ARTICLE 16
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH.
England.
Website: http://www.smphillips.mysite.com
“The first and noblest application of music is in offering the tribute of
praise to the immortals; the next is the purifying, regulating and harmonizing the
soul.”
Plutarch
Abstract
The likelihood that the ancient
Egyptians passed on their knowledge of the seven types of
musical scales to Pythagoras, Plato and others who studied with
them makes the currently accepted view that the musical modes
of ancient Greece were keys rather than octave species highly
implausible. The ten Hebrew Godnames known to Kabbalah are
found to prescribe the interval composition of these scales.
Both Plato’s Lambda and the first four polygons enfolded in the
inner form of the Tree of Life embody the minimum number of
different intervals between the notes of the seven modes.
Remarkable parallels between the interval composition of the
latter and the tetrahedral generalisation of the Lambda are
demonstrated. There is similar correspondence with the
oscillatory form of the superstring. This is because
superstring theory is connected to the mathematics of
octonions, whose properties are analogous to musical intervals.
Remarkable correspondence exists between the 168 permutations
of the seven 3tuples of octonions, the 168 automorphisms of
the Klein Quartic, the 168 repetitions of the Pythagorean
intervals of the seven octave species and the superstring
structural parameter 168. They indicate that the oscillations
performed by superstrings are akin to music played with the
notes of the seven musical scales.

I. Ancient Greek musical modes —
octave species or keys?
Controversy has persisted to this day over the issue of whether the
ancient Greeks classified their different types of music according to keys or to octave
species. Did the names Dorian, Phrygian, Lydian, etc. refer to specific keys conferring a
distinct moral characteristic, or ethos, or were these qualities of music due to
their being played in scales that differed in the ordering of what we would now call their
tones and semitones? The former view is the more accepted today. For example, Rita Steblin,
the worldrenowned musical scholar, states: “The ancient
1
Greeks believed that each of their keys, which they identified by the tribal
names Dorian, Phrygian, etc., possessed a strongly marked ethical character.”^{1} This opening statement in her treatise on the history of the
psychological attributes of musical keys may state what is now commonly accepted by many
(but not all) scholars but it still begs the question. If the ethos of ancient
Greek music was really a matter of higher and lower pitch, one has, like Donald Grout, one
of the giants of American musicology, to ask: “... how could a mere transposition by one
tone up or down account for the difference in ethos which Plato postulates
between the Dorian and the Phrygian modes?” … Its “only audible effect would be a
redistribution of the tones and semitones within one and the same octave range...”^{2} It is quite possible that the different ethical attributes
of the ancient modes had more to do with the nature of the songs and melodies of the
different races from whom the ancient Greeks descended. No one, however, knows how the music
of ancient Greece sounded. In the absence of reliable, surviving evidence like musical score
sheets or playable musical instruments, one can argue either case, citing various
authorities to support one’s position, whilst at the same time ignoring past and present
writers of equal prestige who take the contrary view. As Donald Jay Grout remarked: “…we do
not know, and in all probability we shall never know, exactly what were the modes about
which Plato and Aristotle wrote, or how these may have been different from the modes as
understood by later writers. …Each of the various hypotheses about the modes has claims to
credibility, and each is warmly defended by its champions. No field of musicology has
produced a richer crop of disputation from a thinner soil of fact.”^{3} This position of neutrality seems a more honest attitude to
take than that of Steblin, whose partisanship deceives her readers into thinking that the
argument was long won, whereas controversy persists amongst scholars on this issue.
The problem hinges on the question of whether the Greek musical terms
harmonia and tonos meant different things or the same thing. Some
scholars, who believe that the ancient Greeks knew only keys argue for the latter, whilst
others propose that the Greek modes had a double meaning. For the latter, the use of the word
harmonia indicated a “mode” and signified the order in which the intervals
followed one another, i.e., the species of the octave scale, whilst tonos signified a fixed
pattern of intervals that could be transposed in pitch, i.e., a key. Unless one is wedded to
the implausible view that all the modes — however dissimilar in ethos — were the
same diatonic scale, differing only by pitch, it is hard to believe that two different musical
terms meaning the same would have been in use in the Greek musical lexicon. They are, surely,
evidence that the ancient Greeks did know of both keys and octave species. The
historical evidence cannot decide whether the ethos of their modes was due to the former or to
the latter. Why, however, should it be a case of one or the other? It is possible that
both played a role in defining the distinctions between modes and that this
ambiguity led to the disagreements between writers as to the correct ethos to be attached to a
particular mode. They may have been all correct, because the modality ascribed to the same
pieces of music might not over time have always been determined by the same criterion! This
possibility seems to have been ignored in the longstanding, polarised debate.
Another argument used against the octave species interpretation of Greek
modes is that the church modes had Byzantine origins, their four authentic and four plagal
modes arising from the Byzantine octoechoi, which, too, were grouped into four pairs
and which were not scales but collections of melodic motives expressing more or less the same
quality of feeling. However, the fact that early, Western musical theorists deduced the scales
from which these groups of motives were formed and then incorporated (wrongly, as it turned
out) ancient Greek names of modes into their labelling of these scales does not imply that the
original Greek modes differed by key. This is a non sequitur, although a convenient
one for advocates of the key interpretation of modes to make. Yet it is implicit in the
arguments used by Steblin against the interpretation of the Greek modes as octave species. In
her discussion of the origins of Christian church music, she is satisfied to stop at the
Byzantine era, not bothering to ask whether its music was influenced by that of Greece, Egypt,
Syria and other regions where Christians settled. As Grout
2
points out, the early Christians were Jews and would have borrowed their
music from the chants of the Hebrew synagogues, which had melodies based upon
modes.^{4} Far from being, as she supposes, a false attribution to the
ancient Greeks by early Christian musical theorists, modes as octave species may have been
known to the Greek cognoscenti and may have influenced the creation of the
church modes indirectly through the largely unknown impact of Greek music on
the Byzantine system. In other words, although the church modes were not derived from the
Greek musical modes, they could still have amounted to their rediscovery! This possibility
is not considered by Steblin in her defence of the orthodox position in the recirculates
opinions that, however learned, must remain speculative for the obvious reason that these
authorities were not contemporaneous with the culture of ancient Greece but, instead, were
separated from it by many centuries. No scholar can have read all that was written later
about music in that era. Quotations will inevitably be selective and selfserving, excluding
whatever contradicts firmly held beliefs. Pressure in the academic world to conform to
orthodox opinion can never be overestimated.
2. Ancient evidence for modes being
octave species
There are only seven octave species because they are generated by taking
successive notes of the sevennote Pythagorean scale as their tonic (it makes no difference to
this fact whether musical scales are regarded as ascending or descending):
(T is the Pythagorean whole tone interval of 9/8, which is slightly sharper
than its equaltempered counterpart, and L is the Pythagorean leimma, an interval of 256/243,
which is somewhat flatter than the semitone). It must be emphasized that the modal labels shown
here are those of the church. It is now recognised that the early church musical theorists
assigned the Greek labels Dorian, Phrygian, Lydian, etc to successive octaves of the ascending
musical scale, not realising that in ancient Greece it was always regarded as descending. This
caused mislabelling, so that the ethos that the church associated with the modes
sometimes disagreed with ancient opinion owing to their misnaming. It was also confusing to
number the four Authentic Modes by odd integers and the four Plagal (Hypo) Modes by even
integers because this destroyed all sense of their order of generation from successive notes of
the diatonic scale, as is shown above by the modal numbers being out of sequence. Perhaps the
motivation for this was to make Plato’s ideal mode — the Dorian — the first in the list.
Given that the ancient Egyptians had an 8note scale and that Pythagoras
spent many years studying mathematics, music and astronomy with the Egyptian priesthood, it is
very likely that he did not discover the mathematics of the musical scale, as popular textbooks
on the history of science still inform us, but that he learnt it from his Egyptian teachers. It
is therefore not hard to believe that he would have passed on his acquired knowledge to his
brotherhood of disciples in Croton, Italy, some of whom then imparted it to their students in
ancient Greece itself. Indeed, we know that Plato obtained some of the mathematical secrets of
the early Pythagoreans by purchasing a book called Fragments from the parents of
its author, Philolaus of Tarentum (ca. 480–unknown B.C.E.). Philolaus was educated by Lysis,
one of the two disciples of Pythagoras
3
who escaped the massacre of the Pythagorean community at Croton and who
subsequently became the first to write down his master’s teachings in his
Fragments.^{5} A tradition is recorded by Strabo^{6} that Plato spent thirteen years of study at Heliopolis, making it
highly plausible that he learnt there the mathematics of music. In his The History of
Music, W. Chappell went as far as to say that the “entire Greek system was mainly
derived from Egypt, Phoenicia, Babylon, or other countries of more ancient civilisation than
Greece.”^{7} He pointed out that:
“Diodorus Siculus says that the musicians and poets of Greece
visited Egypt for the purpose of improvement, and that the Egyptian priests had records
of their visits in their books. The first two names so recorded were those of Orpheus
and Musæus, and Homer followed. So, at least, the Egyptians claimed to have taught music
and poetry to the Greeks at a very early period. The later names in Diodorus’s list,
such as those of Solon, Plato, and others, are admittedly authentic.”^{8}
Furthermore, he said:
“Nichomachus, quoting Pythagoras and Plato,
tells us that the Egyptians ascribed twentyeight sounds to the universe, calling it
“twentyeight sounding.” So the Egyptians must have had twentyeight sounds, i.e.,
twentyeight notes, in their scales. That is the precise total number of Greek notes, in
their greater and lesser perfect systems combined, and including all their scales —
Diatonic, Chromatic, and Enharmonic. Neither in Egypt nor in Greece was there an actual
limit to twentyeight sounds, because all scales were transposable, but only twentyeight
notes could be defined, starting from any given pitch. Euclid, Nicomachus, Aristides
Quintilianus, and others, enumerate the Greek scales and their notes, and all authors are
agreed as to the number being precisely twentyeight. This most remarkable coincidence
between Egypt and Greece seems nevertheless to have escaped the observation of historians
of music. If it stood alone, it would almost suffice to prove the origin of Greek music.
The number is too peculiar to have been arrived at by accident, within a compass of only
two Octaves.”^{9}
A simple calculation reveals the significance of this
number. Table 1 below shows the tone ratios for each mode, derived from the patterns of
intervals that were listed above:
Table 1. Tone ratios of the seven Church
modes.
Lydian

