<div id="LayerE10" style="visibility: visible; overflow: visible; z-index:0; left:10px; top:5px; position:absolute; "><?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> <head> <!-- ======================================================= --> <!-- Created by AbiWord, a free, Open Source wordprocessor. --> <!-- For more information visit http://www.abisource.com. --> <!-- ======================================================= --> <meta content="text/html;charset=UTF-8" http-equiv="content-type" /> <title>Harmonics & Mathematics - Number Symbolism.html</title> <style type="text/css"><!-- #toc, .toc, .mw-warning { border: 1px solid #aaa; background-color: #f9f9f9; padding: 5px; font-size: 95%; } #toc h2, .toc h2 { display: inline; border: none; padding: 0; font-size: 100%; font-weight: bold; } #toc #toctitle, .toc #toctitle, #toc .toctitle, .toc .toctitle { text-align: center; } #toc ul, .toc ul { list-style-type: none; list-style-image: none; margin-left: 0; padding-left: 0; text-align: left; } #toc ul ul, .toc ul ul { margin: 0 0 0 2em; } #toc .toctoggle, .toc .toctoggle { font-size: 94%; }@media print, projection, embossed { body { padding-top:1.0000in; padding-bottom:1.0000in; padding-left:1.2500in; padding-right:1.2500in; } } body { text-indent:0in; text-align:left; font-weight:normal; text-decoration:none; font-variant:normal; color:#000000; font-size:12pt; font-style:normal; widows:2; font-family:'Times New Roman'; } table { } td { border-collapse:collapse; text-align:left; vertical-align:top; } p, h1, h2, h3, li { color:#000000; font-family:'Times New Roman'; font-size:12pt; text-align:left; vertical-align:normal; } *.footnote_text { font-size:10pt; } *.footnote_reference { font-size:10pt; vertical-align:super; } --></style> </head> <body> <div> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> <table style="border-collapse: collapse;" width="600" cellspacing="0" cellpadding="3" bordercolor="#000000" border="0"><tbody><tr><td width="100%" valign="top"> <div> <span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> §50. NUMBER SYMBOLISM</span> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§50.1. Harmonic Derivation</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§50.2. Ektypics: the Harmonic Number</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span lang="en-US" xml:lang="en-US"> or 5 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§50.3. Number Symbolism</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§50.4. Methods of Harmonic Number Analysis</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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 </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> with 2, since indeed </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> divides the relevant unit of frequency or string </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">twice</span><span lang="en-US" xml:lang="en-US">. Or we identify the number progression 1 2 3 4 5 6 7 with the scale </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c d e f g a h </span><span lang="en-US" xml:lang="en-US">(</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">), 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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">origin</span><span lang="en-US" xml:lang="en-US"> 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 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c g e </span><span lang="en-US" xml:lang="en-US">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 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">; in this case the number 135 has the tone-value </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">cis</span><span lang="en-US" xml:lang="en-US">ˆ</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">7</span><span lang="en-US" xml:lang="en-US">.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§50.5. Individual Numbers</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Number 1</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Aristotle (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Metaphysics</span><span lang="en-US" xml:lang="en-US"> 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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">one</span><span lang="en-US" xml:lang="en-US">, 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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">duality</span><span lang="en-US" xml:lang="en-US"> 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” (</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> → </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">) and “limited” (</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US"> ← </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">)-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 </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> and continually slower downwards. On Plato's views on this subject, Aristotle writes (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Metaphysics</span><span lang="en-US" xml:lang="en-US"> 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</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">(limiting) </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US"> ← </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> → </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> (unlimited)</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">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 </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">, as purely theoretic and both unending, as indeed the mathematic term </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US"> and </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> allows in any case.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Plotinus seeks to clarify the relationship of the intellect to the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">One</span><span lang="en-US" xml:lang="en-US"> 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: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Plotin</span><span lang="en-US" xml:lang="en-US">, Leipzig 1907, p. 118). Here, also, the value-formal expression of the “P” is obvious: the “One” bathed in light by the </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> that is identical with it, and its “equal-tone lines,” and the “generator-tone line” with its “all-pervading” successive manifestations of the unity </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> ... 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-</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">values</span><span lang="en-US" xml:lang="en-US"> to the divine (</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US">), is found summarized in the following words by Plotinus (6</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Ennead</span><span lang="en-US" xml:lang="en-US">, 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, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Ennead</span><span lang="en-US" xml:lang="en-US">, tr. by O. Kiefer, Jena 1905, vol. I, 129).</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “Instead of the second One of Iamblichus, Proclus has a limited number of unities emanating from the primal Being: the Henads [from </span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">̔</span><span lang="el-GR" xml:lang="el-GR">έν</span><span lang="en-US" xml:lang="en-US"> = Hen = the One], which represent an intermediate transition from the primal One [</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">] to the eternally growing and finally unlimited multitude in the lower regions” (Überweg's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Geschichte der Philosophie</span><span lang="en-US" xml:lang="en-US">, I, 1926, p. 627). Compare to this idea the generator-tone line </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> ... </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">n</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">n</span><span lang="en-US" xml:lang="en-US">, which actually constructs an “intermediate transition” between the ratios in the “P” system!</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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 (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Abhandlungen</span><span lang="en-US" xml:lang="en-US">). Furthermore, see §25 and §55 of this text.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Number 2</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If we take the number </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">two</span><span lang="en-US" xml:lang="en-US"> as an essence in the sense of the polar emergence of the two principles from the harmonic unity:<br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"> <img width="250" height="169" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/ns_0.png" alt="" title="" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"> </p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">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.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Number 3</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The three as a number, and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">three-</span><span lang="en-US" xml:lang="en-US">ness as an essence, as we saw in §30, is localized at a decisive place in the “P” system:<br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="200" height="146" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/ns_1.png" alt="" title="" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Prometheus </span><span lang="en-US" xml:lang="en-US">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.