ARTICLE 38
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
The seven possible types of musical scales contain 14
different types of notes (7 notes and their 7 tonal “complements”). The 91
intervals between these notes are found to consist of 40 Pythagorean intervals
(notes belonging to the Pythagorean musical scale) and 51 nonPythagorean
intervals. As a sequence of monotonically increasing tone ratios, they group
into 65 intervals up to the 7th note and 26 larger intervals that are
complements of some of these intervals. This shows how the Divine Name ADONAI
with gematria number value 65 and the Godname YAHWEH with number value 26
prescribe the composition of the 91 intervals between the basic set of 14 notes.
The Godname EHYEH with number value 21 prescribes the 21 intervals that are not
notes of the seven scales because they are intervals between notes belonging to
different scales. EHYEH also prescribes all 91 intervals because 91 is the sum
of the 21 odd integers making up the squares of the first 6 integers. There are
25 pairs of notes and their complements. The Godname ELOHIM with number value 50
prescribes these 50 intervals. The Godname ELOHA with number value 36 prescribes
the 36 intervals between the eight notes of each scale. The Godname YAH with
number 15 prescribes the 15 intervals that have no complements. The Godname
YAHWEH ELOHIM with number value 76 prescribes the number of remaining intervals
that do have complements. There are 24 pairs of intervals other than 1 and the
octave. EHYEH prescribes the 21 pairs that are notes, as well as the 21 types of
intervals found in them. The 24 pairs of intervals are symbolized by the 24
pairs of vertices and their mirror images outside the shared root edge of the
first (6+6) enfolded polygons of the inner Tree of Life. They are also
symbolized by the 24 vertices that are above or below the equator of the
disdyakis triacontahedron, its 12 vertices representing the 12 basic notes
between the tonic and octave found in the 7 musical scales. The 8 basic
intervals and their 8 complements found in the set of 90 intervals below the
octave are analogous to the 8 simple roots of E_{8} and the 8
simple roots of E_{8}' appearing in E_{8}×E_{8}'
heterotic superstring theory. There are also 8 triplets of notes with tone
ratios in the proportion 1:T:T^{2}, where T (=9/8) is the tone ratio of
the Pythagorean tone interval. As four triplets of intervals and four triplets
of their complements, they are the counterpart of the four trigrams of the I
Ching and their four polar opposites with yang and yin lines interchanged. They
are also the counterpart of the 8 unit octonions. The geometrical realisation of
the 26 unpaired intervals and the 24 pairs of intervals is a polyhedron with 144
faces and 74 vertices, of which 26 vertices belong to its underlying disdyakis
dodecahedron, the remaining 24 diametrically opposite pairs pointing outward
from the 24 pairs of faces of this polyhedron. These 24 pairs of intervals
spanning the octave constitute the source of the 7 musical scales. They group
into 8 sets of 3 intervals and their 3 complements with tone ratios in the
proportions 1:T:T^{2}. The 90 edges in one half of the disdyakis
triacontahedron represent the 90 rising intervals below the octave. The 90 edges
in its other half represent the 90 falling intervals. The 6 edges and their
mirror images in its equator represent the six rising intervals of a perfect
fifth and the six falling intervals of a perfect fifth. The 168 edges outside
the equator and the 168 intervals other than these 12 intervals are both
analogous to, if not actual manifestations of, the 168 symmetries of the group
PSL(2,7), whose centre, SZ(3,2), is isomorphic to the 3rd roots of 1: 1, r &
r^{2}, where r = exp(2πi/3).

1
Table 1. Tone ratios of the notes in the seven musical scales.
Musical scale 
Tone ratio

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

C scale 
1

9/8

81/64

4/3

3/2

27/16

243/128

2

(Tone ratios belonging to the Pythagorean scale are written in black and
nonPythagorean tone ratios are written in red).
The seven species of musical octaves^{1} comprise 14 different notes (Table 1). In order of increasing tone ratios, they are:
1 256/243 9/8 32/27
81/64 4/3 1024/729 729/512
3/2 128/81 27/16
16/9 243/128 2
They form seven pairs of notes x and their complements y, where xy = 2
1. 

1

2

T^{5}L^{2}

1×2 = 2

2. 
L

256/243 
243/128

T^{5}L

256/243×243/128 = 2

3. 
T

9/8

16/9

T^{4}L^{2}

9/8×16/9 = 2

4. 
TL

32/27

27/16

T^{4}L

32/27×27/16 = 2

5. 
T^{2} 
81/64

128/81

T^{3}L^{2}

81/64×128/81 = 2

6. 
T^{2}L 
4/3

3/2

T^{3}L

4/3×3/2 = 2

7. 
T^{2}L^{2} 
1024/729

729/512

T^{3}

1024/729×729/512
= 2

(T = 9/8 is the Pythagorean tone interval and L =
256/243 is the Pythagorean leimma). Let X =
(x_{1},x_{2},x_{3}, …x_{7}) be the set of the first seven notes
(x_{m}>x_{n} for m>n) and Y = (y_{1},y_{2},y_{3},
...y_{7}) be the set of their complements
(x_{7}<y_{n}<y_{m} for m>n), where
x_{n}y_{8n} = 2. There are (^{14}C_{2} = 91) intervals between
the 14 notes. The largest of these is the octave, so that 90 intervals are below it. Their
explicit values can be calculated in three steps:
1. Work out the (^{7}C_{2} = 21) rising
intervals X_{nm} between the notes x_{n} and x_{m} in X (m>n), where
X_{nm} ≡ x_{m}/x_{n}. By definition, x_{n} =
X_{1n};
2. Work out the 21 rising intervals Y_{nm} between
the notes y_{n} and y_{m} in Y (m>n), where Y_{nm} ≡
y_{m}/y_{n}. As y_{m} = 2/x_{8m} and y_{n} =
2/x_{8n}, Y_{nm} = x_{8n}/x_{8m} =
X_{(8m)(8n)}
3. Work out the (7×7=49) rising intervals Z_{nm}
between the notes x_{n} and y_{m}, where Z_{nm} ≡
y_{m}/x_{n} = 2/x_{8m}x_{n}. By definition, y_{n}=
Z_{1n}, so that the octave y_{7} is Z_{17}.
Tables 2, 3 & 4 display the magnitudes of the 90 rising intervals below the octave.
Table 2. Intervals X_{nm}.
n 
m 
1

