<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 lang="en" xml:lang="en" xmlns="http://www.w3.org/1999/xhtml"> <head> </head> <body> <p> <table style="border-collapse: collapse;" bordercolor="#000000" cellspacing="0" cellpadding="3" width="600" border="0"> <tbody> <tr> <td valign="top" width="100%"> <div> <?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//ABISOURCE//DTD XHTML plus AWML 2.2//EN" "http://www.abisource.com/2004/xhtml-awml/xhtml-awml.mod"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:awml="http://www.abisource.com/2004/xhtml-awml/"> <head> <!-- ======================================================= --> <!-- Created by AbiWord, a free, Open Source wordprocessor. --> <!-- For more information visit http://www.abisource.com. --> <!-- ======================================================= --> <meta http-equiv="content-type" content="text/html;charset=UTF-8" /> <title>Cosmology & Cosmogony - Pythagoreanism.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 dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">The Lambdoma</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> The fundamental harmonic diagram, which Thimus called the “Lambdoma” because it is drawn in the shape of a Greek lambda (Λ), and to which I gave the designation “partial-tone coordinates” because it is best for us to notate it in the familiar form of a coordinate grid, is (like all harmonic diagrams) not simply a mathematical-logical scheme that could be drawn with any other symbols. The form of these tone-number groups is also justified optically through the geometric-arithmetical arrangement of their contents, as is shown by the construction of the monochord, on which every tone-value in the diagram can be realized by means of the “equal-tone lines.” In pure numeric terms, this is a division canon that rationally subdivides any linear unit.</span><span class="footnote_reference" id="footnote_ref-1"><a href="#footnote_anchor-1">1</a></span><span xml:lang="en-US" lang="en-US"> Regarding the tonal content, these partial-tone coordinates provide, for the first time in the history of acoustics and music theory, a </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">system of tones</span><span xml:lang="en-US" lang="en-US">, which science has lacked before now. Since this system is based on the natural law of the overtone series, and its group-theoretical form also appears to be based in nature in other ways</span><span class="footnote_reference" id="footnote_ref-2"><a href="#footnote_anchor-2">2</a></span><span xml:lang="en-US" lang="en-US">-and since, on the other hand, all the tone-values in it correspond to forms present in our psyche, as the “monochord test”</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US"> </span><span class="footnote_reference" id="footnote_ref-3"><a href="#footnote_anchor-3">3</a></span><span xml:lang="en-US" lang="en-US"> demonstrates-this fundamental harmonic diagram contains one of those rare coincidences of the natural and the psychic, of matter and spirit, which promises to impart completely different findings from those of a merely logical (mathematical) formulation, or from a merely psychological analysis of inner forms and experiences.</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> “Theology in the form of mathematical figures is taught by Plato, the Pythagorean scriptures, and Philolaus,” says Proclus in his Commentary on Euclid.</span><span class="footnote_reference" id="footnote_ref-4"><a href="#footnote_anchor-4">4</a></span><span xml:lang="en-US" lang="en-US"> With this we return once again to Pythagoreanism itself. It is my firm belief that the only way to orient oneself in the great maze of Pythagorean traditions and their “authentic” and “inauthentic” theorems is to starts with the central idea of Pythagoreanism-namely with the tone-number, and the geometric-tonal configurations developing from it. I made a first attempt at this orientation in my essay on Pythagoras,</span><span class="footnote_reference" id="footnote_ref-5"><a href="#footnote_anchor-5">5</a></span><span xml:lang="en-US" lang="en-US"> and even in those initial harmonic analyses showed that many of the previously “obscurest” Pythagorean theorems could be illuminated with comparative ease; indeed, a few previously considered “foreign” and “un-Greek,” such as the doctrine of the transmigration of souls, can be derived directly from the Lambdoma. The reader will find various such analyses mentioned in this book.</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Pythagoreanism had an enormous influence in all of antiquity with its double concept of number </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">and</span><span xml:lang="en-US" lang="en-US"> harmony. This effect spread both into the most problematic individual phenomena and into the great universal akróatic concepts, as we have already learned (the harmony of the spheres, for example). Pythagoras taught</span><span class="footnote_reference" id="footnote_ref-6"><a href="#footnote_anchor-6">6</a></span><span xml:lang="en-US" lang="en-US"> in two ways. With one section of his students, the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">mathematikoi</span><span xml:lang="en-US" lang="en-US">, he approached problems discursively, working through proofs; with the others, the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">akusmatikoi</span><span xml:lang="en-US" lang="en-US"> (= hearers), he taught symbolically by means of concise, easily remembered epigrams (</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">akusmata</span><span xml:lang="en-US" lang="en-US">). Of the many </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">akusmata</span><span xml:lang="en-US" lang="en-US"> that remained in circulation throughout antiquity, an especially tenacious one has been preserved in different variations.</span><span class="footnote_reference" id="footnote_ref-7"><a href="#footnote_anchor-7">7</a></span><span xml:lang="en-US" lang="en-US"> Reduced to a common denominator, its translation reads: “</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Number</span><span xml:lang="en-US" lang="en-US"> is the wisest of things, and the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">name</span><span xml:lang="en-US" lang="en-US"> of things the next wisest.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Plato's and Aristotle's position on akróasis</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> The position on akróasis of the two great ancient philosophers, Plato and Aristotle, is interesting to us. Today it is agreed that for the ancient Greeks, “music” (understanding this concept in its broadest meaning, i.e. “akróasis”), like poetry (which is also perceived through the medium of the ear), was not merely an art, but that they sensed the singing and the order of the cosmos in the phenomenon of tone.</span><span class="footnote_reference" id="footnote_ref-8"><a href="#footnote_anchor-8">8</a></span><span xml:lang="en-US" lang="en-US"> Physiologically, the eye was more spiritual for Plato; for Aristotle, the ear.</span><span class="footnote_reference" id="footnote_ref-9"><a href="#footnote_anchor-9">9</a></span><span xml:lang="en-US" lang="en-US"> Plato</span><span class="footnote_reference" id="footnote_ref-10"><a href="#footnote_anchor-10">10</a></span><span xml:lang="en-US" lang="en-US"> gives the eye the predicate “sun-like,” and speaks</span><span class="footnote_reference" id="footnote_ref-11"><a href="#footnote_anchor-11">11</a></span><span xml:lang="en-US" lang="en-US"> of an “eye of the spirit.” Aristotle</span><span class="footnote_reference" id="footnote_ref-12"><a href="#footnote_anchor-12">12</a></span><span xml:lang="en-US" lang="en-US"> was of the opinion that the ear is the most spiritual organ of the human senses (!). For both however, Plato and Aristotle, the high value of “music” and the musical in educational and ethical terms is self-evident. The two thinkers only differ regarding their physiological priorities in that Aristotle, especially in his </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Metaphysics</span><span xml:lang="en-US" lang="en-US">, argues against the Pythagorean speculations numbers. The examples he gives show clearly, for those who have worked over the actual Pythagorean tone-number system and analyzed its forms, that Aristotle did not know the concrete basis of Pythagorean harmonics, and therefore did not understand many of the traditional Pythagorica. Plato, on the other hand, only objects