<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 http-equiv="content-type" content="text/html;charset=UTF-8" /> <title>Harmonics & Symbolism - Cosmic Harmony in 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> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"></span><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 align="center"></div> <div> <div align="center"> <span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Cosmic Harmony in Symbolism and Natural Science</span></div> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">By Rudolf Haase</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Translated by Ariel Godwin</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">On the World-View of J.M. Hauer</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Lecture at the Vienna Academy of Music and the Visual Arts</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">in the Josef Matthias Hauer Circle, Nov. 22, 1979</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">The Philosophy of Josef Matthias Hauer</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Josef Matthias Hauer's music, especially his twelve-tone compositions, hardly presents a problem for human hearing; but their entire meaning can only be grasped with the help of additional clarification. The music, namely, is a component, even in a certain sense a result, of the composer's very individual philosophy, which is not easily accessible, especially since its understanding is still hindered by some of Hauer's idiosyncrasies. Thus, we will attempt in the following to provide an access to this world view of Hauer's, which may illuminate his philosophical viewpoint and facilitate further study of his related writings. Since it is impossible to give an exhaustive overview of his world of thought, we will begin with a few characteristic quotations, from which a suitable interpretation will develop.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “The eternal unchanging absolute music is the great writing and language of the universe, the harmony of the spheres, the cosmic order, the art of all arts, the science of all sciences, the culture, ... religion as a connection to eternity, the highest, holiest, most spiritual, valuable, intelligent thing in the world: truth.”</span><span class="footnote_reference" id="footnote_ref-1"><a href="#footnote_anchor-1">1</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “If there were nothing else in the world but the twelve well-tempered tones, we would still have to believe in a wise creator who had built the world on a great plan. And if there is something that lets us at least suspect this plan, it is the melody of these twelve tones.”</span><span class="footnote_reference" id="footnote_ref-2"><a href="#footnote_anchor-2">2</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “The hearing human is therefore at the outset the spiritual, as opposed to the simply speaking, seeing, comprehending. Hearing, perception (reason) is the spiritual act of man, hearing the immutable, untouchable, ungraspable, unchanging, eternal-melody. He who hears can also perceive, interpret, think, speak, comprehend, conceive.”</span><span class="footnote_reference" id="footnote_ref-3"><a href="#footnote_anchor-3">3</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From these three passages, it can be seen that Hauer believed in a cosmic order, a harmony in the sense of the ancient harmony of the spheres whose laws correspond to our earthly music. But how these laws should be specifically obtained is a second set of problems, which we will introduce with some further quotations.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “There are habits of hearing that are rooted in the natural ratios of the overtone series-in the dullness of matter. They can only be overcome by the spirit, i.e. by musical intuition (the non-sensory ... hearing of the interval, which from its highest purity gives forth the atonal melody as its own greatest creation). The spirit, in a certain sense, forms the matter, the physical ear. Only those who set the intervals free, in and of themselves, from their natural functions, who can comprehend them as the seeds of melody, only they truly hear. The ears of those who hear only sensorily, naturally, mechanically, are dead.”</span><span class="footnote_reference" id="footnote_ref-4"><a href="#footnote_anchor-4">4</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “Through equal tempering of tones and intervals, everything grossly sensual (leading-tones, differences in the sounds of the individual tones, etc.) is eliminated as much as possible ... the intuitive human, the musician, suppresses the 'world,' space- naturally only illusionary-he feels himself in the cosmos and lives in the temporal, in the progression, in organic growth.”</span><span class="footnote_reference" id="footnote_ref-5"><a href="#footnote_anchor-5">5</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “The interpretation, the rhythmic arrangement of the pure melody, of the atonal-because only this is free of all sensory things-is therefore a markedly spiritual act, the only one that exists in the scientific and artistic life of man.”</span><span class="footnote_reference" id="footnote_ref-6"><a href="#footnote_anchor-6">6</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From these further passages we see that Hauer understood the laws of music as something very specific, namely tempered tuning. This, as we know, involves a detuning of the interval proportions given by nature, where by “nature” Hauer means the overtone series, whereas today we know</span><span class="footnote_reference" id="footnote_ref-7"><a href="#footnote_anchor-7">7</a></span><span lang="en-US" xml:lang="en-US"> of another much more important natural component: the psychological constitution of human hearing. However, our purpose here is not a critical statement on Hauer's ideas, but as intensive an understanding as possible of his viewpoint. And in this sense it is appropriate to assume that Hauer was right in seeing tempering as something spiritual of material nature; because tempered tuning (by which he means equal tempering) is in fact an invention of the human mind, which thereby uses mathematics, and these mathematical values appear as laws of nature. This is therefore a pointed separation of mind and matter, connected with the idea that only so-called atonal music, based on tempered tuning, is in a position to reproduce the aforementioned cosmic order, the harmony of the spheres.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> This exposition of Hauer's ideas leads us logically to ask two questions:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 1. Where do these ideas come from?</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 2. What truth lies in this philosophy?