by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH.
England.
Website: http://www.smphillips.mysite.com
“Change has an absolute limit:
This produces two modes;
The two modes produce four forms,
The four forms produce eight trigrams;
The eight trigrams determine fortune and misfortune."
Confucius
(commentary on the I Ching) 
“I am the One who becomes Two,
Who becomes Four,
Who becomes Eight,
And then I am One again.”
Coffin of
Petamon
(Ancient
Egyptian text)

Abstract
The eight trigrams of the ancient Chinese divinatory
system of I Ching represent stages or phases of universal cycles. In
Kabbalah, they correspond to Daath and the seven Sephiroth of Construction
of the Tree of Life. Their generation from the interaction of yang and yin
leads to their natural division into two sets of four. This reflects the
ancient view of the number 8 as 2×4. An example of such doubling is how, as
the product of two identical groups E_{8}, the gauge symmetry group
E_{8}×E_{8} of heterotic superstrings has a dimension 496
that is embodied in an octagon, whose eight corners are those of two squares
rotated through 45°. This geometrical property has an arithmetic counterpart
that is manifested in the sacred geometry of the Tree of Life. Another
example of the universal 4:4 pattern in nature are the eight Gregorian
musical modes, which consist of four authentic modes and four plagal modes.
These eight octave species constitute a cycle of successive sequences of
seven intervals spanning the Pythagorean scale. Moreover, comparison between
these patterns and their trigram counterparts indicates that the only
ordering of musical scales consistent with I Ching is the traditional Church
one, although their ancient Greek names are wrong. Comparison of the average
distances of the planets predicted by the author’s modified TitusBode Law
with the population of yin/yang lines in the I Ching table indicates that
the octet of planets Mercury–Uranus splits into two groups of four, the
distances spanned by each set being geometrically embodied in the inner form
of the Tree of Life. Other examples of eightfold cycles include the eight
stages in the cycle of the cell leading to its division into two daughter
cells.