2

3

4

5

6

7


1

256/243

9/8

32/27

81/64

4/3

1024/729

1 
1

1

256/243

9/8

32/27

81/64

4/3

1024/729

2 
256/243


1

2187/2048

9/8

19683/16384

81/64

4/3

3 
9/8



1

256/243

9/8

32/27

8192/6561

4 
32/27




1

2187/2048

9/8

32/27

5 
81/64





1

256/243

65536/59049

6 
4/3






1

256/243

7 
1024/729







1

(Cells highlighted in turquoise are the tone ratios of the first seven
notes. Cells for the falling intervals are left blank).
The 21 intervals X_{nm} consist of 3 Pythagorean
notes, 5 Pythagorean intervals, 3 nonPythagorean notes and 10 nonPythagorean intervals.
2
Table 3. Intervals Y_{nm.}
n

m

7

6

5

4

3

2

1


2

243/128

16/9

27/16

128/81

3/2

729/512

7

2

1

256/243

9/8

32/27

81/64

4/3

1024/729

6

243/128


1

2187/2048

9/8

19683/16384

81/64

4/3

5

16/9



1

256/243

9/8

32/27

8192/6561

4

27/16




1

2187/2048

9/8

32/27

3

128/81





1

256/243

65536/59049

2

3/2






1

256/243

1

729/512







1

(The 7 complements are tabulated in order of decreasing tone ratio in
order to demonstrate that the set of 21 intervals Y_{nm} is identical
to the set of 21 intervals X_{nm}).
The 21 intervals Y_{nm} comprise 8 Pythagorean
intervals and 13 nonPythagorean intervals.
Table 4. Intervals Z_{nm}.
n

m

1

2

3

4

5

6

7


729/512

3/2

128/81

27/16

16/9

243/128

2

1

1

729/512

3/2

128/81

27/16

16/9

243/128

2

2

256/243

177147/131072

729/512

3/2

6561/4096

27/16

59049/32768

4/3

3

9/8

81/64

4/3

1024/729

3/2

128/81

27/16

16/9

4

32/27

19683/16384

81/64

4/3

729/512

3/2

6561/4096

27/16

5

81/64

9/8

32/27

8192/6561

4/3

1024/729

3/2

128/81

6

4/3

2187/2048

9/8

32/27

81/64

4/3

729/512

3/2

7

1024/729

531441/524288

2187/2048

9/8

19683/16384

81/64

177147/131072

729/512

(The cell with tone ratio 2 is coloured black to indicate that it does
not belong to the set of 90 intervals).
The 48 intervals Z_{nm} below the octave consist of
3 Pythagorean notes, 20 Pythagorean intervals, 3 nonPythagorean notes and 22 nonPythagorean
intervals, that is, 23 Pythagorean intervals and 25 nonPythagorean intervals, regarding notes
as intervals.
The 91 intervals consist of 7 Pythagorean notes, 6 nonPythagorean notes, 33
Pythagorean intervals and 45 nonPythagorean intervals, that is, 40 Pythagorean and 51
nonPythagorean intervals. In increasing order of size, their tones ratios are:
(The 21 starred intervals are not notes of the seven
musical scales). In total, there are 40 Pythagorean intervals and 51 nonPythagorean intervals.
The Godname ELOHIM (Table 5) with number value 50 prescribes the latter because
51 is the 50th integer after 1. There are 65 intervals up
to the seventh and last note with tone ratio 1024/729 before the crossover to notes that are complements of the first
seven notes. The Godname ADONAI with number value 65 prescribes how
many
3
Table 5. 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.

