§50. NUMBER SYMBOLISM
§50.1. Harmonic Derivation
This derivation is not necessary here, since the reader has been
informed of the importance and significance of certain numbers, such as
the senary, etc., at many places in his previous study of this text.
Therefore, we will proceed immediately to ektypics, and refer to the
harmonic origin of the numbers in question case by case
§50.2. Ektypics: the Harmonic Number
Whereas all number systems are built upon powers of a certain basal
number, e.g. 10 in our decimal system, and thus follow purely
arithmetical viewpoints, the harmonic number system is already
psychophysically evaluated due to its evolution from the tone-number,
and thus structured, formed, and shaped (§4a, §31a). Such shaping and
evaluation consequently has its effect on the individual number groups
and number progressions, indeed on the individual numbers themselves.
For example, whereas the number 5 indicates in pure mathematical terms
only “five times one,” or the arithmetical quantity “five,” etc., in
harmonics 5 e or 5 as
is also the major third, as well as the fifth frequency or wavelength.
Here, then, a psychic value is added to the pure abstract concept of
quantity. Psychic values, however, are not only in our heads, but also
in the psyche, localized in perception, and precisely for this reason
it will be possible to observe the great disreputable (although
venerable and never completely lost) domain of “number mysticism,”
“number superstition,” and in more objective terms, “number symbolism,”
from an adequate and above all accessible point of view: that of
harmonics.
§50.3. Number Symbolism
All antipathy toward number mysticism, its convenient shelving among
the superstitions, or at best its symbolic redemption, has for so long
been meaningless and stuck in mere historicism that it is only handled
purely intellectually, historically “critically,” or “analytically.” If
we take only the numbers in themselves in their mathematic notation
systems, then it is completely incomprehensible why certain numbers
over millennia have been connected so tenaciously with certain images
and ideas, or indeed why numbers were ever allowed and assigned
meanings in themselves, be they extrasensory, symbolic, or whatever
else.
The problem looks entirely different once we know that in our psyche
itself, definite number proportions are present, indeed must be, since
otherwise we would not be in a position to hear the tones and intervals
correctly and “purely.” But the things we do hear are initially
tone-perceptions; or rather, we perceive the numbers and number-ratios
undoubtedly unconsciously present within us in wakeful consciousness
not as numbers, but as tones. Although we can also say retrospectively
that we consequently hear numbers spontaneously, this act of
apperception of hearing is not conscious for us; it is latent, but
still present.
For these reasons it is obvious that harmonics must have a singular
access to this domain of “number mysticism,” since with harmonics, the
number reaches deeply into the domain of the unconscious, into the very
domain of our psychical life, from which these symbols, images, and
analogies of so-called “number superstition” also emerge.
Since modern harmonics is currently undergoing “rebirth” and is a very
young science, it is not complete enough in itself to handle ektypic
analyses satisfactorily-regardless of the fact that specialists must
work with the relevant domains here. All we can do here is make a first
attempt at a harmonic sounding-out.
§50.4. Methods of Harmonic Number Analysis
There are various methods of harmonic analysis of simple numbers. We
identify the numbers directly with frequencies or string lengths, and
insert the corresponding tone-numbers, whereby we are allowed, in
appropriate cases, to identify 1/2 with 2, since indeed 1/2 divides the relevant unit of frequency or string twice. Or we identify the number progression 1 2 3 4 5 6 7 with the scale c d e f g a h (c),
since this succession undoubtedly corresponds to an inner number
succession of our psyche. Or we see in the numbers more of a
simultaneous expression for group-theoretical forms of the “P”, for
example the progression 1-6 for the “senarius,” which covers the index
6 of the “P”, at least its two reciprocal basal ratios. We must always
seek to trace the findings in question back to a harmonic
wholeness-form, through which it might naturally appear that some
number or another is associated with various such harmonic forms, or
conversely: that a harmonic form is able to interpret various numbers.