1
1. I Ching’s origins and the Tree of
Life
When the historical origins of any religion are examined, it is often
difficult (if not impossible) to separate fact from mythology. However, as in the case of
the ancient oracular system known in the West as I Ching, the two may sometimes coincide.
The scholar Kung MganKwoh in the second century B.C.E credited Fu Hse, the first Chinese
emperor, with its discovery in the following legend: as he was meditating one day by the
River Ho (the Yellow river), Fu Hse noticed a tortoise climb out of the river. Examining the
pattern of lines on the creature’s scales, he had a flash of insight. The shell of a
tortoise (Fig. 1) consists of tessellating hexagons, each one surrounded by six others.
Two hexagons are arranged centrally on the top of the shell and are
surrounded by eight other hexagons.
According to Taoist philosophy, yang and yin, respectively, the positive,
active and negative, reactive phases of nature’s great cycles, emerged from Tai Yi, the
“Great Unity,” symbolized by a circle. As opposite states represented by the positive (+)
and negative (–) signs, they can be paired in four ways: + +, – +, + – & – –.
2
(Fig. 2). Adding a plus or minus sign at the end creates eight possible
combinations:
plus sign:
+
+ +, – + +, + – +, – – +,
minus sign:
+
+ –, – + –, + – –, – – –.
Fu Hse took the central pair of hexagons to be a symbol of the principles of
yang and yin and realized that the eight hexagons surrounding them represented the eight
possible triple combinations of yang and yin attributes. This would mean that there would be
eight different sets of three elements, where each is in either a yang or a yin state. In
the divinatory context of I Ching, the eight outcomes of three successive steps that each
allows the same two possibilities are known as ‘Kwa,’ or ‘trigrams.’ Single lines symbolize
yang and broken lines symbolize yin. Plus and minus signs are read from left to right and
corresponding lines in trigrams are read from top to bottom.
Figure 1 shows that the ten hexagons are arranged in the same way as the ten
Sephiroth on the Tree of Life. The three vertically aligned hexagons on the left correspond
to Binah, Geburah and Hod on the lefthand Pillar of Judgement, the three hexagons on the
right correspond to Chokmah, Chesed and Netzach on the righthand Pillar of Mercy and the
four hexagons in the middle column correspond to Kether, Tiphareth, Yesod and Malkuth on the
central Pillar of Equilibrium. The upper member of the pair of hexagons in the middle of the
tortoise’s shell symbolized yang; it corresponds to Tiphareth. The lower member shown in
black to symbolize yin corresponds to Yesod.
Their similar arrangement might be thought to be fortuitous. However, it is
hardly surprising that a geometrical pattern that is similar to the Tree of Life should,
when contemplated upon by someone in a meditative state, activate deep insights into nature.
After all, Indians have mediated upon mandalas with the same purpose for thousands of years.
Moreover, the pair of hexagons symbolizing for Fu Hse the yang and yin polarities in nature
has properties that similarly match those of the Tree of Life, as now explained. The four
stages in the creation from unity of the eight hexagrams is an example of the Pythagorean
teaching that, whatever its particular realisation, the wholeness of God unfolds in four
steps represented by the four rows of the tetractys:
3
Each of the ten dots of the tetractys will be called a ‘yod,’ which is the
name of the tenth letter of the Hebrew alphabet. When the 16 triangles making up the
threedimensional Tree of Life are turned into tetractyses, they are found to comprise 70
yods. Just as the yods in each row of the tetractys signified for the Pythagoreans the
point, the two endpoints of a line, the three corners of a triangle and the four corners of
a tetrahedron — the emergence of form from the dimensionless point — so the ten Sephiroth of
the Tree of Life are arranged in a similar, geometrical sequence. This socalled ‘trunk’ of
the Tree of Life contains 35 yods (Fig. 3). Hence, there are also 35 yods outside its trunk. Its geometry
implies a hidden, geometrical object discovered by the author and called its inner form. It
consists of two identical sets of seven regular polygons:
triangle, square, pentagon, hexagon, octagon, decagon, dodecagon.
Each set has 36 corners, 35 corners being associated with either set. The
hexagon is the fourth regular polygon:
1
2
3
4 
triangle
square
pentagon
hexagon 
According to the Pythagorean principle mentioned earlier, it therefore has
special significance in nature. Examples are the hexagonal benzene molecule serving as the
template of aromatic hydrocarbons and the hexagonal groupings of baryons and mesons
predicted by the theory of quarks.^{1} Its status is confirmed by the fact that the pair of hexagons in
the inner form of the Tree of Life is found to possess
70 yods when their sectors are converted into tetractyses (Fig. 4), i.e., one hexagon with 35 associated red yods corresponds to its
trunk, whilst the other hexagon has the same number of black yods as those lying outside the
trunk. Notice also that the shapes of both objects are marked out by ten points, leaving 60
yods in either case generated by the transformation of their triangles into tetractyses. The
mirror symmetry of each half of the inner Tree of Life is what expresses the yangyin
polarity. This is embodied in its outer form as its two side pillars, headed by Chokmah, one
of whose Kabbalistic titles is Abba (“Great Father”), and by Binah, one of whose titles is
Aima (“Great Mother”).
According to Kabbalah, three factors were hidden within the three Veils of
Negative Existence from which the ten Sephiroth emanated. They were the Zahzahot,
or the three Hidden Splendours. These principles determined the three pillars of
the
4
Tree of Life symbolising the active, passive and balanced Sephirothic
qualities. Their counterparts in the I Ching are the three rows of a trigram.
Tai Yi, the “Great Unity,” is Kether, the first Sephirah, embodying Divine
Unity. They are represented in both Kabbalah and Taoism by a circle. This bifurcates into
the duality of the male, generative or lifegiving principle, corresponding to Chokmah and
to yang, and the female, reproductive or formative principle, which corresponds to Binah and
to yin. This is shown in Fig. 5 by the line and broken line next to
these Sephiroth. With successive interplays of these fundamental principles,
starting with the Heaven trigram denoting maximum yang quality, all seven Sephiroth of
Construction ensue, the Earth trigram that corresponds to Malkuth (the physical universe)
expressing maximum yin. The last four hexagrams result from the first four by interchanging
yang and yin lines:
This will be discussed later in the context of the Church musical modes.
From a Kabbalistic perspective, this switch over due to progressive reversals of polarity
corresponds to the division of the Tree of Life into the Upper Face (KetherTiphareth) and
the Lower Face (TipharethMalkuth). As mirror to the Upper Face, the Lower Face is always
subjective or passive — the personality or ego is the creation and vehicle of the Higher
Self at Tiphareth, the centre of the Tree of Life map of consciousness.
5
Table 1. Gematria 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 Yesodoth
The Breaker of the
Foundations.
The Elements.
(Earth)
168 
The Sephiroth exist in the four Worlds of Atziluth, Beriah, Yetzirah
and Assiyah. Corresponding to them are the Godnames, Archangels, Order
of Angels and Mundane Chakras (their physical manifestation). This
table gives their number values obtained by the ancient practice of
gematria, wherein a number is assigned to each letter of the alphabet,
thereby giving a number value to a word that is the sum of the numbers
of its letters. Numbers from this table appearing in the article will
be written in boldface.