</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">The Mythos of the Sound-Cosmos</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Josef Matthias Hauer's ideas about a connection between universal order and music are in fact ancient, and Hauer was well aware of this, even if he could not know of the wealth of research on this subject existing today. For an outline of the most important components of this sound-worldview, we refer above all to the works of Marius Schneider, in which much more material is accumulated than we can apply here.</span><span class="footnote_reference" id="footnote_ref-8"><a href="#footnote_anchor-8">8</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the advanced civilizations of antiquity, and also in modern primitive societies, the idea is widespread that the world arose from or is made from sounds. Of course this idea is not illustrated in scientific form, but in myths and/or with the help of symbols, which must be decoded. Through this, we can distinguish three types of symbolism:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 1. Mythological stories with possibly distinguishable forms of representation, especially from India;</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 2. Numeric symbolism, which is partly handed down in great systematic combinations, especially from China;</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 3. Harmonic symbolism as a particular type of numeric symbolism, applied almost exclusively by the Greeks.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> We will give examples from each of these three domains, so that the reader will at least have an approximate idea of what is being debated. For the characterization of the sound myths, we will quote a sentence from Marius Schneider's radio broadcast script, which outlines various simple notions:</span><span class="footnote_reference" id="footnote_ref-9"><a href="#footnote_anchor-9">9</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “The ancient Indian tradition tells of a sound which rose from the perfect silence of nothingness and gradually formed the world of our perceptions. In the Vedanta philosophy, which comments on the ancient Indian Rig-Veda, it is written that the gods and all the rest of the world are contained in the Word, or the rhythm of the Veda-Word, and that the Creator God, while he was forming the world, remembered this rhythm and used it as a guide for creation. The Aitareya Upanishad encloses the primordial sound in an egg, from which the world emerged. As Atman created (incubated) the world, his mouth opened like an egg and speech, fire, and the whole world came out of him. In Javanese cosmology, the creator himself is first created by a hearing entity, which announces itself with bell tones. In the Brihadaranyaka Upanishad, the creator of the world is hungry Death; he creates the world by singing a song of praise and heating himself that way. Thus emerged water, fire, and the song of speech.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> This text contains a few clear indications of the link between world creation and sound; yet still other connections are hidden behind single words, since these words are symbols and must be interpreted differently. For example, the word “water” always means a pure sound, whereas “fire” should be understood as rhythm. The old legends are mostly far more complicated; we must therefore eschew detailed description here and refer to relevant publications.</span><span class="footnote_reference" id="footnote_ref-10"><a href="#footnote_anchor-10">10</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> We will apply ourselves somewhat more closely to the second possibility, relating cosmos and music to one another-numeric symbolism. It played a large role above all in China, and we will use this culture as our example. First, however, it must be emphasized that the basis of this thought is analogical relationships, which were seen in China between all imaginable domains and were illustrated in the application of numbers, whereby an extensive system of logical correspondences emerged. This way of thinking appears even in the most ancient Chinese philosophical writings, e.g. the Hung-fan,</span><span class="footnote_reference" id="footnote_ref-11"><a href="#footnote_anchor-11">11</a></span><span lang="en-US" xml:lang="en-US"> from whose statements the following table was compiled.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"><br /> </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> </colgroup> <tbody><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"><br /> 1</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Winter</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Water</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">North</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">f</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Summer</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Fire</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">South</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">c</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">3</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Spring</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Wood</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">East</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">g</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">4</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Fall</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Metal</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">West</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">d</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">5</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Year</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Earth</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Center</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">a</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">6</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Winter</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Water</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">North</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">e</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">7</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Summer</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Fire</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">South</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">b</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">8</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Spring</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Wood</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">East</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">f#</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">9</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Fall</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Metal</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">West</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">c#</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> To be more exact, we constructed the table incorporating of later stipulations from the Yüeh-ling,</span><span class="footnote_reference" id="footnote_ref-12"><a href="#footnote_anchor-12">12</a></span><span lang="en-US" xml:lang="en-US"> so that each concepts has two numbers assigned to it, the differences of which are always 5. As can easily be seen, the seasons, elements, and directions stand in a significant interrelation of analogies; e.g. winter is cold like the North and the rainy (water), summer is hot (fire) like the South.