<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>Harmonic Theories & Diagrams - The Lambdoma.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: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US"> <table style="border-collapse: collapse;" width="600" cellspacing="0" cellpadding="3" bordercolor="#000000" border="0"><tbody><tr><td width="100%" valign="top"> <div> <span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Lambdoma</span> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The fundamental harmonic diagram, which Thimus called the “Lambdoma” because it is drawn in the shape of a Greek lambda (Λ), and to which I gave the designation “partial-tone coordinates” because it is best for us to notate it in the familiar form of a coordinate grid, is (like all harmonic diagrams) not simply a mathematical-logical scheme that could be drawn with any other symbols. The form of these tone-number groups is also justified optically through the geometric-arithmetical arrangement of their contents, as is shown by the construction of the monochord, on which every tone-value in the diagram can be realized by means of the “equal-tone lines.” In pure numeric terms, this is a division canon that rationally subdivides any linear unit.</span><span class="footnote_reference" id="footnote_ref-1"><a href="#footnote_anchor-1">1</a></span><span lang="en-US" xml:lang="en-US"> Regarding the tonal content, these partial-tone coordinates provide, for the first time in the history of acoustics and music theory, a </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">system of tones</span><span lang="en-US" xml:lang="en-US">, which science has lacked before now. Since this system is based on the natural law of the overtone series, and its group-theoretical form also appears to be based in nature in other ways</span><span class="footnote_reference" id="footnote_ref-2"><a href="#footnote_anchor-2">2</a></span><span lang="en-US" xml:lang="en-US">-and since, on the other hand, all the tone-values in it correspond to forms present in our psyche, as the “monochord test”</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US"> </span><span class="footnote_reference" id="footnote_ref-3"><a href="#footnote_anchor-3">3</a></span><span lang="en-US" xml:lang="en-US"> demonstrates-this fundamental harmonic diagram contains one of those rare coincidences of the natural and the psychic, of matter and spirit, which promises to impart completely different findings from those of a merely logical (mathematical) formulation, or from a merely psychological analysis of inner forms and experiences.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “Theology in the form of mathematical figures is taught by Plato, the Pythagorean scriptures, and Philolaus,” says Proclus in his Commentary on Euclid.</span><span class="footnote_reference" id="footnote_ref-4"><a href="#footnote_anchor-4">4</a></span><span lang="en-US" xml:lang="en-US"> With this we return once again to Pythagoreanism itself. It is my firm belief that the only way to orient oneself in the great maze of Pythagorean traditions and their “authentic” and “inauthentic” theorems is to starts with the central idea of Pythagoreanism-namely with the tone-number, and the geometric-tonal configurations developing from it. I made a first attempt at this orientation in my essay on Pythagoras,</span><span class="footnote_reference" id="footnote_ref-5"><a href="#footnote_anchor-5">5</a></span><span lang="en-US" xml:lang="en-US"> and even in those initial harmonic analyses showed that many of the previously “obscurest” Pythagorean theorems could be illuminated with comparative ease; indeed, a few previously considered “foreign” and “un-Greek,” such as the doctrine of the transmigration of souls, can be derived directly from the Lambdoma. The reader will find various such analyses mentioned in this book.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Pythagoreanism had an enormous influence in all of antiquity with its double concept of number </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">and</span><span lang="en-US" xml:lang="en-US"> harmony. This effect spread both into the most problematic individual phenomena and into the great universal akróatic concepts, as we have already learned (the harmony of the spheres, for example). Pythagoras taught</span><span class="footnote_reference" id="footnote_ref-6"><a href="#footnote_anchor-6">6</a></span><span lang="en-US" xml:lang="en-US"> in two ways. With one section of his students, the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">mathematikoi</span><span lang="en-US" xml:lang="en-US">, he approached problems discursively, working through proofs; with the others, the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">akusmatikoi</span><span lang="en-US" xml:lang="en-US"> (= hearers), he taught symbolically by means of concise, easily remembered epigrams (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">akusmata</span><span lang="en-US" xml:lang="en-US">). Of the many </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">akusmata</span><span lang="en-US" xml:lang="en-US"> that remained in circulation throughout antiquity, an especially tenacious one has been preserved in different variations.</span><span class="footnote_reference" id="footnote_ref-7"><a href="#footnote_anchor-7">7</a></span><span lang="en-US" xml:lang="en-US"> Reduced to a common denominator, its translation reads: “</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Number</span><span lang="en-US" xml:lang="en-US"> is the wisest of things, and the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">name</span><span lang="en-US" xml:lang="en-US"> of things the next wisest.”</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"> See §21a.</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"> E.g. in the growth of crystal surfaces, see §21a.</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"> See §20.</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"> See Diels, op. cit. I, 317.</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"> Abhandlungen zur Ektypik harmonikaler Wertformen, Occidentverlag, Zürich (1938) 1946.</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"> From Porphyry: de vita Pythagorae sect. 36.</span></p> <p style="text-align: justify;" 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"> According to Proclus, Aelian, Theodotus, Iamblichus. H. Steinthal has devoted a special excursion to this akusma in his Geschichte der Sprachwissenschaft bei den Griechen und Römern, 2nd ed., 1890, part 1, p. 153 ff.</span></p> </div></div></td></tr></tbody></table> </span></div> <div style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US"></span></div></div> </body> </html></div>
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