1

9/8

81/64

729/512

3/2

27/16

243/128

2

Phrygian

1

256/243

32/27

4/3

3/2

128/81

16//9

2

Dorian

1

9/8

32/27

4/3

3/2

27/16

16/9

2

Hypolydian

1

9/8

81/64

4/3

3/2

27/16

243/128

2

Hypophrygian

1

256/243

32/27

4/3

1024/729

128/81

16/9

2

Hypodorian

1

9/8

32/27

4/3

3/2

128/81

16/9

2

Mixolydian

1

9/8

81/64

4/3

3/2

27/16

16/9

2

Blue cells contain Pythagorean tone ratios and white
cells contain tone ratios not belonging to any octave of the Pythagorean scale. There are 16
such ‘nonPythagorean’ notes. The (7×7=49^{†}) notes below the octaves of the seven modes comprise
(49–16=33) Pythagorean tone ratios, of which seven are 1’s, leaving
(33–7=26) Pythagorean tone ratios other than 1. Including the tonic and
octave, the seven modes have 28 notes belonging to the Pythagorean scale. This confirms
the quotation above. It proves that the ancient Egyptians must have known about seven
different octave
_______________________________
^{†}Numbers appearing throughout the text in bold type are the number values of
the Hebrew Godnames of the ten Sephiroth of the Tree of Life (see previous articles for their
tabulated values).
4
species, for Pythagoras (556 B.C.E.–unknown) and Plato (428–347 B.C.E.)
certainly knew that the Egyptians had knowledge of 28 musical sounds. As both studied with
the Egyptian priesthood, it is reasonable to infer that they knew about the seven octave
species with 28 Pythagorean notes. This means that the existence of seven musical
scales, if not common knowledge, was at least known to musicians who had visited Egypt, to
Pythagoras, Plato and to mathematicians that came after him, like Euclid (4th–3rd C B.C.E.),
and the Pythagorean, Nicomachus of Geras (about 60–about 120 C.E.). This does not
necessarily imply that the musical modes were octave species. However, given
that the evidence presented here proves that some learned, ancient Greeks
were, indeed, acquainted with seven scales and that musicians visited Egypt to improve
their art, it makes the possibility very likely, if not certain. Scholars
who deny this identity are ignoring the fact that music has never recognised borders. It has
always been influenced and enriched by styles and talent imported from other countries. It
would explain why two ancient Greek musical terms — harmonia and
tonos — were in use, which advocates of the ‘key’ interpretation of modes
have, implausibly, to assume to have been synonymous. It seems that the agreement between
ancient Egyptians and Greeks as to the number of musical notes and its implication that
both knew about the existence of seven octave species have escaped the
attention of modern musical scholars as much now as it did when Chappell wrote his book in
1874! Otherwise, they would not keep dogmatically denying that musicians of ancient Greece
were unacquainted with them. It does not make sense that they knew about different scales
yet played only one of them in different keys!
The possibility of seven octave species would have been in accord with
Pythagorean principles for the following reason: the Pythagorean tetractys* symbolises the three archetypal and seven formative principles inherent
in all holistic systems. The former — the noumenal aspect of a holistic system, are
symbolised by the three points or yods at its corners, which give the tetractys its bare
shape as a triangle; the latter — its phenomenal nature — are symbolised by the seven yods
arranged at the centre and corners of a hexagon (socalled “hexagonal yods”). Just as the
first seven notes of an 8note musical scale are symbolised by the seven hexagonal yods
(shown in Figure 1 as coloured dots, so the 49 notes below
the octaves of the seven octave species are symbolised by the analogous
49 hexagonal yods of the next higherorder tetractys — the 2ndorder
tetractys (Fig. 2). The seven octaves species represent a more differentiated version of
the whole system of musical notes. They constitute another whole.
______________________________
^{*} The author calls this the ‘1storder tetractys,’ the point
symbolising the Pythagorean Monad being the 0thorder tetractys. In general, the nthorder
tetractys has 7^{n} hexagonal yods.
5
6
The yods at the corners of the 1storder tetractys mark out the shape of
this template, from which the other seven yods emerge as independent, phenomenal
qualities of the holistic system. In the same way, the number 1 denoting the tonic, the
number 2 denoting the octave and the number 3/2 denoting the perfect fifth should be
assigned to these three yods because 1 and 2 define the pitch range of the musical scale,
whilst their arithmetic mean: (1+2)/2 = 3/2 is the pitch of the perfect fifth G from which
the other notes of the scale can be generated by jumps of intervals of a perfect fifth
or a perfect fourth — the interval between G and the octave C' (Fig. 3).
As the Pythagorean scale has eight notes, they form
^{8}C_{2} = 28* pairs or intervals. There are
as many intervals in the Pythagorean scale as there are notes in the seven octave species
that belong to the Pythagorean scale. Representing the eight notes of the scale by the
corners of an octagon, these intervals correspond to the straight lines that join pairs of
corners (Fig. 4). An octagon is formed by the corners of two squares rotated relative
to each other by 45°. By joining corners of each square and grouping analogous connections,
the 28 lines group naturally into four sets of seven. Column 1 indicates consecutive edges
of one square (shown by solid lines); column 2 shows consecutive edges of the second square
(shown by dotted lines); column 3 shows opposite diagonals of each square; column 4 shows
corners of one square joined along an edge of the octagon to a corner of the other square;
column 5 indicates corners of the second square joined along an edge of the octagon to a
corner of the first square; column 6 exhibits corners of the first square joined internally
to corners of the other square; column 7 shows corners of the second square joined
internally to corners of the first square. The corresponding intervals are indicated inside
each octagon. The 20 Pythagorean intervals include five tone intervals of 9/8. The eight
nonPythagorean intervals include two leimmas of 256/243. There are five seconds, three
thirds, four perfect fourths, four perfect fifths, two sixths, one seventh and one octave.
No one row contains all the seven Pythagorean one ratios. If we do not join corners in a
clockwise sequence in each column, at least one row (the top) can contain all seven diatonic
notes. It is unimportant which column or row is treated as the first, although it seems
appropriate to start (as shown in Figure 4) with the interval from C to D. All that matters in any
rearrangement is that similar types of lines remain in the same column.
The following passage from Chappell’s book seems to give significance to
this 4×7 pattern of intervals:
“The Greeks copied the Egyptians in associating musical sounds with
the heavenly bodies; but, as they made their computations of time by the lunar month,
they connected the twentyeight notes of the scales with the twentyeight days of the
moon; and the fifteen notes of the Diatonic scale were the fifteen
______________________
* The number of combinations of r objects from a set of n objects is
^{n}C_{r} = n!/(r!(nr)!, where n! = 1×2×3×… n.
7
days of the moon’s increase.^{10} The Egyptians subdivided the lunar month into weeks
through dedicating the first hour of each of the seven days to the seven planets, as the
seven deities, who were supposed to watch over them. This association seems to have
originated in Babylon. The seven planets and the seven days coincided with the seven
notes of the Octave.”^{11}
Quintilianus’s reference to the 15 notes of the
diatonic scale must be an error because the seven octave species have (including the tonic)
14 different notes, of which eight are Pythagorean, whilst, although the first two octaves
of the Pythagorean scale have 15 notes, eight of them belong to the
Pythagorean scale on the next higher octave and so the seven notes above the first octave do
not belong to the 28 notes making up the seven scales, which all belong to the first octave.
The numbers of notes of each type are: 1 (×1), 9/8 (×5), 81/64 (×3), 4/3 (×6), 3/2 (×6),
27/16 (×4), 243/128 (×2) & 2 (×1). Suppose that Quintilianus was confusing the 28 notes
of the seven octave species with the 28 intervals of the Pythagorean scale, among which are
the eight notes and their seven intervals, totalling 15, the number that he
mentions. Figure 4 shows that, by interchanging columns 1 and 2 and then rows 3
and 4, the first octave with tone ratio 2 will go into the 15th member of
the set of 28 intervals. However, this will not work because, among the
15 notes and intervals of the Pythagorean scale, is the
nonPythagorean interval 256/243, a number which is absent from the 28 Pythagorean tone
ratios. Given this, does the set of 28 notes in the seven scales still have a subset made up
of 15 notes defined in some way? Fortunately, the list of numbers of
notes given above provides a natural solution because the number of notes in the list up the
perfect fourth is 15! Intuitively, this seems right because, just as the
15th note is the midpoint of the sequence of 28 notes, so the perfect
fourth with tone ratio 4/3 is the middle note of the scale:
1
243/12
9/8
27/16
81/64
3/2
4/3
As a week corresponds to a row and as there are as many as six notes of the
same type, at least one week must have two days with the same note. This is to be expected
because, whether they are intervals of one scale or notes of seven scales, the set of 28
musical elements are not all different. However, the quotation above definitely refers to
the “twentyeight notes of the scales,” that is, the ancient Egyptians associated the 28
notes of the seven octave species, not the 28 intervals of the Pythagorean scale, with the
28 days of the lunar month. It is the subset of 28 numbers in the turquoise cells of Table 1
that needs to be grouped into four sets of seven.
Table 2. Pythagorean notes corresponding to the 28 Days of the
Egyptian Lunar Month.
Week 1
Week 2
Week 3
Week 4