fundamentally to the “haptification” of those Pythagoreans who treated tone-number like “hair splitting,” and seeks, in the acoustic phenomena in question, an ascent to the spiritual, to the Idea. In the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Philebus</span><span xml:lang="en-US" lang="en-US">, Plato introduces the arrangement of the sound system in speech with these words: “Some god or divine man observed that the voice [</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">phonè</span><span xml:lang="en-US" lang="en-US">] was infinite...” In his later works, he returns openly to Pythagoreanism, presumably as a result of personal contact with Pythagoreans in Sicily and southern Italy. His harmonic derivation of the soul of the world in the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Timaeus</span><span xml:lang="en-US" lang="en-US"> is both famous and infamous. Since Plato was avowedly an initiate (see his Seventh Letter), he gave this derivation only in veiled language. However, for those who know, there is no doubt that the two series given by Plato, 1 2 4 8 and 1 3 9 27, joined in the correct way,</span><span class="footnote_reference" id="footnote_ref-13"><a href="#footnote_anchor-13">13</a></span><span xml:lang="en-US" lang="en-US"> produce what is still the only correct formula for the diatonic scale, and therefore a law on which not only all practical music is based, but also, for the ancients, the music of the cosmos; which is exactly what Plato wished to show. Plato's thoughts are even closer to Pythagoreanism in his enigmatic posthumous work, the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Epinomis</span><span xml:lang="en-US" lang="en-US">. This has long been assumed to be the work of his student Philippos of Opus, but arguments are now being made for its authenticity.</span><span class="footnote_reference" id="footnote_ref-14"><a href="#footnote_anchor-14">14</a></span><span xml:lang="en-US" lang="en-US"> It is possibly a transcription of one of Plato's later lectures. “If the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Epinomis</span><span xml:lang="en-US" lang="en-US"> is authentic, then its special significance lies in the fact that it represents Plato's closest approach to Pythagoreanism.”</span><span class="footnote_reference" id="footnote_ref-15"><a href="#footnote_anchor-15">15</a></span><span xml:lang="en-US" lang="en-US"> The subject of the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Epinomis</span><span xml:lang="en-US" lang="en-US"> is the question: what must mortal men learn in order to be wise? The individual observations of this work continually lead back to number. Everything unintelligent, incalculable, unordered, non-rhythmical, and inharmonic is lacking in number-ratio. Number is the gift of the divine universe. The </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">harmonic</span><span xml:lang="en-US" lang="en-US"> idea of unity rules the forms of the upper and the lower world, and also determines that in which man is absent: at death the multitude of sensory perceptions are extinguished, the dying go from the condition of multiplicity to that of unity and thus attain perfect wisdom and happiness.</span><span class="footnote_reference" id="footnote_ref-16"><a href="#footnote_anchor-16">16</a></span><span xml:lang="en-US" lang="en-US"> Looking at the “Lambdoma”</span><span class="footnote_reference" id="footnote_ref-17"><a href="#footnote_anchor-17">17</a></span><span xml:lang="en-US" lang="en-US"> or simply at the “partial-tone coordinates,” one can see the tendency of all tone-values to return to the unity of the generator-tone line, and from there to achieve the “perfect wisdom and happiness” of the </span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">0</span><span xml:lang="en-US" lang="en-US">/</span><span style="vertical-align:sub" xml:lang="en-US" lang="en-US">0</span><span xml:lang="en-US" lang="en-US">. In the series of mathematical sciences, from arithmetic to geometry and stereometry to the harmonic intervals, the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Epinomis</span><span xml:lang="en-US" lang="en-US"> gives special attention to the arithmetic, geometric, and harmonic proportions. I suggest that the reader look at Fig. 179 in this book. These three proportion types were therefore present in the original Pythagorean diagram, which may serve as further proof of the connection between Plato's later works and Pythagoreanism.</span><span class="footnote_reference" id="footnote_ref-18"><a href="#footnote_anchor-18">18</a></span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:center"><span style="font-weight:bold" xml:lang="en-US" lang="en-US">Pythagoreanism</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> In this section we will trace the influences of Pythagoreanism from ancient times and late antiquity to the Middle Ages, and in the next will show, with the aid of concrete examples, the akróatic elements in various domains in modern times, and finally, especially in poetry, right up to the present. In sections I through V of this Introduction, my primary task is to collect the information on akróasis up until now in a condensed selection, so that one can see and hear how the characteristic peculiarity of this mode of thought and inner attitude has manifested again and again in all periods. This information, together with the summary of the history of harmonics in §55, can then provide future research with the foundations of a universal history of akróasis.</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Hippocrates; Alcmaeon; Empedocles</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Pythagorean tendencies in medicine appearing with Hippocrates (around 400 B.C.) can be traced to Alcmaeon, a doctor from Croton and a student of Pythagoras. A fragment from Alcmaeon tells us:</span><span class="footnote_reference" id="footnote_ref-19"><a href="#footnote_anchor-19">19</a></span><span xml:lang="en-US" lang="en-US"> “Health depends on the correct proportions of the qualities.” The philosopher Empedocles of Agrigentum (490-430 B.C.) gives us some very concrete harmonic rules for these proportions. On the construction of bones, for example, one of his fragments reads:</span><span class="footnote_reference" id="footnote_ref-20"><a href="#footnote_anchor-20">20</a></span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">“But the earth took up lovingly, in spacious crucible,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Of her 8 parts in all, 2 from Nestis [water]</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">And from Hephaistos [fire] 4, which became glowing bones,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Thus glued by harmony into a wondrous image.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US">This is the harmonic octave proportion 8 : 4 : 2, or in string lengths on the monochord, 4 </span><span style="font-weight:bold" xml:lang="en-US" lang="en-US">c</span><span style="font-family:'Tahoma'" xml:lang="en-US" lang="en-US">,,</span><span xml:lang="en-US" lang="en-US"> : 2 </span><span style="font-weight:bold" xml:lang="en-US" lang="en-US">c</span><span style="font-family:'Tahoma'" xml:lang="en-US" lang="en-US">,</span><span xml:lang="en-US" lang="en-US"> : 1 </span><span style="font-weight:bold" xml:lang="en-US" lang="en-US">c</span><span xml:lang="en-US" lang="en-US">. These two octaves are the range of an average human voice! Desire, flesh, and blood obey the proportions 2 : 1 : 1. All beings are put together from the elements through sound proportions (</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">harmosthenta</span><span xml:lang="en-US" lang="en-US">).</span><span class="footnote_reference" id="footnote_ref-21"><a href="#footnote_anchor-21">21</a></span><span xml:lang="en-US" lang="en-US"> The following wonderful fragment from Empedocles-one of those that may have inspired Hölderlin to write his </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Empedocles</span><span xml:lang="en-US" lang="en-US">-tells how, from these primal phenomena, fire, water, and earth (which it is a great mistake to understand in our modern material sense), all form themselves, in hate and love, up to the highest ones, the gods:</span><span class="footnote_reference" id="footnote_ref-22"><a href="#footnote_anchor-22">22</a></span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">“But come, see testimony to the earlier words,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Lest they still be lacking for the making of things,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">See how the sun shines warm and bright on all sides,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">And the immortal bodies, replete with light and warmth,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">And the water, which is in all things, dark and cool, </span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">And springs from the earth, which has foundation and fixed form.