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Typical for ancient China, furthermore, is the use of magic squares, and one of the most important of these is that of the numbers 1 to 9, with the 5 in the middle:</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> </colgroup> <tbody><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">6</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">1</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">8</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">7</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">5</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">3</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: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">9</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">4</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The sums of all the parallel and perpendicular rows, as well as the diagonals, are 15. Now the four pairs of numbers surrounding the 5 are opened, so to speak, forming a cross, which is the core of the following illustration:</span><span class="footnote_reference" id="footnote_ref-13"><a href="#footnote_anchor-13">13</a></span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">North - Winter</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">6</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">11 1</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">4 6 8</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">West - Fall 9 9 4 5 3 8 3 East - Spring</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">2 7 10</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">7 5</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">12</span></p> <p style="text-align: center;" dir="ltr"><span style="font-family: 'Courier New';" lang="en-US" xml:lang="en-US">South - Summer</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If one compares the number values of the bars of the cross with the above table of analogies, it becomes apparent that 1 and 6 mean North and Winter, 3 and 8 East and Spring, 2 and 7 South and Summer, and 4 and 9 West and Fall; a meaningful illustration has thus emerged, which symbolically contains some of the important aspects of the cosmos. This is also a kind of outline of the Ming-Tang, the Chinese calendar house from the later Sung dynasty, whose 9 spaces correspond to the numbers.</span><span class="footnote_reference" id="footnote_ref-14"><a href="#footnote_anchor-14">14</a></span><span lang="en-US" xml:lang="en-US"> There were, furthermore, 12 “stations of grace” (perhaps also windows) in the calendar house, which the Emperor included when he gave his instructions-corresponding to the 12 months. These stations were likewise symbolized by numbers, and they form a circle in our illustration; the arrangement of these numbers has a connection to that of the bars of the cross; yet this is no longer strictly feasible, since there are 12 numbers.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the Yüeh-ling, the tones of the Chinese tone system, which we have already set out in our table of analogies, are added to the aforementioned correspondences, at least up to the 9</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> tone, while the entire tone system has 12 so-called Lü (corresponding to the tones or intervals in our 12-step chromatic scale), which were indicated through a complicated proportion system that will not be discussed here. The assignment of the tones to the numbers</span><span class="footnote_reference" id="footnote_ref-15"><a href="#footnote_anchor-15">15</a></span><span lang="en-US" xml:lang="en-US"> and their completion to the number twelve allows one to see, through comparison with the table of analogies, that a sequence emerges, corresponding to those of the outer number circle in the Ming-Tang, which can be illustrated thus:</span><span class="footnote_reference" id="footnote_ref-16"><a href="#footnote_anchor-16">16</a></span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Harmonics_and_Symbolism_files/0.jpg" /></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If one follows the inscribed twelve-pointed star, beginning with 1, the sequence of the table of analogies emerges as the circle of fifths; if one proceeds in the other direction, clockwise from 1 (past 8, 3, 10, etc.) around the circle, then the familiar chromatic scale emerges.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The result of this observation is a system built of numbers, in which seasons, directions, elements, and tones are unified into a symbolic layer, forming an analogy to cosmic principles. For our viewpoint, the tones are naturally the most important, their succession being identical with the stations or windows of the calendar house, so that they have a direct relationship to the individual months. From this it follows that in Chinese ritual music, their scales (selected pentatonic scales) must each be transposed by month to a different keynote! The music therefore has a symbolic connection to cosmic principles.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The third type of symbolic representation of sound-cosmic connections is harmonic symbolism. It differs from the pure number symbolism of the previous example in that only those numbers that are the acoustic basis of music can be used for application. These are the so-called interval proportions, as they have been used since ancient times in the most varied cultures. In our own cultural domain, they were discovered by the Pythagoreans through experiments with the monochord. Pythagoras, however, played a key role above all with regard to the sound-cosmic ideas we have outlined. It was obviously he who, one might say, rationalized those mythical narratives in combination with the emergence of the scientific mode of thought in ancient Greece, and with the help of the aforementioned study of proportion, brought them into a mathematic and scientific form. Unfortunately, this can only be gathered in fragmentary form from the Pythagorean legacy; yet with Plato, who was indubitably influenced by Pythagoras, we find very clear references to what has since been handed down as “the harmony of the spheres” or “universal harmony.” And these texts of Plato are characteristic evidence for harmonic symbolism.