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EGYPTIAN CONNECTIONS
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NARMER INSCRIPTIONS AS SUMERIAN MUSICOLOGY
Ernest G. McClain
Thesis: Before Greece abstracted a theory of harmony, Egypt embodied its
musical mythology in graphic art that remains vibrantly alive to our eyes today.
Here is adventure in recovering its metaphors and perhaps their origins.
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Treasures from the past
In 1897-98 Egyptian archaeology unearthed some of the earliest arithmetic of the
ancient Near East.
Fig.1
These symbols on the mace head of Narmer—sometimes read as Menes, or “King
Scorpion,” perhaps the first pharaoh of a unified Upper and Lower Egypt-presumably celebrate the capture of 400,000 bulls, 1,422,000 goats, and an army of
120,000 men. Bulls, goats, and men are identified by picturing them.1 The numbers
below each are indicated by glyphs for an early base 10 arithmetic with seven different
symbols for units, tens, hundreds, thousands, myriads, hundred thousands, and
millions. Four tadpoles below the bull at the left each indicate 100,000. At the far
right-- under a bound prisoner (apparently awaiting decapitation like ten others
standing tall with severed heads aligned between their feet)--another tadpole indicates
100,000 men while two “bent” human fingers each indicate 10,000 more for a total of
120,000 soldiers being killed. Behind the symbolic goat in the middle a seated man
with upright arms indicates 1,000,000 in a posture suggesting awe of Egypt’s largest
numerical counter (soon discarded); 4 tadpoles add another 400,000, and two bent
fingers add 20,000, while two “lotus buds” below the man indicate another 1,000
thousand each for a grand total of 1,422,000 goats. Georges Ifrah asks: “Are these real
Copyright © 2011 Ernest G. McClain
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numbers, or are they purely imaginary figures whose sole aim is to glorify King
Narmer?2
Ifrah’s own brilliant work in studying the manipulation of numerical tokens (prior to
the invention of “floating place value” and a simpler notation) helped to expose the
Sumerian “grain pile” of 1,152,000 units presented to ICONEA 2008 as an “Ur-text”
in archaeomusicology.3 That single tablet suffices to integrate all of Narmer’s numbers
and many other symbols. I abstract what is needed for his story, and append a copy its
matrix for readers interested in more detail. Here I assemble for musicology the
metaphors I find most felicitous in English without respect to linguistic sources, and I
regret having to cite any foreign words whatever; more coherent presentations must
be left to translators who recognize their story lines and
D-re
Fig.2
to musicians less crippled linguistically. Pythagoreanism
must remain translatable into any language or it fails of
C- do
E- mi
all purpose.
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A picture worth many thousand words.
Sumer’s grain pile required notation by 32 counters
each worth 36,000 units; Egypt’s 400,000 bulls need
A-la
G- sol
only 4 pollywogs. The tonal model for both maps a
pentatonic “peaceable kingdom” of pentatonic
graciousness that we all know as a musical paradise for children of any age. This
visually provocative symbol is the Egyptian double glyph for heaven as a “humanoid”
five-pointed star set within the all-encompassing circle of the sky in a manner
suggesting “vector analysis” of the scale by a human “fiver.” Two legs, two arms and
a head radiate from a sexless crotch at angles appropriate for singing do re mi sol la.
This star glyph is repeated endlessly in other contexts (with variable positions of
hands and arms), and it decorates the body of sky goddess Nut, arched on fingers and
toes over subjects suckling from her breasts. These five are accepted here as “mantles
of radiance” that Sumer assigned to its own deities—the highest of whom eventually
were identified by numerical “nicknames” that belong to Plato’s musical science as
defined by Narmer’s numbers.4 Here is the planetary subset of his own “World
Soul”—the most seriously studied model of the cosmos ever proposed by the
ancients and still useful to Ptolemy and Kepler as we approach modern times. Meet
yourself as a “human fiver” under a “Tri-morphic” deity limited to products of 2, 3,
and 5 and who reigns throughout eternity in “virtual” models in which feet and head
alone are Platonic constants--permitting Socrates to discover near the end of Plato’s
REPUBLIC that the elusive meaning of “Justice” has
Fig. 3.
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been “circling round our feet” from the beginning.
