<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 & Astronomy - Planetary Harmonics.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; } --></style> </head> <body> <div> <div style="text-align: justify;" 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">§41.5. An Ektypic Example, Taken from Astronomy, for the Theorem of the Melodic (Value-Form: Horizontal Step) </span> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In my </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Hörende Mensch</span><span lang="en-US" xml:lang="en-US">, p. 191 ff., I gave a harmonic analysis, from specific viewpoints, of the distances between the planets, which when reduced by octaves produce a closed scale of major-minor character. There one can read about the results, arising from the enharmonics of two steps of this scale, significant for the speculative knowledge of a “disruption factor” connected </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a priori</span><span lang="en-US" xml:lang="en-US"> to the birth of the solar system.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> As mentioned often in this book, the modern numerical formulation of precise scientific laws and relationships is free of prejudice in itself, but in the majority of cases it obscures the morphological background of these laws more than it illuminates them. Especially typical of this are numbers in modern astronomy.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The voluminous work of the astronomer Wilhelm Kaiser attempts to regain a morphological, formal grasp of the astronomical cosmos, as was mentioned in §24a.2. In this work, especially in the second book of his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Geometrische Vorstellungen in der Astronomie</span><span lang="en-US" xml:lang="en-US"> (Selbstverlag des Verfassers, Subingen, Canton Solothurn, Switzerland, 1933), we harmonists find a wealth of previously unknown proportional relationships that merit deeper harmonic analysis.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> One of the most significant discoveries made by W. Kaiser appears to me to be what he called the “inversion spheres of the planets” (ibid., 143 ff.). If we set the average distance of Earth from the sun as 1 [astronomical] unit, then Jupiter's distance from the sun at aphelion (the point in its orbit at which it is farthest from the sun) is 5.45 = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">60</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US"> times that of Earth. W. Kaiser now produces the following table (ibid., 162) using Earth's average distance from the sun as a unit 1:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Upper limiting sphere with the radius 5.45 = J = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">60</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US"> of Jupiter's domain of movement</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> </colgroup> <tbody> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Asteroids</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">P = 3</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0.03pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">6</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">109</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">= T Saturn</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Mars</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">M = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">180</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">121</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0.03pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">20</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">= U Uranus</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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="border: 0pt solid rgb(0, 0, 0); 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 = 1</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0.03pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">60</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">119</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">4</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">= N Neptune</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Venus</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">V = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">4</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0.03pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">80</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">80</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">119</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">= W Pluto</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Mercury</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">K = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">50</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">131</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0.03pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">66</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">66</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">72</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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">= X Gaea</span></p> </td> </tr> </tbody> </table> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Radial ratios and radii of the planets' orbits (from W. Kaiser)</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Kaiser writes: “In the middle columns are the inverse ratios, which, multiplied by the value J = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">60</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US">, always yield the radius of the corresponding planetary orbit. For Saturn, for the sake of the summary, the fraction </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US"> is written instead of the more precise ratio </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">109</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span><span lang="en-US" xml:lang="en-US">; through this one finds that all these ratios have something to do with the number 11, including the bottommost </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">66</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US">, because 6 × 11 = 66. This summary also contains 11 spheres, 11 special domains in the ether, so that the number eleven is in fact a realistic symbol for the conditions existing here.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> As one can see, the reciprocal symmetry of this table hints towards the inclusion of a hitherto undiscovered trans-Neptunian planet (Gaea); its further justification, as well as the inner meaning of this peculiar mirror-image quality of the inner and outer planets, can be studied further in W. Kaiser's work.