1

9/8

81/64

4/3

3/2

27/16

2

9/8

4/3

3/2

4/3

9/8

81/64

4/3

3/2

27/16

243/128

9/8

4/3

3/2

27/16

4/3

3/2

9/8

81/64

3/2

27/16

243/128



Quintilianus was right about the Egyptians associating lunar days with notes
but erred in thinking that the first 15 days were associated with the
notes of the first two octaves, some of which do not belong to the notes of the seven scales
and so cannot be included with them. Some criterion must determine the ordering of notes
into subsets of 15 and 13, as the former can be neither the
15 notes of two octaves nor the 15 notes and
intervals of one scale. It seems reasonable to assume that the progression of the days of
the lunar month follows the order of the notes in the natural mathematical sequence of the
seven modes as they are generated from
8
successive notes of the Pythagorean scale. There are two possibilities:
Mixolydian–Lydian, if the notes are ascending, and Lydian–Mixolydian, if they are
descending. Considering the former first, the fourth note of the fourth mode (the
Hypolydian), when it is part of the modal sequence of 28 notes, is a perfect fifth, 3/2 (see
Table 1), and it is the 15th note, which make it fitting
that the turning point of the lunar cycle as the full moon begins to wane should coincide
with the very musical centre of gravity of the seven scales, namely, the fourth note of the
fourth scale in the unfolding series of seven octave species. This is shown in Table 2 as the white cell. Considering the second possibility, according
to Table 1, the 15th note in the set of 28 Pythagorean notes is
E, the third note of the fourth mode, the Hypolydian, with tone ratio 81/64. This is not the
middle of the Hypolydian mode. If the days of the lunar month start with the notes of the
Lydian mode, its turning point is not the middle of the middle scale. As symmetry in the
modal spectrum should have its counterpart in the lunar cycle, with the two halves of the
cycle having their counterparts in the seven scales, the correct sequence of modes is
Mixolydian–Lydian. This is indicated in Table 2 by the tone ratios being written in the seven colours of the
spectrum. As both the perfect fourth and the perfect fifth occur six times, it is inevitable
that some notes appear more than once in a week, however they are assigned to the days.
3. Intervallic structure of the seven
modes
The pattern of notes and intervals for each mode are displayed below,
together with their numbers of Pythagorean and nonPythagorean intervals and their
minimum numbers of different intervals.
Table 3. Hypolydian mode.

C

D

E

F

G

A

B

C

C

1

9/8

81/64

4/3

3/2

27/16

243/128

2

D

8/9

1

9/8

32/27

4/3

3/2

27/16

16/9

E

64/81

8/9

1

256/243

32/27

4/3

3/2

128/81

F

3/4

27/32

243/256

1

9/8

81/64

729/512

3/2

G

2/3

3/4

27/32

8/9

1

9/8

81/64

4/3

A

16/27

2/3

3/4

64/81

8/9

1

9/8

32/27

B

128/243

16/27

2/3

512/729

64/81

8/9

1

256/243

C

1/2

9/16

81/128

2/3

3/4

27/32

243/256

1


Blue cells contain rising intervals
belonging to the basic set of 13 intervals/notes. Violet cells contain their
falling intervals. White cells contain repeated intervals.

Number of intervals = 28 (Pythagorean) + 8
(nonPythagorean) = 36.
Minimum number of different intervals = 7 (Pythagorean) + 5 (nonPythagorean) = 12.
Table 4. Mixolydian mode.

C

D

E

F

G

A

B

C

C

1

9/8

81/64

4/3

3/2

27/16

16/9

2

D

8/9

1

9/8

32/27

4/3

3/2

128/81

16/9

E

64/81

8/9

1

256/243

32/27

4/3

1024/729

128/81

F

3/4

27/32

243/256

1

9/8

81/64

4/3

3/2

G

2/3

3/4

27/32

8/9

1

9/8

32/27

4/3

A

16/27

2/3

3/4

64/81

8/9

1

256/243

32/27

B

9/16

81/128

729/1024

3/4

27/32

243/256

1

9/8

C

1/2

9/16

81/128

2/3

3/4

27/32

8/9

1

9
Number of intervals = 25 (Pythagorean) + 11 (nonPythagorean) =
36.
Minimum number of different intervals = 6 (Pythagorean) + 5 (nonPythagorean) = 11.
Table 5. Lydian mode.