</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">In hate, all this is torn asunder and made different,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">But it yearns and comes together again in love.</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">For thence comes all that was and is and will be,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Trees grow from it, men and women grow,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Beasts and birds, and fishes feeding in the water,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">And long-lived gods, foremost in power and noblesse.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">The Number Seven</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> The significance of the number seven is dominant in this early iatro-mathematics, and even more so later. The “tyranny” of this number can predominantly be traced harmonically to the norm of the seven-stepped diatonic scale existing in our psyche. The analogy of the 7 tones to the 7 planets is well known. As an iconographic element, the harmonic scale number 7 has typological significance far into the Middle Ages, and also in art history (e.g. the 7 church modes on the capitals of the 12</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">th</span><span xml:lang="en-US" lang="en-US"> century Cathedral of Cluny).</span></p> <p dir="ltr" style="text-align:justify"></p> <p dir="ltr" style="text-align:justify"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Homer</span></p> <p dir="ltr" style="text-align:left"><span xml:lang="en-US" lang="en-US"> Homer's </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Odyssey</span><span xml:lang="en-US" lang="en-US">, XIX, 456-7, reads:</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">“And as for the wound of the noble godlike Odysseus,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">They bound it up skillfully, and stayed the black blood with a song of healing.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> An ancient commentator already remarked upon “songs of healing”: “One must know that ancient medicine was founded upon singing.”</span><span class="footnote_reference" id="footnote_ref-23"><a href="#footnote_anchor-23">23</a></span><span xml:lang="en-US" lang="en-US"> This refers to staunching bleeding with the magic of tones and words, and presumably this is the oldest form of acoustic therapy: the healing song.</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Athanasius Kircher</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Athanasius Kircher</span><span class="footnote_reference" id="footnote_ref-24"><a href="#footnote_anchor-24">24</a></span><span xml:lang="en-US" lang="en-US"> made the first comparison of nerves with strings: “The nerves receive the same impression through the outer air / as strings have / when they are pulled over a smooth and resonant piece of wood / and just like these, they become excited / not only through the outer air / but also through the inner air / when it is in proportion / thus nervi and musculi are also agitated and moved / through the inwardly implanted air and spirit / which is like the conductor of the motive power in man / and this proportioned form / in that it concerns the soul, works on it all sorts of alterations / of happiness or sadness.” This passage should comfort all those sensitive to drafts! Examples of this theme of tone and healing could easily be multiplied hundredfold; they serve merely as preparatory illustrations.</span><span class="footnote_reference" id="footnote_ref-25"><a href="#footnote_anchor-25">25</a></span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Solon</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> It is interesting, and in a certain sense relevant to this topic, how widespread this theory of the number seven was in ancient times.</span><span class="footnote_reference" id="footnote_ref-26"><a href="#footnote_anchor-26">26</a></span><span xml:lang="en-US" lang="en-US"> Solon, the famous Athenian lawmaker (around 600 B.C.), wrote an elegy on the hebdomads (sevens) of human life: new teeth at age 7, puberty at 14, beard growth at 21, the greatest bodily strength at 28, the time to marry and produce children at 35, the full development of character at 42, the mature development of understanding and speech at 49 and 56 (through 2 hebdomads), a reversion at 63 (= 9 × 7), and finally, at 70, nothing further until death. This poem of Solon later influenced Aristotle and others; but his theory of the number seven received its highest intensification in the pseudo-Hippocratic writing </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">On the Hebdomads,</span><span xml:lang="en-US" lang="en-US"> which abounds with septenary analogies.</span><span class="footnote_reference" id="footnote_ref-27"><a href="#footnote_anchor-27">27</a></span><span xml:lang="en-US" lang="en-US"> The purpose of this single example of numeric symbolism is merely to show that harmonics, in this somewhat disparaged domain of “number superstition,” may provide quite definite clarifications: the first numbers in the whole number series are present in our psyche as tone proportions!</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Plato</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> On the “well-sounding and bad-sounding” in musical rhythm, Plato says:</span><span class="footnote_reference" id="footnote_ref-28"><a href="#footnote_anchor-28">28</a></span><span xml:lang="en-US" lang="en-US"> “Painting and all other works of this type are full of it, as are also weaving, both simple and artistic, and architecture and the manufacture of all other implements; also the nature of the body and all other growing things; because in all these dwells a propriety or an impropriety, and impropriety, boundlessness, and dissonance [!] are kindred to evil babble and evil-mindedness, but the opposite is kindred to the opposite, the rational and good spirit and its representation.” Truly, this is the harmonic ethos reduced to a formula!</span><span class="footnote_reference" id="footnote_ref-29"><a href="#footnote_anchor-29">29</a></span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Heraclitus; Stoics</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Heraclitus</span><span class="footnote_reference" id="footnote_ref-30"><a href="#footnote_anchor-30">30</a></span><span xml:lang="en-US" lang="en-US"> said: “Things repelling each other unify, from multiple tones emerges the most beautiful harmony, and everything emerges from strife.” The Stoics preserved these ideas in their own way: “Perfect serenity will be that” which “is firmly rooted in the knowledge of the divine and human things, through which it holds the opposites in the world as the most beautiful harmony in creation.”</span><span class="footnote_reference" id="footnote_ref-31"><a href="#footnote_anchor-31">31</a></span><span xml:lang="en-US" lang="en-US"> Compare to this the inner “opposites” of our fundamental harmonic diagram, through which the “harmony” (i.e. order) of the cosmos of the tones first emerges! In the Stoa, etymology (the study of the origins of words) was practiced vigorously; indeed, the word “etymology” supposedly originates from it. For example, they derived the word </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">phònè</span><span xml:lang="en-US" lang="en-US"> = </span><span xml:lang="el-GR" lang="el-GR">φωνή</span><span xml:lang="en-US" lang="en-US"> (voice) from </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">phòs nou</span><span xml:lang="en-US" lang="en-US"> = φω̃</span><span xml:lang="el-GR" lang="el-GR">ς</span><span xml:lang="en-US" lang="en-US"> νου̃ (light of the spirit)!</span><span class="footnote_reference" id="footnote_ref-32"><a href="#footnote_anchor-32">32</a></span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Gnosis</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Hippolytus, in his account of the heresies of the Phrygians, reports that they called reborn spiritual people by the name “Papa,” and meant all heavenly, earthly, and subterranean beings when they said: “Paue, paue-stop, stop the dissonance of the cosmos!”