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Most important of all is a statement in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Timaeus</span><span lang="en-US" xml:lang="en-US"> dialog</span><span class="footnote_reference" id="footnote_ref-17"><a href="#footnote_anchor-17">17</a></span><span lang="en-US" xml:lang="en-US"> describing the creation of the so-called Soul of the World. Here, Plato tells us that the world as a whole has a soul-just like all living beings-and describes the almost architectural creation of this structure by the Demiurge, Plato's lower god. The process is described in detail, but in such a way that it makes no sense to the reader. The passage is, in fact, a secret text, only understandable by readers who approach it with certain prior knowledge. This fact already points to the Pythagoreans, whose sect was indeed a kind of mystery religion with a very strict policy of secrecy. But the essential knowledge for the solution of the text is connected with the Pythagoreans; because it relates to the interval proportions, whose mathematical understanding-in no way simple-is assumed here, even though only the subtlest references to it can be found.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The result of the decoding of the text is a scale, namely the Dorian, the central scale of the Greek tone system, in the form of precisely calculated proportions. This is the oldest of the Greek modes in the most ancient diatonic scale, the invention of which is attributed to Pythagoras and which also had the highest rank in the ethical discipline of the Greeks. Plato therefore presents the soul of the world as the basis of the Greek tone system, and with this symbol he doubtless wished to express that musical laws have an important function in the cosmos. The interpretation of this text can be somewhat simplified and thus made more accessible;</span><span class="footnote_reference" id="footnote_ref-18"><a href="#footnote_anchor-18">18</a></span><span lang="en-US" xml:lang="en-US"> but the passage also conceals more complications, which lead to variations of the solution, so that even today new proposals appear relating in part to the accompanying text, which is similarly puzzling.</span><span class="footnote_reference" id="footnote_ref-19"><a href="#footnote_anchor-19">19</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Besides the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Timaeus</span><span lang="en-US" xml:lang="en-US"> scale, there are a few further examples of harmonic symbolism that we cannot discuss here. About a hundred years ago, Albert von Thimus applied himself exhaustively to this domain and wrote a two-volume work on it, entitled </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">The Harmonic Symbolism of the Ancients</span><span lang="en-US" xml:lang="en-US">.</span><span class="footnote_reference" id="footnote_ref-20"><a href="#footnote_anchor-20">20</a></span><span lang="en-US" xml:lang="en-US"> In this work he begins with the assumption that harmonic symbolism was the core of ancient wisdom, and provides extensive and dense material on this topic. Hans Kayser built his ideas upon those of Thimus, and devoted space in his “Kayserian harmonics” to speculative symbolism going much farther than Thimus. Today we know that Thimus proceeded very speculatively,</span><span class="footnote_reference" id="footnote_ref-21"><a href="#footnote_anchor-21">21</a></span><span lang="en-US" xml:lang="en-US"> and therefore we must make many exceptions to his theses; but once this is done, the true core of harmonic symbolism remains undisturbed.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Thus we conclude the outline of the three forms of sound symbolism, and can return to Josef Matthias Hauer. The reader will have already noticed the similarities existing to Hauer's quoted statements. In fact, he was very familiar with the second and third forms of sound symbolism. Interestingly, Hermann Bahr made him aware of the work of Albert von Thimus;</span><span class="footnote_reference" id="footnote_ref-22"><a href="#footnote_anchor-22">22</a></span><span lang="en-US" xml:lang="en-US"> but at this time (1918), Hauer already knew Chinese symbolism, as can be seen from the correspondence, and from this perspective he repudiated Thimus, as we shall see.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> We have now dealt with the question of the origin of Hauer's ideas about a musical order of the cosmos, and must turn to the second question: the truth of the content of these ideas.</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Fundamental Harmonic Research</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The answer to this question should be anticipated: it has been proven that true factual content lies at the basis of the sound symbolism described, because the foundations of music are also provable as general laws of nature. Therefore, a close connection to harmonic symbolism exists; because this is a matter of the interval proportions, which constitute the material of harmonic symbolism and are also present in nature. Thus a circle, so to speak, is completed. If Pythagoras, as can be reconstructed from later sources, presented the idea that identical laws existed in nature (the planetary spheres), in people, and in music, but carefully kept this wisdom secret and only passed it on in symbolic code, then today we are in a position to prove this connection scientifically.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The first and most important proof was made in the Baroque period by Johannes Kepler, the famous astronomer and mathematician, who we now know had the lifelong goal of proving the Pythagorean teachings of universal harmony, handed down only in legendary form. He finally found this proof, and published it in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonices mundi libri V</span><span lang="en-US" xml:lang="en-US">,</span><span class="footnote_reference" id="footnote_ref-23"><a href="#footnote_anchor-23">23</a></span><span lang="en-US" xml:lang="en-US"> the five books of universal harmony, in 1619. The main conclusion is shown in the following table, in which he compares the angular velocities of the planets, measured relative to the sun, at the farthest point from the sun in their orbits (aphelion) and the nearest point (perihelion), and determines that simple proportions emerge corresponding to those of musical intervals. For reasons of simplicity, we have replaced Kepler's measured values with the letters A through M in this table.