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Such pronounced reciprocities are always a sure sign for us harmonists that yet more harmonic relationships must be present-considering that the “P” system itself is reciprocal through its < 1 and > 1 sectors. If one now examines the average distances of the planets from the sun (where the Earth-sun distance = 1) by means of octave potentiation and reduction-a typical harmonic operation that Kepler used often in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">-one obtains the following values:</span><span style="color: rgb(255, 0, 0);" lang="en-US" xml:lang="en-US"> </span></p> <table style="border: 0.8pt solid ; width: 100%; border-collapse: collapse; empty-cells: show; table-layout: fixed;" rules="all" cellpadding="0" border="1"> <colgroup> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> </colgroup> <tbody> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Planets</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">av. dist. from sun</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">Octave operations</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 9pt;" lang="en-US" xml:lang="en-US">(reduction to the octave 1.000-2.000)</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="3"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">nearby tone-values</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">tem-pered</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Mercury </span><span style="font-family: 'MS Mincho';" lang="en-US" xml:lang="en-US">☿</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.39</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">0.39 → 0.78 → 1.56</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.58 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">as</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">128</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">8</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 678</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">gis</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Venus ♀</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.72</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">0.72 → 1.44</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.44 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">ges</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">36</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">25</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 526</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">ges</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">fis</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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><span style="font-family: 'MS Mincho';" lang="en-US" xml:lang="en-US">♁</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">1</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">c</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 000</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Mars ♂</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1.52</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">1.52</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.52 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">x</span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">g</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">32</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">21</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 608</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Asteroids</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">3.04</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">3.04 → 1.52</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.52 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">x</span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">g</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">32</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">21</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 608</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Jupiter </span><span style="font-family: 'MS Mincho';" lang="en-US" xml:lang="en-US">♃</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">5.2</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">5.2 → 2.6 → 1.3</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.3 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">ø</span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">f</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">13</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">10</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 378</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Saturn </span><span style="font-family: 'MS Mincho';" lang="en-US" xml:lang="en-US">♄</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">9.54</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">9.54 → 4.77 → 2.385 → 1.192</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.2 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">es</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 263</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">es</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Uranus </span><span style="font-family: 'MS Mincho';" lang="en-US" xml:lang="en-US">♅</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">19.19</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">19.19 → 9.585 → 4.7975 → 2.39875 → 1.19937</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.2 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">es</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 263</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">es</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Neptune </span><span style="font-family: 'MS Mincho';" lang="en-US" xml:lang="en-US">♆</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">30.11</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">30.11 → 15.055 → 7.5275 → 3.76375 → 1.881875</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.875 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">h</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">15</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">8</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 907</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">h</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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">Pluto </span><span style="font-family: 'MS Mincho';" lang="en-US" xml:lang="en-US">♇</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">39.6</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">39.6 → 19.8 → 9.9 → 4.95 → 2.475 → 1.2375</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.25 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">e</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">4</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0pt 0pt 0.03pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 322</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">Gaea (Kaiser) hypothetical</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">72</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); 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: 10pt;" lang="en-US" xml:lang="en-US">72 → 36 → 18 → 9 → 4.5 → 2.25 → 1.125</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1.125 </span><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">d</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">9</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">8</span></p> </td> <td style="border-style: solid; border-color: rgb(0, 0, 0); border-width: 0.5pt 0pt 0pt; padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">log 170</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span></p> </td> </tr> </tbody> </table> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Astronomy/20.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 428</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> As one can see and hear, the distances of the planets, under this harmonic analysis, approximately and with the addition of the hypothetical “Gaea,” form a definite “harmonic” minor scale </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c d es a g as h c</span><span lang="en-US" xml:lang="en-US">, with the variants </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e </span><span lang="en-US" xml:lang="en-US">and </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">ges</span><span lang="en-US" xml:lang="en-US">. The reciprocity of W. Kaiser's “inversions,” </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">11</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">6</span><span lang="en-US" xml:lang="en-US"> etc., is expressed in this scale in the equal interval ratios upwards and downwards from the “middle” </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span lang="en-US" xml:lang="en-US"> (Jupiter).