</span><span class="footnote_reference" id="footnote_ref-33"><a href="#footnote_anchor-33">33</a></span><span xml:lang="en-US" lang="en-US"> Again and again, the mysterious power of the call, the voice, and the name breaks through. “Simon Magus the Gnostic teaches that there is an unlimited power and calls it the origin of everything, stating the following: 'This writing of the proclamation of a voice and a name comes from the decree of the great, unlimited power.'”</span><span class="footnote_reference" id="footnote_ref-34"><a href="#footnote_anchor-34">34</a></span><span xml:lang="en-US" lang="en-US"> The verse at the beginning of Genesis: “And God made two great lights: a great light to rule the day and a lesser light to rule the night,” is interpreted by Simon thus: the sun and moon are the powers of the voice and the name, the speaking and the spoken.</span><span class="footnote_reference" id="footnote_ref-35"><a href="#footnote_anchor-35">35</a></span><span xml:lang="en-US" lang="en-US"> According to the Gnostic Marcus, the hidden, beingless deity opens its mouth and emits the word (Logos) that is its likeness. This Logos now speaks various archetypical names: “Each of the letters [of the names] has its own sign, its own character, its own pronunciation, its own form, its own appearance, and none of them knows the form of that of which it is only a letter, indeed it does not even comprehend the utterance of its neighbor; with the sound that it makes itself, it believes that it is naming everything.</span><span class="footnote_reference" id="footnote_ref-36"><a href="#footnote_anchor-36">36</a></span><span style="vertical-align:super" xml:lang="en-US" lang="en-US"> </span><span xml:lang="en-US" lang="en-US">Marcus is the “Greek classicist of letter-metaphysics.”</span><span class="footnote_reference" id="footnote_ref-37"><a href="#footnote_anchor-37">37</a></span><span xml:lang="en-US" lang="en-US"> Thus the “dieresis” of the letters goes forth into infinitude; but everything returns to the letters from whose pronunciation it emerged, back into the lost unity, and sounds therein as on the first day: one day everything will resound in a World-Amen.</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> F. Dornseiff</span><span class="footnote_reference" id="footnote_ref-38"><a href="#footnote_anchor-38">38</a></span><span xml:lang="en-US" lang="en-US"> believes that this idea “is not lacking in grandeur,” and as harmonists, we see in this an astonishing concordance with the value-formal content of the fundamental Pythagorean diagram of the “P.” The reader will be able to judge this for himself after studying the relevant chapters of this book.</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Proclus; Plotinus; Prudentius</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> The last great synthesist of the Greek philosophers, the Neoplatonist Proclus (410-485), believed (like Kepler, who quotes him extensively) that the psyche contains the “harmonically making” (</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">harmonika</span><span xml:lang="en-US" lang="en-US">) before the “harmonically made” (</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">harmosmenoi</span><span xml:lang="en-US" lang="en-US">);</span><span class="footnote_reference" id="footnote_ref-39"><a href="#footnote_anchor-39">39</a></span><span xml:lang="en-US" lang="en-US"> or as we would express it: some psychic prototypes must be within us </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">a priori</span><span xml:lang="en-US" lang="en-US">, otherwise we would not be able to tell whether a harmony is in tune or not. </span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> “Since it is thus that the whole world attains existence, look upon it: then perhaps you will hear its voice,” says Plotinus</span><span class="footnote_reference" id="footnote_ref-40"><a href="#footnote_anchor-40">40</a></span><span xml:lang="en-US" lang="en-US"> (205-270), the most important Neoplatonist. And Prudentius, the master of Christian poetry, tells his spirit: “Let loose your voice, sonorous spirit, free your noble tongue, tell of the triumphal signs of the Passion.”</span><span class="footnote_reference" id="footnote_ref-41"><a href="#footnote_anchor-41">41</a></span><span xml:lang="en-US" lang="en-US"> Both are typical akróatica, referring repeatedly to the toning magic of voice and word. “Just as the individual string is set in its proper place corresponding to the ratio of its tones, just as it appears with its ability to sound,” so the soul enters into the poetry of the world: “Every soul sings the song from its place, harmonizing both with the place and with the collective all.”</span><span class="footnote_reference" id="footnote_ref-42"><a href="#footnote_anchor-42">42</a></span><span xml:lang="en-US" lang="en-US"> “The unutterable names of the Gods, so the theurgists say, fill the whole world, and not only this world but all powers over the world. Before the souls come into being, they see that the Gods fill the whole world with themselves and their names. They long to be like them when they have come into being,” says Proclus</span><span class="footnote_reference" id="footnote_ref-43"><a href="#footnote_anchor-43">43</a></span><span xml:lang="en-US" lang="en-US"> in his Commentary on Plato's </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">First Alcibiades</span><span xml:lang="en-US" lang="en-US">.</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Kepler</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Kepler emphasizes in his </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Harmonice Mundi</span><span xml:lang="en-US" lang="en-US"> that in opposition to the “meaningless symbolizations” of Ptolemy and the old speculations about harmonies of the spheres, he has reestablished harmonics upon scientific theorems, in astronomy: “Therefore, with an improved astronomy that establishes the true and simple movements of the planets, eliminating the apparent, the things that rest upon the deception of our sense of sight, I have shown that all harmonic proportions appear in the heavens according to a true and real proportion, quantitative and measurable, not according to a mere meaningless symbolization, as well as the keys, the musical system or the scales and most of their notes, the differences of the modes, the imitation of polyphonic music by the planets, and finally the collective counterpoint of the six primary planets, which varies according to the keys and modes.”</span><span class="footnote_reference" id="footnote_ref-44"><a href="#footnote_anchor-44">44</a></span><span xml:lang="en-US" lang="en-US"> In the dedication of his work to King James of England, he supports this, and his investigations, with the words:</span><span class="footnote_reference" id="footnote_ref-45"><a href="#footnote_anchor-45">45</a></span><span xml:lang="en-US" lang="en-US"> “The reasons for thinking of this </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">patrocinium </span><span xml:lang="en-US" lang="en-US">in terms of my harmonics originated from that manifold dissonance in human affairs that is too well-known not to affect one, but which is built from true and plainly perceptible intervals, whose nature it is to soothe the hearing amidst dissonance by promising a pleasant concord to follow, and to keep it in happy expectancy.” Throughout the entire work the numbers found in the sky are harmonized, i.e. converted to tone-numbers and “scientifically” analogized:</span><span class="footnote_reference" id="footnote_ref-46"><a href="#footnote_anchor-46">46</a></span><span xml:lang="en-US" lang="en-US"> “As the simple or monophonic song known as plainsong, which alone was known in ancient times, is to the polyphonic so-called figured song, which is an invention of the last century, so the harmonies made by the individual planets are to the harmonies of the planet pairs.” “Harmonies” is no vague concept for Kepler, no “hint,” no </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">epitheton ornans</span><span xml:lang="en-US" lang="en-US"> as is has been considered by all of modern science since Whewell: instead, it conforms to tone-numbers, intervallic proportions, scale steps. It was by means of these tone-numbers, i.e. through harmonic analyses, that he discovered his Third Law!