</span><span class="footnote_reference" id="footnote_ref-24"><a href="#footnote_anchor-24">24</a></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> <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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Aphelion 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">A : B = 4 : 5 (major third)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Perihelion B</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">A : D = 1 : 3 (duodecimal)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">C : D = 5 : 6 (minor third)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Aphelion 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">B : C = 1 : 2 (octave)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">c</span><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">′</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Perihelion 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">C : F = 1 : 8 (3 octaves)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">c</span><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">′</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">E : F = 2 : 3 (fifth)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Aphelion 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">D : E = 5 : 24 (minor 3rd + 2 oct.)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Perihelion F</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">E : H = 5 : 12 (minor 3rd + octave)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Earth</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Aphelion 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">G : H =15 : 16 (diatonic semitone)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">b</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">c</span><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">′</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Perihelion 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">F : G = 2 : 3 (fifth)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">G : K = 3 : 5 (major sixth)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Venus</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Aphelion I</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">I : K =24 : 25 (chromatic semitone)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Perihelion K</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">H : I = 5 : 8 (minor sixth)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">c</span><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">′</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">I : M = 1 : 4 (2 octaves)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">c</span><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">′</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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Aphelion L</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">L : M = 5 : 12 (minor 3rd + octave)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">Perihelion M</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" lang="en-US" xml:lang="en-US">K : L = 3 : 5 (major sixth)</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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: left;" dir="ltr"><span style="font-size: 9pt; font-family: 'Courier New';" 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);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It should also be noted that the values given by Kepler have remained constant up to the present, despite ongoing changes in the orbits; because these are angle measurements that are hardly affected by these changes.</span><span class="footnote_reference" id="footnote_ref-25"><a href="#footnote_anchor-25">25</a></span><span lang="en-US" xml:lang="en-US"> Of course, the intervals do not really make sounds, since these are ideal proportions, but they can be transposed into the domain of hearing and be made audible there. The same goes for all other natural harmonic laws.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> At the beginning of the 20</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> century, the crystallographer Victor Goldschmidt also found proportional laws that he recognized as harmonic in the structure of crystals,</span><span class="footnote_reference" id="footnote_ref-26"><a href="#footnote_anchor-26">26</a></span><span lang="en-US" xml:lang="en-US"> and later Hans Kayser successfully expanded this fundament of nature in his “Kayserian harmonics.” Modern-day “fundamental harmonic research,” established by the Author in Vienna in 1965, stems from this tradition. Since then, numerous further harmonic laws have been proven by inductive methods, building upon empirically confirmed facts, in astronomy, crystallography, chemistry, physics, botany, zoology, and anthropology.</span><span class="footnote_reference" id="footnote_ref-27"><a href="#footnote_anchor-27">27</a></span><span lang="en-US" xml:lang="en-US"> There are similar interval proportions in significant places everywhere in nature, so that a great analogical interrelation emerges between the most varied domains and sciences.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> But an analogy also exists, as has been mentioned many times, between these natural laws and the foundations of music; because intervals also appear there, and these have been based on simple number ratios since ancient times. Here an argument often heard today presents difficulties: that these mathematical foundations are mere convention, and are based above all on acclimatization. If this were so, then the relationship between nature and music would simply be coincidental, or entirely arbitrary. However, this is not the case; because fundamental harmonic research can also provide the proof that there is a natural psychophysiological predisposition of human hearing, on the basis of which the simple proportions are favored and have therefore emerged as fundaments from the laws of nature. This predisposition of hearing will now be discussed more thoroughly.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Our investigations into the physiological predisposition of hearing are based on experiments and their interpretation by Heinrich Husmann.</span><span class="footnote_reference" id="footnote_ref-28"><a href="#footnote_anchor-28">28</a></span><span lang="en-US" xml:lang="en-US"> He points out that, based on the non-linearity of the human ear, supplementary tones emerge whose vibrations cause interferences. These are the subjective overtones, which have the same laws of structure as the objective, and-with at least two simultaneous tones-combination tones (also of higher orders). From Husmann's specifications, we have prepared the following table, from which it can be seen that the interferences are different for each interval, since in each case a different number of combination tones is identical with overtones. On the basis of the differing identities, a series of intervals emerges.