C

D

E

F

G

A

B

C

C

1

9/8

81/64

729/512

3/2

27/16

243/128

2

D

8/9

1

9/8

81/64

4/3

3/2

27/16

16/9

E

64/81

8/9

1

9/8

32/27

4/3

3/2

128/81

F

512/729

64/81

8/9

1

256/243

32/27

4/3

1024/729

G

2/3

3/4

27/32

243/256

1

9/8

81/64

4/3

A

16/27

2/3

3/4

27/32

8/9

1

9/8

32/27

B

128/243

16/27

2/3

3/4

64/81

8/9

1

256/243

C

1/2

9/16

81/128

729/1024

3/4

27/32

243/256

1

Number of intervals = 27 (Pythagorean) + 9
(nonPythagorean) = 36.
Minimum number of different intervals = 7 (Pythagorean) + 6 (nonPythagorean) = 13.
Table 6. Dorian mode.

C

D

E

F

G

A

B

C

C

1

9/8

32/27

4/3

3/2

27/16

16/9

2

D

8/9

1

256/243

32/27

4/3

3/2

128/81

16/9

E

27/32

243/256

1

9/8

81/64

729/512

3/2

27/16

F

3/4

27/32

8/9

1

9/8

81/64

4/3

3/2

G

2/3

3/4

64/81

8/9

1

9/8

32/27

4/3

A

16/27

2/3

512/729

64/81

8/9

1

256/243

32/27

B

9/16

81/128

2/3

3/4

27/32

243/256

1

9/8

C

1/2

9/16

16/27

2/3

3/4

27/32

8/9

1

Number of intervals = 26 (Pythagorean) + 10
(nonPythagorean) = 36.
Minimum number of different intervals = 6 (Pythagorean) + 5 (nonPythagorean) = 11.
Table 7. Hypodorian mode.

C

D

E

F

G

A

B

C

C

1

9/8

32/27

4/3

3/2

27/16

16/9

2

D

8/9

1

256/243

32/27

4/3

3/2

128/81

16/9

E

27/32

243/256

1

9/8

81/64

4/3

3/2

27/16

F

3/4

27/32

8/9

1

9/8

32/27

4/3

3/2

G

2/3

3/4

64/81

8/9

1

256/243

32/27

4/3

A

81/128

729/1024

3/4

27/32

243/256

1

9/8

81/64

B

9/16

81/128

2/3

3/4

27/32

8/9

1

9/8

C

1/2

9/16

16/27

2/3

3/4

64/81

8/9

1

10
Number of intervals = 25 (Pythagorean) +11 (nonPythagorean) =
36.
Minimum number of different intervals = 6 (Pythagorean) + 5 (nonPythagorean) = 11.
Table 8. Phrygian mode.

C

D

E

F

G

A

B

C

C

1

256/243

32/27

4/3

3/2

128/81

16/9

2

D

243/256

1

9/8

81/64

729/512

3/2

27/16

243/128

E

27/32

8/9

1

9/8

81/64

4/3

3/2

27/16

F

3/4

64/81

8/9

1

9/8

32/27

4/3

3/2

G

2/3

512/729

64/81

8/9

1

256/243

32/27

4/3

A

81/128

2/3

3/4

27/32

243/256

1

9/8

81/64

B

9/16

16/27

2/3

3/4

27/32

8/9

1

9/8

C

1/2

128/243

16/27

2/3

3/4

64/81

8/9

1

Number of intervals = 28 (Pythagorean) + 8
(nonPythagorean) = 36.
Minimum number of different intervals = 7 (Pythagorean) + 5 (nonPythagorean) = 12.
Table 9. Hypophrygian mode.

C

D

E

F

G

A

B

C

C

1

256/243

32/27

4/3

1024/729

128/81

16/9

2

D

243/256

1

9/8

81/64

4/3

3/2

27/16

243/128

E

27/32

8/9

1

9/8

32/27

4/3

3/2

27/16

F

3/4

64/81

8/9

1

256/243

32/27

4/3

3/2

G

729/1024

3/4

27/32

243/256

1

9/8

81/64

729/512

A

81/128

2/3

3/4

27/32

8/9

1

9/8

81/64

B

9/16

16/27

2/3

3/4

64/81

8/9

1

9/8

C

1/2

128/243

16/27

2/3

512/729

64/81

8/9

1

Number of intervals = 27 (Pythagorean) + 9
(nonPythagorean) = 36.
Minimum number of different intervals = 7 (Pythagorean) + 6 (nonPythagorean) = 13.
INTERVALLIC COMPOSITION OF SEVEN
MODES Total number of intervals = 7×36
= 252.
Total number of intervals other than 1 = 252 – 7×8 = 196.
Total number of inverse intervals = 7×28 = 196.
Total number of intervals and their inverses = 252 + 196 = 448.
Total number of intervals and their inverses other than 1 = 2×196 = 392.
Total number of Pythagorean intervals = 186 (186 – 56 = 130 Pythagorean intervals other than
1). Total number of Pythagorean intervals and their inverses = 2×130 + 7×8 = 316.
Total number of Pythagorean intervals and their inverses other than 1 = 2×130 = 260.
Total, minimum number of different intervals = 46 (Pythagorean) + 37 (nonPythagorean) =
83
All the pitches of these intervals are pitches of tones
of the seven Church modes. Including the seven tonics, the total, minimum number of Pythagorean
intervals = 7 + 46 = 53. Total, minimum
11
number of different intervals (including that between
each note and itself) = 53 (Pythagorean) + 37 (nonPythagorean) = 90.
Table 10. Tone ratios of the notes of the seven Church
musical modes.
Lydian 
1