His “musical proportion” 12:9::8:6 quantifies
beginning and end as the “head” at the unambiguous
D : G :: A : D rising
D : A :: G : D falling
zero hour of 12:00 as cyclic perspective. Equal
12 : 9 :: 8 : 6 symmetry
Temperament ensures that feet at 7:00 and 5:00 are in
Copyright © 2011 Ernest G. McClain
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perfect agreement. We live still with this Egyptian heritage and its Platonic trinity that
all major religions share. Leaving hands free to explore other possibilities, it is feet that
guard that throne of deity. Sumer pictured this schema as naked Ishtar (Sumerian
Inanna) standing on the backs of two lions lying peacefully head-to-tail under her
naked feet, and on each foot we see three giant talons as “toes” as if belonging to
some great bird of prey. Egypt pictures every detail in its own way, with this glyph as
supreme sophistication. Meaning was never lost; New Testament Jesus washes the
feet of his own disciples (to their consternation), to embed this metaphor in
Christology forever, and Homer’s great “sure-footed” chariot horses, some with both
minds and voices, are indelible in Greek mythology, inspired by this same arithmetic.
Any five consecutive tones in the universal “Spiral” of musical fifths and fourths must
be tuned aurally from nearest neighbors, and so I name this pentatonic subset (C G
D A E) as if paired “from the middle” (representing the “sun of insight”) with pitch
class “D” as center of symmetry, preserving A and G (our first and seventh letters) as
Plato’s “means” and “first pair of twin sons,” named “Atlas and Gadirus” (as if to
anticipate the modern roles of A and G as first and seventh).16
Figure 4 The Great Serpent of the “Spiral” of musical 5ths and 4ths”

pentatonic model
 
A E B
  
heptatonic model
F
C
G
D
A
E
B
F
C G
This naming minimalizes and equalizes sharps and flats and proves indifferent to
physics, for quantification that defines frequency ratios in one direction naturally
defines idealized string and pipe lengths in the other. On Narmer’s palette two giant
four-footed “Serpopards” confront each other face to face with their very elongated
necks curled around each other, hovering over a circular depression presumably
meant for the preparation of ritual “eye paint.” Stare on the Great Serpent understood
in the metaphor of “lion of the ground,” for a first intimation of the meaning of
ancient Egypt’s strange beasts; eventually we learn to recognize them as ourselves—
astonishing both “self” and “other.”
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Embodiment
To help arouse feeling to its own great responsibility in intellection stand in front of
your own clock with feet at 7:00 and 5:00 and hands at 10:00 and 2:00 and be assured
that you constitute a vector analysis of the pentatonic model (any five consecutive
pitch classes) that modern 12-tone equal temperament equates with 12 hours of
demonic regularity.5 Sumer relieves Egypt of a “13th pitch class.” Narmer is
12-tone,” and the interesting question is how it happens.
3:4
2:3
4:3
9:8
3:4
Fig. 5. New intervals in Spiral 5ths must be tuned aurally and can be “mapped” similarly in pairs in.either direction as
7 or 5 ‘semitone hours’ respectful of the “four winds” that establish cosmic balance. Radial and rectilinear measure
are correlated, however stubborn. Celestial convergence means within 30 degrees, intolerable to musicology.
Raise your right hand overhead (as if pointing toward Isaiah’s location of the “throne
of heaven”) and bring it down in a great arc circling to the right through seven hours to
point toward your left foot and thus ‘semaphore’ seven undefined “semitone hours” as
Plato learned (perhaps from 5th c. BCE Philolaus whom he was accused of
plagiarizing. (He knew that a teacher needed to transform into wonder what his
culture took for granted.) This “first position” of the musical 5th D:A as 2:3 rising in
modern frequency ratios identifies Babylon’s EA-CREATOR string as symbolizing
“god 40” in its musical pantheon and meaning 40:60=2:3. He is the first of Narmer’s
400,000 “bulls” as “god 40” leading myriads (ten thousands). Now do the opposite
with your left arm to end up crossing the right and pointing to the inverse (reciprocal)
fifth D:G as 2:3 falling to the left. Your crossed arms embody the glyph for a
“crossroad” leading four ways to civilization. We have mapped Plato’s musical
proportion 12:9::8:6 without doing any arithmetic. (Serpopard necks are entwined.)
Now, remaining conscious of the four directions (god Marduk’s “four winds” given
him to play with at birth, and preserved by our own ears as “seat” of balance) move
each arm separately seven more hours in the same directions to return to the positions
of Figure 2. Feet map the Greek tonal “constants” framing tetrachords of 3:4 as
“perfect fourths” awaiting whatever we please to insert, and arms map the “cosmic”
meaning of Egypt’s double hieroglyph for heaven in the ratios of Plato’s “World
Soul” as a planetary subset of “What 12 is.” Clocks serve both tone and time with the
graciousness of sister muses not yet quarreling over each other’s curricular priority.
The difference between fourths and fifths vocally is routinely experienced as between
“small” intervals and “large” intervals associated with physical control of the voice.