</span></p> <p style="text-align: center;" dir="ltr"><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Combination tones identical with overtones:</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> </colgroup> <tbody><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Number</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Percent</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Interval</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Proportion</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">72</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">100</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Octave</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 : 2</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">42</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">58</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Fifth</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">2 : 3</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">28</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">39</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Fourth</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">3 : 4</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">24</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">33</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Major sixth</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">3 : 5</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">18</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">25</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Major third</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">4 : 5</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">14</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">19</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Minor third</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">5 : 6</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">10</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">14</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Minor sixth</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">5 : 8</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">8</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">11</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Minor seventh</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">5 : 9</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">3</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Major second</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">8 : 9</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">3</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Major seventh</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">8 : 15</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Minor second</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">15 : 16</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Tritone</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">32 : 45</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> This series also shows the degree of purity of the respective intervals, since the combination tones falling in the gaps in the overtone series can be viewed as dullings of the natural sound. The third column now shows that the sequence yielded corresponds very precisely to the differentiation of consonances and dissonances, known for centuries, and that each interval has a specific degree of sonance, which is expressed in percentages (2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">nd</span><span lang="en-US" xml:lang="en-US"> column).</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The differentiation of consonances and dissonances is thus based on a predisposition given as a law of nature by the physiology of the ear. This “mechanism” of overtones and combination tones, however, only works with the specified exactitude when the intervals are formed from proportions of whole numbers (4</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> column), since otherwise an identity of frequencies cannot be obtained. But this means that these same proportions are perfect, because only they lead to the optimal clarity of perception. This is the true reason why simple number ratios were favored in the construction of intervals in ancient times, especially among the Greeks. This predisposition-determined explanation of the interval proportions is most important for our subject, since our aim is to establish the analogy between natural harmonic laws and musical fundamentals on a scientifically confirmed basis. But further important musical foundations can also be derived from the predisposition of hearing, which we will indicate only briefly: the 12-step chromatic scale, the diatonic scale, the favoring of the major and of other central scales in tone systems. The complex reasoning can be found in other places.</span><span class="footnote_reference" id="footnote_ref-29"><a href="#footnote_anchor-29">29</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It must be mentioned, however, that for some conditions of this musical predisposition</span><span class="footnote_reference" id="footnote_ref-30"><a href="#footnote_anchor-30">30</a></span><span lang="en-US" xml:lang="en-US"> of hearing, a psychical predisposition works along with the explained physiological one. The 12 familiar interval perceptions are obviously anchored psychically; they are inborn sensory qualities, just like colors, for example. The investigation of this psychical predisposition, however, yields another important result. Namely, the fact emerges that on this psychical plane, none of the kind of exactitude is assessed as with the physiological disposition, indeed quite the contrary: the interval perceptions have margins allowing for deviations of up to 40% of the distance between two semitones to be assigned the correct perception (these distances are the logarithms of the proportions, since as is known we hear logarithmically). This “adjusted hearing” of deviations is also an important fundament of music, and is of great significance for our topic.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Those detunings of pure-tonal intervals consciously induced by people, such as the tempering of instruments requires, also belong in these domains of adjusted hearing; and here we are reminded immediately of Josef Matthias Hauer, for whom tempering is the most important fundament of music. The mathematical values of tempered tuning are not natural laws, as we have already said, but the possibility of tempering in principle is anchored in nature, in as much as adjusted hearing assures that deviations from pure-tonal tuning are corrected and do not bother us.