is highly unlikely that they
neither learnt of the existence of the seven octave
species nor played music in various modes based upon them.
The musical modes that have served as the basis of plainsong in the Roman
Catholic Church originated not from the ancient Greek modes described by Plato and Aristotle but from the Byzantine
octoechoi, which were four pairs of melodic motives that expressed different feelings, or
ethos. The church modes consist of four ‘Authentic Modes’ (socalled because they were
given the ancient Greek names of Dorian, Phrygian, Lydian and Mixolydian) and four ‘Plagal
Modes’ (Fig. 3). The latter are separated from the former by a perfect fourth
(Fig. 4). By dividing a circle into seven arcs (Fig. 5) representing successive intervals of the Pythagorean scale and
selecting consecutive sequences of seven intervals as each scale, it is readily seen that
there can be only seven different musical scales because there are seven intervals between
successive notes in a scale and so the eighth sequence merely repeats the first one:
Moreover, as all intervals are selected, their sequences do not depend upon
the starting
4
point on the circle. The eight Church modes are labelled 1–8 (see
Fig. 4) and start with the D scale. It is called the ‘Dorian mode,’ although
whether it was what the ancient Greeks understand as this mode is unknown. It turns out that
the numbering of modes is not arbitrary because, starting with the tonic of the D scale, the
last of the different scales is the Pythagorean scale (C scale), the only scale with
all its notes having tone ratios with Pythagorean values. As it displays the most
mathematical harmonies, it is intuitively natural that the last of any sequence of these
scales should be the Pythagorean scale. The D scale, however, is the only scale whose
pattern of intervals is symmetric. Fig. 3 indicates that it is the same as its mirror image and that Modes 3, 5
& 7 are the mirror image of Modes 6, 4 & 2. This unique property of the D scale
gives it a pivotal role in establishing the connection between the musical scales and the
polyhedral Tree of Life, as will be explained later. If each rising interval in a scale is
replaced by its falling counterpart, i.e., T T^{–1} and LL^{–1}, an ascending scale, e.g., TLTTLTT, becomes
T^{–1}L^{–1}T^{–1}T^{–1}L^{–1}T^{–1}T^{–1},
which is the descending version of the scale TTLTTLT. Inverting each interval of a scale
creates another scale whose pattern of intervals is its mirror image. Only the D scale is
unchanged when its intervals are inverted.
The eight C–C' scales shown above (or eight consecutive scales starting with
any note) span 15 successive notes C–C'' and 14 intervals, showing how the
Godname YAH with number value 15^{4} prescribes the range of notes needed to define the complete set of
four Authentic Modes and four Plagal Modes. Their (8×7=56) intervals comprise 16 leimmas (L)
and 40 whole tones (T), where
and
This demonstrates how the Tetrad (4) symbolised by the square expresses the
tone interval and leimma composition of the eight scales. Notice that, as each scale has the
same number of tone intervals and the same number of leimmas, this composition is the same
whatever the scale chosen to start the sequence, e.g., 40 whole tones and 16 leimmas are
spanned by the eight scales that start and end with the Dscale.
Table 1. The tone ratios of the eight scales C–C'.
Musical scale 
Tone ratio

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

128/81

16/9

2

C scale 
1

9/8

81/64

4/3

3/2

27/16

243/128

2

(White cells denote Pythagorean notes; red cells denote nonPythagorean
notes).
5
The eight C–C' musical scales have 48 notes with
Pythagorean tone ratios (white cells in Table 1) and 16 notes with nonPythagorean tone ratios (red cells). Table 2 lists their numbers in each scale. These notes can be represented by
an octagon whose sectors are tetractyses (Fig. 6) because 48 yods surround its centre.
The Pythagorean notes consist of eight tonics (1), eight octaves (2) and 32
notes between the tonic and octave with intermediate tone ratios. They comprise
26 notes of the seven scales C–B and the six notes of the repeated C'
scale.
Table 2. Number of Pythagorean/nonPythagorean tone ratios in the eight
modes.
Musical scale

Number of Pythagorean
tone ratios (white cells)

Number of nonPythagorean
tone ratios (red cells)

C' scale

8

0

B scale

3

5

A scale

5

3

G scale

7

1

F scale

7

1

E scale

4

4

D scale

6

2

C scale

8

0

Total
=
48
Total = 16
The number of independent (nonrepeated) Pythagorean notes is
26, which is therefore prescribed by the Godname YAHWEH (YHVH), the values of
the letters of the Godname denoting the numbers of tones with tone ratios of two possible
values:
The letter values of YAHWEH also define the numbers of Pythagorean notes
between
6
the tonic and octave of the seven independent scales:
Including their octaves, there are 33 Pythagorean notes above the tonic of
the seven scales. The numbers 26 and 33 measure, respectively, the number of
combinations and the number of permutations of 10 objects a, b, c, ...j arranged in a
tetractys:
Indeed, Table 1 indicates that those types of Pythagorean and nonPythagorean tones
above the tonic that occur more than once are themselves 10 in number:
The seven distinct octave species have notes above the tonic with 13
different tone ratios listed below in increasing order of magnitude:
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
(Numbers in red denote notes not belonging to the Pythagorean scale
(Cscale)). Including the tonic, they consist of seven pairs that, as intervals, span an
octave. In order of increasing magnitude, they are:
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 14 notes consist of two pairs of Pythagorean notes, four pairs of a
Pythagorean and a nonPythagorean note and one pair of nonPythagorean notes. The significance
of this structure will emerge later.
As demonstrated in Article 16^{5}, every one of the 28 intervals between two notes belonging to each of
the seven different musical scales is equal to the tone ratio of one of the 13 different
notes above the tonic that can be found in these scales. In other words, all their intervals
are just repetitions of this basic set.
Table 3 lists the tone ratios of the eight D–D' scales. They comprise 30
Pythagorean
7
notes and 18 nonPythagorean notes. Table 4 compares the number of notes in the C–C' scales and the D–D'
scales.
Table 3. The tone ratios of the notes of the eight D–D' musical scales.
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).
Both have the same numbers of notes (24) and their octave complements (24).
It is straightforward to verify that all of the other scales except the F scale (Lydian mode)
exhibit this symmetry possessed by the C scale (Pythagorean) and D scale (Dorian musical mode).
The latter is unique among the scales in that its pattern of intervals — TLTTTLT — is its own
mirror reflection.
Table 4. Numbers of different notes in the eight C–C' and the eight D–D'
scales.
C scale–C' scale

6(9/8)

5(16/9)

4(81/64)

3(128/81)

7(4/3)

7(3/2)

4(32/27)

5(27/16)

2(256/243)

3(243/128)

1024/729

729/512


D scale–D' scale

6(9/8)

6(16/9)

3(81/64)

3(128/81)

7(4/3)

7(3/2)

5(32/27)

5(27/16)

2(256/243)

2(243/128)