Furthermore, attention must be paid to the origin
of the numbers in question. For example, there is no sense in analyzing
the absolute dates of years, months, etc., since for example the year
1939 is dependent in its numeric value on the arbitrarily fixed year 1,
the birth of Christ. On the other hand, we can very well take the
proportions of various dates among themselves for the occasion of a
harmonic analysis, since this is a matter of comparing time periods. It
likewise makes no sense, for example, to place the tone-values c g e in
the number progression 135 and then to believe that this number
one-hundred-thirty-five has something to do with the triad. With such
isolated numbers we must first examine what relationship they have to
the generator-tone 1 c; in this case the number 135 has the tone-value cisˆ7.
§50.5. Individual Numbers
The Number 1
Aristotle (Metaphysics
I, 5) writes of Parmenides: “Since he is of the opinion that it is
impossible to speak of a non-being beside being, he finds himself
forced to assume that being is one,
and there is nothing else ... Since he sees himself as forced, however,
to get involved with the appearances, and grasps the One as the
concept, the Many as associated with the sensory apperception, he once
again sets up a duality
of causes and a duality of principles, the warm and the cold ... on the
side of being he places the warm, on the side of non-being the cold.”
If one substitutes the evolution of the “P” from unity according to the
duality of “unlimited” (1/1 2/1 → ∞/1) and “limited” (1/∞ ← 1/2 1/1)-the
well-known Pythagorean concept which can only be understood at all in
harmonic terms-then one understands “being” as the > 1 sector of the
“P”, “non-being” as the > 1 sector, if we grasp the “P” as a whole
as the inner psychic structure of our psychical capability. Even
“warmth” and “cold” can be thought of as unconscious ektypics of the
movement that is continually faster upwards from the 1/1 and continually slower downwards. On Plato's views on this subject, Aristotle writes (Metaphysics
I, 16): “Thus he sets the large and small as principles in the sense of
matter, and the One in the sense of substance; from this emerges the
participation in the One as a result of the ideas ... the curious thing
with him [Plato] is that he grasps the unlimited not as a unity, but as
a duality, and turns the large-small into his elements.” This section,
apparently contradicting the Pythagoreans, redeems itself immediately
as a different aspect of the harmonic psychic form that is at the basis
of both viewpoints. Whereas the Pythagoreans grasped the polarity
(limiting) 1/∞ ← 1/1 → ∞/1 (unlimited)
in
the sense of the “large and small,” the “unlimited” and “limiting,”
thus purely geometrically, Plato sees the whole, i.e. the two
principles that emerge from the unity 1/1, as purely theoretic and both unending, as indeed the mathematic term 1/∞ and ∞/1 allows in any case.
Plotinus seeks to clarify the relationship of the intellect to the One
with images. He compares the generated nature of the intellect through
the One with a brilliance erupting around the One, while the One
remains quiet “like the brilliant light that surrounds the sun...” “The
One is like the life of a mighty tree, it pervades the All in which the
beginning remains and is not entirely destroyed, simultaneously solidly
founded in the roots” (A. Drews: Plotin, Leipzig 1907, p. 118). Here, also, the value-formal expression of the “P” is obvious: the “One” bathed in light by the 0/0
that is identical with it, and its “equal-tone lines,” and the
“generator-tone line” with its “all-pervading” successive
manifestations of the unity 1/1 2/2 3/3
... as well as the overall continually emerging idea of the “tree” as
one of the primal expressions of the “P”. The theorem of the equal-tone
lines, i.e. the return of all being-values to the divine (0/0), is found summarized in the following words by Plotinus (6th Ennead,
Book 9): “Every soul is an Aphrodite, which is darkly indicated by the
myth of the birth of Aphrodite and of Eros born with her. In its
natural condition, the soul aspires toward God, so as to lovingly
become one with him, as a virgin of high birth longs for a noble love.
But when it has fallen down into generation, and is at the same time
beguiled by the intoxication of sensual love, it has traded for
another, mortal love, and behaves impudently in the separation from its
father. But finally, it begins to hate the mischief down below, it
cleanses itself of its earthly shackles; absolved, it returns to its
father and finds its true salvation with him” (Plotinus, Ennead, tr. by O. Kiefer, Jena 1905, vol. I, 129).