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The mention of adjusted hearing has another result for our debate. Namely, we can assume that the ancient Chinese (as well as the Greeks) were aware of this quality of the sense of hearing. In the diagram of the 12 Lü in the form of a twelve-pointed star, the twelve tones have exactly equal distances, and this equality cannot be attained with pure-tonal proportions (or their logarithms)! Only the logarithms of equally vibrating tempered intervals have the same distance in the psychical hearing space, and the Chinese diagram can be perceived as a hearing space bent into a circle.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> But here we must warn of a related fallacy. The existence of the equidistant representation by the Chinese must in no way lead to the conclusion that they knew of the mathematical or practical method of the tempering of instruments! Their mathematics would not have made this possible at all. It was sufficient to know that deviations from the proportions (which the Chinese did know of, and applied) were tolerated by hearing in order to allow the fascinating circular diagram of the tone system to enter the plane of symbolism. (The Chinese actually were the first to calculate equal tempering mathematically, but that was some 2,000 years later: Chu Tsai-yü discovered it in 1584, a year before it was discovered in Europe by Simon Stevin.)</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Hauer's Spiritual Position</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Our investigations into sound symbolism and the harmonic laws of nature have provided the requisite foundations for a presentation of Hauer's philosophy, and we can now give an assessment of his viewpoint.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">1. Hauer is completely right in principle when he describes music as “...the harmony of the spheres, the cosmic order...”;</span><span class="footnote_reference" id="footnote_ref-31"><a href="#footnote_anchor-31">31</a></span><span lang="en-US" xml:lang="en-US"> because an interrelation of intervals with a multitude of harmonic natural laws can be factually proven. Admittedly, these analogies rest upon simple proportions of whole numbers, so that only those intervals correspond to the conditions named that also emerge from proportions of whole numbers-and Hauer dismisses this. For the clarification of his viewpoint, we will let him speak for himself again, through his comments on Thimus's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonic Symbolism of the Ancients </span><span lang="en-US" xml:lang="en-US">in a letter to Hermann Bahr</span><span class="footnote_reference" id="footnote_ref-32"><a href="#footnote_anchor-32">32</a></span><span lang="en-US" xml:lang="en-US"> from April 11, 1918:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “In the latter work, what Thimus says matters less than the directions of thought and perception of musical humanity stipulated there. The good Baron von Thimus leans more to the side of the mathematician; he has no hearing-oriented person within him to overcome. All mathematicians have long combated tempering, that can be seen plainly from Thimus's book; and all practical musicians have heard a finer play of colors in tempering than in the pure intervals with the rational numbers.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Hauer therefore sees only the mathematical side of the interval proportions and obviously undervalues their significance as a law of nature, of which Thimus admittedly also made little mention, since at that time he could only refer to Kepler, albeit very frequently.</span><span class="footnote_reference" id="footnote_ref-33"><a href="#footnote_anchor-33">33</a></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">2. If a connection of Hauer to harmonic research is impossible, given how much its modern results contradict his ideas, then it will work all the better with symbolism. Since Hauer continually places the spiritual in the foreground and pits himself against the material rooting of music in the overtone series, he ends up on the same plane as symbolism, this being also a product of the human spirit. At best, a connection with Chinese number symbolism is possible, since it is on an astonishingly high spiritual level and is also fully cosmically oriented. Hauer's entirely logical conclusion is that this type of sound symbolism corresponds at best to those musical foundations that are also created entirely spiritually, and such is the case for tempering, the mathematical principles of which are the product of human thought. This unification is highly logical-only on this plane is the true “musical” order of the cosmos not perceived, but only its symbolic image.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">3. The two viewpoints are, in fact, not so incompatible, as Hauer often pointed out polemically. And had our modern-day knowledge been at his disposal, his opinion would probably have been less extreme. Disregarding the fact that sound symbolism and natural harmonic research are only two different sides of the same issue, a unification of the two mathematical foundations of music is easily possible, namely through adjusted hearing, the spiritually-mathematically produced detuning that tempering tolerates, in the following manner:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">a) the tempered atonal music assumed by Hauer could just as well be played on pure-tonally tuned instruments, without other fundamental perceptions emerging, and</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">b) the natural harmonic laws, formed from proportions of whole numbers, if played on tempered instruments, would also yield the same sensory qualities.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Josef Matthias Hauer cannot know the modern results of the harmonic research that his philosophical ideas confirmed in principle, albeit in a different form from what he assumed. His life's work will thus be incorporated into the great framework of our knowledge of an all-embracing musical universal order; how that can happen, we have indicated in our studies. But precisely because he knew nothing of the real existence or truth of a harmonic plane of creation, all the more credit is due to Hauer for his own belief in a great spiritual connection in which music plays a leading role. To this idea he dedicated his life, with great personal sacrifice.</span></p> <p> </p> <div> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-1"><a href="#footnote_ref-1">1</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> J.M. Hauer: Kardinalsätze zur Zwölftonmusik, Plan 1945, p. 14</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-2"><a href="#footnote_ref-2">2</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> J.M. Hauer: Vom Melos zur Pauke, Vienna 1925, p. 19</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-3"><a href="#footnote_ref-3">3</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> J.M. Hauer: Deutung des Melos, Leipzig etc. 1923, p. 15</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-4"><a href="#footnote_ref-4">4</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> J.M. Hauer: Vom Wesen des Musikalischen, Leipzig etc. 1922, p. 34 ff.