(All numbers from this table that are referred to in the article are written in
boldface).
4
independent intervals there are between the 14 notes making up the
seven musical scales. They are independent in the sense that all other larger intervals
complete the octave as their complements and so are determined by them. The Godname
YAHWEH with number value 26 prescribes the number of these complementary
intervals. The Godname EHYEH with number value 21 prescribes the
21 asterisked intervals (eight types) between notes in different
scales.
As 65 is the sum of the first 10 integers after 1:
we see how the Decad, given the title “All Perfect” by the ancient
Pythagoreans, defines the number of independent intervals between the 14 different notes in the
seven musical scales.
Excluding the octave leaves 25 complements. The 65:25
division of intervals below the octave between the 14 notes is represented in the Lambda
Tetractys (Fig. 1). The sum of the four numbers forming its base is 65
and the sum of the six remaining numbers is 25. That this is no coincidence is the fact
that the 65 intervals are made up of 27 Pythagorean intervals, 18 that
are not notes in the seven scales, 12 nonPythagorean intervals and eight leimmas of
256/243, all of which are the numbers
forming the base of the Lambda Tetractys. Indeed, the central number
6 denotes the number of perfect fifths, the number 4 denotes the number of
the note A with tone ratio 27/16, the number 2 is the number of the note B with tone ratio
243/128, the number 1 denotes the largest tone interval 59049/32768 not belonging to the seven scales, the
number 3 is the number of intervals 128/81 and the number 9 is the number of intervals
729/512, 6561/4096 and 16/9. In other words, every number in
the Lambda Tetractys denotes the number of different subsets of intervals in the set of
90 intervals between the notes of the seven musical scales. This reveals the amazing,
archetypal quality of the Lambda Tetractys in quantifying such holistic systems, as well
as in defining the tone ratios themselves as ratios of its numbers.
The same 65:26 division as that exhibited
by the intervals between the 14 different notes of the seven scales is expressed arithmetically
by the fact that 91 is the sum of the squares of the first six integers:
91 is the sum of 21 odd integers, showing how this number
is prescribed by the Godname EHYEH with number value 21 (the sum of the first
six integers). The sum of the six integers within the central blue triangles is
26, which is the number of intervals with tone ratios that takes them past the
crossover between notes and their complements. The sum of the 15 integers on
its boundary is 65, which is the number of intervals that are not notes and
which are not paired with their complements (see below). 15 is the number
value of YAH, the older version of the Godname YAHWEH.
EHYEH determines the 21 intervals that are not notes of the
seven scales. This leaves 40 Pythagorean intervals and 30 nonPythagorean intervals that are
notes, that is, 70 intervals. YAHWEH determines the 27 Pythagorean intervals before the
crossover into complementary notes because 27 is the 26th integer after 1.
Let us now examine those intervals that do or do not have complements. There
are 15 intervals without complements, none of which are found as notes in the
seven musical scales. They are of the type TL^{2}, TL^{1}, T^{2}L^{1} and T^{3}L^{1}. This means that
5
there are (91–15=76) intervals, some of
which are paired as an interval and its complement. This shows how the Godname YAHWEH ELOHIM
with number value 76 prescribes these intervals. Six of these are not notes of
the scales, leaving 70 intervals that are notes. Some of them, however, cannot be paired with
their complements because the number of complements for a given interval is not always equal to
the number of intervals of that type.
Tabulated below in order of increasing tone ratio are the number of
intervals of each type that are left after the 24 pairs of intervals and their complements are
subtracted from the complete set of 91 intervals between the 14 notes of the seven scales:
The Godname YAH with number value 15 prescribes the number
of unpaired intervals that are not notes and the full Godname YAHWEH with number value
26 prescribes the number of unpaired notes before the crossover point. There
is one note T^{3} and the octave
T^{5}L^{2} after the crossover point. There are 24 pairs of notes and
their complements (see below), so that the set of 76 intervals consists of
26 unpaired notes before the crossover point and 50 other
intervals. This reflects the number values 26 and 50 of the
words YAHWEH and ELOHIM in the Godname YAHWEH ELOHIM.
Listed below are those intervals between the tonic and octave and their
numbers that do form pairs of intervals and their complements:
There are 48 intervals forming 24 pairs. Including the
tonic and octave, there are 25 pairs, i.e., 50 intervals. The Godname ELOHIM
with number value 50 prescribes how many of the intervals between notes in the
seven scales actually group together as complementary pairs. Including the tonic and octave,
there are 25 Pythagorean intervals and 25
6
nonPythagorean intervals. The 25:25 split exists not only for the intervals
and their complements but also for Pythagorean and nonPythagorean intervals! There are
49 intervals above the tonic that form pairs, showing how EL CHAI, the Godname
of Yesod with number value 49, prescribes the spectrum of intervals between
the 13 notes above the tonic. There are (49–13=36) intervals
that are not notes (i.e., 18 pairs), showing how ELOHA, Godname of Geburah with number value
36, prescribes these intervals. The 50 intervals therefore
become 36 intervals. This illustrates how the musical potential defined by ELOHIM, Godname of
Binah, becomes restricted by ELOHA, the Godname of the Sephirah below Binah on the
Pillar of Severity.
This 50→36 reduction is geometrically
represented in the inner form of the Tree of Life (Fig. 2). The seven separate polygons have 48 corners
symbolizing the 48 intervals that can form complementary pairs. The two
endpoints of the root edge, which formally are corners, symbolize the unit interval and the
octave. Together, they constitute 50 corners. The 12 notes in the seven
scales other than the octave are symbolized by the 12 corners of the dodecagon. The
36 corners of the first six separate polygons symbolize the intervals
forming pairs that are not notes. These extra musical intervals are symbolized by the
36 corners of the seven enfolded polygons.
The intervals in three pairs are not notes of the seven scales, leaving
21 pairs that are such notes. EHYEH prescribes those pairs of
intervals and their complements that are notes of the scales. There are (8+8=16) types of
intervals, 12 of which are notes of the seven scales and four of which are not. Taking into
account the four types of intervals that have no complements, the 14 notes of the seven scales
have (16+4+1=21) types of intervals. EHYEH prescribes how many kinds of
intervals there are in the 91 intervals between the 14 notes.
There are 16 types of rising intervals below the octave (6 Pythagorean, 10
nonPythagorean). Including the octave, there are 17 types (7 Pythagorean, 10 nonPythagorean).
Similarly, there are 16 types of falling intervals with tone ratios that are the reciprocal of
those of the rising intervals. Including the interval 1, there are (16+1+16=33) rising and
falling types of intervals between the 24 pairs of intervals. 33 = 1! + 2! + 3! + 4! and 24 =
1×2×3×4. This demonstrates how the Pythagorean integers 1, 2, 3, 4, which are symbolized by the
tetractys and whose ratios define the octave, perfect fifth and perfect fourth, express the
number of pairs of intervals and the number of types of intervals in them.
Including the unit interval and octave, the (9+9=18) types of intervals form
25 pairs:
1

2/1 
(×1) 
L 
2/L 
(×2) 
L^{2}* 
2/L^{2}* 
(×1) 
T 
2/T 
(×2) 
TL 
2/TL 
(×4) 
TL^{2}* 
2/TL^{2}* 
(×2) 
T^{2} 
2/T^{2} 
(×3) 
T^{2}L 
2/T^{2}L 
(×6) 
T^{2}L^{2} 
2/T^{2}L^{2} 
(×4) 
(As before, the tone ratios of intervals written in red are not those of
notes in the Pythagorean scale, and asterisked intervals are not notes of the seven musical
scales). Figure 3 shows how they constitute a Tree of Life pattern. The first (6+6)
enfolded polygons are a subset of the (7+7) enfolded polygons that constitute such a pattern
in themselves because they, too, are prescribed by the Godnames of the ten
Sephiroth.^{2}
The 50 intervals are symbolized by the
50 corners of the first (6+6) enfolded polygons (Fig. 3). The unit interval and the octave are denoted by the two endpoints of
the shared root edge. The 24 intervals and
7
their complements are symbolized by the 24 corners on each side of this
edge. The mirror symmetry of the two sets of polygons is the geometrical counterpart of the
complementarity between certain pairs of notes. The detailed correspondence between intervals
and corners is set out below:

Interval 
Complement 
Corner of triangle 
1×L^{2}* 
1×2/L^{2}* 
Two corners of square 
2×TL^{2}* 
2×2/TL^{2}* 
Three corners of pentagon 
3×T^{2} 
3×2/T^{2} 
Four corners of hexagon 
2×L + 2×T 
2×1/L + 2×2/T 
Six corners of octagon 
6×T^{2}L 
6×2/T^{2}L 
Eight corners of decagon 
4×TL + 4×T^{2}L^{2} 
4×1/TL + 4×2/T^{2}L^{2} 
The three intervals that are not notes of the seven scales are symbolized by
the corners of the triangle and square. This means that the 21 intervals that
are notes are naturally symbolised by the 21 corners of the next four
polygons.
The eight kinds of intervals between the notes of the seven scales that form
pairs correspond to the eight trigrams of the Taoist I Ching:
This is another example of the eightfold way discussed in Article
19^{3} They divide into two sets of four trigrams that express the two
Yang/Yin halves of a cycle. A musical octave is such a cycle and its eight notes, symbolised
by the eight trigrams, are created by leaps of four perfect fifths and four perfect fourths
(Fig. 4). The ancient Greeks regarded the eightnote musical scale as two
joined tetrachords, or groups of four notes. The fact that eightfold cyclical systems
divide into two sets of four phases raises the question of whether the eight types of
intervals naturally split into two quartets.
We pointed out in Article 32 that the 12 notes between the tonic and octave
that create the seven musical scales form two triplets: (T, T^{2}, T^{3}) and (T^{2}L^{2}, T^{3}L^{2}, T^{4}L^{2}), whose tone ratios are in the proportions
1:T:T^{2}, and two triplets (L,
TL, T^{2}L) and (T^{3}L, T^{4}L,
T^{5}L), whose tone ratios are in the same proportions. There are therefore
four triplets with the same proportions of their tone ratios. In each pair of triplets,
one triplet contains notes that are the complement of their corresponding notes in the
other triplet. These double and triple relationships can be represented by two Stars of
David (Fig. 5), one nested inside the other. The three points of one red or blue
triangle denote a triplet of notes and the three points of the inverted blue or
red triangle denote the ‘antitriplet’ of its complementary notes. The tonic and octave
may be thought of as the centre of the star nest. There are two triplets of intervals and
two antitriplets of their corresponding complements. Adding the two intervals
L^{2}* and TL^{2}* that do not belong
to any scale to the former and their complements to the latter will create
two quartets of intervals, so
8
that the eight basic intervals can be divided into two halves, thus
upholding the ancient view of the number 8 as “twice 4.”^{4}
According to Tables 2, 3 & 4, L^{2}* =
6536/59049* appears twice either as X_{57} = 1024/729 ÷81/64 or as Y_{31} =
128/81÷729/512. In either case, the pair of tone ratios does not appear within the same
scale. TL^{2}* = 8192/6561
appears three times either as X_{37} = 1024/729÷9/8, Y_{51} = 16/9÷729/512
or as Z_{53} = 128/81÷81/64. In all three cases, the two tone ratios do not appear
in the same scale. This means that the extra two intervals L^{2}* and TL^{2}* and their six complements added
to the six intervals and their complements are between two notes in different
scales. In other words, they do not appear when music is played in any one scale, only if
the available notes are all 14 notes.
The eight basic intervals L, L^{2}*, T,
TL, TL^{2}*, T^{2}, T^{2}L
& T^{2}L^{2}
and their eight complements have their counterpart in superstring theory as the eight
roots of E_{8} and the eight roots of E_{8}'. Musically speaking, the
division of the octave into notes and their complements corresponds to the distinction in
the E_{8}×E_{8}' heterotic superstring theory between superstrings of
ordinary matter governed by E_{8} and superstrings of shadow matter governed by
E_{8}'. In music, the distinction between notes and their complements is the
manifestation in tones of the duality of Yang and Yin. The same can be said for the
fundamental difference between ordinary and shadow matter. The musical counterpart of the
group distinction between E_{8} and its exceptional subgroup E_{6} with
six roots is the difference between the eight distinct intervals, of which six are actual
notes. It may not be coincidental that the dimension 78 of E_{6} is the number of
intervals between the 13 notes of the seven musical scales above the tonic, as
^{13}C_{2} = 78.
The sequence of nine basic intervals^{1}:
1, L, L^{2}, T, TL, TL^{2}, T^{2}, T^{2}L,
T^{2}L^{2}
can be written
(1, L, L^{2})
T(1, L, L^{2})
T^{2}(1, L, L^{2})
Successive triplets of intervals have the same proportion 1:L:L^{2}
in the tone ratios of the members of each triplet. We discussed earlier that triplets of notes
in the seven scales can be found that have the same proportion of 1:T:T^{2} of the
first three notes C, D & E of the Pythagorean scale (C scale). Let us therefore carry out
an exhaustive analysis of triplets of intervals drawn from the complete set of 18 intervals
that exhibit proportions of the form 1:X:X^{2}, where X = L, T, TL, T^{2} or
T^{2}L (the only possible values, because the largest interval is
T^{5}L^{2} = 2).
X = L.
1. ×1: 
(1, L, L^{2}) 
(T^{5}, T^{5}L, T^{5}L^{2}) 
2. ×T: 
(T, TL, TL^{2}) 
(T^{4}, T^{4}L, T^{4}L^{2}) 
3. ×T^{2}: 
(T^{2}, T^{2}L, T^{2}L^{2}) 
(T^{3}, T^{3}L, T^{3}L^{2}) 