“Instead of the second One of Iamblichus, Proclus has a limited number
of unities emanating from the primal Being: the Henads [from ̔έν = Hen = the One], which represent an intermediate transition from the primal One [1/1] to the eternally growing and finally unlimited multitude in the lower regions” (Überweg's Geschichte der Philosophie, I, 1926, p. 627). Compare to this idea the generator-tone line 1/1 2/2 3/3 ... n/n, which actually constructs an “intermediate transition” between the ratios in the “P” system!
A substantial book could be written on the unity, duality, and the
dieresis of ideas and numbers in ancient times, from the point of view
of harmonic akróasis, and I believe that from this true
number-harmonics, not only could many theorems, previously only
examined purely philosophically, be interpreted in a new way, and the
spiritual and psychical life of the ancients be classified in general
terms, but above all, at least a ray of light could be thrown on the
jungle of ancient number-magic that is almost impenetrable today,
especially that of late antiquity (Gnosis etc.), which in many cases
can be traced back to Pythagoreanism and Orphism. As for the
illumination of the Pythagorean number theorems, the first harmonic
attempt since Thimus was made in my essay on Pythagoras (Abhandlungen). Furthermore, see §25 and §55 of this text.
The Number 2
If we take the number two as an essence in the sense of the polar emergence of the two principles from the harmonic unity:
then
a whole world of number-symbolic observations appears to us. I am
reminded of the dualistic religion systems and philosophical doctrines,
the Dyas of Ormuzd and Ahriman, the Chinese Yin-Yang, the symbolism of
sexual polarity in various cults and epochs in art, etc. Compare this
with §23a.2.
The Number 3
The three as a number, and three-ness as an essence, as we saw in §30, is localized at a decisive place in the “P” system:
and
it is thus understandable that this akróatic form presses towards
realization especially powerfully in the human consciousness as the
image-concept of the “trinity” and the manifold ideas connected with it
(body-soul-spirit; time-space-causality; thesis, antithesis, synthesis;
thinking, willing, feeling; truth, goodness, beauty; triads of gods
like Brahma, Shiva, Vishnu; the threefold burnt sacrifice of the Jews;
the thrice holy, holy, holy is the Lord Zebaoth; the Brahmanic trinity
of the Trimurti; the Greek triglyphs; etc.). Poets, also, cannot elude
this inner significance of the trias. In the Prometheus of
Aeschylus (515-516), Prometheus answers Oceanus's daughter's question:
“Who steers necessity?” by saying: “Threefold Moira (fate).” Even if we
knew nothing about harmonic psychophysical anchoring, we would have to
accept an image-concept latently present in our subconscious
determining the various trinitarian realizations in our waking
consciousness.
The Number 4
In the symbolism of the number four,
I finally see an expression of a “coordinate consciousness,” which is
based physiologically on the balance organs in our ears and is thus
harmonically founded. The four, then, plays a role especially in the
philosophical and religious systems,
or predominantly appears where there is a more realistic thought and
perception, more oriented towards the world and the here and now. The
four and the square are known in all possible variations in Chinese
culture; the Romans not only laid their cities out mostly at right
angles, but also imagined the Earth as a right-angled coordinate system
(M. Cantor: Vorlesungen über Geschichte der Mathematik, 4th
ed., 1922, p. 533). Since four is the number of matter, the Four
Horsemen of the Apocalypse destroy (Revelation, Ch. 6; here the four
colors are white, red, black, and sallow, i.e. deathly green-cf. the
Tarot colors) turning that which deserves it back into chaos, i.e. to
simply formless space. The ancient Indian book of the Laws of Manu
divides humans into 4 categories and teaches 4 sciences. If this
“Quadrivium,” albeit different in content, continues so tenaciously up
until the Middle Ages, that is only proof for the “topological”
significance of the four, which naturally can only be an ektypic
expression of the four as a psychical value-form.