9/8

81/64

729/512

3/2

27/16

243/128

2

Phrygian 
1

256/243

32/27

4/3

3/2

128/81

16/9

2

Dorian 
1

9/8

32/27

4/3

3/2

27/16

16/9

2

Hypolydian 
1

9/8

81/64

4/3

3/2

27/16

243/128

2

Hypophrygian 
1

256/243

32/27

4/3

1024/729

128/81

16/9

2

Hypodorian 
1

9/8

32/27

4/3

3/2

128/81

16/9

2

Mixolydian 
1

9/8

81/64

4/3

3/2

27/16

16/9

2

Table 10 shows the tone ratios of the notes of the seven octave species.
White cells indicate nonPythagorean tone ratios. The modes have notes with thirteen
different tone ratios other than 1 (the “basic set”), as well as their thirteen inverses,
totalling 26 (27, including 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
243/256 8/9 27/32
64/81 3/4
729/1024 512/729 2/3
81/128 16/27
9/16 128/243 1/2
The number value 26 of
Yahweh ('הוה), the Godname of Chokmah, is the number of
different rising and falling tone intervals in the seven modes. The Godname
prescribes their tonal range. This is consistent with the location of Chokmah in the Tree
of Life at the head of the Pillar of Mercy, for this Sephirah denotes the creative power
of God, and Yahweh lays down the fullness or scope of the archetypal ideas
of the Divine Mind as abstract possibility. The number 26 is also
the number of Pythagorean tone ratios other than 1 in the seven modes:
The letter values of YHVH denote the numbers found in individual modes: Y =
10 denotes the number of Pythagorean notes other than 1 in the Dorian, Hypodorian, Phrygian
and Hypophrygian modes, H = 5 denotes the number of such Pythagorean notes in the Mixolydian
mode and in the Lydian mode and V = 6 denotes the number of such Pythagorean notes in the
Hypolydian mode.
The Godname Yah ('ה) with number value
15 prescribes the eight notes of each mode and their seven
intervals. This is consistent with its Sephirah Chokmah for the same reason as that given
above for Yahweh, the later, full version of this Godname. The Godname
Eloha with number value 36 prescribes the
36 notes and intervals of an 8note musical scale. The Godname
Ehyeh with number value 21 prescribes the seven notes
below the octave because they have 21 intervals. Of the 56 separate
notes in the seven modes, 14 are different notes, leaving (56–14=42) repetitions, where
42 is the 21st even integer. Ehyeh therefore also
prescribes those notes that are repeated. Six of them are the tonic 1, leaving
(42–6=36) repeated notes above the tonic. Eloha prescribes
those notes of the seven modes above the tonic that are repeated. The numbers of
intervals of each type in the seven octave species are:
The numbers of intervals of each type in the seven
octave species are:
____________________________
* The rising intervals are listed in ascending order of pitch.
12
There are 130 Pythagorean intervals other than 1 and 130 of their inverses,
totalling 260 (=26×10). Yahweh with number value
26 prescribes the number of Pythagorean intervals and their inverses. This
intervallic composition is encoded in the inner form of the Tree of Life as the 260 yods in
the seven enfolded polygons outside their shared root edge (Fig. 2). It demonstrates the fact that the seven musical modes constitute an
archetypal musical whole. The 36 intervals of each mode are
symbolised by the 36 corners of these polygons (Fig. 3).
130 is the 129th integer after 1. The number value
129 of Yahweh Sabaoth, Godname of Netzach, is the number of
rising, Pythagorean intervals other than 1 in the seven octave species.
Including the intervals 1, the total number of Pythagorean intervals and
their inverses = 260 + 56 = 316. The number of Pythagorean intervals and their inverses
other than ½, 1 and 2 = 316 – 7(8+1+1) = 316 – 70 = 246. This is the number value of
Gabriel, Archangel of Yesod. It is the number of true intervals and their inverses
in the seven modes other than whole octaves.
There are 66 nonPythagorean intervals and 66 of their inverses,
totalling 132. 66 is the 65th integer after 1, where
65 is the number value of Adonai, Godname of Malkuth. This
also
13
prescribes the number of Pythagorean intervals other than 1 because 130 is
the 65th even integer. The number of intervals other than 1 = 130 + 66 =
196.
The number of intervals (including 1) = 56 + 196 = 252. The number of
intervals (including 1) and their inverses = 56 + 2×196 = 448. As a check, this is 7×8×8,
the number of entries in the seven 8×8 arrays of intervals.
Table 11 indicates that the Pythagorean scale has 13 repetitions of
Pythagorean notes and three repetitions of nonPythagorean intervals shown in white cells,
totalling 16 (purple cells). This is the 15th integer after 1.
Table 11

C

D

E

F

G

A

B

C

C

1

9/8

81/64

4/3

3/2

27/16

243/128

2

D


1

9/8

32/27

4/3

3/2

27/16

16/9

E



1

256/243

32/27

4/3

3/2

128/81

F




1

9/8

81/64

729/512

3/2

G





1

9/8

81/64

4/3

A






1

9/8

32/27

B







1

256/243

C








1

It shows how the Godname Yah with number value
15 prescribes this pattern of intervals of the Pythagorean scale. It
more fundamentally prescribes the 8×8 array of intervals (including 1) and their inverses in
terms of the 15 entries forming two adjacent sides of this square
array:
The number value 36 of Eloha, Godname of
Geburah, is the number of intervals between the eight notes of any scale, counting the
intervals 1 between the notes themselves:
14
The number value 49 of the Godname El
Chai is the number of intervals between successive notes of the seven octave
species:
The number of different intervals between the 56 notes of the seven modes is
90. Remarkably, this is the sum of the integers in the tetractys formed by Plato’s
Lambda:
1
2 3
4 6 9
8 12 18 27.
27, the largest integer in the Lambda, is the number of different tone
intervals and their inverses between the notes of the seven modes. The sum of the integers
1, 8 and 27 at the corners of the tetractys is 36, which is the number of
intervals in each mode.
The four simplest polygons in the inner form of the Tree of Life have 90 yods
outside their shared root edge (Fig. 4). This is a remarkable illustration of the Tetrad Principle defining a
set of geometrical objects that embody a number of universal significance — in this case the
total minimum
number of different intervals between the notes of the seven musical modes.
As confirmation that this is not just coincidence, the numbers of yods in the triangle,
square, pentagon and hexagon outside their root edge are, respectively, 15,
21, 27 and 27, which are the sums of the integers along diagonal rows of the Lambda
tetractys:
This beautiful parallel between the inner geometry of the Tree of Life and
the arithmetic properties of the Lambda serves to demonstrate their archetypal
character.
15
The minimal set of 90 different intervals in the seven modes includes ones
where an interval occurs once in a mode, e.g. the interval 1024/729 in the Lydian mode. The
basic set has 14 different ones, so (90–14=76) intervals are repeated. The
number value 76 of Yahweh Elohim, Godname of Tiphareth,
is the minimum number of these repeated intervals.
Inspection of the tables of intervals reveals that, of the 66
nonPythagorean intervals, 16 (=4^{2}) are between the tonic C and other notes.
There are (66–16=50) nonPythagorean intervals between notes other than C.
This shows how the Godname Elohim with number value
50 prescribes nonPythagorean intervals between notes other than the
tonic. The number of nonPythagorean intervals and their inverses between the notes and the
tonic C or octave C' is 32, which is the 31st integer after 1. This shows
how the Godname El assigned to Chesed with number value
31 prescribes the nonPythagorean intervals between notes and opposite
ends of each mode.
Similarly, the tables show that, of the 130 Pythagorean intervals, 33 (= 1!
+ 2! + 3! + 4!) intervals are between C and other notes, leaving (130–33=97) Pythagorean
intervals between notes other than the tonic. 97 is the 49th odd integer,
showing how the Godname El Chai assigned to Yesod with number value
49 prescribes this number. 97 is also the number value of
Haniel, the Archangel of Netzach. Including the unit intervals between the 56 notes
themselves, the number of Pythagorean intervals between notes other than the tonic = 97 + 56
= 153. This is the number value of Elohim Sabaoth, the Godname
assigned to Hod. The number of Pythagorean and nonPythagorean intervals between C and other
notes = 33 + 16 = 49. Therefore, El Chai prescribes the
intervals in the seven modes between the tonic and other notes, as well as the Pythagorean
intervals between notes above the tonic.
Below is a summary of how the Godnames prescribe properties of the seven
octave species.
Kether: 21. 
21 intervals between 7
notes below the octave; 42 repetitions of basic set of 14 notes forming the
7 modes, where 42 = 21st even integer. 
Chokmah: 15.
26.

8 notes in each mode separated by 7 intervals, totalling
15 notes & intervals.
number of different intervals other than 1 and their
inverses; number of harmonics in 7 cycles of 7 modes.
260 (=26×10) Pythagorean intervals and their inverses.