Copyright © 2011 Ernest G. McClain
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Head and feet map Plato’s musical
Dance of the “semitone” hours
around “D” at 12:00
proportion as any three consecutive tones in
7:00
A:G
5:00
the great “Winged Serpent” of Spiral Fifths
2:00
E:C
10:00
tuning: they lie automatically at the angles
9:00
B:F
3:00
of Figure 2. Paired “feet” locate the “head” for
4:00
8:00
F : B
us and Sumerian Inanna (Ishtar’s prototype) as she
11:00
1:00
C : E
carefully nurses the broken roots of the “tree of life”
6:00
6:00
GA
(damaged by the flooding Euphrates) back to life with her
(contested societal unity)
Table 1
feet, not her hands to grow into a magnificent “tree of life”
from which Gilgamesh eventually fabricates her “bed and
moveable chair.” Pantomime this story while you learn it. Share its clocked geometry with a
child preparing for kindergarten.
Now learn to draw this Pythagorean symbol in one continuous line starting from 7:00
to 12:00 to 5:00 to imprint “good footing” on the mind. The pattern completes to
twelve tones in six paired steps with the “musical proportion of 12:9::8:6 as rotor of
eternal time—hiding a trivial but accumulating deficiency that must be discovered by
calculation to demonstrate the near-perfection of this pentatonic “paradise,” so
certain to disintegrate with time. Patterns continue through the hours. The five tones
of a pentatonic model are defined by four new pitch classes as 3x3x3x3 (“take 3 four
times,” an old Chinese algorithm for 1-3-9-27-81) to establish the 2-digit 81 as
sufficiently “bullish” for pentatonic musicology East and West. In Egyptian
128
Table 2
“counting board” vertical arrays these five could be merely
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doubled sufficiently to let any one of the first four “corral the
32 96
bull” in its own mode, defined by the smallest “double” that
16 48 144
exceeds 81. We can only guess whether others knew the second
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24 72
part of the Chinese algorithm, completed by progressing
3
12 36 108
“boustrophedon” (as the ox plows, reversing course at the end of
2
6
18
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a furrow) under the rubric: “Add or subtract one-third.” This
1
3
9
27 81
casual indifference in completing quantification naturally exhausts C G D A E
factors of the three in the first column.
double any generator to
Fig. 6
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161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
152
153
154
155
156
exceed the “bull” of 81
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Computing the entire Spiral 5ths system.
Table 3 Computing triples to the comma at 312 = 531,441 for twelve successive musical fifths
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185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
6
9
18
27
54
81
162
243
486
729
1,458
2,187
4,374
6,561
13,122
19,683
39,366
59,049
118,098
177,147
354,294
531,441
3
9
27
81
243
729
2,187
6,561
19,683
59,049
177,147
531,441
312 or 96
Not until an interested scribe discovered a convergence between the 1st and 13th pitch
classes (A-flat and G-sharp) could the relative cyclic perfection of pentatonic have
been fully appreciated. A Chinese algorithm for “tripling” leads naturally to Apollo
Smintheus as Homer’s “mouse god” nibbling his way toward our “Pythagorean
comma” at 312 =531,441.Sumerian scribes corralled him by doubling the
“cornerstone” unit to 220 (with Shamash the sun as “god 20” who “sees
Table 4
everything looking. This helps to demonstrate how an only slightly
Sumer’s 20
smaller “JUST” comma of 81:80 provides a suitable 2-digit surrogate in
doublings
our “pentatonic bull.” Apollo’s nibbling in Table 3 is the technology of
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Egyptian heaven. In the immortal words of Chapter 42 of the TAO Tê
2
4
CHING he proceeds: “The way begot one, And the one two; Then the
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two begot three, And three all else,” meaning (in any language) the “ten
16
thousand things.”7 What else could “mouse god” mean in this arithmetic?
32
Half-way to the mis-named “P” comma is its square root of 729 (Plato’s
64
“cube of 9” with its dialectical double meanings). Suppressing it limits
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necessary octave doubling to only 218=262,144 whose square root is 29 as
256
half-way. The language of arithmetical and proportional measure
512
1,024
converge toward each other.
2,048
Egyptian doubling can be summed to any integer to avoid the intricacy of
4,096
multiplication and division. Find the largest double within the limit, then
8,192
add remaining largest doubles to secure the result; if no perfect
16,384
summation is possible then we are confronted with a “continued
32,768
fraction” that can be truncated with a remainder (a “trick” that explains
65,536
131,072
Narmer’s “goats” that only appear to exceed his bulls). Table 3 displays
262,144
the resources needed in Narmer’s time to frame all necessary bulls, and his
524,288
various limits necessarily lie somewhere between them. This list surfaces
1,048,576
explicitly within a scribal text of 30 doublings from Semitic Mari that was
destroyed by Hammurabi in 1757 BCE. (A doubling of “9” ten times in an Old
Babylonian text displays familiarity with 320 defining the 21st tone in Spiral 5ths
tuning.)8 Iterations produce a “log” series with periodic coincidences with
arithmetical fractions. Another counter means, “Do it again.”