The Number 5
The five
as an image-concept and essence again has an entirely different
character. Here, harmonic analysis must insert or underlay the 5th ratio, i.e. the third interval, the “gender interval” (see my Harmonia Plantarum,
p. 201 ff.). The ektypic analogy is especially close. Five is the
number of Ishtar, the Babylonian Venus, and its symbol is the
pentagram. There are five-sided temples of Venus from Hellenistic times
(Baalbek). The spiritual tendencies of Christianity converted these
ancient cults to the worship of the Virgin Mary. The pentagram is worn
as an amulet as a symbol of the spiritual Eros, the love of God, and
this aspect reminds one of the colossal pentalpha-rose windows in the
west end of various Gothic cathedrals. Not only in the Orient, but also
in modern Italy, the spread five-fingered hand serves as magical
protection against the “evil eye.” Consider, further, the quinta essentia of the Alchemists, which was the actual generating means, only to be used correctly by a loving and humble hand, to create the lapis philosophorum,
the philosopher's stone. All these symbols, image-concepts, and
connections, with their special characters, must have a deeper
reference-point in our psyche.
The Number 6
We once again enter an entirely different sphere when examining the symbolism of the number six.
Harmonically, six signifies the “senary,” i.e. the pure chord in the
partial-tone series, the index of the pure major-minor chord domain in
the “P” system, and the senary ratios in the index of the audible, as
material for normative music-making. One should consider inwardly what
this restriction of the major and minor chord to the frequency or
wavelength numbers 1 2 3 4 5 6 means for the importance of this number
group and its signifier 6.
Thus it is no wonder that in theological arithmetic, for example, the
number six is defined as the symbol “of animation and the nature of the
living,” and if we remember the various further ektypic meanings of the
senary, we can write this definition as a motto. There is a
considerable symbolism of six-pointed stars (hexagrams), which finds
its most beautiful artistic expression in the great cathedrals even
more commonly than the Pentalpha. The significance of the hexadecimal
system has already been discussed (§26a.2). The “six days' work” in the
Bible is in a certain sense the most marvelous expression of the
senarius with regard to its ektypic expression, and one needs only to
look at the “skeleton” of the “P”:
Figure 455
to
be able to read off the “six to the right, six to the left” of the
Christian (and other) symbolism with all its accompanying ideas. The
Chinese cultural attaché in Switzerland, Lin-Tsiu-Sen, writes in his
excellent book China und Japan
(vol. I, Zürich-Erlenbach 1944, pp. 161-162): “The oldest record of
religious perception of the Chinese is in the first book of the Shu
King, in which it is said of the legendary Emperor Shun that after his
coronation in 2255 b.c., he sacrificed
to the Shang-ti, to the highest,
to the six venerable ones,
to the mountains and waters, and
to the hundred spirits.
Western
savants have often wanted to see a highest personal God in the
Shang-ti; the six venerable ones are almost harder to interpret.” Here
if we set the Shang-ti as our symbol 0/0
(Eidos, impersonal deity), the six venerable ones as the senary of the
primary pure chord, the mountain and water as the polarity 1/∞ ← 1/1 → ∞/1,
and the hundred spirits (which, as Lin-Tsiu-Sen remarks, means nothing
more than “all”) as the multitude of “P” ratios, then in this ancient
Chinese religious document we have a precise ektypic expression of
primal prototypical harmonic value-forms, which manifested themselves
in those ancient times when thought was still intuitive, in such
image-concepts as Shang-ti, six venerable ones, etc.
Then there is the Zodiac, indeed our whole system of time measurement
and angle division, which have the “six” fully in their backgrounds as
a value-formal gestalt. As a psychical expression in harmonics, the 6
and the number-forms connected with it (double series for every 6, the
number 12, the hexadecimal system, the senary, etc.) will always be the
symbol for a something closed, “effable,” capable of realizing itself.
Thus it is merely a sign of deeper intuition when Jakob Böhme
identifies his “sixth nature-form” with the “reverberation,” i.e. with
effability, the “voice” of the world.
The Number 7
The number seven
in harmonics has the significance of the seven-step diatonic scale. At
the seventh place in harmonic series-development, an “ekmelic” ratio
appears for the first time, i.e. a ratio no longer applicable in
practical musicianship:
1/1 c 2/1 c 3/1 g 4/1 c 5/1 e 6/1 g 7/1 xb
We tolerate this so-called “pure seventh” in the chord c e g xb as still “in tune,” if a bit too low; but as a tone in itself, it does not fit in our tone-system. As a step, however, something new and strange appears, and if we observe its position between 6 g and 8 c,
we can say: at the “location” seven the tone-apperception “reposes,” or
at least stops to think, to then begin a new octave again at the eight
(8 c). This stopping to think, however, has a completely different character akróatically. In the seventh chord c-e-g-xb
that it defines, our psyche perceives a summons to a continuation, an
advancement into another domain of perception, which is expressed
musically through the bringing of this chord to some kind of resolution:
Figure 456
In any case, outside the seven-step scale, the seven in the harmonic
ratio-development signifies the first value that one can view as
“strange,” “not in tune,” indeed “wrong” or as the step of a transition into another world, or a summoning thereto.