1024/729

729/512


Total = 24

Total = 24


Total = 24

Total = 24

The tonic, octave and the 24 pairs of notes and their complements can be
assigned to the 50 corners of the first (6+6) regular polygons of the inner
Tree of Life (Fig. 7). Article 4 showed that this set of polygons constitute a Tree of Life
pattern because the number values of the ten Godnames define its geometrical
properties.^{6} The two endpoints of the shared root edge denote the tonic and
the octave, and a corner and its mirror image in
8
the other similar set of polygons denote a note and its complement.
Suppose that the eight notes of a scale are denoted by points on a great
circle of a sphere passing through the South Pole (tonic) and North Pole (octave)
(Fig. 8). As the intervals of the E, F & G scales are the mirror
images of, respectively, the C, B & A scales, a note is diametrically
opposite every other note on the circle. As the pattern of intervals of the D scale is its
own mirror image, its notes can be represented by points on the central vertical axis of the
sphere, each point being the same distance from
the centre as its mirror image. This means that the eight scales D–D' can be
represented by (2+8×6 = 50) notes on the sphere and its axis, 12 of them lying
on the latter as six different pairs of identical notes assigned to the six points between the
North and South Poles. It demonstrates how the Godname ELOHIM with number value
50 prescribes this sequence of scales. It also demonstrates the power of the
Tetrad (4) to express properties of holistic systems such as the seven musical scales, for the
number of points symbolising their notes is 44. The Divine Name ELOHA assigned to Geburah with
number value 36 prescribes the (6×6=36) notes between the
tonic and the octave of the E–C scales that lie on the surface of the sphere and surround the
14 notes on its axis.
The 42 notes between the tonics and octaves of the seven different musical
scales consist of the following pairs:
The Godname EHYEH (AHIH) with number value 21 prescribes
the 21 notes and their 21 complements, its letter values
specifying different combinations of notes. As pointed out earlier, the Godname YAHWEH
prescribes the 26 Pythagorean notes in this set of 42 notes.
The 50 notes of the eight scales D–D' represented in
Fig. 8 consist of seven sets of
9
notes and their seven sets of complementary notes:
There have to be at least seven pairs of complementary notes
because there are seven notes in a scale below the octave. The fact, however, that the eight
scales contain only seven pairs is nontrivial, because, although they comprise 14
different notes
when the tonic is included, there is no obvious reason why they should
group into seven pairs of complementary notes. It becomes obvious only when all possible
intervals between the notes in each scale are calculated and found to consist of the
same set of tone ratios of their notes.^{7} This is because every interval has its complementary interval and
therefore 14 different intervals must be grouped into seven pairs. As there is a note in the
seven scales whose tone ratio equals one of these intervals, all their types of notes can be
grouped into seven pairs.
As pointed out earlier, the D scale is uniquely symmetric in its sequence of
intervals. It is also the only scale whose 36 intervals have the values of
26 Pythagorean tone ratios.^{8} This prescription by the Godnames ELOHA with number value
36 assigned to Geburah and YAHWEH with number value 26
assigned to Chokmah reflects its status in Church music as the Dorian Mode, labelled Mode 1
to indicate that it is the scale from which all the other scales (ending with Mode 7, the
mathematically most harmonious Pythagorean scale) can be generated by shifting one interval
at a time. Furthermore, the D scale is the only one with 10 intervals with nonPythagorean
values.^{9} As the seven octave species have 66 nonPythagorean
intervals,^{10} the eight DD' scales have (66+10=76)
nonPythagorean intervals. This is how the Godname YAHWEH ELOHIM with number value
76 prescribes the set of eight Church musical modes: it measures all its
nonPythagorean intervals.
Excluding the interval 1 between a note and itself, the seven different
scales D–C have 130 Pythagorean intervals.^{11} Excluding their seven octaves, they have 123 Pythagorean
intervals. The eight notes of the D scale have 26 Pythagorean intervals.
They include eight ‘1’s and one '2', that is, (26–8–1=17) such intervals
between 1 and 2. The number of intervals between 1 and 2 in the eight scales D–D' = 123 +17
+ 76 = 216, which is the number value of Geburah.
48 of these are their notes between their tonic and octave, leaving
168 intervals between the notes of the eight D–D' scales above the
tonic. This is the number of Cholem Yesodeth, the Mundane Chakra of Malkuth.
It is the superstring structural parameter discussed in many previous articles. They are
symbolised in the first six enfolded, regular polygons as the 168 yods
within them other than their 26 corners (Fig. 9), which, as we saw in Fig. 7, symbolise notes of the eight
10
scales. They are symbolised in the spherical representation of the eight
D–D' scales as the 168 intervals between the seven notes above the tonic in
each scale.
We found in Article 16^{12} that the seven scales D–C have 84 rising, Pythagorean intervals that
are repetitions of the six Pythagorean notes between the tonic and the octave. They have 84
falling intervals as their counterparts, making 168 in total. They are
symbolised by the 84 yods along the edges of the first six enfolded polygons and their 84
mirror images in the other set of polygons (Fig. 10). Mirror symmetry of corresponding corners denotes the difference
between a rising and a falling interval. In terms of
notes of the C scale, the composition of the 84 rising, Pythagorean
intervals is
D E
F
G A
28 11 23 17
5
Therefore, there are (2×(28+11) = 78) rising and falling seconds and thirds
and 2×(23+17+5) = 90 rising and falling fourths, fifths and sixths (the seventh never appears
more than once in a scale). 78 is the number value of the Kabbalistic word ‘Cholem’ and 90 is
the number value of the word ‘Yesodeth.’
The seven different scales have 12 basic notes with different tone ratios
between 1 and 2. All the intervals between notes in these scales have values found in this set.
Hence, there are in the eight scales (216–12=204) repetitions of these notes
and intervals. Including the tonic and the octave, the 50 notes of these
scales arranged as in Fig. 8 and starting with the same note have 206 notes/intervals other than
the basic set. Of these, 136 are Pythagorean notes/intervals and 70 are nonPythagorean
notes/intervals. It was shown in Article 29^{13} that the triakis tetrahedron, the simplest Catalan solid, has 136
vertices, edges and triangles surrounding its centre when its internal triangles are
regarded as single tetractyses and its faces are divided into three tetractyses. There are
as many geometrical elements surrounding this polyhedron as there are Pythagorean notes and
intervals in the eight scales in addition to its basic set. The reason for this
correspondence is that both the eight D–D' musical scales and the triakis tetrahedron
conform to the archetypal pattern of the Tree of Life, as proved for the latter in Article
29, and therefore must display properties that numerically correspond to each other.
Ten Pythagorean notes (tonic, octave & two identical sets of the four
notes E, G, A & B of the D scale) lie on the central axis of the spherical representation
of the eight D–D' scales. Hence, the 206 intervals consist of (136–10=126) Pythagorean
intervals of the six scales other than the two D scales and 80 intervals made
up of 70 nonPythagorean intervals and 10 Pythagorean notes arranged along the axis. As,
according to Table 2 of Article 29, the triakis tetrahedron has 132 geometrical elements
surrounding its axis, six of which are its vertices, 126 other vertices, edges and triangles
surround its axis.
11
Once again, correspondence appears between different features of each
system.
The counterpart in the triakis tetrahedron of the fact discussed above that
168 intervals exist between the notes above the tonic of the eight D–D' scales
is that, when its internal triangles are divided into three tetractyses, it has
168 geometrical elements surrounding its axis. As the Pythagoreans taught,
“music is geometry.”