6
2. The octad as
2×4
As Fig. 2 shows, the sequence of eight trigrams is built up in four stages. The
four pairs of yin/yang lines, or Hsiang, create two sets of four trigrams — one set by the
addition of a yang line and the other set by the addition of a yin line. As trigrams
symbolise not things but everchanging phases of a cycle (these may, of course, correspond
to different objects), the sequence is better represented by the figureeight flow chart
shown in Fig. 6. Starting with the Heaven trigram at the top of the octagon and
labelled ‘0,’ the next three stages 1, 2 & 3 are passed through (indicated by the blue
flow curve), then it crosses over the centre of the octagon to trigram 4, flowing through 5
and 6 to trigram 7 (indicated by the red flow curve) and, finally, returning vertically
through the centre again to the starting point 0 to commence another cycle. The curves of
the Tai Chi symbol reproduce these flow lines. The very Arabic numeral for the number 8
aptly represents the flow. Laid on its side, it is the mathematical symbol for infinity.
However, this symbol measures quantity; it does not symbolize the indefinite duration of
cyclic activity. A more appropriate term that the closed loop signifies would be
perpetuity.
In the context of I Ching, the number 8 has a natural factorization as 2×4.
According to Michael Schneider in his “A Beginner’s Guide to Constructing the Universe”:
“The IndoEuropean names for “eight” emphasize it as a “doubling of four,” most deriving
from the Sanskrit ocatasrah (“twice four”), which became okta in Greek
and octo in Latin. These references to one, two, and four tell us that the archetype of
eight, called Octad, weaves together the principles of Monad’s unity, expansion, and cycles,
Dyad’s polarity, and tetrad’s materialization, that is, material forms precipitate from
their archetypes through polarity and pulsing cycles.”^{2} Referring to this number, W. Wynn Westcott in his “The Occult Power of
Numbers” said “The Greeks thought it an allpowerful number; they had a Proverb ‘all things
are eight.’”^{3} One of the titles given to the number 8 by the ancient Pythagoreans
was “All Harmonious.” This referred to their understanding that it measures a complete
cycle. For example, the eight notes of the musical scale span a complete octave and every
eighth day of a week is the same one.
The Kabbalistic significance of the octagon is that, when its sectors are
converted into tetractyses (Fig. 7), it is found to contain 49 yods, where
49 is the number value of the Godname El Chai assigned to Yesod (Table 1^{4}). This property of the octagon is a sign that it embodies a number of
universal importance. This number is revealed by considering the next higherorder
generalisation of the tetractys (the 1storder tetractys), namely, the 2ndorder tetractys
(Fig. 8). An octagon whose sectors are 2ndorder tetractyses has
80 corners of 1storder tetractyses surrounding its centre and
496 coloured yods (Fig. 8). Each yod denotes one of the 496 10dimensional
gauge fields of the gauge
7
8
symmetry group E_{8}×E_{8} governing the unified
interactions of heterotic superstrings. The number value 80 of the Sephirah
Yesod is the number of 1storder tetractyses making up the octagon. As a square has half the
number of sides of an octagon, it contains 496/2 = 248
coloured yods when its four sectors are transformed into 2ndorder tetractyses.
248 is the number of gauge fields of the superstring gauge symmetry group
E_{8}.
The ancient view of the number 8 as twice four was a profound insight that
has manifested again in modern times in superstring theory, for the discovery that the
heterotic superstring gauge symmetry group is the product of two identical groups simply
reflects the geometrical fact that the corners of two similar squares rotated through 45°
coincide with the corners of an octagon, that is, that 8 = 2×4.
The superstring number 496 is connected to the number 8
arithmetically as well as geometrically through its tetractys encoding in the octagon. It is
known to mathematicians as the third, socalled ‘perfect number.’ This type of number is one
whose factors (including 1) add up to the number itself. For example, 6 is the first perfect
number because 6 = 1 + 2 + 3, where 1, 2 and 3 are the factors of 6, and 28 is the second
perfect number because 28 = 1 + 2 + 4 + 7 + 14, where 1, 2, 4, 7 and 14 are its factors. As
shown in Fig. 10a, 496 has eight factors other than 1 that can be
assigned to the corners of an octagon. Moreover, these factors group naturally into two sets
of four: (2, 4, 8, 16) and (31, 62, 124, 248) (Fig. 10b). Each set can be assigned to the corners of a square. Again, the
number 8 splits into two 4s. The numbers of divisors other than 1 (that is,
nontrivial ones) for the first three perfect numbers 6, 28 and
496 are 2, 4 and 8:
These are the very digits in 248 — the dimension of the
gauge symmetry group E_{8}!
9
10
As shown in Fig. 2, the eight trigrams are built up in four stages, starting with Tai Yi,
the “Great Unity.” As discussed in Article 18, the I Ching table of hexagrams is built up in