Precisely these interpretations can be found expressed in the
number-mystics and symbolism of the seven in manifold forms. The
seven-step scale reminds me of its identification with the seven
“planets,” meaning the sun, the moon, Mercury, Venus, Mars, Jupiter,
and Saturn. The “harmony of the spheres,” connected with this and with
harmonic proportions, constructs a typical image of a circle, which
reaches far into domains of cultural history and art history, and has
remained influential up to modern times. The parallel scale-the days of
the week-also belongs here. Dio Cassius writes that the Egyptians
brought the order of the planets in agreement with the days of the
week, and arranged the series-progression “according to every fourth
step of their scale.”
With this arrangement:
Saturday |
Sunday |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
Saturn |
Sun |
Moon |
Mars |
Mercury |
Jupiter |
Venus |
h |
e |
a |
d |
g |
c |
f |
these fourths, put in order, yield the diatonic scale:
c d e f g a h
As
for the “evil seven,” i.e. the uncomfortable, “out of tune” feeling
arising with the perception of this number, folklore is full of
examples. There are notable temporal periods in history where the seven
was an unlucky number. For example, the following kings of France were
taken prisoner: Louis IX, in the year 1250, by the Saracens in Egypt;
John, the seventh king after Louis IX, after a battle in 1356; Francis
I, the seventh after John, by Emperor Karl V in 1525 at Pavia; and the
next two seventh kings after that, Louis XIV, and then Louis Philippe,
by his three sons. The seven plagues of Egypt, the seven deadly sins of
Catholic dogma, the seven days spent mourning the dead for the Jews,
Romans, and many Oriental peoples, the seven avenging angels in John's
Apocalypse, the seventh phrase of the Lord's Prayer: “deliver us from
evil”; and so on.
And regarding the significance of the seven as the turning toward
another (seventh chord) or its position as something special, indeed
holy, Sunday is the best known ektypic realization, as the seventh day
of the week, the day of rest and sanctification (for the Jews Saturday,
the Sabbath, is the seventh day). This “modulation power” of the seven,
in the form of intervals of seven, has so many being-values of various
types, such an enormous number of applications, in every case
meaningful, special, and important, that one could fill pages with
them: seven mountains, seven hills of Rome, Siebenbürgen,
the seven-armed menorah of the Jews, the Seven Sisters (Pleiades), the
seven Swabians and seven dwarves in folk tales, seven-league boots, the
Seven Sleepers of Ephesus, “in seventh heaven,” etc. Franz Boll, in Aus der Offenbarung Johannis
(1914, p. 21), writes of a “tyranny of the number seven,” and the
Hebrew word for swearing (to take an oath) is, translated literally,
“to be sevened,” or “to seven oneself.” The nature of the seven as a
psychical form can hardly be denied any longer.
Harmonically, we take up the trail of the most varied aspects of this
psychical form of the seven, which we would otherwise have to shrug off
as something unexplainable, simply given. The fact that on the one hand
we speak of an “evil seven,” on the other hand of the seven as
something special and holy, has its origin in the seven-step scale
(harmony of the spheres), then in the setting of the psychical form of
the seventh interval as something alien, outside the framework, impure;
and finally, the seventh interval is the psychical expression for
modulation, i.e. the transition into another, novel, “better” world, or
however else one wants to express this. If we simply compare the “evil
seven” with the “good seven”-for this there is also a wealth of
examples (seven fat cows and seven thin cows, etc.)-as a theoretical
background, we can use the conjugated double series:
1/7 xd,,, 1/6 f,,, 1/5 as,,, 1/4 c,, 1/3 f,, 1/2 c, 1/1 c 2/1 c 3/1 g 4/1 c 5/1 e 6/1 g 7/1 xb
as a basis for interpretation, analogous to the double series of the senarius.