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Number 4</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the symbolism of the number </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">four</span><span lang="en-US" xml:lang="en-US">, 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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">systems</span><span lang="en-US" xml:lang="en-US">, 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: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Vorlesungen über Geschichte der Mathematik</span><span lang="en-US" xml:lang="en-US">, 4</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Number 5</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">five</span><span lang="en-US" xml:lang="en-US"> as an image-concept and essence again has an entirely different character. Here, harmonic analysis must insert or underlay the 5</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> ratio, i.e. the third interval, the “gender interval” (see my </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonia Plantarum</span><span lang="en-US" xml:lang="en-US">, 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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">quinta essentia</span><span lang="en-US" xml:lang="en-US"> of the Alchemists, which was the actual generating means, only to be used correctly by a loving and humble hand, to create the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">lapis philosophorum</span><span lang="en-US" xml:lang="en-US">, the philosopher's stone. All these symbols, image-concepts, and connections, with their special characters, must have a deeper reference-point in our psyche.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Number 6</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> We once again enter an entirely different sphere when examining the symbolism of the number </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">six</span><span lang="en-US" xml:lang="en-US">. 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.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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”:<br /> <br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="485" height="306" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/ns_2.png" alt="" title="" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 455</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">China und Japan</span><span lang="en-US" xml:lang="en-US"> (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</span></p> <p style="text-align: left; margin-left: 1.5583in;" dir="ltr"><span lang="en-US" xml:lang="en-US">to the Shang-ti, to the highest,</span></p> <p style="text-align: left; margin-left: 1.5583in;" dir="ltr"><span lang="en-US" xml:lang="en-US">to the six venerable ones,</span></p> <p style="text-align: left; margin-left: 1.5583in;" dir="ltr"><span lang="en-US" xml:lang="en-US">to the mountains and waters, and</span></p> <p style="text-align: left; margin-left: 1.5583in;" dir="ltr"><span lang="en-US" xml:lang="en-US">to the hundred spirits.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">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 </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> (Eidos, impersonal deity), the six venerable ones as the senary of the primary pure chord, the mountain and water as the polarity </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US"> ← </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> → </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">∞</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">, 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. </span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Number 7</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The number </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">seven</span><span lang="en-US" xml:lang="en-US"> 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:</span></p> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">4</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">7</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">x</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">We tolerate this so-called “pure seventh” in the chord </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c e g</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">x</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> 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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">step</span><span lang="en-US" xml:lang="en-US">, however, something new and strange appears, and if we observe its position between 6 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> and 8 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US">, 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 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US">). This stopping to think, however, has a completely different character akróatically. In the seventh chord </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">-</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span lang="en-US" xml:lang="en-US">-</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">-</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">x</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> 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:<br /> <br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="500" height="93" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/ns_3.png" alt="" title="" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 456</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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” </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">or</span><span lang="en-US" xml:lang="en-US"> as the step of a transition into another world, or a summoning thereto.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> With this arrangement:</span></p> <table style="border: 0.8pt solid ; width: 100%; border-collapse: collapse; empty-cells: show; table-layout: fixed;" rules="all" cellpadding="0" border="1"> <colgroup> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> </colgroup> <tbody><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Saturday</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Sunday</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Monday</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Tuesday</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Wednesday</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Thursday</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Friday</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Saturn</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Sun</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Moon</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Mars</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Mercury</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Jupiter</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Venus</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">h</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">a</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">these fourths, put in order, yield the diatonic scale:</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c d e f g a h</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">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.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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,</span><span lang="-none-" xml:lang="-none-" class="footnote_reference">*</span><span lang="en-US" xml:lang="en-US"> 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 </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Aus der Offenbarung Johannis</span><span lang="en-US" xml:lang="en-US"> (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.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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:</span></p> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">7</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">x</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,, </span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,, </span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">4</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">4</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">7</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">x</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">as a basis for interpretation, analogous to the double series of the senarius.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§50.6. The “Duplicity of Events”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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</span></p> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> ← </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> → </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">which is in turn closely connected with the number two from the “double,” i.e. same event (1 + 1 = 2), its place is here.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If we write out a tone-series, no matter which one, we will always find the model of the overtone series:<br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="300" height="108" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/ns_4.png" alt="" title="" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">or its mirror image, the model of the undertone series:<br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="300" height="109" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/ns_5.png" alt="" title="" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c-c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> or </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c-c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US">. 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 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">-values). As I remarked earlier (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Grundriß</span><span lang="en-US" xml:lang="en-US"> 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.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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. </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">ad infinitum</span><span lang="en-US" xml:lang="en-US">) 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”: </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c-c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> or </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c-c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US">.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 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 </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> as a reference value).</span></p> <p> </p> <div> <p style="text-align: left;" dir="ltr" awml:style="Footnote Text" class="footnote_text"><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US">* Translator's note: the German name for Transylvania, literally “seven castles”</span></p> </div></div></td></tr></tbody></table><br /> </span></p></div> </body> </html></div>
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