4. ×T^{3} : 
(T^{3}, T^{3}L, T^{3}L^{2}) 
(T^{2}, T^{2}L, T^{2}L^{2}) 
5. ×T^{4}: 
(T^{4}, T^{4}L, T^{4}L^{2}) 
(T, TL, TL^{2}) 
6. ×T^{5}: 
(T^{5}, T^{5}L, T^{5}L^{2}) 
(1, L, L^{2}) 
As (1) is the same as (6), (5) is identical to (2) and (4) is the same as
(3), there are three different triplets: (1), (2) & (3).
X = T.
1. ×1: 
(1, T, T^{2}) 
(T^{3}L^{2}, T^{4}L^{2},
T^{5}L^{2}) 
2. ×L: 
(L, TL, T^{2}L) 
(T^{3}L, T^{4}, T^{5}L) 
3. ×L^{2}: 
(L^{2}, TL^{2}, T^{2}L^{2}) 
(T^{3}, T^{4}, T^{5}) 
4. ×T: 
(T, T^{2}, T^{3}) 
(T^{2}L^{2}, T^{3}L^{2},
T^{4}L^{2}) 
5. ×TL: 
(TL, T^{2}L, T^{3}L) 
(T^{2}L, T^{3}L, T^{4}L) 
6. ×TL^{2}: 
(TL^{2}, T^{2}L^{2}, T^{3}L^{2}) 
(T^{2}, T^{3}, T^{4}) 
7. ×T^{2}: 
(T^{2}, T^{3}, T^{4}) 
(TL^{2}, T^{2}L^{2}, T^{3}L^{2}) 
8. ×T^{2}L: 
(T^{2}L, T^{3}L, T^{4}L) 
(TL, T^{2}L, T^{3}L) 
9. ×T^{2}L^{2} 
(T^{2}L^{2}, T^{3}L^{2},
T^{4}L^{2}) 
(T, T^{2}, T^{3}) 
Multiplying by the remaining intervals just replicates the pairs above
because they are the complements of the first eight intervals. As (7) is the same as (6), (8)
is the same as (5) and (9) is identical to (4), there are six different pairs of triplets:
(1)(6).
X = TL.
1. ×1: 
(1, TL, T^{2}L^{2}) 
(T^{3}, T^{4}L, T^{5}L^{2}) 
________________________________
^{1} The asterisk and red lettering for nonPythagorean
intervals are dropped from now on.
9
2. ×T: 
(T, T^{2}L, T^{3}L^{2}) 
(T^{2}, T^{3}L, T^{4}L^{2}) 
3. ×T^{2}: 
(T^{2}, T^{3}L, T^{4}L^{2}) 
(T, T^{2}L, T^{3}L^{2}) 
4. ×T^{3}: 
(T^{3}, T^{4}L, T^{5}L^{2}) 
(1, TL, T^{2}L^{2}) 
As (1) & (4) are the same and as (2) and (3) are the same, there are two
different triplets: (1) & (2).
X = T^{2}.
1. ×1: 
(1, T^{2}, T^{4}) 
(TL^{2}, T^{3}L^{2}, T^{5}L^{2}) 
2. ×L: 
(L, T^{2}L, T^{4}L 
(TL, T^{3}L, T^{5}L) 
3. ×L^{2}: 
(L^{2}, T^{2}L^{2}, T^{4}L^{2}) 
(T, T^{3}, T^{5}) 
4. ×T: 
(T, T^{3}, T^{5}) 
(L^{2}, T^{2}L^{2}, T^{4}L^{2}) 
5. ×TL: 
(TL, T^{3}L, T^{5}L) 
(L, T^{2}L, T^{4}L) 
6. ×TL^{2}: 
(TL^{2}, T^{3}L^{2}, T^{5}L^{2}) 
(1, T^{2}, T^{4}) 
There are three different triplets: (1), (2) & (3). For X =
T^{2}L, there is only the triplet: (T, T^{3}L, T^{5}L^{2}).
Hence, this case is of no interest.
There are, therefore, 14 independent triplets and their complements:
1 
(1, L, L^{2}) 
(T^{5} T^{5}L, T^{5}L^{2}) 
(13: X = L) 
2 
(T, TL, TL^{2}) 
(T^{4}, T^{4}L, T^{4}L^{2}) 
3 
(T^{2}, T^{2}L, T^{2}L^{2}) 
(T^{3}, T^{3}L, T^{3}L^{2}) 
4 
(1, T, T^{2}) 
(T^{3}L^{2}, T^{4}L^{2},
T^{5}L^{2}) 
(49: X = T) 
5 
(L, TL, T^{2}L) 
(T^{3}L, T^{4}L, T^{5}L) 
6 
(L^{2}, TL^{2}, T^{2}L^{2}) 
(T^{3}, T^{4}, T^{5}) 
7 
(T, T^{2}, T^{3}) 
(T^{2}L^{2}, T^{3}L^{2},
T^{4}L^{2}) 
8 
(TL, T^{2}L, T^{3}L) 
(T^{2}L, T^{3}L, T^{4}L) 
9 
(TL^{2}, T^{2}L^{2}, T^{3}L^{2}) 
(T^{2}, T^{3}, T^{4}) 
10 
(1, TL, T^{2}L^{2}) 
(T^{3}, T^{4}L, T^{5}L^{2}) 
(1011: X = TL) 
11 
(T, T^{2}L, T^{3}L^{2}) 
(T^{2}, T^{3}L, T^{4}L^{2}) 
12 
(1, T^{2}, T^{4}) 
(TL^{2}, T^{3}L^{2}, T^{5}L^{2}) 
(1214: X = T^{2}) 
13 
(L, T^{2}L, T^{4}L) 
(TL, T^{3}L, T^{5}L) 
14 
(L^{2}, T^{2}L^{2}, T^{4}L^{2}) 
(T, T^{3}, T^{5}) 
Seven triplets [(1)(6) & (10)] have intervals in the first half of the
octave (they are all notes in four of them). There are even triplets of intervals [(2), (3)
& (5)(9)] other than 1 with X = L or T. There are also seven such triplets with X = T or
T^{2}. Of these, five [(5)(9)] show the proportions 1:T:T^{2} and two
[(13) & (14)] show the proportions 1:T^{2}:T^{4}. They are shown below:
1 
(L, TL, T^{2}L) 
(T^{3}L, T^{4}L, T^{5}L) 
(15: X = T) 
2 
(L^{2}, TL^{2}, T^{2}L^{2}) 
(T^{3}, T^{4}, T^{5}) 
3 
(T, T^{2}, T^{3}) 
(T^{2}L^{2}, T^{3}L^{2},
T^{4}L^{2}) 
4 
(TL, T^{2}L, T^{3}L) 
(T^{2}L, T^{3}L, T^{4}L) 
5 
(TL^{2}, T^{2}L^{2}, T^{3}L^{2}) 
(T^{2}, T^{3}, T^{4}) 
6 
(L, T^{2}L, T^{4}L) 
(TL, T^{3}L, T^{5}L) 
(67: X = T^{2}) 
7 
(L^{2}, T^{2}L^{2}, T^{4}L^{2}) 
(T, T^{3}, T^{5}) 
They contain the intervals L^{2}, TL^{2} and their
complements. These are not notes of the seven scales, merely intervals between notes in
different scales. There are six triplets [(4), (5), (7), (8), (12) & (13)]
with X = T or T^{2} whose intervals are all notes. There is one triplet (3) with X
= L whose intervals are notes. Hence, there are seven triplets all of whose intervals are notes
with X = L, T or T^{2}.
However the seven triplets be defined, they bear a striking correspondence
to the seven 3tuples of octonions, as now explained. The octonions are the numbers of the
fourth and last class of division algebras. They are linear combinations of the eight unit
octonions e_{i} (i = 0, 1, 2, … 7) that consist of the real unit octonion
e_{0} = 1 and seven unit imaginary octonions e_{j} (j = 17) whose
multiplication is nonassociative and noncommutative:
e_{i}e_{j} = δ_{ij}e_{0} +
Σf_{ijk}e_{k} (i, j, k = 1, 2,….7)
where f_{ijk} is antisymmetric with respect to the indices i,
j, k and has values 1, 0, & 1. The seven unit imaginary octonions form seven 3tuples
(e_{i}, e_{i+1}, e_{i+3}) with the cyclic property of
multiplication
e_{i}e_{i+1} = e_{i+3}.
Their explicit forms are listed below:
(e_{1}, e_{2}, e_{4})
10
(e_{2}, e_{3}, e_{5})
(e_{3}, e_{4}, e_{6})
(e_{4}, e_{5}, e_{7})
(e_{5}, e_{6}, e_{1})
(e_{6}, e_{7}, e_{2})
(e_{7}, e_{1}, e_{3})
Their multiplication is geometrically represented by the Fano plane
(Fig. 6), which is the simplest projective plane. A projective plane of order