As commented upon earlier, the 14 different notes of the seven musical
scales consist of seven pairs of notes and their complements:
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 ordering is in terms of increasing size of the first member of each
pair). Let us write, firstly, the octave 2 as ‘1’ (using boldface to remind
ourselves that 1 is not the number 1), secondly, each note of one set as
a_{i} (i = 1–7) and, thirdly, its complement as a_{i}^{–1}. The product
of an interval and its complement may then be expressed as:
a_{i}a_{i}^{–1} = 1.
The complement of a given note is as far below the octave as that note is
above the tonic — they are equally spaced from each end of the scale (Fig. 11). The complement can be regarded as the inverse of a note in the
sense that their intervals span the octave. A rising musical interval a = n (n>1) has a
complement with interval 2/n, not 1/n, which is the value of the falling interval.
Every member of the set of 14 notes has a complement, or inverse, which is also a member of
that set. As the D scale is unique in having a pattern of intervals that is identical to its
mirror image, this means that it alone amongst the scales is made up of four pairs of notes
and their complements:
In terms of the Pythagorean tone T and leimma L, the seven pairs of notes
making up the musical scales have the form:
1 
1 
T^{5}L^{2} (= 2) 
2 
L 
T^{5}L 
3 
T 
T^{4}L^{2} 
4 
TL 
T^{4}L 
5 
T^{2} 
T^{3}L^{2} 
6 
T^{2}L 
T^{3}L 
7 
T^{2}L^{2} 
T^{3} 
12
Eight notes are Pythagorean and six notes (written above in red) are
nonPythagorean. By arranging the intervals between successive notes of the Pythagorean scale
on a circle (Fig. 12), all possible scales are the sequences of seven intervals generated
by starting at successive intervals until all of them have been selected. Their notes and
intervals are all possible different combinations of consecutive intervals. They
have the forms listed above. The complement of any note represented by an arc of the circle
in
Fig. 12 is that created by the opposing arc that completes the circular
pattern of intervals. Just as every colour has its complementary colour, so, too, do the
notes.
The eight notes of a musical scale have ^{8}C_{2} = 28
intervals. They include the seven notes above the tonic. There are 21
intervals between these notes (Fig.13). The eight DD′ scales have
(21×8=168) such rising intervals between their (8×7=56)
notes:
Similarly, there are 168 falling intervals. They are
symbolised in the inner Tree of Life by the 168 yods in each set of the first
six enfolded polygons (Fig. 14), whose corners denote the 50 notes of the eight
D–D' scales arranged on the surface of the sphere and along its axis, as discussed earlier
and shown in Fig. 8.
13
This remarkable analogy between the intervallic composition of the eight
church musical modes and the inner Tree of Life would remain valid whatever set of eight scales
was considered. The eight scales starting with the D scale have been chosen because: 1. it was
the Dorian mode that was, historically speaking, made the first of the Authentic Modes, and 2.
starting with the D scale means that the Pythagorean scale is the last different scale that can
be generated from it — a sequence that is natural in view of its special significance as the
most mathematically harmonious scale and its historical importance as the basis of Western
musical theory. The interval composition of the scales does, of course, depend upon which set
of eight scales is considered.
The seven pairs of notes and their complements consist of two
pairs of Pythagorean notes (tonic/octave & perfect fourth/fifth) and five
pairs of notes that each includes at least one nonPythagorean note. The 2:5 division is
another example of the basic
pattern that starts in the seven scales, each with five tone intervals T
with magnitude 9/8 and two leimmas L with magnitude 256/243.
The last of the regular polygons in the inner Tree of Life is the dodecagon.
As the tenth regular polygon, it represents the completion of the Pythagorean paradigm of
wholeness symbolised by the tetractys. When its 12 sectors are each divided into three
tetractyses, it contains 181 yods (Fig. 15). 13 yods are at the corners and centre of the dodecagon and
168 yods are generated by this transformation. The 12 types of musical
notes between the tonic and the octave in the seven types of scales can be
assigned to its corners, with the octave assigned to its centre. The
168 rising intervals between them can be assigned to the 168
yods. The 168 yods in the other dodecagon symbolise the 168
falling intervals. This demonstrates the remarkable connection between the sacred geometry of
the inner Tree of Life and the musical potential of the eight scales.
As
13^{2} – 1 = 168 = 3 + 5 + & + … + 25,
168 is not only the number of yods in a dodecagon other
than its centre and corners but also the sum of the first 12 odd integers that can be assigned
to its corners (Fig. 16).
14
This shows the arithmetic, geometric and musical connection between the
numbers 12 and 168. It appears in the first four Platonic solids ending with
the icosahedron with 12 vertices as the average number of yods needed to build them out of
tetractyses.^{14}
Inspection of Tables 3, 4, 5, 7, 8 & 9 in Article 16^{15} shows that the C, E, F, G, A & B scales have (including octaves)
112 Pythagorean intervals and 56 nonPythagorean intervals, that is,
168 intervals. Once again, the number value of Cholem Yesodeth
measures the number of intervals — this time for the six scales other than the D scale and
including octaves.
2. Connection
between the 144 Polyhedron and the eight scales
We found earlier that the eight D–D' scales have 216 intervals other than
octaves between their 64 notes. We also found that they have 12 types of
notes other than the octave and 168 intervals between notes
above the tonic, totalling 180 intervals other than the octave. Compare this with the
facts that the 144 Polyhedron has 216 edges and that the disdyakis triacontahedron has 180 edges (Fig. 17). In the first case, there are (8×6 + 168 =
216) intervals. The (8×6=48) notes between the tonic and
octave of
the eight scales contain the basic set of 12 notes, that is, there are
36 repetitions of some of these notes, where 36 is the
number value of ELOHA, Godname of Geburah. These repetitions are represented by the
36 extra edges of the 144 Polyhedron. We see that the edges of the 144
Polyhedron represent all possible intervals other than octaves between the notes of the
eight Church musical modes and that the edges of the disdyakis triacontahedron represent the
basic set of 12 intervals and their 168 repetitions:
15
The disdyakis dodecahedron that generates the 144 Polyhedron is itself
generated from the rhombic dodecahedron by attaching pyramids to its 12 rhombic faces
(Fig. 18). This increases the number of vertices from 14 to
(14+12=26), the number of edges from 24 to (24+12×4=72)
and the number of faces from 12 to (12×4=48). The change in the number of
edges: 24→(24+48=72)→(72+144=216), has
its counterpart in the eight Church modes, whose 216 intervals below the
octave comprise the 48 notes between the tonic and octave, (8×3=24)
intervals between the octave and the second, third and fourth notes in each scale, and 144
other notes (Fig. 19).
Notice that the 216 intervals split also into the
48 notes and 168 other intervals, where 168
is the number value of Cholem Yesodeth, the Mundane Chakra of Malkuth and a basic
structural parameter of the E_{8}×E_{8} heterotic superstring. The number
48 is represented in the 144 Polyhedron as the 48 peaks of
the tetrahedra that have to be added to the faces of the disdyakis dodecahedron in order to
create it. The 24 notes and their 24 complements correspond to the 24 peaks and their 24 mirror
images. The latter originate in the four faces of a pyramid that is attached to each of the
(6+6) faces of the rhombic dodecahedron. This means that each set of 24 peaks is six sets of
four. Similarly, the 24 notes and their 24 complements can be regarded as six sets of eight
notes, one note from each of the four Authentic Modes and one note from each of the four Plagal
Modes.
The 48 notes are copies of the basic set of 12 notes
between the tonic and octave of the seven types of musical scales:
256/243 9/8
32/27 81/64
4/3 1024/729
729/512 3/2 128/81 27/16 16/9 243/128
Eight notes comprise four pairs of Pythagorean notes and their