(4+4) stages. Starting with the eight trigrams (Fig. 11), the fifth stage is the addition of the Ch'ien trigram to each to
form the first row of hexagrams. The sixth stage is the addition of the Ch' ien and Tui
trigrams to form the first two rows of hexagrams, then the seventh stage is the addition of
the Ch' ien, Tui, Li and Chen trigrams to form the first four rows and, finally, the eighth
stage is the addition of all eight trigrams to form the I Ching table itself. The 4:4
division in the eight trigrams results from
interchanging yang and yin lines, so that, as displayed in Fig. 5, trigrams at the corners of an octagon and diametrically opposite each
other are polar opposites. The first four trigrams are the essential ones because the last
four result from interchanging yang and yin lines. The 4:4 division in the eight stages of
creation of the table of hexagrams reflects the distinction between the whole table and the
eight trigrams as its building blocks. In both cases, we see that the first four
stages generate the basic set of trigrams.
Figure 12 indicates how the square and the octagon are present in the
geometry of the Tree of Life. Its inner form consists of two sets of seven regular polygons,
each set having 36 corners. The Godname Eloha of the Sephirah Geburah with
number value 36 prescribes each
half of the inner Tree of Life. If the first eight integers are assigned to
the corners of an octagon (Fig. 13), the sum (20) of the first four even integers at the corners of a
square is the number of corners of the decagon and dodecagon and the sum (16) of the first
four odd integers at the corners of a second square is the number of corners of the
triangle, square, pentagon, hexagon and octagon.
11
The lowest Tree of Life of any set of overlapping Trees of Life (the
‘1tree’) has 19 triangles. If these are transformed into tetractyses, they comprise
80 yods. This is the number of corners of the 94 triangular sectors of the
two sets of seven, enfolded regular polygons (Fig. 14).
If the first eight odd integers after 1 are assigned to the corners of an
octagon, their sum is 80. The four integers arranged in one square add up
to the number (44) of corners of the 50 sectors of the two sets of the
first five polygons and the four integers at the corners of the other square sum to the
number (36) of corners of the 44 sectors of both pairs of the last two
polygons (Fig. 15). Whether the
first eight integers or the first eight odd integers are considered, their
separation into two sets of four creates sums that differentiate the first five polygons
from the last two polygons.
The sum of the first six odd integers after 1 is 48. This
is the number of Kokab (Mercury), the Mundane Chakra assigned to Hod. It is the
number of yods in the 1tree up to Chesed, the first Sephirah of Construction. In other
words, the emanation of the seven Sephiroth of Construction generates 48
yods. Therefore, the sum (32) of the two last odd integers 15 and 17 in Fig. 15 is the number of yods above Chesed.
12
The seven separate polygons have 48 corners. Two squares
rotated through 45° require 48 yods to delineate their boundary (Fig. 16).
The reason for this beautiful harmony between number and the sacred geometry
of the Tree of Life and its inner form is that a Divine Name — El Chai — is associated with
the octagon, its number vale of 49 being the number of yods in this polygon
when divided into tetractyses. No wonder that the Pythagoreans called the octad “All
Harmonious”!
3. I Ching & the eight Church
musical modes
The Pythagorean octave of eight musical notes is spanned by successive
intervals of a perfect fifth and a perfect fourth. or Ascending descending the scale through these two intervals generates the
same note an octave higher or lower. Played in succession, these two intervals therefore
complete a musical cycle, whatever the note that was played first. They may be regarded as
the yang and yin phases of a cycle that ends on the same note, albeit higher or lower in
pitch by one octave. Rather than make any assumptions about the ‘masculine’ or ‘feminine’
characters of these intervals, we shall use the convention that a perfect fifth is the
yang phase and the perfect fourth is the yin phase (Fig. 17). Yang and yin have equal and complementary status and neither
musical interval can be said to be the more fundamental.
The octave CC' can be divided into either four perfect fourths or four
perfect fifths (Fig. 18).
Compare this with how the eight trigrams consist of two sets of four, one
set being derived from the other by interchanging yang and yin lines:
13
The same 4:4 division is found in the intervals of the notes of the
Pythagorean scale as with the eight trigrams. This is evidence of course that this scale is
truly the fundamental musical scale, unlike many of its more modern imitations.
It was argued in Article 16^{5} that the current view held by many musical scholars that the
ancient Greek musical modes were not octave species but different keys of the same scale is
wrong. Two reasons were given:

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.

Seven octave species have 28 notes with tone ratios belonging to the Pythagorean
scale. ^{6} As the ancient Egyptians, according to the
Pythagorean mathematician, Nichomachus ( quoting Pythagoras and
Plato), ascribed 28 sounds to the universe, they must have been referring
to these notes, implying that they were well aware of the existence of the
seven types of musical scales. It is highly implausible that this knowledge
was not shared with musicians and with great minds like Pythagoras and Plato,
both of whom lived some years in Egypt, learning music, and therefore that the
ancient Greeks never learnt of the existence of the seven octave species.
The Pythagorean scale consists of five tone intervals T of 9/8 and two
leimmas L of 256/243 (the modern semitone):
T T L
T T
T L
By starting successive sequences of intervals displayed in Figure 19 on the circumference of a circle, it is readily seen that there
can be only seven different octave species:
T L T T T L
T
L T T T L T
T
T T T L T T
L
T
T L T T L
T
T
L T T L T T
L
T T L T T T
T
T L T T T L

Dorian (mode 1)
Phrygian (mode 3)
Lydian (mode 5)
Mixolydian (mode 7)
Hypodorian (mode 2)
Hypophrygian (mode 4)
Hypolydian (mode 6) 
The three ‘hypo’ versions of the Dorian, Phrygian and Lydian modes are
separated from the latter by a perfect fourth (Fig. 20).
14
The Roman Catholic Church added an extra mode called the Hypomixolydian
(mode 8) to be the counterpart of the Mixolydian. It is the same octave species as the
Dorian but has a distinct ethos because its finalis (ending note) and dominant (reciting
note) are different. Figure 21 shows the four Authentic Modes and the four Plagal Modes.
The doubleheaded arrows link an Authentic Mode and a Plagal Mode whose
patterns of intervals are mirror images of each other.
15
Two modes — the Dorian and the Hypomixolydian — are trivial mirror images of
each other because their identical sequences of intervals are mirror images of themselves (T
= tone interval, S = semitone):
Fig. 22 is a threedimensional way of depicting the mirror symmetry of the
eight Church modes. Along each of three great circles of a sphere, spaced 60° apart, are
eight points denoting the eight notes of the Phrygian, Lydian and Mixolydian modes, the
South Pole of the sphere denoting the tonic and the North Pole denoting the octave.
Along the vertical axis of the sphere between the poles are six red points
representing the six notes between the tonic and the octave of the Dorian mode. As the
Hypomixolydian mode has the same notes as the Dorian mode, its notes are represented by the
same points. These are depicted in Fig. 22 as red circles. Points on a great circle representing a note of a
mode and its counterpart in the mirror image of that mode are diametrically opposite each
other.
16
The eight modes with the same tonic consist of 50 notes.
This is the number value of Elohim, the Divine Name assigned to Binah. This is a very
important property because it shows that the eight Church modes form an archetypal musical
pattern even though the Hypomixolydian mode — the last mode — has the same interval pattern
as the Dorian mode — the first mode. It amounts to the difference between the first seven
notes and all eight notes completing the cycle of the octave. The eight modes conform to the
Tree of Life blueprint because the two sets of the first six of the seven enfolded polygons
constituting the inner Tree of Life have 50 corners (Fig. 23), the 24 corners on one side symbolizing the notes of the Authentic
Modes between the tonic and the octave and the 24 corners on the other side symbolizing the
corresponding plagal notes. The four Authentic modes correspond to the Ch’ien, Tui, Li and
Chen trigrams and the four plagal modes — their mirror images — correspond to the Sun, K’an,
Ken and K’un trigrams — the counterparts of the first with yang and yin lines interchanged.
The Heaven trigram corresponds to the Dorian mode, the Earth trigram corresponds to the
Hypomixolydian mode and — very interestingly — the seventh trigram (Ken)
corresponds to the Hypolydian mode, which is the Pythagorean scale. In other words, the last
trigram before the end of the cyclic sequence of trigrams has as its counterpart the
mathematically perfect Pythagorean scale. The fact that in the Church classification it
turns out to be a plagal mode, not (as would be expected of such an influential musical
scale), an authentic mode, demonstrates the errors of nomenclature in the Gregorian modes.
This can be seen independently by comparing the known ethos of some ancient Greek modes with
those of Church modes with the same name: they do not match.^{7}
Is it necessary to follow the Church numbering system for modes that start
with the Dorian mode? After all, this mode might have been chosen by human beings for merely
historical reasons, being the scale that was most favoured by Plato for playing music in his
ideal society. It might not be what God conceived. Indeed, how can a particular mode be
deemed to generate all the others when the ring of intervals shown in Fig. 19 favours no starting point? As now shown, comparison of the 4:4
system of modes with the eight trigrams upholds the traditional list that starts
with the Dorian mode.
If any musical scale might be considered most apt to start the sequence of
modes, it would have to be, of course, the Pythagorean scale in view of its being the C
scale. Suppose that it corresponds to the first trigram, then (considering the possibilities
of either ascending or descending scales):
Whether the scale ascends or descends, two scales in the same quartet (shown
above linked by brackets) are mirror images of each other. The Pythagorean scale cannot
therefore correspond to the first trigram. In view of the way the ancient Greeks understood
octaves to be created from two tetra chords, let us next suppose that the Pythagorean scale
corresponds to the fourth trigram. Then:
In either case, two scales belonging to the same set of four are
mirror images, whereas
17
what is wanted is that only scales belonging to different sets of
four be mirror images. This means that the Pythagorean scale cannot correspond to the fourth
trigram. Finally, suppose that it corresponds to the seventh trigram. Then either (case A):
or (case B):
Case B is forbidden because pairs of sequences of intervals that are mirror
images belong to the same set of four scales. In case A, however, the interval
patterns corresponding to the first four trigrams are the mirror images of those
corresponding to the last four trigrams. This is the required result.
There is therefore only one possible scheme of correspondence
between trigrams and scales that permits mirror symmetry in both. The Heaven trigram with
three yang lines has to correspond to the Dorian mode and the Earth trigram with three yin
lines must correspond to the Hypomixolydian mode. These two trigrams are special in that
they are the first and last of the trigrams. Their corresponding modes are also special in
that, as well as being mirror images of each other, their interval patterns are also mirror
images of themselves. Although the early musical theorists erred in their naming of the
Gregorian modes after the ancient Greek modes, they were correct in the way they ordered
them. Far from being a matter of arbitrary choice, there is only one
ordering of octave species that is consistent with the sequence of generation of trigrams
and which preserves the correlation between mirror image patterns of intervals and symmetry
with respect to interchange of yin and yang lines. It starts with the mirrorsymmetric
Dorian mode.^{8}
Figure 5 shows that trigrams at the corners of the octagon that are
diametrically opposite each other have, as representations of binary numbers, the following
integers as their values:
Number values of trigrams opposite each other always add up to 7. The
Pythagoreans
18
regarded the odd integers as male and the even integers as female. Although
the ‘gender’ of each integer alternates as one goes through the sequence of trigrams lying
on the figureofeight (Fig. 25), pairs of integers diametrically opposite each have opposite
polarity. Although the ethos attributed by the ancient Greeks to their musical modes is
poorly known and contradictory, some of their characteristics do fit this alternating
pattern. Phrygian music was exciting and passionate, generating a masculine energy, whilst
the Lydian mode was thought a ‘wailing mode suitable for women.’ However, if strictly
pursued, this would make the Hypolydian mode a feminine mode because it corresponds to the
even integer 6. Music played according to the Pythagorean scale does not have feminine
emotional qualities. As one might expect, it is clearly wrong to assign psychological
qualities to modes based merely upon the genders that the ancient Pythagoreans assigned to
even and odd integers!
4. The planetary
octet
It was shown in Article 17^{9} that the two assumptions:

the nebula which condensed into the Solar System had the shape of a logarithmic
spiral that expanded twice with each revolution;

the fractallike, scaleinvariant geometry of the spiral groups the planets into
octets
led to a modified form of the famous TitiusBode Law governing distances of
planets from the Sun that accurately applies to Neptune and Pluto as well, unlike the
earlier version. The predicted average distances (in terms of 1/10 A.U.) of the planets from
the asymptotic centre of the spiral, Mercury and the Sun are:
The first eight planets Mercury–Uranus group into an octet. The three
planets Uranus–Pluto are the first three members of a second octet whose mean distances from
Uranus are determined by the same mathematical relation as that governing the distances of
the first octet of planets from the asymptotic centre, albeit with different parameters.
Notice that the values of n appearing in the TitiusBode Law for the distance:
D_{n} = 4 + 3×2^{n1}

(n = 1–7)

of the nth planet after Mercury in the first octet are the same as the
number values of the seven trigrams after the Heaven trigram shown in Fig. 5. Mercury plays the role of the tonic of the musical octave. Just as
its pitch can have any value that, once fixed, determines the pitches of the remaining
notes, so the average distance of Mercury from the Sun is a free variable* that is not part of the geometric series:
3, 6, 12, 24, 48, 96, 192
______________________________
*In the author’s theory modifying this law, it ceases to be an ad
hoc constant but becomes predicted.
19
occurring in the TitusBode Law. This series determines the average
distances of the planets not from Mercury, as has always been assumed by
astronomers, but from the asymptotic centre of the logarithmic spiralshaped planetary
nebula that spawned the Solar System. This explains why the mathematical equation for
D_{n} given above could accommodate Mercury only by assuming for it the singular
value n = –∞. Astronomers have not realized that this determines the position of the
asymptotic centre of a logarithmic spiral that doubles in size with every revolution, i.e.,
that the TitiusBode numbers refer actually to the distance of planets from this centre, not
from Mercury.
Just as the 64 hexagrams of the I Ching table build up in
two sets of four stages
20
(Fig. 11), so the average distances of the octet of planets from the
asymptotic centre, which are the numbers of yin and yang lines in the trigrams making up
these stages (Fig. 26), group into two sets:
The average distances of these planets from Mercury are 4, 7, 13, 25,
49, 97 and 193. The fourth ‘planet’ after Mercury is the Asteroid Belt with
an average distance of 25 (actually 23.8) and the average distance between this and Uranus,
the last member of the octet, is 168 (actually, 164.2). This pair of
distances is encoded in the first six regular polygons enfolded in the inner form of the
Tree of Life because, when their 35 sectors are converted into tetractyses, one finds that
there are 25 corners of polygons and 168 other yods associated with these
polygons (Fig. 27).
The distinction between the shapemarking corners and the other
yods corresponds to the ‘turning of the corner’ at the Asteroid Belt, which plays the role
of the perfect fifth G of the Pythagorean scale:
As Fig. 24 indicates, this corresponds to the completion of the eight trigrams
with 24 yang and yin lines. The completion of the octet of planet spans a further distance
of 168. This is the number of additional yang and yin lines need to
construct the first half of the I Ching table. The number of lines in the whole table is
384, which is the average distance of the last planet Pluto from the asymptotic centre
predicted by the logarithmic spiral theory.
21
22
This is simply the number of yods in both sets of six polygons other than
the two endpoints of the root edge (Fig. 28). Alternatively, there are 384 yods intrinsic to each pair of six
polygons enfolded in successive, overlapping Trees of Life. This property is a remarkable
demonstration of the precise correspondence or isomorphism between the Tree of Life and its
Chinese counterpart — the 64 hexagrams of the I Ching.
The square embodies the numbers 25 and 168 both
geometrically and arithmetically.
23
The square with its four sectors turned into tetractyses has 25 yods, whilst
the twelve odd integers 3, 5, 7, …25 lying on the sides of a square, four to a side, add up
to 168:
The number 168 is the number value of Cholem
Yesodoth (“breaker of the foundations”), the Mundane Chakra assigned to Malkuth and
astronomically represented by the planet Earth. Its fundamental significance as the
structural parameter of superstrings has been discussed in many earlier articles by the
author.
The number 25 is the number of spatial dimensions predicted by quantum
mechanics for spinless strings. Embodied in the distances spanned by the four planets after