n consists of (1+n+n^{2}) points and (1+n+n^{2}) lines. The Fano plane is of
order n = 2 because it comprises seven points and seven lines. The eight notes of the
Pythagorean scale are analogous to the eight unit octonions.
As a note, the tonic can have any pitch, being simply the base with respect
to which the tone ratios of the other notes are measured. It corresponds to e_{0} = 1,
the base of the real numbers. The seven rising intervals n_{i} above the tonic
correspond to the seven unit imaginary octonions, their falling intervals (their reciprocals
1/n_{i}) corresponding to the conjugates of the imaginary octonions e_{i}* =
e_{i}, so that e_{i}e_{i}* = 1 = n_{i}×1/n_{i}.
Alternatively, the counterpart of conjugate octonions may be thought of as the complement m of
a note n, where nm = 2.
The counterparts of the seven 3tuples are the seven musical scales.
Table 1 shows the tone ratios of their notes. Table 6 shows their composition in terms of the T and L.
Table 6. Intervallic composition of the notes of the seven musical scales.
The 17 triplets that show a 1:T:T^{2} scaling of their tone ratios
are not all different. Including the triplet (1, T, T^{2}), there are eight distinct
triplets (four triplets of intervals and four triplets of their complements):
11
This is another musical counterpart of the eight trigrams, the Yang/Yin
polarities of the lines and broken lines in each one corresponding to the intervals and their
complements. It is also the musical counterpart of the eight unit octonions, with (1, T,
T^{2}) being equivalent to the real unit octonion e_{0} and the seven other
triplets being equivalent to the seven imaginary octonions.
Each musical scale is unchanged under interchange of each note and its
complement. Similarly, the Fano plane is invariant under interchange of its points and lines
and the eight trigrams remain the same set when their Yang and Yin lines are interchanged. The
seven scales have 168 rising and falling intervals that are repetitions of the
basic set of 12 notes between the tonic and octave. In the 64 hexagrams of the
I Ching table, there are 28 pairings of different trigrams with 168
Yang/Yin lines. The Fano plane has 168 symmetries described by SL(3,2),
the special linear
group of 3×3 matrices with unit determinant over the field of complex
numbers. The trigrams are the expression of the 3×3 matrices and their pairing is the
counterpart of this field of order 2. SZ(3,2), the centre of SL(3,2), is the set of scalar
matrices with unit determinant and zero trace. It is isomorphic to the third roots of 1. The
three roots are 1, exp(2πi/3) and exp(4πi/3). Plotted in the Argand diagram, they are located at
the three corners of an equilateral triangle. The cyclic group of order 3 is
C_{3} = (1, r, r^{2}), where the generator r = exp(2πi/3) is the primitive third root of 1. It is the
counterpart of the generation of the nine basic types of intervals in the seven
scales:
(1+T+T^{2})(1+L+L^{2}) = 1 + L + L^{2} + T + TL +
TL^{2} + T^{2} + T^{2}L + T^{2}L^{2}.
It is known that 1 + X + X^{2} is the only irreducible polynomial of
degree 2 on the finite field of order 2. This plus the fact that the algebra of the octonions
can be represented by the Fano plane of order 2, which is the simplest of the projective
planes
12
of order n that have (1+n+n^{2}) points and lines, is strong
evidence that the mathematical analogy between the octonions and the notes of the seven musical
scales is significant. It exists because the Pythagorean mathematics of music and the
mathematics of octonions are parallel manifestations of a universal paradigm.
Polyhedral geometrization of the seven musical
scales It was found earlier that the maximum number of intervals between the
14 basic notes of the seven scales that have complements is 24, leaving 26
unpaired intervals before the crossover into the complements of these intervals. There are
therefore 74 such intervals. Table 5 in Article 26^{5} indicates that there are no Archimedean or Catalan solids with 74
vertices. However, the disdyakis dodecahedron has 48 faces and
26 vertices. If a tetrahedron is attached to each face, the resulting
polyhedron has (3×48=144) faces and
(48+26=74) vertices (Fig. 7). This polyhedron was discussed in Article 24 as being the ‘yang’
counterpart of the ‘yin’ disdyakis triacontahedron, i.e., they constitute a dualistic
whole. Remarkable evidence of this is that the 264 faces of both polyhedra are
symbolised by the 264 yods of the inner Tree of Life, the 120 yods on their boundaries
symbolising the 120 faces of the disdyakis triacontahedron and the 144 yods inside them
denoting the 144 faces of the polyhedron (Fig. 8). The 48 peaks of the tetrahedra correspond to the 24
pairs of complementary intervals and the 26 original vertices correspond to
musical intervals that are left over, so to speak, unable to form such pairs, and therefore
not actively participating in the embodiment of the seven musical scales in the disdyakis
triacontahedron. These play a dynamic, generative role because creation is a cyclic
interplay of Yang and Yin represented by intervals and their complements making up the
octave cycle and no more than 24 tonal intervals (yang) have their complementary opposites