Copyright © 2011 Ernest G. McClain
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219
220
221
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225
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228
229
230
231
232
233
234
235
236
237
238
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240
241
242
243
244
245
246
247
248
249
250
251
252
253
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255
This “P” comma requires tripling 12 times to 312=531,441 to
Table 5
show that leading digits of 53 exceeds those of 219 at 524,288,
Computing the comma
and so their difference of 7,153 units now must be doubled to
312 =
531,441
the maximum to discover its relative worth. Any interested
219 =
-524,288
scribe could have proceeded via the technology of Tables 2 and
1
=7,153
+
2
14,306
3 (let’s call him Anonymous I, since we know that Pythagoras
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28,612
couldn’t prove anything) toward the result in Table 5
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57,224
+
illustrating the comma as the ratio of about 1/73—within a
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114,448
digit of 1/72 that determined the 5 “holidays” of the Egyptian
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228,896
64
457,792
+
360-day year. There is simply no way to divorce the sister
sum
521,169
=73
muses of music, arithmetic, geometry, astronomy, poetry,
dance, etc. from their early gracious collaboration which
grounds our cultures (until they must fight each other for academic survival today).
A plumb line from the throne above maps the tritone square root of 2 in equal
temperament at 6:00 o’clock, cutting the model (i.e., the first) wholetone of 9:8 in half
as two “semitones”--never computed, and always appearing as remainders from
computing consonances. All equal divisions of the octave cycle are naturally irrational
“roots” of the ratio 2:1, but no intervals smaller than 10:9 are actually computed or
drawn, and so early musical geometry is merely illustrative, not definitive; “knowing”
meant knowing the right number. Within wholetones (easily mapped in E.T. with a “rusty
compass” in which the radius as half the diameter cuts off semitone hours along the
circumference, it is easy to approximate by eye the positions of semitone half-hours
and “quarter-tone” quarter-hours (minimal time for observing stellar movement by
naked eye astronomy), potentially halved further by metaphorical “commas” not
worth the trouble of defining and often subliminal to ears. “Third-tones” (unpopular
in the West can be approximated by two “infixes” within the wholetone.
256
257
258
Taming the Serpent as “Lion of the Ground”
Fig.7
1
4:3
3:4
4:3
9:8
3:4
Within the middle “third-tone” it is easy to approximate the positions of two more
infixes to display the blackened comma as about one-ninth of a wholetone. And in
Table 5 we notice that a wholetone of 9:8 appears with 9 units between 72 and 81
between D and E. Any scribe might welcome a “little bull of 81” as an evasion of the
six digits required for the “P” comma (between which we do not aurally notice a
difference). No further computation is necessary for musical understanding.
A very different matrix alignment cuts the first paired musical fifths into “Just” major
and minor thirds of 4:5 and 5:6 indicated here as A-f-D falling and G-b-D rising, and
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these new major thirds are smaller than C:E shown here by a comma that we
approximate numerically from Table 2 as 81:80 and can estimate geometrically by eye
from Figure 7. In the chromatic environment of Figure 8 we are in a new aural
context mapped as a “winged cartouche” in which the serpent as “great wing” is
severed into three parts that are reassembled as its own “body” with no more than the
original five tones in common. New pitches differ only by a comma so that it proves
convenient to assign the same letter names, but presented in lower case as reminders
that in this “triadic” alignment those in the outer rows lie a comma closer to the
middle row. In tuning practice we are taught to find them first in Spiral 5ths (that
enjoy stronger resonances) and then modify them slightly to the weaker resonances of
5:4. The result is Plato’s quantified “societal” model that we meet in Sumer’s “base
60” arithmetic that Narmer graphics exploit.
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280
281
282
283
284
285
286
287
288
289
Higher order abstraction of a “Just” alternative tuning realigned as “body” of
the great hawk “watching over” the throne of Egypt
The fifteen elements within the cartouche belong to Ishtar/Inanna (“god 15”) as her
“bed and moveable chair” constructed by Gilgamesh from the restored “tree of life”
that she “tended with her feet.” In Egypt the farthest corners of the rhomb are cut off
to isolate twin sons Horus and Seth, contending from childhood for the future
throne; as a consequence the six-sided figure remaining takes on the appearance of a
royal cartouche framing a ruler’s name. The visual result schematically imitates a
“flying” cartouche” within which the king is sometimes pictured as double-faced. In
matrix array, tones in the upper row are paired opposites of those in the row below;
in Sumer the nearest water is discovering by digging “straight dow” to a semitone of
24:25, both “left-overs.” This new tuning correlates three slightly different parallel
spirals with an intricate new internal symmetry. The new tritones in the corners are
indistinguishable aurally from the Spiral Fifth tritones (capitalized), but in opposite ways
(a  with G , and g with A ), and the new Just tritones (lower case)
lie slightly closer to the square root of 2 in the middle to justify
suppressing the P. tritones. Plato’s 3 fixed “means” in the middle now
alternately function as major and minor triadic thirds; G, D and A are
potential members of six different triads . Rows 1 and 2 together (and
also 2 and 3) are now intertwined as two “small serpents” visible on the
two crowns of Upper and Lower Egypt, unified by a common calendar.