nonPythagorean complements. These four pairs can be assigned to eight of the 12 vertices of an
icosahedron denoting these notes (Fig. 20). These are the 12 B vertices of the
16
disdyakis triacontahedron. The eight vertices are those of an
octahedron.^{1}
Including the tonic and the octave, the seven scales contain eight different
Pythagorean notes and six different nonPythagorean notes. The former cannot be assigned to an
octahedron because they do not consist of four pairs of notes and their complements. This means
that its vertices must denote four pairs of notes and their complements, where each
pair is a Pythagorean note and a nonPythagorean note. The four other notes comprise a pair of
Pythagorean notes (perfect fourths & fifths) and a pair of nonPythagorean notes:
Each diametrically opposite pair of B vertices denotes a note and its
complement. The inversion symmetry of the icosahedron has its musical counterpart in the fact
that the complement of any note in the seven different musical scales with tone ratio n is one
with tone ratio 2/n that is also one of these notes. Notice that the four Pythagorean notes
that can be assigned to four vertices of an octahedron: 9/8, 27/16, 81/64& 243/128, are
each separated by a perfect fifth. Similarly for their respective complements 16/9, 32/27,
128/81 & 256/243. The four remaining B vertices of the icosahedron signify the pair of
Pythagorean notes (4/3, 3/2) & the pair of non Pythagorean notes (1024/729, 729/512) at the centre of the basic set of 12 different
notes making up the seven different musical scales:
256/243 9/8
32/27
81/64 4/3 1024/729 729/512
3/2 128/81
27/16 16/9 243/128
The 30 Pythagorean notes of the eight D–D' scales between the tonic and the
octave are assigned to the 30 A vertices of the disdyakis triacontahedron and its 18
nonPythagorean notes, the tonic and the octave are assigned to the 20 C vertices of the
dodecahedron.
The musical correspondence is summarised below:
________________________________
^{1}It may be relevant that an octahedron with eight faces has
48 rotational and mirror symmetries.
17
3. Comparison
with the Tree of Life
Given that previous articles have demonstrated that the disdyakis triacontahedron is the
polyhedral form of the inner Tree of Life, it should come as no surprise that other aspects of
this form should appear in the musical context, as now explained.
16 separate triangles combine by joining 41 of their edges to create the
outer form of the Tree of Life with 10 vertices of 16 triangles with 22 edges (Fig. 21). 64 geometrical elements (38 vertices,
26 edges) disappear in this union, where 64 =
4^{3}. It comprises 48 geometrical elements made up of 10 vertices,
22 edges and 16 triangles. As the geometrical representation of the integers 1, 2, 3 &
4, the ‘trunk’ of the Tree of Life consists of the point (Kether), line (ChokmahBinah),
triangle (ChesedGeburahTiphareth) and tetrahedron (NetzachHodYesodMalkuth). Fig. 22 shows that it is
18
19
made up of 26 geometrical elements. This trunk is
prescribed by the Godname YAHWEH with number value 26, its letter values
denoting different groups of elements. The counterpart of the 48 elements in
the inner form of the Tree of Life is the 48 corners of the seven regular
polygons. The counterpart of the trunk in the inner Tree of Life is the set of the first five
polygons, which have 26 corners.
Transformed into tetractyses, the 19 triangles of the lowest Tree of Life of
any set of overlapping Trees of Life contain 80 yods. There are
48 yods up to the level of the path joining Geburah to Chesed — the first
Sephirah of Construction (Fig. 19). This further demonstrates that 48 fundamental
degrees of freedom are needed to express just that part of the Tree of Life that contains
the seven Sephiroth of Construction. This number is the basic structural
parameter of the Tree of Life — whether geometrical
elements in the case of its outer form or corners of polygons in the case of its
inner form. It quantifies whatever holistic system is designed according to this
blueprint. In the case of the eight Church musical modes, the number 48 is
the number of notes
between the tonic and octave. It should therefore come as no surprise that
it figures so prominently in the construction of the geometry of the disdyakis triacontahedron.
We saw earlier that, if the 12 notes of the eight musical scales are assigned to the corners of
the dodecagon, the extra 168 yods needed to construct it from 12 tetractyses
symbolise the 168 intervals between their notes. Thus formed, its centre is
surrounded by 12 vertices, 24 edges and 12 triangles, that is, 48 geometrical
elements are needed to construct the dodecagon, starting from a point. Once again, the number
48 defines the minimal number of bits of information required to
create a complete Tree of Life pattern.
When its faces are divided into three tetractyses, the simplest Platonic
solid — the tetrahedron — has 48 hexagonal yods in its 12 tetractyses
(Fig. 25). The basic building block of polyhedra is composed of
48 yods that symbolise the seven Sephiroth of Construction. Any system
whose structure conforms to the universal blueprint of the Tree of Life must necessarily
possess this characteristic number of degrees of freedom.
We found in Article 27 that, when its interior triangles and its faces are
each divided into three triangles, the disdyakis triacontahedron comprises 2400 vertices, edges
& triangles.^{16} As
49^{2} – 1 = 3 + 5 + 7 + … +
97,
this number is the sum of the first 48 odd integers after
1. Yet again, the number 48 defines arithmetically the very geometrical
composition of the disdyakis triacontahedron — the polyhedral manifestation of the inner Tree
of Life.
Just as the rhombic triacontahedron underlies the disdyakis triacontahedron,
so the Catalan solid with the next larger number of faces — the disdyakis dodecahedron — is the
foundation of the 144 Polyhedron, which is the result of adding tetrahedra to its
48 faces with 26 vertices. In fact, it may be seen as the
‘trunk’ of this polyhedron. Just as YAHWEH (YHVH) with number value 26
prescribes the trunk of the Tree of Life because the latter comprises 26
geometrical elements (Fig. 22) and the disdyakis dodecahedron because it has 26
vertices, so YAH (YH) with number value 15 marks out this member of the
family of 26 Archimedean and Catalan solids. This is because,
20
starting from the simplest Catalan solid — the triakis tetrahedron — and
counting back and forth between pairs of dual polyhedra, the disdyakis dodecahedron is the
15th polyhedron (see Fig. 24), whilst, starting from the simplest Archimedean solid, the
26th solid is the disdyakis triacontahedron, excluding
chiral partners. The Godname YAHWEH therefore determines
both polyhedra in the polyhedral Tree of Life whilst YAH prescribes the
family of 15 Archimedean solids, the family of 15 Catalan
solids and
the disdyakis dodecahedron as the 15th in the two families,
including their chiral versions.
When the outer Tree of Life is projected onto the plane containing its inner
form (Fig. 26), eight of the 70 corners of the two sets of enfolded polygons
coincide with the projections of Sephiroth and Daath. The 62 unshared
corners represent new degrees of freedom. They are the 62 vertices of the
disdyakis triacontahedron. The hexagon is the fourth regular polygon. Constructed from
tetractyses, a pair of separate hexagons is made up of 74 yods (Fig. 27). They symbolise the 74 vertices of the 144 Polyhedron.
Its 48 ‘new’ vertices correspond to the 48
yods on the boundaries of the pair of hexagons and at the centres of their sectors, whilst its
26 other vertices (vertices of the underlying disdyakis dodecahedron)
correspond to the 26 radial yods. Joined together in the inner Tree of Life,
the pair of hexagons has 62 yods other than those shared with the outer Tree
of Life. They symbolise the 62 vertices of the disdyakis triacontahedron. The
12 yods at the centres of each sector of the hexagons denote its 12 B vertices, the 20 yods
lying on their boundaries symbolise the 20 C vertices and the 30 radial yods symbolise its 30 A
vertices.
The isomorphism between the pair of hexagons and the disdyakis
triacontahedron provides a natural representation of the 12 different notes in the seven
musical scales
21
because we found earlier that they correspond to the 12 B vertices of the
icosahedron. The six yods at centres of sectors in one hexagon (Fig. 28) denote (in order of increasing tone ratios) the six notes 256/243,