</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From the octave operation applied to this analysis, as with all octave operations, one might begin to think that maybe any seven random numbers, packed into an octave, would produce seven tones and thus a scale. The former idea, the 7 tones, is definitely the case, but the latter, that this would produce a scale, is absolutely wrong, as anyone can easily find from experimenting; the starting ratios must be chosen so that they produce an approximate scale or an image similar to a scale. Admittedly, we must not forget that we have used the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">average</span><span lang="en-US" xml:lang="en-US"> distances as a measure. The distance of a planet from the sun varies during its elliptical orbit between the limits of its perihelion and aphelion. These are often significant, depending on the “eccentricity” of orbit; e.g. for Mercury:</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="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" rowspan="7"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Potentiated by octaves</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">↓</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Perihelion</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Average</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Aphelion</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.31</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.39</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.47</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.62</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.78</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">0.94</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1.24</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1.56</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1.88</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">e</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">gis-a</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">h</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="3"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">└─────────────────────┘</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="3"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">fifths</span></p> </td> </tr> </tbody> </table> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"><br /> The characteristic (tempered) value is a full fifth; in the above scale there must therefore be the “tolerance” of a fifth around the tone </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span><span lang="en-US" xml:lang="en-US">. Regardless of this, it can be no coincidence for those versed in morphology that the average distances of the planets yield a complete scale when the distance between sun and Earth is set equal to 1, approximate enough if not pure-tonal. The reader can now easily perform the monochord testing of the exact average planetary distances, reduced by octaves, or better still the perihelions and aphelions reduced by octaves, whose values can be found in any collection of mathematical formulae. In testing this he will convert the decimal values into monochord lengths, according to the formula:</span></p> <p style="text-align: left; margin-left: 0.5194in;" dir="ltr"><span lang="en-US" xml:lang="en-US">Decimal fraction</span></p> <p style="text-align: left; margin-left: 0.5194in;" dir="ltr"><span lang="en-US" xml:lang="en-US">(octave 1.00 - 2.00) : 2 = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</span><span lang="en-US" xml:lang="en-US"> : 1200 (Monochord)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">x</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> = (Decimal fraction · 1200)</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> For </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x </span><span lang="en-US" xml:lang="en-US">we obtain the string segment in millimeters on a 1200-mm monochord, for example: for the average octave-reduced distance of Mercury, 1.56:</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="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">x =</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">1.56 × 1200</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">= 936 (position on the 1200-mm monochord)</span></p> </td> </tr> </tbody> </table> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> As one can see from Table 481, this position 936 is near to the position 937.5, which represents the tone </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">25</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">32</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">fes</span><span lang="en-US" xml:lang="en-US">. We have obtained the above scale from the harmonic analysis of the average octave-reduced planetary distance, assuming W. Kaiser's “Gaea” exists; this last trans-Neptunian planet has not yet been discovered, but due to the surprising inversion symmetry, its existence is very likely-if it can be discovered at all with our current optical methods. The distance from Earth to the sun has been used as the unit for this, as is usual in astronomy textbooks.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Another planetary distance scale is given in my </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Hörende Mensch</span><span lang="en-US" xml:lang="en-US">, p. 191 ff. The average distances are analyzed there, and their logarithms on base 10 are compared with our partial-tone logarithms on base 2, while the distance from Mercury to the sun is used as a measuring unit (= 1). The familiar planets sufficed there, without the hypothesis of an undiscovered one. The juxtaposition of these two logarithmic systems, and the emergent scale, yielded a series of interesting results; the problem of Lucifer especially seemed to be surprisingly clarified, at least in cosmic terms, and to become understandable in mythological and psychic terms. This will be discussed in §53 and §54.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> To recapitulate: all these harmonic analyses of planetary distances that lead to scales or similar images begin with space (distances) and lead both to a spatial coexistence (intervals) and a temporal succession (scales). The reader who has “time” should perform further tests with other distances as units, e.g. Sun-Jupiter = 1, and play the planet-tones with their characteristic intervals (perihelion and aphelion) on the monochord, noting the locations on separate sheets of paper and attempting to hear the various possible scales.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§41.6. Kepler</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In this book, it is impossible to thoroughly discuss Kepler's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">-the work that was most important to him, and a treasure in many ways for modern harmonics. After the completion of this textbook, I published a more thorough digest of the content of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US"> in the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Schweizer Rundschau</span><span lang="en-US" xml:lang="en-US"> (Oct.-Nov. 1946, pp. 545-553).</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> What could be said in summary I attempted to say in </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Hörende Mensch</span><span lang="en-US" xml:lang="en-US">, p. 171 ff. The plea there for a new collected edition of the works of this genius has been answered in the excellent translations of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Prodromus</span><span lang="en-US" xml:lang="en-US">, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Marswerk</span><span lang="en-US" xml:lang="en-US">, and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, by Max Caspar (see the Bibliography for this chapter). These new editions, revisions, and translations give us the duty as harmonists to view Kepler's works as he would have wanted them to be seen: not as dry formulaic articles on astronomy, but as the brilliant unveiling of a speculative thinker and scholar, believing in the harmony of the universe, to whom the formulae were not ends in themselves, but merely means and tools for the proof of his great harmonic formal ideas.