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292
293
294
295
296
e
A E B
Fig. 8
F
C
b
G
a e b
f
D
c g
A
f
E
B
F
C
G
c
Copyright © 2011 Ernest G. McClain
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The more carefully we look at Narmer’s arithmetic the more interesting
his paired giant Serpopards become.
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301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
Plato’s arithmetic for the Egyptian calendar
Narmer’s numbers must be read with Egyptian technology , and so I do it
with Plato’s examples in mind, trusting Ptolemy’s assertion in the second
century AD that music still was understood as metaphor for ratio theory
performed with integers and by manipulating only one integral part (as if
using unit fractions of one-half, one-third, one-fourth etc.). This idea is
strongly backed by the SECTIO CANONIS, perhaps in Euclid’s time (c.
300 BCE), by computing the wholetone of 9:8 by first halving a line into
8 parts so that one unit length can be added with the compass as
“8+1=9. But in THE REPUBLIC Plato’s formula for his base 60 harmonic theory
is presented a century earlier in far more powerful algebraic language as: “three
multiplied by four and by five, and raised to the fourth power.”9 He has doubled
the “demiurgic” unit from 2 to 4 to reach both 3 and 5 in the Egyptian
manner of 4-1=3 and 4+1=5. An authentic reading of N armer must heed
these early constraints still respected 3,000 years later in Gree k
musicology.
Plato writes that “times of fertility or barrenness of soul and body”
come when their revolutions “complete for each species the
circumferences of circles,” cycles linked to the calendar of 360 days for
which 3, 4, and 5 are divisors. But he has specified in advance that all
patterns (“everything”) suffer “dissolution” with time , and his model
illustrates it at the 3 r d power.
a)
b)
5
3
4
5
20
3
L1=60
25
16
c)
15
12
L2=602
125
100
9
80
64
d)
75
60
48
500
45
36
400
27
L3=603
Figure 9
625
“Knowledge of good and evil”
320
256
375
300
240
192
225
180
144
135
108
81
L4=604 = 12,960,000
Notice that the model itself (see Fig. 9a) suppresses reciprocals (mere
scribal economy) so that “corner” values are always factors of successive
powers of 60. In Fig. 9c we notice ratios of 5:4 along the ascent from
the left (64-80-100-125) failing to double to 2x64=128 by only three
units in three upward steps, each computed as “4”+1 part=5 to arrive at
Copyright © 2011 Ernest G. McClain
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339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
5 3 =125 where indeed we learn both “good and evil” (in explicit Bible
metaphor about the same model).
We accumulate a comma of 81:80 at every step. (Notice how the fourth
Figure 10
125
25
5
1
625
375
75
15
3
The Egyptian calendar
constant of 360 units
625
500
225 675
45
9
135
27
400
405
81
243
640
512
600 450
480
384
375
720
360
675
540
576 432
?
405
648
486
Rotation of the matrix outline exposes paired
reciprocals within at equal distances on straight lines
through the center, and isolating the “cornerstone.”
step along the base to 81 in L 4 exceeds the first step upward to 80 in L 3
by one unit.)) We dare not accept this accumulation of commas in a
“science” mapping aural consonances as “veils of the temple” within the
soul. (The first “throne” appears at 60 in the second row of L 3 ; in L 4
three candidates appear at 180, 240, and 300 . No more than one step up
or down can be permitted from the throne in harmonic mythology although
this ratio theory extends to infinity both to the right and upward among
powers of 3 and 5.
Powers of 2 and 4 that define “different” doubles also coincide
“cyclically” forever and thus giv e the “cornerstone” a prestige as
“demiurge” clouded by its “blindnes s” to consequences. By a coincidence
that only Deity could arrange, the shortage at 100 can be transformed
into a “cube root of 2” by the addition o f “one part” meaning 1/125 but
computed in Egypt (and apparently in Greece) as 100+1/4+1/100. Here
is a trap for the unwary about which Plato did his best to warn posterity
but that Platonists, beginning with Boethius , suppressed by “dumbing
down” his model, never publishing correlations of 3 primes. No English
text known to me prints Figure 9 whose pattern Pla to develops with
elaborate care and conscious pretentiousness.