9/8, 32/27, 81/64, 4/3 & 1024/729 and their mirror images in the other hexagon denote
the complements of these notes. The 30 Pythagorean notes in the eight D–D' scales are
symbolised by radial yods and the 20 notes consisting of the tonic, octave and 18
nonPythagorean notes are represented by the 20 boundary yods.
That this parallelism between the pair of hexagons and the disdyakis
triacontahedron is not merely due to coincidence is indicated by the fact that they require
168 more yods to construct their sectors from three tetractyses (Fig. 29). As discussed earlier (see Fig. 10), this is the number of rising and falling, repeated
intervals between the notes between the tonic and the octave of the seven different types of
musical scales.^{17} The 84 yods associated with one hexagon denote their 84 rising
intervals and the 84
yods in the other hexagon denote their 84 falling intervals.
4. The holistic character of the number 33 The
62 vertices of the disdyakis triacontahedron are arranged in either
seven sheets perpendicular to an AA axis, 11 sheets perpendicular to a BB axis or
15 sheets perpendicular to a CC axis. The correspondence between
this pattern and the note composition of the seven different musical scales was mapped
out in Article 26.^{18} It was pointed out in Section 1 that they comprise
26 Pythagorean notes between the tonic and the octave and seven
octaves, that is, 33 Pythagorean notes. The tonal composition of the former is:
The letter values of YAHWEH denote the numbers of notes of one or two types.
Alternatively, the 26 Pythagorean notes between the tonic and the octave may
be
22
divided up in terms of the seven musical scales/church modes listed in
Table 1:
However the composition of the 26 Pythagorean notes is
decided, the crucial point is that the 7:15:11 pattern of sheets of vertices
is identical to the pattern of different notes in the seven different scales, the
seven sheets of vertices corresponding to their seven octaves. The parallelism exists because
both systems — one geometrical, one musical — are Tree of Life patterns that are prescribed by
Divine Names such as YAH and YAHWEH. The number 33 is always a measure of a whole.
The same pattern can also be seen in the building up of the regular and
semiregular polyhedra. The first stage of creation of form is the straight line, the second
stage is the triangle and the third stage is the tetrahedron, the simplest of the five regular
polyhedra, or Platonic solids. The dodecahedron, the last of the Platonic solids, is formed at
the seventh stage and the disdyakis triacontahedron, the last of the 26
semiregular polyhedra, represents the 33rd stage:
23
Listed above are, firstly, the Archimedean solids and then the Catalan
solids. Just as it exhibits 33 sheets of vertices, so the disdyakis triacontahedron is the 33rd
stage in the generation of the semiregular polyhedra. The 15th stage after
the seventh generates the rhombic dodecahedron with 26 vertices & faces,
36 edges & faces and 50 vertices, edges & faces. With
its geometry prescribed by four Godnames (YAH = 15, YAHWEH =
26, ELOHIM = 50 and ELOHA = 36), it should
not be surprising that this polyhedron plays a fundamental role in the generation of the 144
Polyhedron. Sticking rhombic pyramids to its 12 faces creates the disdyakis dodecahedron with
48 faces and 26 vertices (shown above linked by a red arrow
to the rhombic dodecahedron). It is this polyhedron that is fundamental to the
generation of the 144 Polyhedron because it generates the latter in a way analogous to how the
rhombic triacontahedron creates the disdyakis triacontahedron. Sticking tetrahedra to its
48 faces creates the 144 Polyhedron with (48+26=74) vertices
and (3×48=144) faces, whilst adding pyramids to the faces of the rhombic
triacontahedron generates the disdyakis triacontahedron.
It was shown in Article 24^{19} that the geometry of the Tree of Life generates the rhombic faces of
the rhombic dodecahedron and the golden rhombic faces of the rhombic triacontahedron.
Confirmation that the rhombic dodecahedron is, as the 15th
stage in the generation of the disdyakis triacontahedron, the polyhedral root of the polyhedral
Tree of Life is provided by the letter values of YAHWEH. In the last 11 stages, the rhombic
triacontahedron is the fifth stage and the disdyakis dodecahedron is the sixth stage and the
first of the last five stages. The rhombic dodecahedron is picked out by the first two letters
of YHVH and the two polyhedra that determine the two polyhedra in the polyhedral Tree of Life
are selected by the number values of the last two letters H and V of YHVH:
Notice also that the disdyakis dodecahedron is 21st in the
list of semiregular polyhedra and so is prescribed by the EHYEH, Godname of Kether with number
value 21. The 21st stage marks the triakis tetrahedron, the
simplest of the Catalan solids. It was shown in Article 29^{20} that, if the holistic disdyakis triacontahedron is likened to the
tetractys — the Pythagorean symbol of holistic systems — the triakis tetrahedron is its
‘yod.’ This is because the geometrical properties of the former are exactly ten
times the corresponding one for the latter. The triakis tetrahedron is built from 137
vertices, edges & triangles, where 137 is the 33rd prime number and the number known to
physicists to define the fine structure constant e^{2}/ħc ≈ 1/137. The number 33 not
only determines the stage of generation of the disdyakis triacontahedron — the last Catalan
solid — but also the geometrical composition of the first Catalan solid, whose properties
are multiplied tenfold in the latter!
The reason why 33 appears in the Pythagorean note composition of the seven
musical scales and in the mathematical description of the disdyakis triacontahedron is that
both
24
are holistic systems whose components are 33 in number. Or, better, that they
are manifestations of a universal cycle that requires 33 stages to complete itself. This
is demonstrated par excellence in the tree level structure of ten overlapping Trees of
Life, each representing one of the ten Sephirah of the Tree of Life. Fig. 30 indicates that the latter has seven horizontal divisions marking the
stages of emanation of Sephiroth of Construction. Although Daath is not a Sephirah, this
point in the Tree of Life is Yesod of the next higher, overlapping tree and therefore counts
as a division unless there is no higher tree. These divisions are called ‘tree levels.’
Fig. 31 shows that ten overlapping Trees of Life have 33 tree levels (in
general, n trees have (3n+3) tree levels). They measure the complete cycle of emanation of
10 Sephiroth. In an abstract sense, the number 33 denotes the number of stages in the
completion of a cycle of development of a system that culminates in an object that fully
embodies the universal blueprint of the Tree of Life. The relationship between the 33 vertex
sheets of the disdyakis triacontahedron and the 33 tree levels of 10 overlapping Trees of
Life was analysed in Article 25.^{21}
Also discussed there is what the 7:15:11 vertex sheet
composition of the disdyakis triacontahedron means for superstring physics. The 10dimensions
of spacetime that are predicted by superstring theory are mapped by 10 overlapping Trees of
Life. The Godname ADONAI (ADNI)
25
26
assigned to Malkuth prescribes the lowest 10 trees because its number value
65 is the number of their Sephirothic emanations. Fig. 31 shows how the letter values of ADNI specify different types of
emanations. The number 33 appears here because 65 is the 33rd odd integer
after 1. Once again, it quantifies a holistic system, because each tree represents one of
the Sephiroth.
Counting from the tonic of the first octave of the Pythagorean scale, the
tone ratio 24 is the 33rd note and the perfect fifth of the fifth octave (Table 5). Counting from the latter, the 33rd note is 576 =
24^{2} and still the perfect 5th of the new fifth octave. This is the
65th note from the tonic of the first octave. The Godname Adonai with
number value 65 prescribes sequences of 33 notes in the Pythagorean scale
whose last note has a tone ratio always 24 times that of the first note. This does not
depend on which note is the starting point. Any sequence of 33 notes ends in one whose
frequency is 24 higher than that of the first, thus demonstrating the cyclic character of
this number.
Table 5. Every 33rd note in the Pythagorean scale increases in pitch by a
factor of 24.