</span><span class="footnote_reference" id="footnote_ref-47"><a href="#footnote_anchor-47">47</a></span><span xml:lang="en-US" lang="en-US"> For Kepler, the formative idea lies in these harmonies, not just in the geometric figures: “Thus I came gradually, especially in recent years, upon the harmonies, tolerating very small variations in the spatial figures. The thought came to me that on the one hand, the harmonies played the role of form, applied by the final touch, whereas the figures played the role of the material that is in the number-world of the planetary bodies and is the gross extent of spatial domains. The harmonies, on the other hand, also provided eccentricities, which the spatial figures never offered.”</span><span class="footnote_reference" id="footnote_ref-48"><a href="#footnote_anchor-48">48</a></span><span xml:lang="en-US" lang="en-US"> And precisely through this </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">harmonic</span><span xml:lang="en-US" lang="en-US"> and </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">not</span><span xml:lang="en-US" lang="en-US"> “physical” observation-that the tone-numbers and tone proportions play a decisive role in the eccentricities of the planetary orbits, and that through them the Creator “applied the final touch”-Kepler found his famous Third Law. Because for him, the harmonist, the discovery of the exponent </span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">3</span><span xml:lang="en-US" lang="en-US">/</span><span style="vertical-align:sub" xml:lang="en-US" lang="en-US">2</span><span xml:lang="en-US" lang="en-US"> was the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">fifth</span><span xml:lang="en-US" lang="en-US">, the interval that generates all diatonic steps, and in the formula p</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">2</span><span xml:lang="en-US" lang="en-US"> = cr</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">3</span><span xml:lang="en-US" lang="en-US"> (p = orbital period, c = a constant, r = radius vector) he saw, behind the relationship of orbital period and orbital radius, the background of the psychic form of the “dominating” interval, the fifth! Later in this book, the reader will discover other “harmonies” in this main work of Kepler's, as well as other “Kepleriana.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Thimus</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Speaking of Kepler, here is a quote from the second great harmonist of modern times, A. von Thimus,</span><span class="footnote_reference" id="footnote_ref-49"><a href="#footnote_anchor-49">49</a></span><span xml:lang="en-US" lang="en-US"> which gives, in a certain sense, a concentrated summary of his way of thinking: “Only by means of thought, and only with the inner ear of a God-enlightened sense, can the indescribable sound of this harmony, eternally exalted above human earthly music due to its impressiveness and beauty, be surmised. Only by the Creator himself, and by the blessed spirits joined with him, is it beheld and known in its entirety. Its sounds come together from the opposition and gradation of powers concurring and unifying harmonically in a higher accord, as well as from the diversity and yet the strict order of the movement that forms itself through the action and reaction of these powers in colorful multiplicity according to the laws of an entirely musical number, faster or slower, greater or smaller, more closely bounded or reaching into the outermost distance.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Jakob Böhme</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> In Jakob Böhme's theosophy, “reverberation” or “resonance” is the sixth nature-form, concluding the evolution of the “eternal nature.” The </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Aurora</span><span xml:lang="en-US" lang="en-US">, Ch. 10, 1, reads: “The sixth source spirit in the divine power is the resonance or tone / that clangs and rings everything within / from there comes speech and difference of all things / as well as sound and song of holy angels / and therein is the formation of all colors and beauty / and the heavenly domain of joy.” In this doctrine of nature forms or “qualities”-a type of metaphysical principle-“tone” therefore means grasping and becoming conscious of things. From this comes the “speech and the difference” of all things, articulation and “formation” in the broadest sense.</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Baader; Schelling</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Franz Baader, the great commentator on Böhme and profound theosophist, part of that German idealism which continually surpasses itself intellectually, carries this idea of Böhme's further, in his own way: “All movement [e.g. of the stars] is figure writing, and these natures </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">write</span><span xml:lang="en-US" lang="en-US"> because they cannot </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">speak</span><span xml:lang="en-US" lang="en-US">.”</span><span class="footnote_reference" id="footnote_ref-50"><a href="#footnote_anchor-50">50</a></span><span xml:lang="en-US" lang="en-US"> Here, the replacement of speech with “figure writing,” mute speech, changes through movement into form! Schelling, who was strongly influenced by Böhme and Baader in his later work, writes</span><span class="footnote_reference" id="footnote_ref-51"><a href="#footnote_anchor-51">51</a></span><span xml:lang="en-US" lang="en-US"> of a “wonderful” explanation of the “resonance-principle” as of an “inner light.” However, the only modern philosopher who has declared possible the restitution of harmonics as a science is Schopenhauer.</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">Schopenhauer</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> After deep reflection upon the nature of music in the 3</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">rd</span><span xml:lang="en-US" lang="en-US"> book of his main work, </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Die Welt als Wille und Vorstellung</span><span xml:lang="en-US" lang="en-US">,</span><span class="footnote_reference" id="footnote_ref-52"><a href="#footnote_anchor-52">52</a></span><span xml:lang="en-US" lang="en-US"> Schopenhauer writes: “One could call the world embodied music just as much as embodied will.” His view is that “the apparent world, or nature, and music are two different expressions of the same thing,”</span><span class="footnote_reference" id="footnote_ref-53"><a href="#footnote_anchor-53">53</a></span><span xml:lang="en-US" lang="en-US"> and in conclusion he summarizes as follows:</span><span class="footnote_reference" id="footnote_ref-54"><a href="#footnote_anchor-54">54</a></span><span xml:lang="en-US" lang="en-US"> “If I am given the task, in this whole illustration of music, of making it clear that it declares in a most universal language the inner nature, the In-Itself of the world (which we think of, according to its most usual utterance, as the concept of Will) in a unique medium, namely simple tones, and does so with great certainty and truth; if, moreover, in my opinion and aspiration, philosophy is nothing other than a complete and correct repetition and enunciation of the nature of the world in universal concepts, since only thus is a sufficient and correct overview of this whole being possible, then those who follow me and have entered into my way of thinking will find it not so paradoxical when I say: supposing that a complete, correct, and detailed clarification of music, hence an exhaustive repetition of that which it expresses, were given in concepts, this would also immediately be a sufficient recapitulation and clarification of the world in concepts, or something equal to that, and would therefore be the true philosophy; and that consequently we can parody the saying of Leibniz quoted above [music is a mysterious arithmetical exercise of the soul = '</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Musica exercitium arithmeticae occultum nescientis se numerare animi</span><span xml:lang="en-US" lang="en-US">'], which is completely correct from a lower viewpoint, in the sense of our higher viewpoint of music, as follows: 'Music is a mysterious metaphysical exercise of the spirit, unconscious of its philosophizing' ['</span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Musica est exercitium metaphysices occultum nescientes se philosophari animi</span><span xml:lang="en-US" lang="en-US">']. For </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">scire</span><span xml:lang="en-US" lang="en-US">, to know, always means to have set down in abstract concepts. Since further, by virtue of the much-supported truth of Leibniz's pronouncement, music, disregarding its aesthetic or inner meaning and observing it merely outwardly or empirically, is nothing other than the means of grasping, directly and </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">in concreto</span><span xml:lang="en-US" lang="en-US">,</span><span style="font-style:italic" xml:lang="en-US" lang="en-US"> </span><span xml:lang="en-US" lang="en-US">greater numbers and more complicated number-ratios than we would otherwise be able to know indirectly through grasping them in concepts; thus, through unification of these two so different and yet correct views of music, we can make for ourselves a concept of the possibility of a number philosophy, like that of Pythagoras or the Chinese in the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">I Ching</span><span xml:lang="en-US" lang="en-US">, so as to then interpret, in this sense, that saying of Pythagoras which Sextus Empiricus</span><span class="footnote_reference" id="footnote_ref-55"><a href="#footnote_anchor-55">55</a></span><span xml:lang="en-US" lang="en-US"> presents: '</span><span xml:lang="el-GR" lang="el-GR">τω</span><span xml:lang="en-US" lang="en-US">̃ </span><span style="font-family:'Tahoma'" xml:lang="en-US" lang="en-US">̕</span><span xml:lang="el-GR" lang="el-GR">αριθμω</span><span xml:lang="en-US" lang="en-US"> </span><span xml:lang="el-GR" lang="el-GR">δε</span><span xml:lang="en-US" lang="en-US">̀ </span><span xml:lang="el-GR" lang="el-GR">τα</span><span xml:lang="en-US" lang="en-US">̀ </span><span xml:lang="el-GR" lang="el-GR">πάντ</span><span xml:lang="en-US" lang="en-US">' </span><span style="font-family:'Tahoma'" xml:lang="en-US" lang="en-US">̕</span><span xml:lang="el-GR" lang="el-GR">επέοικεν</span><span xml:lang="en-US" lang="en-US"> = all is assimilated through number.'” Here one must naturally abstract from the familiar concept of “music” and understand it in its broadest sense: just as broadly as Schopenhauer does.</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> We modern harmonists admittedly consider our discipline as </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">one</span><span xml:lang="en-US" lang="en-US">, but not </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">the </span><span xml:lang="en-US" lang="en-US">“true philosophy”; the possibility of a “number philosophy,” on the other hand, emerges of itself within harmonic symbolism.</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left"><span style="font-weight:bold;font-style:italic" xml:lang="en-US" lang="en-US">The Harmony of the Spheres</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> The concept and worldview of the </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">harmony of the spheres</span><span xml:lang="en-US" lang="en-US"> is common to almost all classical and pre-classical peoples. But in it we find not only the “cosmic” content of akróasis, expressed in myths and legends and interwoven in numerous forms, but even more, we find that specialized harmonic research since the earliest antiquity repeatedly sought concrete connections between the stars and the laws of tones, until Kepler and modern harmonics proved these connections. I will give a few examples so that a few of these mythological forms of the celestial world can be “heard.” Lucian writes: “Thus the lyre served Orpheus, its inventor, as the noblest instrument of his clandestine religion; but this lyre, which had seven strings, was to him a symbol indicating the harmonies of the planets. It was with this secret science that he charmed and mastered all: his concern was not the lyre he had made himself, nor what one generally thinks of as music.”</span><span class="footnote_reference" id="footnote_ref-56"><a href="#footnote_anchor-56">56</a></span><span xml:lang="en-US" lang="en-US"> In the Orphic hymns, Helios Apollo is entreated: “You who with golden lyre guide the harmonic progression of all”; Pan is addressed as “under the stars playing / the harmonies of the world on a jesting flute”; and Apollo is sung to thus: “With your bright playing you guide / the whole pole; now changing to the lowest string / now to the highest, and now, in the Dorian mode / completely harmonizing the pole”-the celestial pole, of course.</span><span class="footnote_reference" id="footnote_ref-57"><a href="#footnote_anchor-57">57</a></span><span xml:lang="en-US" lang="en-US"> Franz Cumont</span><span class="footnote_reference" id="footnote_ref-58"><a href="#footnote_anchor-58">58</a></span><span xml:lang="en-US" lang="en-US"> found depictions of the Muses on seven Roman sarcophagi from the 1</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">st</span><span xml:lang="en-US" lang="en-US"> to the 4</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">th</span><span xml:lang="en-US" lang="en-US"> centuries, and comments on them as follows: “The sister goddesses who oversee the harmony of the spheres awaken in people's hearts, through music, the passionate longing for those divine harmonies and the yearning toward the heavens. At the same time the daughters of Mnemosyne recall to consciousness the memory of the truths she knew in an earlier life. They share their wisdom with her, the pledge of immortality. Thanks to them, thought rises up to the ether, is initiated into the secrets of nature, and reaches the circle of the choir of the stars. It is relieved of the worries of this world, is transported to the world of ideas and of beauty, and cleansed of material passions. And after death the heavenly maidens summon the soul they have consecrated in their service to the celestial sphere, and allow it to take part in the blissful life of the immortals.” Of the eight heavenly spheres, Plato</span><span class="footnote_reference" id="footnote_ref-59"><a href="#footnote_anchor-59">59</a></span><span xml:lang="en-US" lang="en-US"> writes that on each circle sits “a siren, who goes round with them, hymning a single tone or note. The eight together form one harmony; and round about, at equal intervals, there is another band, three in number, each sitting upon her throne: these are the Fates, daughters of Necessity, who are clothed in white robes and have chaplets upon their heads, Lachesis and Clotho and Atropos, who accompany with their voices the harmony of the sirens-Lachesis singing of the past, Clotho of the present, Atropos of the future.”</span></p> <p dir="ltr" style="text-align:left"><span xml:lang="en-US" lang="en-US"> The singing swan also reaches the stars. Virgil</span><span class="footnote_reference" id="footnote_ref-60"><a href="#footnote_anchor-60">60</a></span><span xml:lang="en-US" lang="en-US"> tells of the legend:</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">“For it is told that Cygnus [the swan], mourning for beloved Phaeton,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Under budding poplar branches and the shadows of the sisters,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">As he sought to ease the sorrow of his love by singing songs,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Old age hastened in upon him, silver-gray with tender down,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">And flying up from earth, he pursued the stars with chanting.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> There are many more examples of ancient poetry and speculation relating to the harmony of the spheres. We will mention only a few others.</span></p> <p dir="ltr" style="text-align:left"><span xml:lang="en-US" lang="en-US"> Pindar, a contemporary of Pythagoras (6</span><span style="vertical-align:super" xml:lang="en-US" lang="en-US">th</span><span xml:lang="en-US" lang="en-US"> century B.C.), sang:</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">“Golden lyre,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Apollo plays you above in heaven,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">and you rule the dance and song</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">of the violet-ringlet Muses.</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">Below on Earth the choirmasters</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">hear these sounds,</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">and the singers follow the directions</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">when you strike up the prelude</span></p> <p dir="ltr" style="text-align:left;margin-left:0.5194in"><span xml:lang="en-US" lang="en-US">giving beat and tone to the song.”