He has warned readers earlier that “marriages are made in the prime of
life,” and so Figure 10 heeds this warning by developing the calendar
limit (“What 12 is”) as 12x60=720 “days plus nights”--laid out first as
1x3x5 (to avoid octave redundancy), and then “corrected” by doubling all
Copyright © 2011 Ernest G. McClain
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379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
values to the maximum within the limit of 720. We know this total as
“factorial 7” meaning 1x2x3x4x5x6x7 (transforming most “odd males”
into “married men.” But notice that its “bull on the mountain” as largest
factor of 5 (always reached by doubling the “unit” twice before adding
one part)is 5 5 =625 as in L 4 . Interior symmetries are predictable from the “peak”
as “triangular five,” but the last three elements on the right belong to the “wilderness”
of ratio theory, meaning that each must be computed separately.
These calendar integers are projected into the tone circle of Figure 11, copied from
THE MYTH OF INVARIANCE that attributed them to India’s great creation hymn
with lovely Vedic description:
Twelve spokes, one wheel, navels three.
Who can comprehend this?
On it are placed three hundred and sixty like pegs;
They shake not in the least.
………………………………….
O Agni, there stand seven times hundred
And a score of sons in pairs.10
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398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
……………..
“Mantels of radiance” for Sumerian gods as “vector analysis” of
“regular numbers” in “base 60” framed by the Egyptian calendar
Figure 11
The pentatonic glyph in solid lines is supplemented by four pairs of “Just” chromatic symmetries.
The “forgotten” a-symmetric “demiurgic cornerstone” is the nearest convergence to the square root
of 2 in the middle of the octave. This structure integrates Narmer’s 400,000 bulls. Notice the Just
commas of 81:80 at E:e and C:c, and a-symmetry of the “cornerstone” a-flat beyond (but near” the
square root of 2 a 6:00 o’clock. Numbers without spokes have reciprocals only within larger limits.
But Egypt’s “pentatonic” heaven is Everybody’s. All major religions converge on this single
comprehensive calendar model—that could have served to help locate celestial events computed
with the same base 60 arithmetic. Zodiacal tolerance for “convergence” was 30 degrees for 12 arc
values; commas of 1/9th were easily estimated, have halved further. Whether the model had a
practical purpose, however, is speculation; it is intrinsically fascinating enough in itself have attracted
great attention, and sponsored endless stories. And the arithmetic travelled between cultures more
easily than metaphors. Thus a vast array of cultural mythologies remain to be studied from this
perspective. Any numerical element is potentially helpful in revealing its model. Rotation to the right
maps frequency ratios, and to the left maps their reciprocals as ancient pipe and string lengths.
Alexandrian authors eventually gave up trying to correct Plato’s model to fit newer zodiacal
measures. But astronomers found it useful, and poets found it inspiring. Modern equal temperament
ratios extend outward from the rim as very short spikes, never in perfect agreement with the spokes
except at throne “D” as universal referent. “Firstborn twins” at A and G determine its place.
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446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
The integration of Upper and Lower Egypt
Narmer’s 400,000 bulls are imagined here as led by Babylon’s and Sumer’s EA
CREATOR as “god 40” leading myriads (ten thousands) symbolized by 57 integers of
6-digits each. They frame, however, only seven symmetries (powers of 5) along the
ascent, so that the other 50 have reciprocals easily mapped by geometric pattern but
necessarily computed with some larger limit (in this case L5 =605= 777,600,000) that I
suppress until later. As a visual aid to imagination I present the result first in matrix
array as products of 3 and 5, indicating future reciprocals with +, and replacing factors
with tone-names only where relevant. The matrix itself is ruled by “triangular 9” and
Narmer is pictured appropriately enthroned above 9 steps defined by 58 and with his
sandal bearer behind and below—as if implying that his feet are properly bared. The
central horizontal axis contains 9 tones as the reciprocal meanings of only five
integers, while the base displays all twelve tones as foundational in the Spiral Fifths
tuning of the great serpent as “lion of the ground” not only in both Sumer and Egypt
but in India, China, and the modern world. The notion of “ground bass” helps keep
the metaphor alive.
Copyright © 2011 Ernest G. McClain
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464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
Figure 12 The musical union of Upper and Lower Egypt
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
B F
Horus
+
+
+
+
e
C
+
+
+
+
5
5
5
1
3
3
3
3
3
c g
E
c
3
3
3
g
3
F
3
3
3
3
3
Seth
B
3
3
3
Paired “Just” tritones of a and g as
“twin sons” Horus and Seth contend
for the future throne.
3
A
a e b f
+
+
5
D
+
5
f
b
G
+
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Data base for 400,000 as 57 products of 2, 3, and 5
9
8
7
6
5
4
3
2
1
Table 6. Factors of 3 and 5 are only male generative integers for 12-tone musicology.