C

D

E

F

G

A

B

Number of
overtones

1

1

9/8

81/64

4/3

3/2

27/16

243/128

0

2

2

9/4

81/32

8/3

3

27/8

243/64

2

3

4

9/2

81/16

16/3

6

27/4

243/32

4

4

8

9

81/8

32/3

12

27/2

243/16

7

5

16

18

81/4

64/3

24

27

243/8

11

6

32

36

81/2

128/3

48

54

243/4

15

7

64

72

81

256/3

96

108

243/2

20

8

128

144

162

512/3

192

216

243

26

9

256

288

324

1024/3

384

432

486

32

10

512

576

648

2048/3

768

864

972

38

11

1024

1152

1396

4096/3

1536

1728

1944

39

The significance of the perfect 5th of the fifth octave is that it is the
last of eight successive fifths (not all perfect fifths):
G_{1}

D_{2 }

A_{2}

E_{3 }

B_{3}

F_{4}

C_{5}

G_{5}

3/2

9/4

27/8

81/16

243/32

32/3

16

24

(Subscripts denote the octave number)

In general, every 33rd note is every eight successive fifth. Table 5 indicates that the 33rd note is the tenth overtone. 22 of
the notes up to the perfect 5th of the fifth octave are fractional. They comprise 16 notes
in the first three octaves (separated from the rest of the table by a thick line) and six
notes in the fourth and fifth octaves up to the last fifth. This 6:16 division corresponds
in the Tree of Life to the six Paths that are edges of the tetrahedron whose vertices are
the four lowest Sephiroth and to the 16 Paths outside it (Fig. 33). The 32 fractional notes and overtones up to G_{5} conform
to the geometrical pattern of the Tree of Life, the ten overtones corresponding to the
ten
27
Sephiroth and the 22 fractional notes corresponding to the 22 Paths that
connect them. The ordering of notes in Fig. 33 follows the traditional Kabbalistic numbering of Paths. As the tenth
overtone, G_{5} with tone ratio 24 corresponds to Malkuth, the lowest Sephirah,
signifying the physical manifestation of any holistic system conforming to the Tree of Life
blueprint. The significance of this number to superstring theory is discussed in Article
12.^{22}
The number of corners of the 7n regular polygons enfolded in n overlapping
Trees of Life is
C(n) = 35n + 1,
where “1” denotes the uppermost corner of the hexagon enfolded in the tenth
tree (the hexagon is the only one of the seven regular polygons that shares a corner with
polygons enfolded in adjacent trees. The shared corner is denoted by ‘1’ in the above formula).
Therefore, C(10) = 351 = 1 + 2 + 3 + …. + 26. This shows how
YAHWEH with number value 26 prescribes the inner form of 10 overlapping Trees
of Life. 351 is the number value of Ashim (“Souls of Fire”), the
Order of Angels assigned to Malkuth. The number of corners of the 7n polygons outside their n
root edges = C'(n) – 2n = 33n + 1. C'(n+1) – C'(n) = 33. In other words, there are 33 corners
per set of polygons outside
28
their root edge (Fig. 34). The emanation of successive Trees of Life generates 33 new
geometrical degrees of freedom associated with their inner form. This again
demonstrates the cyclic aspect of the number 33 in measuring the number of entities
(polyhedra, musical notes, etc) in a sequence that represents a holistic
system conforming to the archetypal pattern of the Tree of Life.
When its triangles are turned into tetractyses, the number of yods in the
lowest n overlapping Trees of Life is given by
Y(n) = 50n + 30.
Therefore, Y(33) = 1680. As discussed in many previous articles, this is the
number of turns in each helical whorl of the UPA (Fig. 35), the indivisible unit of matter described by Annie Besant and C.W.
Leadbeater 114 years ago with the aid of a yogic siddhi.
29
In other words, the lowest 33 Trees of Life are made up of as many yods as
there are circularly polarized waves in a whorl. This is how the number 33 determines the
structural parameter of superstrings inside atomic nuclei. Malkuth of the 33rd Tree of Life is
the 65th emanation on the central Pillar of Equilibrium, showing how ADONAI,
the Godname of Malkuth with number value 65, prescribes this number.
As pointed out in Article 22,^{23} there are seven steps leading to the generation of the
disdyakis triacontahedron, starting with the tetrahedron. The first five are the Platonic
solids and the sixth is the rhombic triacontahedron, the Catalan solid that is compounded
from the dodecahedron and the icosahedron. When constructed from tetractyses, the numbers of
hexagonal yods in their faces are listed below:
As discussed earlier, the disdyakis triacontahedron is the 33rd stage in the
cycle of development of regular and semiregular solids. Once again, the number 33 is
associated with the superstring structural parameter 1680. The five Platonic solids and the two
semiregular solids in this list are the polyhedral counterpart of the five whole intervals and
two leimmas in the Pythagorean musical scale:
T T L
T T T
T L
The 62 vertices of the disdyakis triacontahedron define the
vertices of 28 regular (Platonic) and semiregular (Catalan) solids: ten tetrahedra (two
vertices at each C vertex), five cubes (two vertices at each C vertex), five octahedra (one
vertex at each A vertex), one icosahedron (one vertex at each B vertex), one dodecahedron (one
vertex at each C vertex), five rhombic dodecahedra (two vertices at each C vertex and one
vertex at each A vertex) and one rhombic triacontahedron (vertices at B and C vertices).
Constructed from tetractyses, the hexagonal yod populations of the faces of the 28 Platonic
& Catalan solids definable within the disdyakis triacontahedron are:
There are 21 Platonic solids of the four types that the
ancient Greeks believed were the shapes of particles of the elements Earth, Water, Air and
Fire. Their hexagonal yod population is 1680 — the same as the number of hexagonal yods in the
seven polyhedra listed above. This is remarkable evidence of beautiful, mathematical design.
EHYEH, the Divine Name associated with Kether at the top of the Tree of Life has the number
value 21. It prescribes this polyhedral embodiment of the superstring
structural parameter 1680, its letter values specifying the numbers of polyhedra of each
type:
30
The remaining three types of polyhedra also have 1680 hexagonal yods in
their faces, giving a total number of 3360. This is astounding for two reasons:
1. Divine Unity symbolised by the Pythagorean Monad, or mathematical point
("0thorder tetractys"), differentiates, firstly, into the familiar tetractys ("1storder
tetractys") with 10 yods (three corners, seven hexagonal yods), secondly, into the “2ndorder
tetractys” with 85 yods (15 corners, 70 hexagonal yods), and so on:
3360 is the number of yods in the seven enfolded, regular polygons
constituting the inner form of the Tree of Life when their 47 sectors are each turned into
2ndorder tetractyses (Fig. 36).^{24}
2. 3360 is the number of circularly polarised wave oscillations made during
each of the five revolutions of the 10 whorls of the UPA (see Fig. 35). Each yod making up the seven enfolded polygons constructed from
2ndorder tetractyses symbolises one oscillation made in the 10 whorls during one complete
revolution around the axis of the UPA. This is not coincidence but a demonstration of the
equivalence between the 2dimensional form of the inner Tree of Life — the seven enfolded
polygons — and its
polyhedral form — the disdyakis triacontahedron.
Binah, the third Sephirah in the Tree of Life and the one embodying the
cosmic feminine principle (Kabbalists call it Aima, the “Great Mother”), has a gematria number
value of 67. The number of yods in n overlapping Trees of Life is
Y'(n) = 50n + 20.
31
Therefore, Y'(67) = 3370. There are 10 yods in the
uppermost triangle whose base is the path joining Chokmah and Binah of the highest tree. Below
Binah of the 67th tree are 3360 yods. Amazingly, the hexagonal yod population
of the 28 polyhedra within the disdyakis triacontahedron is the number of yods below Binah of
the very number of trees that equals the number value of Binah!
5. The musical
counterpart of G_{2}
The octonions form the fourth (and last) class of division algebras. An octonion has the
form:
N = a_{0}e_{0} + a_{1}e_{1} +
a_{2}e_{2} + …. + a_{7}e_{7},
where the a_{i} (i = 0–7) are real numbers, e_{0} is the
identity element and the seven unit octonions e_{i} (i = 1–7) are imaginary numbers:
e_{i}^{2} = –1. Their multiplication is anticommutative:
e_{i}e_{j} = – e_{j}e_{i} (i≠j)
nonassociative:
e_{i}(e_{j}e_{k})
≠(e_{i}e_{j})e_{k}
and follows the rule
e_{i}e_{i+1} = e_{i+3}.
Every unit imaginary octonion has an inverse, or conjugate, e_{i}* =
–e_{i}, where e_{i}e_{i}* = 1. Similarly, the conjugate of an octonion
N is N*, where
N* = a_{0} – a_{1}e_{1} – a_{2}e_{2}
– … – a_{7}e_{7}.
The norm of an octonion is ││N││= √N*N, where
││N││^{2} = a_{0}^{2} + a_{1}^{2} +
a_{2}^{2} + … + a_{7}^{2}.
The inverse of N is N^{–1} ≡ N*/││N││^{2}, so that
N^{–1}N = NN^{–1} = 1. It was shown in Article 28 that there are 13 classes of
real, linear combinations of two or more unit octonions. It means that there are 13 classes of
octonions that are inverses to the former, so that YAHWEH with number value 26
prescribes the 26 classes of octonions and their inverses.
This pattern is analogous to the 13 Archimedean solids and their duals — the
13 Catalan solids. The counterpart of polyhedral duality in the world of octonions is the
32
relationship of an octonion N' = N^{–1} being the inverse of the
octonion N, where N'N = 1. Its counterpart in music is the relationship between a note of tone
ratio n and its complement of tone ratio m, where nm = 2. What is sufficiently remarkable in
the analogy to take it seriously is that:
1. just as there are 13 classes of octonions and 13 classes of their
inverses, so the seven musical scales are made up of 13 different notes above the tonic, that
is, they have 13 rising intervals and 13 falling intervals;
2. just as the seven imaginary octonions have seven inverses e_{i}*
= –e_{i}, so the 14 different notes of the seven scales are divided into seven notes
and their complements:
imaginary octonion