Mercury (formally regarding the Asteroid Belt as a planet) and by the last four planets of
the octet are, respectively, the dimension of space and the number of circularly polarised
oscillations made by each of the ten closed, stringlike whorls constituting the subquark
state of a superstring during onehalf of its revolution.^{10}
Figure 29 lists examples of the eightfold character of biological,
astronomical, musical and subatomic systems. The eight stages of activity in the life of a
cell are as follows:
1. Interphase: cell grows and performs other life
processes
cell rests (no division)
cell enlarges, produces RNA and synthesizes protein
chromosomes replicate
centrioles replicate, cell grows
Mitosis (cell division) begins at the end of Interphase.
2. Prophase: nucleolus fades/chromatin (replicated DNA and
associated proteins) condense into chromosomes/centrioles migrate into opposite sides of
cell/spindle fibre network form;
3. Prometaphase:nuclear envelope disintegrates/some mitotic
spindle fibres elongate from the centrioles and attach to kinetichores, protein bundles
located on the chromosomes;
4. Metaphase: spindle fibres attach to chromosomes at
kinetichores/chromosomes align on metaphase plate;
24
5. Anaphase: spindle fibres
shorten/kinetichores separate/chromatids (daughter chromosomes),
pulled apart by contracting fibres, move to opposite poles/poles move apart;
6. Telophase: daughter chromosomes arrive at poles/spindle
fibres disappear/nuclear envelope reforms (this is the completion of mitosis);
7. Cytokinesis: cell pinched into two daughters by ring of
actin protein/microtubules reorganise into new cytoskeleton;
8. Interphase.
Figure 30 displays the cyclic nature of the life of a cell. Notice that the
crossing over to the Anaphase stage marks the separation of pairs of chromosomes to opposite
sides of the cell. Once again, the eight stages of a cyclic sequence are divided into two
sets of four. In the Anaphase, Telophase and Cytokinesis stages, there is physical movement
of the DNA material and splitting of the cell. This 4:4 differentiation corresponds to the
division of the eight trigrams into two sets of four:
with the Sun trigram (the fifth one in the sequence) corresponding to the
Anaphase. One may characterize the first four stages as the yangdominated part of the cycle
and the remaining stages as the yindominated, formative phase.
The eight unit octonions e_{i} and their relation to the eight zero
roots of the superstring gauge symmetry group E_{8} were discussed in Reference 7
and in Article 15.^{11}
Finally, the most obvious manifestation of the pattern of twelve yang lines
and twelve yin lines of the eight trigrams is the Earth’s day of 24 hours, which is divided
at the spring and autumn equinox into twelve hours of sunlight and twelve hours of night.
For the ancient Chinese, light was yang and darkness was yin. The 12:12 division of lines in
the two sets of four trigrams is represented in the last pair of polygons making up the
inner form of the Tree of Life — the pair of mirrorimage dodecagons, each of which has
twelve corners. This
polygon is associated with the last Sephirah, Malkuth, because it embodies
the number value 168 of Cholem Yesodoth, the Mundane Chakra of
Malkuth, 168 yods other than corners of the dodecagon surrounding the
centre when each sector
25
is divided into three tetractyses (Fig. 31). That this is not a coincidence is indicated by the fact that the
sum of the twelve odd integers after 1 that can be assigned to the corners of a dodecagon is
also 168. For triple division of a sector, this polygon is unique^{12} in having such a property (in the case of each sector being a
tetractys, the triangle — the simplest polygon — uniquely has this property). The
cycle of seven days of the week lasts for 7×24 = 168 hours. The dodecagon
therefore embodies the numbers of hours in both a day and a week. The
twelve yang lines correspond to one dodecagon and the twelve yin lines
correspond to its counterpart in the mirrorimage set of seven polygons making up the inner
form of the Tree of Life. Yang/yin polarity is incorporated in this universal blueprint
through the two chiral sets of seven polygons.
At the beginning of this article, it was pointed out that a pair of adjoined
hexagons is isomorphic to the Tree of Life because both contain 70 yods, of which ten are
corners. When their sectors are divided into three tetractyses, 168 yods
are generated (Fig. 32). This fundamental superstring structural parameter is therefore
embodied in the very pair of hexagons that the first Chinese emperor Fu Hse realised
symbolised yang and yin when he noticed it in the pattern of scales on the shell of a
tortoise.
We saw earlier that the eight trigrams have their astronomical counterpart
in the first eight planets of the Solar System, which complete an octet. The TitiusBode
distances of these planets from the asymptotic centre of the logarithmic spiral:
0