Binah:
50. 
number of nonPythagorean intervals
between notes above the tonic. 
Chesed:
31. 
number of nonPythagorean intervals
and their inverses between notes and the tonic or octave = 32, which is the
31st integer after 1. 
Geburah:
36. 
number of repetitions of basic set
of 14 notes above the tonics of the 7 modes; each mode has 8 notes and 28
intervals, totalling 36 notes and intervals. 
Tiphareth:
76. 
minimum number of repeated
intervals. 
Netzach:
129. 
130 Pythagorean intervals, where 130
= 129th integer after 1. 
Hod:
153. 
number of Pythagorean intervals
(including those between the notes themselves) between notes above the
tonic. 
Yesod:
49. 
number of intervals between
successive notes of 7 modes; number of Pythagorean and nonPythagorean
intervals between the tonic and other notes; 97 Pythagorean intervals between
notes other than the tonic, where 97 = 49th odd
integer. 
Malkuth:
65. 
number of nonPythagorean intervals
= 66, where 66 = 65th integer after 1; 130 Pythagorean
intervals, where 130 = 65th even integer. 
16
4. The Lambda
Tetrahedron
The number of Pythagorean and nonPythagorean intervals between notes other
than the tonic = 97 + 50 = 147. Similarly, these have 147 inverse
intervals on the opposite sides of the diagonals of Tables 3–9. Therefore, there are (147+147=294) Pythagorean and nonPythagorean
intervals between notes other than C together with their inverses. Including
the 56 intervals 1 on the diagonals, there are (56+294=350) such intervals. They comprise
50 nonPythagorean intervals and their
50 inverses, that is, 100 nonPythagorean intervals, leaving
(350–100=250) diagonal intervals, Pythagorean intervals between notes other than the tonic
and their inverses (check: 250 = 56 + 97 + 97). The division:
350 = 100 + 250
between the 100 nonPythagorean intervals and their inverses and the 250
Pythagorean ones and their inverses has its parallel in the tetrahedral version of the
Lambda tetractys discussed in Article11.^{12} The 20 integers arranged in tetractyses on the four faces of
the tetrahedron add up
to 350 (Fig. 5), the sum of the cubes of 1, 2, 3 & 4 at its four corners is
100 and the sum of the 16 other integers is 250. The tetrahedron embodies the interval
composition of the seven modes.
5. Pattern of modal intervals in
superstring oscillations
The seven modes have 130 rising, Pythagorean intervals other than 1, of
which 46 comprise the minimum number of different Pythagorean intervals, these varying from
mode to mode. There are (130–46=84) such Pythagorean intervals that are repetitions of the
basic set. Similarly, there are 84 falling, Pythagorean intervals with reciprocal tone
ratios that are repetitions. There are (84+84=168) repetitions of the
Pythagorean intervals and their inverses. The number value 168 of
Cholem Yesodeth, Mundane Chakra of Malkuth, is the number of
Pythagorean intervals and their inverses
17
that are repetitions of any of the seven basic Pythagorean intervals other
than 1 and their inverses. This 84:84 division is embodied in the inner Tree of Life as the
168 yods lying on the sides of the first (6+6) enfolded polygons, 84
on either side of the root edge (Fig. 6).
The 2ndorder tetractys has 84 yods surrounding its centre (Fig. 7), where 84 = 1^{2} + 3^{2} +
5^{2} + 7^{2} . This means that any aspect of physical reality
that embodies cosmic wholeness symbolised by the tetractys is expressed by 84 degrees of
freedom in this reality, which is symbolised by the central yod of the central 1storder
tetractys of the 2ndorder tetractys. This number accords with the Tetrad Principle because
1, 3, 5 & 7 are the first four odd integers.
The 84:84 division corresponds in the subquark superstring to the inner and
outer halves of the UPA, each of which comprises 2½ revolutions of ten whorls with 840
1storder spirillae in each one (Fig. 8). Every ten spirillae can be said to correspond to a diatonic interval
between the notes of the seven Greek modes, the outer and inner halves corresponding to the
intervals and their inverses. It is as if the circularly polarised oscillations of each
whorl reproduce all the diatonic intervals between different notes of the seven musical
modes.
6. Comparison of modal intervals with
the octonions
This conclusion in the context of superstring structure returns us full
circle to the discussion in Article 15 of the similarity between the 8note scale and the
eight unit octonions, as now explained. The octonions, or Cayley numbers, form the fourth
class of division algebras. There are one real unit octonion, 1, and seven imaginary unit
octonions e_{i} (i = 1–7) with the property e_{i}^{2} = 1
(i = 1–7). Their multiplication is anticommutative:
e_{i}e_{j} = e_{j}e_{i}
(i ≠ j)
18
Table 12 is the multiplication table for the 8tuple of unit octonions:
(1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5},
e_{6}, e_{7}).
There are 28 octonions in the violet triangle and 28 octonions in the blue
triangle. The 21 octonions formed by multiplication of pairs of
e_{1}, e_{2} … e_{7} in the dashed triangle on one side of
the diagonal of the square array are the negatives (that is, reciprocals, because
e_{i}^{2} = 1, so that e_{i} = 1/e_{i}) of their
21 counterparts in the dashed triangle on the other side of the
diagonal. There are 21 octonions with positive signs and
21 with negative signs. The Godname El Chai with number
value 49 prescribes the number of products of imaginary octonions, the
Godname Ehyeh with number value 21 prescribing how many
of them have positive signs and how many have negative signs, that is, how many are the
reciprocals of them.
Compare this with the intervals between the notes of the Pythagorean scale
shown below in Table 13.
The pitch decreases of the 28 falling intervals (DC, etc) in the violet
triangle are the reciprocals of the pitch increases of the 28 corresponding rising intervals
(CD, etc) in the blue triangle. The eight diagonal entries 1 denote the zero pitch
difference between each note and itself.
19
In both tables, the diagonal set of entries is like a mirror that reflects
each entry into its image. Each entry in a triangle in Table 12 corresponds to an interval, a negative octonion being the
counterpart of a falling interval because it is the reciprocal of an octonion, just as the
decrease in pitch of a falling interval is the reciprocal of the increase in pitch of the
same rising interval.
Multiplication of two imaginary octonions:
e_{i}×e_{j} (j>i) corresponds to the interval between two rising
notes i and j. Multiplication in the reverse order corresponds to the interval between the
notes j and i in order of descending pitch. Noncommutativity of octonion multiplication:
e_{i}e_{j} =  e_{j}e_{i} corresponds to the pitch
decrease of a falling interval being the reciprocal of that of the same rising interval.
The seven imaginary octonions form seven 3tuples:
 (e_{1},
e_{2}, e_{4})
 (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})
It was proposed in Article 15^{13} that these 3tuples are the octonion equivalent of the seven
Greek octave species because they each form seven combinations of one, two and three
octonions in analogy to the seven notes of each species. Furthermore, it was shown that the
seven 3tuples have 84 permutations of pairs and triplets and that the seven 3tuples of
negative octonions (inverses of octonions) have 84 permutations of their pairs and triplets,
a total of 168 permutations. Each 3tuple corresponds to an octave
species. Each mode has on average 12
repetitions of its diatonic intervals (see below) and each of these
intervals correspond to a permutation.
The same pattern is displayed in the Klein Configuration (Fig. 9) of the 168 elements of the group of
transformations of the Klein Quartic Equation, which is well known to
mathematician.^{14} Each of the 14 slices shown numbered has 12 coloured triangles.
The slices are of two types, each type containing 12 triangles arranged in the whole
configuration as 24 heptagons. The seven slices of one type correspond to the repeated,
rising intervals of the seven octave species and the seven slices of the other type
correspond to their repeated, falling intervals, the 12 coloured triangles in a slice
corresponding to the average number of 12 repetitions of diatonic intervals or their
inverses.
Such precise parallelism between the seven musical modes and an object that
is regarded in
20
the mathematical world as possessing great beauty is evidence of the
archetypal Ideas of the Divine Mind being realised in two contexts that appear superficially
to have nothing in common, namely, music and the Mtheory underlying superstring physics,
which physicists are currently seeking in order to unify their understanding of the forces
of nature in a single theory. However, sound is a vibration of the air and, just as a
plucked string creates musical notes of different frequencies, so superstrings vibrate in a
variety of ways to create all the subatomic particles of which they are components. The form
of a superstring is therefore generated by a process analogous to playing the intervals —
both rising and falling — between (as shown below) the notes D, E, F, G & A of the seven
musical modes.
By inspecting the table of intervals for each mode, counting its number of
Pythagorean intervals and subtracting the number of different Pythagorean intervals
indicated under each table, the numbers of repetitions of these intervals for individual
modes are found to be:
Hypolydian