inverse

musical notes

complement

e_{1}

e_{1} ^{1}

tonic (= 1)

octave T^{5}L^{2} (= 2)

e_{2}

e_{2}^{1}

L

T^{5}L

e_{3}

e_{3}^{1}

T

T^{4}L^{2}

e_{4}

e_{4}^{1}

TL

T^{4}L

e_{5}

e_{5}^{1}

T^{2}

T^{3}L^{2}

e_{6}

e_{6}^{1}

T^{2}L

T^{3}L

e_{7}

e_{7}^{1}

T^{2}L^{2}

T^{3}

Moreover, just as the seven e_{i} form seven ordered sets of three,
or 3tuples: (e_{i}, e_{i+1}, e_{i+3}), where the
product of two ordered members is the third in sequences represented by the circle and six
lines of the Fano Plane shown in Fig. 37, so, too, the seven notes and their complements form seven different
musical scales. The analogy between the imaginary octonions and the musical intervals is not
perfect because multiplication of the former is noncommutative, whereas multiplication of
the values of the latter is commutative. However, it is not the imaginary octonions per se
that should be compared with musical intervals but, rather, the seven combinations of one,
two and three octonions in each 3tuple:
e_{i}, e_{i}_{+1},
e_{i}+3
e_{i}e_{i}_{+1},
e_{i}_{+1}e_{i}_{+3},
e_{i}_{+3}e_{i} (6
permutations)
e_{i}e_{i}_{+1}e_{i}_{+3}
(6 permutations)
Each 3tuple has 12 permutations of two or three imaginary octonions.
Similarly, as e_{i+3}e_{i+1}= – e_{i+1}e_{i+3} = –
e_{i} = (–e_{i+3})(–e_{i+1}) and e_{i1} = –e_{i}, then
(e_{i+3})^{–1}(e_{i+1})^{–1} = (e_{i})^{–1},
that is, the inverses of each octonion in a 3tuple also form a 3tuple with 12 permutations of
pairs or triplets. The seven octonions therefore have (7×12=84) permutations, as do
33
their seven inverses. Each permutation reduces to one of the octonions. The
seven 3tuples of octonions and the seven 3tuples of their inverses display
(84+84=168) distinct permutations of their members. Compare this with the
fact^{25} that the seven musical scales have 84 rising intervals between
their notes that are repetitions of the basic set of Pythagorean intervals, which means that
they also have 84 falling intervals, that is, intervals between two notes falling in pitch.
The number of rising intervals in each scale that are repetitions of the Pythagorean
intervals varies from scale to scale:^{26}
C scale

D scale

E scale

F scale

G scale

A scale

B scale

13

12

13

12

11

11

12

Total = 84.
Average = 84/7 =
12.
This is unlike the number of permutations of pairs and triplets of imaginary
octonions in each 3tuple, which is the same (12) in each one. In terms of notes, the
composition of the 84 Pythagorean intervals is
The interval 243/128 for note B and the interval 2 for the octave are absent
because they are the only ones that never appear more than once in a scale (in the former case,
the C, E, F & B scales).
The smallest of the five exceptional groups, the rank2 group G_{2}
has 14 roots. It has two simple roots (0,1,–1) and (1,–2,1) denoted by black dots in
Fig. 38 and 12 roots consisting of six pairs:
They form two nested Stars of David. The red star shown in Fig. 39 denotes the first triplet of roots and their inversions and the blue
star denotes the second triplet of roots and their inversions.
Compare this with the seven pairs of intervals and their complements between
notes in the seven musical scales:
1

T^{5}L^{2} (= 2)