3

6

12


48

96

192

Asymptotic
centre

Venus

Earth

Mars


Jupiter

Saturn

Uranus


imply that the Asteroid Belt (the fourth ‘planet’ after Mercury) is 24 units
from this centre and that Uranus (the last member of the octet and the fourth from the
Asteroid Belt) is 168 units from the latter. The lengths of the Earth’s day
(24 hours) and week (168 hours) cannot be based upon arbitrary
manmade units of time when they appear in the Solar System as numbers expressing distances
of planets that, in their musical counterpart, correspond to the perfect fifth and octave.
Hypothetical astronomers on another planet of the Solar System would discover the same
TitiusBode numbers accurately measuring planetary distances, as would astronomers on a
planet belonging to another Solar System even if their days were ten times the length of the
terrestrial day. They would have to admit that Earthlings have a unique calendar system
based upon the rotation of their planet that measures cycles of time in accordance
with the
26
proportions of the Solar System. The problem
that this poses for Earth astronomers is how people in ancient times could have devised such a
calendar if they had no knowledge of the distances between the planets. Uranus, which is not
bright enough to be visible to the human eye, was not discovered until 1781 by the astronomer
William Herschel. Hard though it may be for astronomers to admit it, such correlation suggests
that a source of knowledge existed in ancient times which transcended the limits of the human
senses, even when aided by a telescope, and which led to time being measured in units whose
duration and numbers could not be arbitrary because they had to harmonize with the cyclic
movement of the celestial spheres — in particular, with the TitiusBode Law.
_________________________
References
^{1} Phillips, Stephen M. Article 17: “The
Logarithmic Spiral Basis of the TitiusBode Law,” (WEB, PDF), p. 23.
^{2} A Beginner’s Guide to Constructing the Universe, by Michael Schneider
(HarperPerennial, 1994), p. 268.
^{3} The Occult Power of Numbers, by W. Wynn Westcott
(Newcastle Publishing House Co. Inc., North Hollywood, California, 1884), p. 85.
^{4} All numbers from this table that appear in the text will be written
in boldface.
^{5} Phillips, Stephen M. Article 16: “The Tone Intervals of the Seven Octave
Species and Their Correspondence with Octonion Algebra and Superstrings,” (WEB, PDF).
^{6} Ibid, p. 4.
^{7} Phillips, Stephen M. Article 14: “Why the Ancient Greek Musical Modes are
Sacred,” (WEB, PDF).
^{8} According to the Egyptologist, Moustafa Gadalla, the term for Saturday, the
first day of the week in ancient and presentday Egypt, is Sabt. This means Sirius, also
known as the Dog Star. It “is referred to as the First, and the Dscale is also referred to
as the First scale. All generated music is a variation of this First scale” (Egyptian
Rhythm, Moustafa Gadalla (Tehuti Research Foundation, 2002), p. 36).
^{9} Ref. 1.
^{10} Phillips, Stephen M. Article 18: “Encoding of Planetary Distances and
Superstring Structural Parameters in the I Ching Table,” (WEB, PDF).
^{11} Phillips, Stephen M. Article 15: “The Mathematical Connection between
Superstrings and their Micropsi Description: a Pointer towards M theory,” (WEB, PDF).
^{12} Proof: an nsided, regular polygon with its sectors each divided into three
tetractyses has (15n+1) yods, that is, 15n + 1 – (n+1) =
14n yods other than corners of the polygon surround its centre. The sum of the first n odd
integers after 1 = (n+1)^{2} – 1 = n^{2} + 2n. Therefore,
n^{2} + 2n = 14n, that is, n^{2} – 12n = 0. Therefore n = 12. This
is the dodecagon. With single tetractyses as sectors, an nsided polygon has (6n+1) yods,
that is, 5n yods other than corners surround its centre. Therefore, n^{2} + 2n
= 5n, that is, n^{2} – 3n = 0, so that n = 3. This is the triangle.
27