Mixolydian

Lydian

Dorian

Hypodorian

Phrygian

Hypophrygian

13

11

12

12

11

13

12

Total = 84.
Average = 84/7 = 12.
In terms of notes, the composition of the 84 Pythagorean intervals is:
The interval 243/128 for note B and the interval 2 for note C' are absent
because they are the only ones that never appear more than once in a mode (Phrygian, Lydian,
Hypophrygian and Hypolydian) and so have no repetitions. Five different intervals comprise
the set of 84. This 5:1 division has the following analogy with the Tree of Life: the lowest
five Sephiroth of Construction:
Malkuth, Yesod, Hod, Netzach & Tiphareth (Fig. 10) constitute what in Kabbalah is called its ‘Lower Face,’ or
‘Microposopus.’ They formally correspond to the five types of intervals that are repeated 84
times. The octave and note B, which are not repeated, formally correspond, respectively, to
Chesed and Geburah, which are unshared with adjacent members of a set of overlapping Trees
of Life, that is, they are intrinsic to their own trees.
This 5:2 division is reflected also in the five tone intervals (T) and two
leimmas (L) of each mode:
21
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
Phrygian
Lydian
Mixolydian
Hypodorian
Hypophrygian
Hypolydian 
It is also manifested by the seven modes themselves because, regarding the
14 notes that they span as a cycle that starts again on the second octave, five successive
cycles generates 15 harmonics and seven cycles creates
26 harmonics, where 15 is the number value of
the Godname Yah, the older version of the Godname Yahweh, whose number
value is 26.
In Kabbalistic traditions of esoteric knowledge, the lowest five Sephiroth
of Construction have the following correspondence with the five Elements:
Tiphareth 
↔

Aether 
Netzach 
↔

Fire 
Hod 
↔

Air 
Yesod 
↔

Water 
Malkuth 
↔

Earth 
The very diatonic intervals that correspond to the five Elements are
therefore those whose repetitions and their inverses number 168, the number
value of Cholem Yesodeth, Mundane Chakra of Malkuth, the lowest Sephirah.
The 168 automorphisms of the Klein Quartic are
represented by the 168 coloured, hyperbolic triangles of the Klein
Configuration:
168 = 7×24 = 7×[(1 + 2) + (4 + 3) + (4 + 3) + (4 + 3)]
= 7(4 + 4 + 4) + 7[(3 + 3 + 2) + (3 + 1)]
= (28 + 28 + 28) + (56 + 28)
= 84 + 84.

Seven slices of one type have 84 triangles made up of 28 red, 28 yellow and
28 green triangles and the seven slices of the other type have 84 triangles made up of
21 yellow, 21 green, 21 red
and 21 blue triangles. Compare the former with the fact that the 84
Pythagorean intervals between notes D, E, F, G & A comprise 28 seconds, 11 thirds, 23
perfect fourths, 17 perfect fifths and 5 sixths, that is, 28 seconds, 28 thirds and fifths and
28 perfect fourths and sixths. For the
22
84 coloured triangles belonging to seven slices of one type, the same
pattern of three groups of 28 exists in both cases, although which sets of 28 coloured
triangles correspond to which groups of 28 intervals cannot be ascertained at this stage.
The four important features to notice are:
1. the Klein Configuration naturally divides the number
168 into two sets of 84 as its two types of slices have a total of 84
hyperbolic triangles;
2. the 168 Pythagorean intervals between notes D, E, F, G & A of the
seven modes divide into 84 ascending and 84 descending intervals (the 84 entries on one side of
the diagonal of their tables of intervals and 84 entries on the other side;
3. the seven 3tuples of octonions have 84 permutations of two and three octonions and 84
permutations of two and three of their inverses;
4. the 168 1storder spirillae in each halfrevolution of a whorl of the
UPA comprise 84 in its outer half and 84 in its inner half.
They indicate the following correspondences:
The division of the Klein Configuration into two types of slices corresponds
to the distinction between rising and falling intervals and to the difference between
octonions and their inverses. The profound consequences of this analogy will be explored in
a future article.
It was found earlier that the first four polygons of the inner form of the
Tree of Life have 90 yods outside their root edge that correspond to the different intervals
(including 1) of the seven Greek musical modes. Four of these yods are their centres and two
(the top and bottom corner of the hexagon) coincide with Chokmah and Netzach (Fig. 11).
There are (90–4–2=84)other yods in these polygons outside their root edge,
both sets of four polygons having (84+84=168) such yods. The yods in one
set of polygons correspond to the coloured triangles in the seven similar slices of the
Klein Configuration, repeated, rising Pythagorean intervals between the notes D, E, F, G,
& B in the seven modes, permutations of two and three octonions in the seven 3tuples
and 1storder spirillae in onehalf of a revolution of the inner or outer half of the UPA.
The yods in the other, mirror image set of polygons correspond to the coloured triangles in
the other set of
23
similar slices of the Klein Configuration, repeated, descending Pythagorean
intervals, permutations of two and three inverse octonions in their seven 3tuples and
1storder spirillae in a half of a revolution of the outer or inner half of the UPA. The
musical counterpart of the two identical halves of the inner form of the Tree of Life is the
musical distinction between rising and falling intervals. Its octonion counterpart is the
difference between the imaginary octonions and their inverses. Its counterpart in the Klein
Configuration is the difference between its two types of slices (their mathematical
difference is too technical to discuss here).
Remarkable confirmation that it is not coincidental that the first (4+4)
polygons should have 168 yods other than their centres and those
coinciding with the positions of Sephiroth comes from the way in which the number values of
the Hebrew words ‘Cholem’ and ‘Yesodeth’ summing to 168 are themselves
embodied in the yod populations of these polygons (Fig. 12). The numbers of yods in the triangle, square, pentagon and hexagon
that are outside the root edge and which are not centres of polygons or coincident with the
positions of Sephiroth of the Tree of Life when the latter is projected onto the plane
containing the polygons are listed below:
The triangle and pentagon have 39 yods and the square and hexagon have 45
yods. Hence, both sets of triangles and pentagons have 78 yods and both sets of squares and
hexagons have 90 yods. The number value of Cholem is 78 and the number value
of Yesodeth is 90. The very numerical values of the Hebrew words of the
Mundane Chakra of Malkuth are encoded geometrically in the first (4+4) polygons! This is an
example of the Tetrad Principle,^{15} wherein the fourth member or the first four members of a
sequence of mathematical objects always embody a number that has cosmic significance, such
as the number of spacetime dimensions, the number of particles transmitting superstring
forces or the oscillatory form of a superstring.
Set out below are the ways in which properties of the first four
polygons and the first (4+4) polygons are prescribed by the number values of the Godnames of
the ten Sephiroth:
Kether: 21

21 geometrical elements in the first
(4+4) polygons are shared with the 1tree;*

Chokmah:
15
26

the first 4 enfolded polygons have
15 sides and 15 corners of their
17 tetractyses. The number of yods on the boundaries of the 34 tetractyses
of the first (4+4) polygons = 150 = 15×10. The first
(4+4) polygons share 15 corners and sides with the
Tree of Life;
the first (4+4) enfolded polygons have 26 corners outside
their root edge. Every first 4 polygons enfolded in successive Trees of
Life have 26 corners & sides. The first (4+4)
enfolded polygons share 26 corners & sides with the
Tree of Life;

Binah: 50 
number of corners & sides of the first (4+4)
enfolded polygons = 51 = 50th integer after 1. The first
(4+4) enfolded polygons have 50 sides unshared with the
1tree;

Chesed: 31 
number of sides of 17 tetractyses in the first 4
enfolded polygons = 31. Tree of Life has
31 geometrical elements unshared with the first (4+4)
enfolded polygons;

Geburah: 36 
the first 4 separate polygons have 36
corners and sides and 36 sides of their 18 tetractyses.
The polygons have 36 hexagonal yods on their boundaries.
The first (4+4) separate polygons have 36 corners and
36 sides;

Tiphareth: 76 
the first 4 separate polygons have 76
geometrical elements. The first (4+4) enfolded polygons have
76 yods outside their root edge on their boundaries and
152 yods unshared

____________________________________________
*The 1tree is the lowest Tree in a set of overlapping Trees of Life.
24

with the 1tree, where 152 = 76th even integer;

Netzach: 129 
the first (4+4) enfolded polygons have 130 yods on their boundaries other
than corners outside their root edge, where 130 = 129th
integer after 1;

Hod: 153 
the first (4+4) enfolded polygons have 154 hexagonal yods outside their
root edge, where 154 = 153rd integer after 1;

Yesod: 49 
every first (4+4) polygons enfolded in successive Trees have
49 corners and sides;

Malkuth: 65 
66 hexagonal yods in the first 4 enfolded polygons are unshared with the
1tree, where 66 = 65th integer after 1;

155

the first (4+4) enfolded polygons have 156 yods outside their root edge
unshared with the Tree of Life, where 156 = 155th integer
after 1.