L

T^{5}L

T

T^{4}L^{2}

TL

T^{4}L

T^{2}

T^{3}L^{2}

LT^{2}

T^{3}L

T^{2}L^{2}

T^{3}

The tonic and the octave play the role of the two simple roots and the six
pairs of intervals and their complements correspond to the six pairs of roots of G_{2}
and their inversions. This correspondence permits a nested Star of David representation of the
12 intervals (Fig. 38). Complementarity of intervals corresponds to spatial inversion of
roots of G_{2}. Diametrically opposite vertices of each star signify an interval and
its
34
complement. Whatever the assignment of intervals to star points, the members
of one triplet have tone ratios in the same relative proportions as those of the other triplet
because the latter are their complements with tone ratios that, being reciprocally related, are
in the same proportions, although reversed in order. The two red triplets (T, T^{2},
T^{3}) and (T^{2}L^{2}, T^{3}L^{2}, T^{4}L^{2}) have tone ratios in the relative proportions
1:T:T^{2}, i.e., those of the perfect 4th, the perfect 5th and the major 6th of the C
scale. What is nontrivial, however, is that the two blue triplets (L, LT, LT^{2}) and (T^{3}L,
T^{4}L, T^{5}L) have tone ratios in the same proportion.
If Mtheory is isomorphic to music even more than what previous articles
have revealed, this appearance of the same relative proportions between intervals represented
by different Stars of David may be telling us that subgroups of rank 3 such as SU(3) play a
role that extends beyond that already known for it as the gauge symmetry group governing the
colour force between quarks. Perhaps the analogy between the root structure of G_{2}
and the 14 basic notes of the seven musical modes is indicating that there is
a fundamental representation of SU(7) with 48 gauge fields
consisting of an SU(3) triplet, an SU(3) antitriplet and two SU(3) singlets?
The Klein Configuration is the {7,3} tiling in the hyperbolic plane of the
168 automorphisms of the Riemann surface of the famous ‘Klein Quartic’:
x^{3}y + y^{3}z + z^{3}x = 0.
As a surface of genus 3, the Klein curve is a Hurwitz curve having the
maximum number of automorphisms for a surface of genus 3. Figure 39 depicts these transformations as black hyperbolic triangles
distributed over 14 sectors of the Klein Configuration, 12 per sector, to create 24
heptagons with 168 triangular sectors that cover a 3torus (Fig. 40). As the 168 element PSL(2,7) of the Klein
Configuration is isomorphic to the symmetry group SL(2,3) of the Fano Plane that describes
octonion multiplication and as the 14dimensional G_{2} is the group of
automorphisms of the octonions, the 12 triangles in each of the sectors of the Klein
Configuration represent each of the 14 elements of G_{2}. These sectors correspond
to the 14 notes of the seven musical scales and their 168 triangles
correspond to the 168 repeated, rising and falling Pythagorean intervals
between these notes (or, alternatively, to the 168 intervals between the
notes above the tonic of the eight D–D' scales).
The (2,3) torus knot is a knot that winds twice around a 1torus as it makes
three
35
revolutions. It can wind on a 3torus as well. (Fig. 41). The former is the surface around which each whorl of the
UPA/heterotic superstring winds 1680 times. The latter is the surface in which
the Klein Configuration can be embedded. As the octonions provide a natural basis for
the Lie algebra of E_{8} and are connected through the Fano Plane to the Klein
Quartic, the (2,3) torus knot determines both the toroidal form and the winding
number 1680 of the E_{8}×E_{8} heterotic superstring.
Because it is wellknown to mathematicians that G_{2} is the
automorphism group of the octonions, perhaps it should not come as a surprise that the pattern
of basic intervals in the seven musical scales is similar to the root structure of
G_{2}, given that we have already demonstrated a correspondence between intervals and
the properties of octonions that is too rich to be merely
coincidental. Nevertheless, it is still remarkable that the
same mathematical structure should appear in the 8component
Table 6. Number values of the Sephiroth in the four Worlds.
Sephirah 
Title

Godname

Archangel

Order of
Angels

Mundane
Chakra

Kether 
620

21

314

833

636

Chokmah 
73

15,26

248

187

140

Binah 
67

50

311

282

317

Chesed 
72

31

62

428

194

Geburah 
216

36

131

630

95

Tiphareth 
1081

76

101

140

640

Netzach 
148

129

97

1260

64

Hod 
15

153

311

112

48

Yesod 
80

49

246

272

87

Malkuth 
496

65,155

280

351

168

36
octonions and in the seven 8note musical scales — even as far as the number
of their repeated rising and falling intervals being the order of the symmetry group SL(3,2)
for octonions! Moreover, the pattern of intervals of the musical scales is independent of the
pitch of a particular octave, remaining true whether the pitch of the tonic is several hundred
cycles per second or 10^{24} cycles per second — the frequency associated with subatomic
processes, although such fast vibrations would no longer amount to the kind of sound that is
detectable to the human ear, which is only sensitive to sounds spanning up to about ten
octaves. The remarkable analogy between the musical scales and octonions strongly suggests
that, far from being a now defunct musical scale that is of interest only to historians of
science and music, the Pythagorean musical scale shares with octonions a vital role in the
unified, holistic description of the real world. This is further strongly indicated by the
analogous structures of the polyhedral Tree of Life and the ancient musical modes of the
Roman Catholic Church. The universality of this system, whose 2dimensional counterpart is
the outer and inner forms of the Tree of Life, means that it has many different layers of
interpretation. Music is one such layer, and it provides insight into the general meaning of
the polyhedral Tree of Life.
References
^{1}Phillips, Stephen M. Article 14: “Why the Greek Musical Modes Are
Sacred,” (WEB, PDF), pp. 4–5.
^{2} Chappell, W. The History of Music, vol. 1 (London, 1874), pp.
50–51.
^{3} Ibid, p. 60.
^{4} Numbers in boldface are the number values of the Sephiroth, their Godnames,
their Archangels and Mundane Chakras, shown in Table 6 above. They are calculated by means
of 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 associated
with its letters.
^{5} Phillips, Stephen M. Article 16: “The Tone Intervals of the Seven Octave
Species and Their Correspondence with Octonion Algebra and Superstrings,” (WEB, PDF), pp. 9–12.
^{6} Phillips, Stephen M. Article 4: “The Godnames Prescribe the Inner Tree of
Life,” (WEB, PDF), pp. 4–5.
^{7} Ref. 5, p. 13.
^{8} Ibid. Compare Tables 3–9, pp. 9–11.
^{9} Ibid, p. 10.
^{10} Ibid, p. 13.
^{11} Ibid.
^{12} Ibid, pp. 17, 21.
^{13} 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. 4–5.
^{14} Phillips, Stephen M. Article 3: “The Sacred Geometry of the Platonic Solids,”
(WEB, PDF), p. 10.
^{15} Ref. 5, pp. 911.
^{16} Phillips, Stephen M. Article 27: “How the Disdyakis Triacontahedron Embodies
the Structural Parameter 1680 of the E_{8}×E_{8} Heterotic
Superstring,” (WEB, PDF), pp. 14–16.
^{17} Ref. 5, pp. 17, 21.
^{18} Phillips, Stephen M. Article 26: “How the Seven Musical Scales Relate to the
Disdyakis Triacontahedron,” (WEB, PDF), pp. 9–10.
37
^{19} Phillips, Stephen M. Article 24: “More Evidence for the Disdyakis
Triacontahedron as the 3dimensional Realisation of the Inner Tree of Life and its
Manifestation in the E_{8}×E_{8} Heterotic Superstring,” (WEB, PDF), pp. 6–7.
^{20} Ref. 13, pp. 3, 8.
^{21} Phillips, Stephen M. Article 25: “The 33 Vertex Sheets of the Disdyakis
Triacontahedron Signify the 33 Tree Levels of Ten Overlapping Trees of Life,” (WEB, PDF), pp. 1315, 20.
^{22} Phillips, Stephen M. Article 12: “New Pythagorean Aspects of Music and Their
Connection to Superstrings,” (WEB, PDF), pp. 24–32.
^{23} Phillips, Stephen M. Article 22: “The Disdyakis Triacontahedron as the
3dimensional Counterpart of the Inner Tree of Life,” (WEB, PDF), p. 14.
^{24} For proof, see ref. 4 in ref. 5.
^{25} Ref. 5, pp. 17–18.
^{26} Ibid, p. 21.
38