</span></p> <p dir="ltr" style="text-align:left"></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> Willamowitz-Moellendorf, from whom this translation is quoted,</span><span class="footnote_reference" id="footnote_ref-61"><a href="#footnote_anchor-61">61</a></span><span xml:lang="en-US" lang="en-US"> recognizes in this poem a poetic veiling of the harmony of the spheres. The dance, so closely bound up with music for the Greeks, is a symbol of the heavenly dance of the stars: “For what is this round dance of the stars, this regular interwoven movement of the planets in relation to the fixed stars, and the rhythmical unification and beautiful harmony of their movements, if not proof of a great primal dance?” writes Lucian.</span><span class="footnote_reference" id="footnote_ref-62"><a href="#footnote_anchor-62">62</a></span><span xml:lang="en-US" lang="en-US"> Cicero</span><span class="footnote_reference" id="footnote_ref-63"><a href="#footnote_anchor-63">63</a></span><span xml:lang="en-US" lang="en-US"> moves entirely in the akróatic realm of ideas when he writes of soul, tone, and cosmos: “Indeed, Socrates asks Xenophon from whence we have conceived the soul, if there is none in the world. And I ask, whence speech, whence the regular harmony of speech, whence song? We would have to assume that the sun converses with the moon when they approach each other, or that the world sings in harmony, as Pythagoras says. These are works of nature, Balbus, not of an artificially intrusive nature, as Zeno expresses it, but one that stimulates and drives everything through its own motions and changes.”</span></p> <p dir="ltr" style="text-align:justify"><span xml:lang="en-US" lang="en-US"> We will remark only in passing that the ancients closely studied the analogies of the elements of speech, of vowels and consonants to the tones of the planets and celestial spheres, through which they regained a connection to the most ancient cosmic meaning of sound and of the word itself; we will return to this later.</span><span class="footnote_reference" id="footnote_ref-64"><a href="#footnote_anchor-64">64</a></span><span xml:lang="en-US" lang="en-US"> But the reader will agree with me that a name, indeed an entire realm of concepts, can now no longer be avoided: </span><span style="font-style:italic" xml:lang="en-US" lang="en-US">Pythagoras</span><span xml:lang="en-US" lang="en-US">.</span></p> <p> </p> </div> <div> <p></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-1"><a href="#footnote_ref-1">1</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See §21a.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-2"><a href="#footnote_ref-2">2</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> E.g. in the growth of crystal surfaces, see §21a.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-3"><a href="#footnote_ref-3">3</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See §20.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-4"><a href="#footnote_ref-4">4</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See Diels, op. cit. I, 317.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-5"><a href="#footnote_ref-5">5</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Abhandlungen zur Ektypik harmonikaler Wertformen, Occidentverlag, Zürich (1938) 1946.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-6"><a href="#footnote_ref-6">6</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> From Porphyry: de vita Pythagorae sect. 36.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-7"><a href="#footnote_ref-7">7</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> According to Proclus, Aelian, Theodotus, Iamblichus. H. Steinthal has devoted a special excursion to this akusma in his Geschichte der Sprachwissenschaft bei den Griechen und Römern, 2nd ed., 1890, part 1, p. 153 ff.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-8"><a href="#footnote_ref-8">8</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> For readers interested in pursuing this subject further, here are three sources: 1) Ambros: Geschichte der Musik, vol. 1, 2nd ed., 1880; 2) Erich Frank, Plato und die sog. Pythagoreer, Halle 1923; 3) Werner Jäger, Paideia, vol. 1, 2nd ed., Berlin 1936, vol. II ibid., 1944.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-9"><a href="#footnote_ref-9">9</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> W. Jäger, op. cit., p. 302.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-10"><a href="#footnote_ref-10">10</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Republic 508 B.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-11"><a href="#footnote_ref-11">11</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Symposium 219 A.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-12"><a href="#footnote_ref-12">12</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> De sensu I, 437 a 5.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-13"><a href="#footnote_ref-13">13</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See p. 000, formula 355 in this book, and p. 000.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-14"><a href="#footnote_ref-14">14</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> For example, Stenzel, op. cit., p. 91 ff. and Überweg, Grundriß, part I, 1926, p. 324 ff.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-15"><a href="#footnote_ref-15">15</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Überweg, op. cit., p. 327.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-16"><a href="#footnote_ref-16">16</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Überweg, op. cit., p. 324 ff.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-17"><a href="#footnote_ref-17">17</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> For example, Figure 248 in this book.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-18"><a href="#footnote_ref-18">18</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See “Plato” in the Index for further discussion of Platonica.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-19"><a href="#footnote_ref-19">19</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Diels, op. cit., I, p. 136.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-20"><a href="#footnote_ref-20">20</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Diels, op. cit., p. 257 fragment 96-this is the metric translation from Karl Goebel, Die vorsokratische Philosophie, 1910, p. 196.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-21"><a href="#footnote_ref-21">21</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Diels, op. cit., fragment 107, p. 261.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-22"><a href="#footnote_ref-22">22</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Diels, op. cit., fragment 21, p. 233, tr. by Karl Goebel, op. cit., p. 197.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-23"><a href="#footnote_ref-23">23</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Jules Combarieu, La musique et la magie, Paris 1909, p. 81.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-24"><a href="#footnote_ref-24">24</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Extract from Musurgia Universalis, German tr., Schwäbisch Hall, 1662, p. 172.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-25"><a href="#footnote_ref-25">25</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See “Medicine” in the Index for further information.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-26"><a href="#footnote_ref-26">26</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See Franz Boll, Die Lebensalter, 1913.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-27"><a href="#footnote_ref-27">27</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See §50, and esp. p. 000.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-28"><a href="#footnote_ref-28">28</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Laws, III, 10, Phaidon-Ausgabe II, p. 169 ff.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-29"><a href="#footnote_ref-29">29</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Compare to this the illustration in Jacob Burckhardt's Griechischer Kulturgeschichte, vol. II: “Die Musik,” which reveals a deep understanding of the universal meaning of “music” for the Greeks.