(yin). Indeed, we are countenancing here the patterndetermining character of the number 24,
as explained in other contexts in Article 37.^{6} The number 48 is the number value of
Kokab, the Mundane Chakra of Hod (Table 1), and its 24:24 division is characteristic of holistic systems that
embody the divine paradigm, as the first (6+6) enfolded polygons of the inner Tree shown in
Fig. 3 illustrate.
The polyhedron with 144 faces and 74 vertices spatially represents the
musical potential in terms of intervals between the basic set of 14 notes that make up
the seven scales, whilst the disdyakis triacontahedron represents their organisation
into the patterns recognisable as the seven musical scales — the very basis of Western music
itself. Its (90+90) edges are the geometrical counterpart of the 90 rising and 90 falling
intervals between these notes, i.e., the mirror symmetry in the distribution in space of their
edges is the counterpart of the distinction between a rising interval and a falling interval.
The 12 edges (six edges & their six mirror images) along the equator of the disdyakis
triacontahedron when its ‘north’ and ‘south’ poles are diametrically opposite A vertices denote
the six rising and six falling perfect fifths found in the 24 intervals and their complements
(the only type of interval to have six copies — see the list on page 7). These 6:84 divisions
in each set of 90 intervals and in the edges of the disdyakis triacontahedron are,
respectively, the musical and geometrical manifestation of the mathematical archetype
represented by Plato’s Lambda Tetractys:
Its ten integers add up to 90, that is, the nine integers surrounding the
central integer 6 add up to 84. The 168 remaining intervals that are
not perfect 5ths (84 rising and 84 falling) correspond to the 84 edges above the equator and
the 84 edges below the equator. The 24 vertices above the equator symbolize the maximal set of
24 intervals, which are matched by their complementary intervals denoted by the 24 vertices
below the equator. The two remaining vertices (the poles of the disdyakis triacontahedron)
denote the tonic and the octave — the beginning and the end of the musical scale.
Figure 8 shows how this information is embodied in the dodecagon — the last
of the regular polygons enfolded in the inner Tree of Life. When its sectors are divided
into three tetractyses, there are 180 yods surrounding its centre. They symbolize the 180
edges of the disdyakis triacontahedron — the polyhedral realisation of the inner Tree of
Life. They also denote the 180 rising and falling intervals below the 14 notes of the seven
musical scales. The 12 vertices of the 36 tetractyses making up the
dodecagon correspond to the 12 edges in the equator of the disdyakis triacontahedron and, in
the musical context of the intervals, the six rising perfect fifths (3/2) and the six
falling perfect fifths (2/3). The 84 remaining yods in six sectors are the counterpart of
the 84 edges above the equator and the 84 rising intervals. The 84 yods in the other six
sectors are the counterpart of the 84 edges below the equator and the 84 falling intervals.
The central yod signifies the tonic as the starting note. Its counterpart in the
disdyakis
13
triacontahedron is the imaginary, internal line joining two
diametrically opposite A vertices — the axis of the polyhedron.
Of the (24+24) intervals, there are 21 notes and
21 complements with tone ratios of notes in the seven scales. Interestingly,
Table 1 indicates that there are actually just 21
notes with these tone ratios! This demonstrates how the Godname EHYEH with number
value 21 prescribes the composition of the 91 intervals between the 14
different notes of the seven scales. The three remaining intervals (one L^{2}* and
two TL^{2}*) are not notes. This 3:21 differentiation was found in
Article 21^{7} in the context of the 24 lines and broken lines
making up the eight trigrams. The positive and negatives lines of each
trigram denote the positive and negative directions with respect to a rectangular coordinate
system of the three perpendicular faces of a cube whose intersection is one of its eight
corners. If such cubes are stacked together, any one corner of a cube coincides with the
corners of seven other cubes (three on the same level and four either above or below it). This
means that a cubic lattice point is defined by the intersections of three faces belonging to
eight cubes, three belonging to the cube itself and 21 belonging to the seven
cubes that surround it.
The same 3:21 division appears in the Klein
Configuration.^{8} This is the hyperbolic mapping of the 168
automorphisms of the equation known to mathematicians as the “Klein quartic”:
x^{3}y + y^{3}z + z^{3}x = 0.
These symmetries of its Riemann surface can be mapped onto the hyperbolic
surface of a 3torus in a number of different ways. Figure 9 shows the {7,3) tiling that requires 24 heptagons divided into
168 coloured triangles. It also has 168 antiautomorphisms
represented by the 168 grey triangles of 24 other heptagons. These two sets
of 24 heptagons are the counterpart of the 24 intervals and their 24 complements. The three
intervals and their complements that are not notes of the seven scales correspond,
respectively, to the three cyan triangles in Figure 9 at the corners of a halfsector and to the three grey triangles at