</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-5"><a href="#footnote_ref-5">5</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> J.M. Hauer, Deutung des Melos, op. cit., p. 22 ff.</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-6"><a href="#footnote_ref-6">6</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> ibid., p. 56</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-7"><a href="#footnote_ref-7">7</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> R. Haase: Über das disponierte Gehör, Vienna 1977</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-8"><a href="#footnote_ref-8">8</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> M. Schneider: Klangsymbolik in fremden Kulturen, Vienna 1979</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-9"><a href="#footnote_ref-9">9</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> M. Schneider: Vom tieferen Sinn des Klanges in Sprache und Gesang der Naturvölker, Köln 1953 (ms.)</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-10"><a href="#footnote_ref-10">10</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> M. Schneider: a) “Das Morgenrot in der vedischen Kosmogonie,” in: Symbolon, vol. 5, Basel 1966</span></p> <p><span lang="en-US" xml:lang="en-US"><br /> b) “Die Grundlagen der Kultsprache in der vedischen Kosmologie,” in: Sprache und Sprachverständnis, 1972</span></p> <p><span lang="en-US" xml:lang="en-US"><br /> c) “Das Schöpfungswort in der vedischen Kosmologie,” in: Musicae scientiae collectanea, Festschrift für K.G. Fellerer, Köln 1973</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-11"><a href="#footnote_ref-11">11</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> M. Granet: Das chinesische Denken, Munich 1963, p. 121</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-12"><a href="#footnote_ref-12">12</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> ibid., p. 124</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-13"><a href="#footnote_ref-13">13</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> ibid., p. 146</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-14"><a href="#footnote_ref-14">14</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> ibid., p. 132 ff.</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-15"><a href="#footnote_ref-15">15</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> ibid., pp. 170, 185.</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-16"><a href="#footnote_ref-16">16</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> At the indication of M. Schneider, the tone star was moved over by one semitone; no reason for this has yet been found. This constitutes a correction.</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-17"><a href="#footnote_ref-17">17</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> Plato: Timaeus, 35B-36B</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-18"><a href="#footnote_ref-18">18</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> R. Haase: Harmonikale Synthese, Vienna 1980, p. 25 ff.</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-19"><a href="#footnote_ref-19">19</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> In: Festschrift Rudolf Haase (ed. W. Schulze), Eisenstadt 1980:</span></p> <p><span lang="en-US" xml:lang="en-US"><br /> a) G. Jahoda: “Die Tonleiter des Timaios - Bild und Abbild”;</span></p> <p><span lang="en-US" xml:lang="en-US"><br /> b) W. Schulze: “Logos - Mesotes - Analogia. Zur Quaternität von Mathematik, Musik, Kosmologie und Staatslehre bei Platon”</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-20"><a href="#footnote_ref-20">20</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> A. v. Thimus: Die harmonikale Symbolik des Alterthums, 2 vols., Köln 1868-1876, reprint Hildesheim 1972</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-21"><a href="#footnote_ref-21">21</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> L. Spitzer: Die harmonikale Symbolik des A. Frh. v. Thimus, Vienna 1978</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-22"><a href="#footnote_ref-22">22</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> R. Haase: “Hermann Bahr und das harmonikale Denken,” in: Österreichische Musikzeitschrift, yr. 31, vol. 3, Vienna 1976 (see preceding essay)</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-23"><a href="#footnote_ref-23">23</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> J. Kepler: Harmonices mundi libri V, Linz 1619 (German translation by M. Caspar: Weltharmonik, Darmstadt 1982)</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-24"><a href="#footnote_ref-24">24</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> J. Kepler, Weltharmonik, p. 301</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-25"><a href="#footnote_ref-25">25</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> R. Haase: Der messbare Einklang. Grundzüge einer empirischen Weltharmonik, Stuttgart 1976, p. 40</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-26"><a href="#footnote_ref-26">26</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> V. Goldschmidt: Über harmonie und Complication, Berlin 1901</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-27"><a href="#footnote_ref-27">27</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> See footnote 25</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-28"><a href="#footnote_ref-28">28</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> H. Husmann: Vom Wesen der Konsonanz, Heidelberg 1953</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-29"><a href="#footnote_ref-29">29</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> See footnote 7</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-30"><a href="#footnote_ref-30">30</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> R. Haase: “Musikdisposition des Gehörs,” in: Die Psychologie des 20. Jahrhunderts, vol. 15, Zürich 1979</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-31"><a href="#footnote_ref-31">31</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> See footnote 1</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-32"><a href="#footnote_ref-32">32</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> See footnote 22</span></p> <p style="text-align: left;" class="footnote_text" awml:style="Footnote Text" dir="ltr"><span class="footnote_reference" id="footnote_anchor-33"><a href="#footnote_ref-33">33</a></span><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US"> A. v. Thimus, op. cit., vol. I, pp. 46, 49, 51, 52; vol. II, pp. 27, 43ff, 157, 165ff, 214, 266, 287ff, 302ff</span></p> </div></div></td></tr></tbody></table> </span></div> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"></span></div></div> </body> </html></div>
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