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-30"><a href="#footnote_ref-30">30</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Diels, op. cit., Fragment 8, vol. I, p. 79.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-31"><a href="#footnote_ref-31">31</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Clement of Alexandria: Stromata, IV, §40.3.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-32"><a href="#footnote_ref-32">32</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Steinthal, op. cit., I, 285. This etymology of the word phonè = voice is in no way “childish,” as is foolishly believed in modern times-as in, for example, Paul Barth's Stoa, 1908, p. 119. Instead, it is supported by the deepest knowledge of a metaphysical relationship between tone and light.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-33"><a href="#footnote_ref-33">33</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> H. Leisegang, Die Gnosis, 1924, p. 127.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-34"><a href="#footnote_ref-34">34</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 67, from Hippolytus.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-35"><a href="#footnote_ref-35">35</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 72.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-36"><a href="#footnote_ref-36">36</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 327, from Hippolytus.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-37"><a href="#footnote_ref-37">37</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Dornseiff, op. cit., p. 128.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-38"><a href="#footnote_ref-38">38</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 129.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-39"><a href="#footnote_ref-39">39</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Nicolai Hartmann, Des Proclus Diadochus philosophische Anfangsgründe der Mathematik, Habil. Schrift. 1909, p. 22.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-40"><a href="#footnote_ref-40">40</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Plotinus's writings tr. by Harder, vol. V (1937), p. 25: “Von der Vorsehung.”</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-41"><a href="#footnote_ref-41">41</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> From E.R. Curtius, Europäische Literatur und lateinisches Mittelalter, Bern 1948, p. 329.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-42"><a href="#footnote_ref-42">42</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Plotinus, op. cit., pp. 48-50.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-43"><a href="#footnote_ref-43">43</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> In: Die angeblichen Schriften des Dionysius Areopagita, tr. by Engelhardt, 1823, vol. II, p. 286.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-44"><a href="#footnote_ref-44">44</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Kepler, Weltharmonik (Harmonice Mundi), Book V. Appendix. Tr. by M. Caspar, 1939, p. 360.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-45"><a href="#footnote_ref-45">45</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 6.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-46"><a href="#footnote_ref-46">46</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 304.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-47"><a href="#footnote_ref-47">47</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See “Kepler” in the Index.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-48"><a href="#footnote_ref-48">48</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 349.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-49"><a href="#footnote_ref-49">49</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Die harmonikale Symbolik des Alterthums, 2 vols., Cologne, 1868-1876, vol. II, p. 190, note.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-50"><a href="#footnote_ref-50">50</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Gesamtausgabe, vol. 2 (1851), p. 396.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-51"><a href="#footnote_ref-51">51</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Philosophie der Offenbarung I, p. 121.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-52"><a href="#footnote_ref-52">52</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Insel-Ausgabe, vol. I, p. 353.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-53"><a href="#footnote_ref-53">53</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., p. 352.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-54"><a href="#footnote_ref-54">54</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Op. cit., pp. 355-356.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-55"><a href="#footnote_ref-55">55</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Adv. Math., VII.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-56"><a href="#footnote_ref-56">56</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Lucian, Sämtliche Werke, tr. by Wieland, Propyläen-Verlag, Berlin 1922, vol. 4, pp. 328-329.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-57"><a href="#footnote_ref-57">57</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Die Hymnen des Orpheus, tr. by Dietsch, 1822, pp. 23, 33, 86-87.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-58"><a href="#footnote_ref-58">58</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> In his last master-work, Recherches sur le symbolisme funéraire des romains, Paris 1942-quoted from the translation by E.R. Curtius in europ. Literatur und lateinisches Mittelalter, Bern 1948, p. 239.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:justify"><span class="footnote_reference" id="footnote_anchor-59"><a href="#footnote_ref-59">59</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Republic X, 617, tr. by Benjamin Jowett, New York: P.F. Collier, 1901.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-60"><a href="#footnote_ref-60">60</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Aeneid, X, 198, from Voß: Mythologische Briefe, vol. 2 (1794), pp. 98-99.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-61"><a href="#footnote_ref-61">61</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Die Harmonie der Sphären, Berlin 1915, p. 130.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-62"><a href="#footnote_ref-62">62</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> Von der Tanzkunst, op. cit., vol. 4, pp. 89-90.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-63"><a href="#footnote_ref-63">63</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> De Natura Deorum, tr. by Kühner 1862, Book 3, Ch. 11.</span></p> <p class="footnote_text" awml:style="Footnote Text" dir="ltr" style="text-align:left"><span class="footnote_reference" id="footnote_anchor-64"><a href="#footnote_ref-64">64</a></span><span style="font-size:12pt" xml:lang="en-US" lang="en-US"> See, among others, Franz Dornseiff: Das Alphabet in Mystik und Magie, 1922, p. 82 ff.</span></p> </div> </body> </html> </div></td></tr></tbody></table></p> </body> </html></div>
History of Harmonics
Ancient Civilizations
Haase - Hermann Bahr & Harmonic Thought
Haase - Leibniz & Kayser
Summary of History
Harmonics & Mathematics
Magic Squares
Number Distribution
Number Symbolism
Fractal Harmonies
Harmonics & Geometry
Musical Ratios
Parabola, Hyperbola, Ellipse
The Proportions
Tone Spirals & Tone Curves
Harmonics & Astronomy
Kepler
Haase on Plato's Harmony of The Spheres
Harmony of The Spheres
Kepler`s Platonic Solids
Planetary Harmonics
Cosmology & Cosmogony
Harmonic Cosmogony
I Ching
Pythagoreanism
Robert Fludd
Robert Fludd Images
Octaves & Spirals
Harmonics & Architecture
Harmonics & Architecture
Architectural Examples
Harmonics & Language
Harmonics & Language
Harmonics & Symbolism
Cosmic Harmony in Symbolism
Doctrine of Correspondences
Harmonics as Symbolism
Hermann Hesse & The Glass Bead Game
Gann & Financial Markets
Partial Tone Diagram & Square of 9
Polar Diagrams & Spiral Charts
Harmonics in Nature
Human Canon
Animals
Plants
Snow Crystals
Harmonics & Music Theory
Harmonics as a Science
Interference
Harmonic Theories & Diagrams
Thought as a Sense
Color & Partial-Tone Coordinates
The Lambdoma
Lambdoma Applications
The Monochord
Library & Resources
Hans Kayser Biography
Hans Kayser's Horoscope
Rudolf Haase Biography
Bibliography of Kayser's Works
Bookstore
Law of the Cosmos
Natural Architecture
The Textbook of Harmonics
News - May 2010
February 2008 Newsletter
Links