1
2
3
4
5
6
7
8
9
10
11
390,625 “Bull on the mountain”
78,125 234,375
“double and add” to multiply by 3 to the right
15,625 46,875 140,625
“double twice before adding” to multiply by 5 upward
3,125
9,375 28,125 84,375 253,125
625
1,875
5,625 16,875 50,625 151,875
125
375
1,125
3,375 10,125 30,375 91,125 273,375
25
75
225
675
2,025
6,075 18,225 54,675 164,025
5
15
45
135
405
1,215
3,645 10,935 32,805 98,415 295,245
1
3
9
27
81
243
729
2,187
6,561 19,683 59,049
12
177,147
Doubling to the maximum within the limit confounds appearances to produce a “tower of Babel” but numbers
then can be realigned into naturally increasing or decreasing sequential and cyclic scale order for further display as
“vectors” (“mantles of radiance”) in a “tone circle.”
9
8
7
6
5
4
3
2
1
484
485
486
390,625
312,500
250,000
400,000
320,000
256,000
204,800
327,680
262,144
234,375
375,000
300,000
240,000
384,000
307,200
245,760
393,216
Table 7. (These 6-digit “bulls” were labor intensive and offensive
to reason in Egyptian duplatio; few were likely computed.)
281,250
This ratio value belongs to Seth as “scapegoat” in narrative.
225,000 337,500 253,125 Single digits 4 and 3 lead hundred thousands
360,000 270,000 202,500 303,750 2 digits lead myriads
288,000 216,000 324,000 243,000 364,500 273,375 3 digits lead thousands
230,400 345,600 259,200 388,800 291,600 218,700 328,050 4 digits lead hundreds
368,640 276,480 207,360 311,040 233,280 349,920 262,440 393,660 295,245
294,912 221,184 331,776 248,832 373,248 279,936 209,952 314,928 236,196 354,294
Seth’s value as 281,250 among the bulls reduces to 45:32 within the octave, but that
role belongs to Horus in the calendar subset when it is abstracted as in Fig 10. In
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15
487
488
489
490
491
492
493
494
other Narmer matrices both twins appear together so that Seth’s ratio is inflated to
64:45 (i.e., as excessive a square root as Horus is defective.
Table 8 follows with the completed matrix for 120,000 male prisoners as these 48 are
studied in close proximity. Results are developed in Egyptian “counting board
columns” more easily, but meaning is seen more elegantly displayed in the hexagonal
arrays of Sumer that follow.
8
7
6
5
4
3
2
1
78,125
Table 8. Doubling to the upper limit of 120,000 male prisoners.
62,500 93,750
100,000 75,000 112,500 84,375
80,000 120,000 90,000 67,500 101,250
Tetrachord leaders
64,000 96,000 72,000 108,000 81,000 60,750 91,125
Pentatonic leaders
102,400 76,800 115,200 86,400 64,800 97,200 72,900 109,350
Heptatonic leaders
81,928 61,446 92,160 69,120 103,680 77,760 116,640 87,480 65,610 98,415
65,536 98,304 73,728 110,592 82,944 62,208 93,312 69,984 104,976 78,732 118,098
495
496
497
498
Reduction to an “army” of 120,000 lowers each column by one value and exposes the
Platonic musical proportion of 12:9::8:6 leading myriads, plus pentatonic and
heptatonic scale ratios of 2 and 3 digits each.
499
500
501
502
Realignment in Sumerian “brick order” facilitates comparison with the union of
Upper and Lower Egypt in Figure 9. The throne is not centered between the
appropriate scale integers. Here a powerful leadership of heptatonic “lower classes”
Figure
13
Why an opposing army should not be enslaved
7
8
78,125
7
6
5
4
3
503
504
505
62,500 93,750
100,000 75,000 112,500 84,375
80,000
120,000
90,000 67,500 101,250
60,000
the musical proportion
pentatonic
thousands
heptatonic
102,400 76,800 115,200 86,400 64,800 97,200 72,900 109,350
hundreds
64,000 96,000 72,000 108,000 81,000 60,750 91,125
2
81,928 61,446 92,160 69,120 103,680 77,760 116,640 87,480 65,610 98,415
1
65,536 98,304 73,728 110,592 82,944 62,208 93,312 69,984 104,976 78,732 118,098
endangers the throne. (Power is more wisely centered on the pentatonic leader of only
72,000, doubled to 144,000. (This happens in the Bible among the male choir of
REVELATION.) This speculative reading is supported by another Narmer graphic.