The Godnames define that section of the inner form of the Tree of Life
blueprint that encodes the structural parameter 168 of superstrings.
Actually, they prescribe other sections as well that embody the same number (e.g., the first
(6+6) polygons).^{16} This is because of the fractallike quality of the inner
form of the Tree of Life, in which the same information about the physical
manifestation of the blueprint is encoded in different but equivalent sections of it.
Conclusion
In coming to the consensus (or almost so) that the ancient Greek musical
modes were likely to have been different keys of a single scale, scholars have either
ignored or been oblivious of ancient writers who told of how Pythagoras, Plato and Greek
musicians learned from the ancient Egyptians, who, according to Nichomachus, ascribed 28
sounds to the universe. This meant that the latter — and almost certainly also the former —
knew about the seven octave species with 28 notes. This makes it highly likely, if not
certain, that the Greek modes were different scales and that their patterns of intervals
were known to those with musical learning, for why would ancient Greeks have played all
their music in different keys of one scale if some of them knew of seven such scales? The
fact that early Christian musical theorists appropriated Greek modal labels in their
derivation of the church modes from the Byzantine octoechoi does not have the
implication that the ancient Greeks did not understand their modes as scales. All it means
is that the church modes were not directly inherited from Greek octave species. To conclude
from this fact that the Greek modes were not different musical scales, as musical scholars
like Rita Steblin do, is simply a non sequitur.
The ten Godnames prescribe different sets of intervals between the notes of
the seven scales. The reason for this is that the seven octave species constitute a
whole — both mathematically and spiritually. The tetractys version of Plato’s
Lambda has ten numbers that sum to 90. This is the number of different intervals between the
notes of the seven scales. Just as four numbers — 1, 2, 4/3 & 3/2 — generate the
Pythagorean scale and therefore all seven scales with a minimum number of 90 different
intervals, so four yods lie on the shared base of the first four polygons of the inner form
of the Tree of Life, having 90 yods outside it. The fact that the four inclined rows of the
Lambda tetractys add up to the yod population of each polygon confirms that this remarkable
correspondence is not coincidental. The tetrahedral generalisation of the Lambda tetractys
has 20 numbers that sum to the number of intervals between notes above the tonic of each
scale. There are 84 copies of the rising, Pythagorean intervals in the seven scales and
similarly for the falling, Pythagorean intervals. This 84:84 pattern is present in the first
(4+4) polygons as their (84+84) yods that are neither centres of polygons nor coincident
with the positions of Sephiroth of the Tree of Life. Confirmation that this is not
coincidental is the fact that the number values of the two words making up the Hebrew name
of the Mundane Chakra of Malkuth with number value 168 are the yod
populations of pairs of polygons in the set of four and their counterparts on the other side
of their shared edge. The ten Godnames prescribe the first (4+4) polygons as a Tree of
Life pattern encoding the superstring structural parameter 168.
The eight unit octonions are the counterpart in number theory of the eight
notes of the musical scale. The real unit octonion corresponds to the octave and the seven
imaginary unit octonions
25
correspond to the seven notes below it. Multiplication of pairs of octonions
is equivalent to the rising interval between two notes. Division is equivalent to their
falling interval, that is, to the interval between them in reverse order. Anticommutivity of
their multiplication corresponds to the pitch difference of two notes being the reciprocal
of that of the two notes in reverse order. The counterpart in the Klein Configuration of the
168 copies in the seven musical scales of the basic set of rising and
falling Pythagorean intervals is its 168 triangles denoting the
automorphisms of the Klein Quartic. Its seven slices of one type with 84 triangles
correspond to the set of 84 rising, Pythagorean intervals in the seven scales and the seven
slices of the other type with 84 triangles correspond to their set of 84 falling,
Pythagorean intervals. The triangles denote the 168 elements of
PSL(2,7). As every element of a group has its inverse, the rising and falling Pythagorean
intervals correspond to group elements of PSL(2,7) and their inverses. The 84 permutations
of two and three octonions in the seven 3tuples and the 84 similar permutations of their
inverses are the octonion counterparts of the 84 copies of rising intervals and the 84
copies of falling intervals that are Pythagorean.
The seven notes above the tonic of the Pythagorean scale and the seven unit
imaginary octonions in the fourth division algebra are analogous in ways beyond coincidence
because they are examples of the sevenfold nature of the phenomenal aspect of God
manifested in the world, as abstractly symbolised by the seven hexagonal yods of the
tetractys and as expressed in the Tree of Life by its seven Sephiroth of Construction. Music
is patterns of sound frequencies, or real numbers (the first division algebra), ordered in
time, i.e., pairs of numbers (frequency, time coordinate) that are analogous to the complex
numbers (the second division algebra). Sound is a vibration of a threedimensional medium
taking place in time, i.e., sets of four numbers define its displacement, analogous to the
fourdimensional quaternions (the third division algebra). The eightdimensional octonions
associated with vibrating superstrings are the fourth and last division algebra. In
conformity with the title “holding the key of nature” assigned by the Pythagoreans to the
Tetrad, the vibrations of the macrocosm and the microcosm are connected by
four classes of number.
References
^{1} A History of Key Characteristics,
Rita Steblin (University of Rochester Press, 2003), p. 13.
^{2} ^{}A History of Western Music (3rd ed.), Donald Jay Grout
(J.M. Dent & Sons Ltd, 1980), p. 32.
^{3 }Ibid, p. 34.
^{4} Ibid, pp. 1415.
^{5} ^{}The Pythagorean Sourcebook and Library, Kenneth Sylvan Guthrie
(Phanes Press, 1987), p. 167.
^{6} ^{}Strabo, lib. xvii, p. 29.
^{7} The History of Music, vol. 1, W. Chappell (London, 1874),
p. 1.
^{8} ^{}Ibid, p. 60.
^{9} Ibid, pp. 5051.
^{10} ^{}Aristides Quintilianus, lib. iii, p. 136.
^{11} Ibid, p. 52.
^{12} ^{}Phillips, Stephen M: Article 11: “Plato’s Lambda — Its Meaning,
Generalisation and Connection to the Tree of Life,” (WEB, PDF), pp. 716.
^{13} ^{}Phillips, Stephen M: Article 15: "The Mathematical Connection between
Superstrings and
26
their Micropsi Description: a Pointer Towards Mtheory,” (WEB, PDF), pp. 2830.
^{14} ^{}Ibid, pp. 2427.
^{15} ^{}Phillips, Stephen M: Article 1: “The Pythagorean Nature of Superstring
& Bosonic String Theories,” (WEB, PDF), p. 5.
^{16} ^{}Phillips, Stephen M. Article 4: “Godnames prescribe inner Tree of Life,”
(WEB, PDF), p. 4.
27