the corresponding corners of the other halfsector. Notice that the one L^{2}* and
the two TL^{2}* intervals in the set of 24 match, respectively, the innermost
triangle and the two outermost triangles in a halfsector. They correspond in the 3×3×3
array of cubes displaying an isomorphism with the Klein configuration to the three faces of
the central cube intersecting at one corner.^{9} The 168 automorphisms of the Klein quartic
correspond to the 168 rising and falling intervals other than the six
perfect fifths and to the 168 edges above and below the equator of the
disdyakis triacontahedron,10 its six edges and their inverted images corresponding,
respectively, to the six rising perfect fifths and to the six falling perfect fifths in the
90 intervals below the octave between the 14 notes of the seven scales. Both are the
manifestation of the projective, special linear group PSL(2,7), which is the quotient group
SL(2,7)/{1,1}, where 1 is the identity matrix, and SL(2,7) consists of
all 2×2 matrices with unit determinant over F_{7}, the finite field with 7 elements.
These elements can be the seven types of intervals between notes of the seven musical scales
and the seven unit imaginary octonions ei, whose algebra is represented by the Fano plane
with the symmetry group SL(3,2) that is isomorphic to PSL(2,7). Their seven conjugates
e_{i}* = e_{i}, where e_{i}e_{i}* = 1, correspond to the
complements y_{i} of the seven notes x_{i}, where x_{i}y_{i}
= 2, whilst their seven 3tuplets (e_{i}, e_{i+1}, e_{i+3}) and the
seven 3tuplets of their conjugates (e_{i}*, e_{i+1}*, e_{i+3}*)
correspond, respectively, to the seven triplets of intervals and to the seven triplets of
their complements that display the same relative proportions 1:T:T^{2}. of their
tone ratios.
Whether the 168 rising and falling intervals are
actual elements of PSL(2,7) is irrelevant except to one who cannot see the larger
picture. Anyone who demands a formal proof that they form this group before he takes the
analogy seriously is missing the crucial point. Such proof is necessary only if one makes the
stronger claim that the intervals are such elements. However, judging the similarity to be
significant evidence of a universal principle because it is too implausible to be due to chance
does not require this
14
stronger version to be made. What is sufficient is to demonstrate that:
1. the mathematical properties of the two sets of seven basic intervals
found in the seven musical scales are at least analogous to the properties of PSL(2,7) in too
many ways for this to be coincidental;
2. these properties can be represented by the polygonal and polyhedral forms
of the outer and inner Trees of Life in too much detail and in too natural a way either for the
matching to be contrived, i.e., for it to indicate anything other than that PSL(2,7) and the
musical intervals between the notes in the seven scales embody the same, essential
Tree of Life pattern. This is what has been done here.
If such matching cannot plausibly be attributed to coincidence because it is
too detailed, two systems can be mathematically analogous only because they are both holistic
in nature and therefore manifest in their own way — physically or conceptually — the
same, universal paradigm. The mathematical patterns in a system and in some symmetry
group need only be similar in appearance. The former does not necessarily have to amount
formally to a group symmetry that is isomorphic to the latter in order to constitute evidence
of such a paradigm. The fact that such extensive analogy exists between topics as diverse as
octonions, the eight simple roots of E_{8}, musical scales and acupuncture meridians,
as demonstrated in this and previous articles, is not an illusion due to some contrived
selection of features that match and the ignoring of those that do not. The remarkable, natural
appearance of at least eight Godname numbers to prescribe the properties of the 90 intervals
totally discredits such a suggestion and confirms the status of the seven musical scales as a
holistic system that embodies the Tree of Life pattern. It indicates that a universal principle
connects all these systems as different facets of a pervading Unity hidden within diversity.
Its polyhedral realisation is the disdyakis triacontahedron.
References
^{1} Phillips, Stephen M. Article 14: "Why the Seven Greek Musical Modes are
Sacred," (WEB, PDF).
^{2} Phillips, Stephen M. Article 4: "The Godnames Prescribe
the Inner Tree of Life," (WEB, PDF), pp. 4–5.
^{3} Phillips, Stephen M. Article 19: "I Ching and the
Eightfold Way," (WEB, PDF).
^{4} Ibid, p. 6.
^{5} Phillips, Stephen M. Article 26: "How the Seven Musical
Scales Relate to the Disdyakis Triacontahedron," (WEB, PDF), Table 5, p. 13.
^{6} Phillips, Stephen M. Article 37: "The Seven Octaves of
the Seven Musical Scales are a Tree of Life Pattern Mirrored in the Disdyakis Triacontahedron,"
(WEB, PDF).
^{7} Phillips, Stephen M. Article 21: "Isomorphism between
the I Ching Table, the 3×3×3 Array of Cubes and the Klein Configuration," (WEB, PDF), pp. 2–4.
^{8} Phillips, Stephen M. Article 15: "The Mathematical
Connection Between Superstrings and Their Micropsi Description: a Pointer Towards Mtheory,"
(WEB, PDF), pp. 24–28; also ref. 3, pp. 29–30.
^{9} Ref. 6, p. 8.
^{10} Ibid, Fig. 35, p. 34.
15