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506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
Glance now at the seated man symbolizing 6,000 and connected by a “nose rope” held securely to
the proud hawk guarding Narmer’s throne. The poverty of this matrix when reduced to its “naked”
male factors of 3x5 is obvious—promising nothing more than a “Trinitarian” leadership over the
similar pentatonic and heptatonic forces in the two rows directly below, but now weakened to only
one-tenth their previous threat. A surprise, however, follows their doubling in Figure 14.
+
+
+
+
+
+
Figure
13
+
+
+
+
6,000 disciplined from Above
+
125
25
+
+
+
+
+
Figure
14
+
6,000 disciplined from Above
+
75
1,125 3,275
225
45
675
135
2,025
405
1,215 3,645
3
9
27
81
243
+
+
+
+
+
+
+
+
+
375
15
+
+
+
625 1,875 5,625
+
5
+
3,125
+
1=n0
+
3,125
+
729 2,187 6,561
+
5,000 3,750 5,625
4,000
6,000
4,500 3,375
3,000
3,200 4,800 3,600 5,400 4,050
5,120 3,840 5,760 4,320 3,240 4,860 3,645
4,096 3,072 4,608 3,456 5,184 3,888 5,832 4,374 6,561
The eight elements in the smallest matrix for base 60 now are “leaders” of hundreds and thousands
under the divine leadership of Mesopotamian divinities as “gods 30-40-50-60” that felicitously
“explain” the “nose-rope” held by guardian hawk. This matrix seems a charming companion for
Babylon’s nine string lyre, raising questions about the present dating of artifacts and the direction of
influence that it seems wise to leave to linguists as they recover more of the epigraphic record from
thousands of tablets yet to be decoded and published.
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554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
Microtonal content
Figure 15
The calendar arithmetic provides for the Greek
enharmonic tunings that Plato bans within its normal
chromatic limits of 720, but their reciprocals mandate
another octave doubling to 1440:720. Just tunings by 5:4
within the perfect fourth of 4:3 leave an oversized
semitone of 16:15 that can be simply halved
arithmetically to produce quartertones of 30:31 followed
by 31:32 that performers would estimate by ear. Table 9
computes the enharmonic genera as ratio theory requires.

e
C
(a
)
b
G
e
f
D
A
f
b
c
(g)
E
c

Hebrew harmonic allegory
Table 9 Enharmonic genera in tetrachords of 40-32-31-30 vs 24-30-31-32
falling
D
b
b
A
G
e
e
D
rising
D
f
f
G
A
c
c
D
decreasing
120
96
93
90
80
64
62
60
increasing
48
60
62
64
72
90
93
96
1440
1152
1116
1080
960
768
744
720
720
900
930
960
1080
1350
1395
1440
self-symmetry
in 1440::720
1440:720 frames reciprocal diatonic, chromatic and enharmonic tetrachords in duplatio.
* indicates a quartertone lower  or higher .
Conclusion of Part I
The Narmer mace-head and palette inscriptions fuel a tremendous display of
mythological events in the Pyramid texts that follow in the next millennium and the
twelve tone base of Spiral 5ths among his bulls becomes the scene of the Sun’s nightly
journey in the “moon boat” through the 12 hours of the night in the base of the mountain
guarded by serpents at ever portal. In that elaborate ritual it seems a pity that
Serpopards have disappeared for they seem a perfect complement to a tuning theory
that possesses paired 13-tone development and with competing pairs of tritones plus
an intricate internal symmetry integrates a multitude of tonal perspectives. Plato’s
“five pairs of twin sons,” for instance, displayed in figure 15, are twinned in the
Hebrew Magen David as mapped in Figure 16 that excludes all complicating tritones.
Those symmetrically flat-footed beasts with intertwined necks and heads overlooking
the ritual circle intended for preparing “eye paint” could be ourselves as
“archaeoethnomusicologists.” We have never yet fully plumbed “what God hath
wrought” in the magic of the musical octave.
Figure 16
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I stop here to forward this progress report to Dumbrill and give us both a chance to assess how things are going. The
formatting is now giving me much trouble, but he must cope with it anyway in his own way, and may have some useful
advice. The big question is: Is the message getting through in a believable way? Here is a potential ending, but many
supporting documents can be added if judged important; readers may feel battered enough by new ideas.
References will be added more carefully in the next draft.
1 Marshall
2 Ifrah
3McClain, E.G.
44god numbers
5Plato, REPUBLIC
6Plato, CRITIAS
7 R.B.Blakney, Mentor, 1955
8 Friberg, Jöran, UNEXPECTED LINKS BETWEEN EGYPTIAN AND BABYLONIAN MATHEMATICS, World
Scientific, Singapore, 2005, pp.1 and 14-18.
9 Lindsay, A.D., London, Dent, 1964, p. 241 (Stephanus 546a-d).
10 M.O.I. p. 34-36
Copyright © 2011 Ernest G. McClain