The subject threatens to be boundless, but we will cease here with the
analysis of the individual numbers, which could multiply itself many
times, and turn to some special problems of number symbolism.
§50.6. The “Duplicity of Events”
In the strict harmonic sense, this is a problem of intervals,
specifically interval continuance. Since this is a continuance of the
octave, of the first interval of the partial-tone series
1/2 ← 1/1 → 2/1
c, c c
which is in turn closely connected with the number two from the “double,” i.e. same event (1 + 1 = 2), its place is here.
If we write out a tone-series, no matter which one, we will always find the model of the overtone series:
or its mirror image, the model of the undertone series:
That is, in the primary harmonic series-development, which is anchored
both objectively in nature (the overtone series) and subjectively in
our psyche (tone-perception), we have, right at the beginning, a
repetition of the generator-tone in its octave c-c or c-c,.
This is the only “duplicity of events” in the succession of the series.
If we grasp this phenomenon more dynamically, and empathize with it,
then we can conclude that both in nature and in our psyche, a form must
be present that somehow postulates this “duplicity” in the
temporal-spatial existence of the being-values. This last, then, will
stand in the succession 1 : 2 as differing in time and space, but will
be inwardly identical in terms of value (two c-values). As I remarked earlier (Grundriß
204), this repetition of the event “in itself” among completely
different situations is the most puzzling element of the “phenomenon of
duplicity of events, obviously elicited by experience,” but this puzzle
is finally explainable and understandable through the analogous
harmonic theorem.
But how about the objective condition? As for my experience, I have
been able to establish the factual existence and incidence of such
duplicities (as most of my readers probably have also) with many
examples-train and plane crashes, important spiritual events, even the
so-called “parallel events” of any type. Karl Marbe, professor of
philosophy at Würzburg, published an extremely interesting two-volume
work between 1916 and 1919 (Oskar Beck, Munich) on “uniformity in the
world,” and grasped the problem in the most general way by
investigating the “return of the same” in all domains of life, science,
and history. The corresponding harmonic configuration for this
universal concept of uniformity would be identical, or at least
similar, situations in the harmonic “P” system and its selections,
whereby the duplicity of events (the duplicate happenings just named,
the simultaneous discovery of infinitesimal calculations by Newton and
Leibniz, parallel movements in culture and history, Hitler-Mussolini
etc. ad infinitum)
represents only a special case of universal identities (equal
intervals, identical tones on the equal-tone lines, enharmonic
reincarnations, etc.): namely the special case of the narrower
duplicity in the sense of the “up-beat” of the “P”: c-c or c-c,.
Marbe, after questioning the reasons for this uniformity (I, 405),
writes further (I, 410): “The crown of a universal theory of uniformity
would be in the proof that a uniformity of conditions leading to
uniformity is necessary.” He adds to this (ibid.): “But the question
still remains, why precisely this initial condition or even earlier
conditions, which must lead under the influence of laws of nature to
uniformities, would be present, and not others that do not lead to
uniformities. This question applies in many cases as unsolvable on
principle. In the oldest initial conditions accessible to our
observations, one sees a historical fact that, if one does not want to
attribute it to some creator, one must simply accept as given.
According to this, the conditions of uniformity would also not be
explainable from simple laws, but only from laws in connection with the
assumption of a definite given initial condition of the world.” Thus,
in the akróasis of this “initial condition of the world,” we have
arrived upon the law of harmonic quantization. But the initial
“uniformity” of this law in the position of the octave, the symbol for
the primary uniformity (duplicity), lifts this law out of the simply
material and merely legalistic into a sphere of values; because the
identity lies not in the numbers, but in the tones. So if one
substitutes the harmonic constitution of this “initial condition,” the
question of the reasons does not seem to me to be “insoluble in
principle,” but on the contrary: here we have the explanation for why,
on the one hand, the law must apply both psychically (psychologically)
and physically, and on the other hand, the proof for a background that
is psychical, and in a further sense spiritual (the “P” system with the
0/0 as a reference value).