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Here I will remark only that Kepler, in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, attempted to find harmonies in the planetary distances. The idea that guided him in this was to coordinate the planetary orbits with the circumferences of the so-called “five regular solids,” to “box” them together so that they corresponded to the approximate distances. Kepler tried to realize this in his first work, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Prodromus</span><span lang="en-US" xml:lang="en-US"> (later reissued by him with corrective notes as </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Mysterium Cosmographicum</span><span lang="en-US" xml:lang="en-US">), and he approached it anew under a harmonic aspect in </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, but no longer found the precision he was seeking, although the morphological value of the five-solid scheme as the background of the distances of the planetary orbits from the sun still seemed correct to him. He then set about analyzing the distances of the aphelions and perihelions of the individual planets harmonically amongst themselves. Regarding the extreme distances, i.e. the greatest and smallest distances from the sun (since the planetary orbits are ellipses), for the individual planets, Kepler found harmonic intervals only for Mars and Mercury; “however, if one compares the extreme distances of the various planets to each other, the first light of harmonics dawns” (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, Book V, Ch. 4). There, in close approximation, are the ratios</span></p> <p style="text-align: center;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">4</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">27</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">20</span><span lang="en-US" xml:lang="en-US"> </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">12</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span><span lang="en-US" xml:lang="en-US"> and </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">243</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">160</span><span lang="en-US" xml:lang="en-US">.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> But this did not satisfy him either. He then sought harmonies not in the distances, but in the velocities, in the daily heliocentric movements. Here, in the extremes of the individual planets (see Caspar's translation, p. 301, as well as my </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Hörende Mensch</span><span lang="en-US" xml:lang="en-US">, p. 177), he found harmonic ratios everywhere, with such close approximation that it could not be coincidental. A further investigation of these ratios, using our familiar operation of interval potentiation, then led him to discover his famous Third Law, which he used not as an end in itself, but as foundation for the proof of the existence of his heavenly harmonics. We know this from the fact that this law was given as the eighth of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">thirteen</span><span lang="en-US" xml:lang="en-US"> fundamental theorems (Book V, Ch. 3), which are “necessary in the observation of heavenly harmonies.” Furthermore, regarding this law, he writes (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Mysterium Cosmographicum</span><span lang="en-US" xml:lang="en-US">, tr. by Caspar, 2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">nd</span><span lang="en-US" xml:lang="en-US"> ed., Ch. 21, note 8, p. 137): “An explanation for the eccentricities has been found, not from the ideas just presented, but </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">from the harmonies</span><span lang="en-US" xml:lang="en-US">.” For the reader of this book, no further proof is necessary that for Kepler, the main issue was not the (modern) formula of his Third Law: f</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> : f</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> = r</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3 </span><span lang="en-US" xml:lang="en-US">: r</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> (“the squares of the orbital periods have the same ratio as the cubes of the major axes”) but the fact that this formula should be the exact proof for the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">harmonies</span><span lang="en-US" xml:lang="en-US"> existing </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">de facto</span><span lang="en-US" xml:lang="en-US"> in the planetary movements and distances-harmonies that achieve their simplest sensory and psychical realization in music, which for Kepler is therefore an image of the heavens, i.e. of the planetary system. Why else would he devote pages in the conclusion of Book V to hundreds of these harmonic ratios, as the crowning of his work, with minute precision and love? And why else would he give, in the same chapter, all the laws of music, chords, melodies, major and minor, the scales, and finally the entire chord of creation in notes?</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> And all this, not before the harmonic derivation of his law, but after it, as proof of the great harmonic form-idea of the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, the “harmony of the world.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The question of whether Kepler's “harmonies” are still correct today can be answered absolutely in the affirmative. A comparison of Kepler's aphelions and perihelions in regard to the distances from the sun with the current values shows, with the exception of Saturn's aphelion, only very minor differences; the same is the case for the daily heliocentric movements, since these fluctuate around average values, despite the “perturbations” that have since become known. Uranus and Neptune also show Kepler's harmonic ratios, and looking through the abovementioned work of W. Kaiser, one discovers so many further typical harmonic ratios that one will find Kepler's “</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice</span><span lang="en-US" xml:lang="en-US">” proven once again, under aspects of which Kepler himself knew nothing. And as for the “failure” of his polyhedron theory, i.e. the theoretically possible existence of only 6 planets through the existence of only 5 regular solids, this is only connected with the actual </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">harmonic</span><span lang="en-US" xml:lang="en-US"> ratios insofar as he derived musical intervals from the polyhedron ratios in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, and there found the theoretical anchor for the harmonies suspected to exist in the planetary system. These are present above all in the planetary movements; and Kepler would have been the first to abandon the polyhedron theory, and would have concentrated completely on harmonics, whose laws he would have proven for the other planets with equal delight and enthusiasm, as he did for the planets known to exist during his time. Kepler's harmonics are still just as valid today as they ever were-for those with ears to hear them. Indeed, we can construct them much further and more broadly with current harmonic resources, especially in terms of the symbolic-spiritual side; the “P” system was unknown to Kepler.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> With this, however, Kepler's significance for modern harmonics is in no way exhausted. In his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, especially in Book IV, there is a series of fundamental and exceptionally important observations for the foundation of harmonic starting positions (if I may thus express myself); in any case, the entire work is saturated with a wealth of principial harmonic observations, examples, ektypic references, etc. We must fittingly and justly view Kepler's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US"> and A. von Thimus's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Symbolik</span><span lang="en-US" xml:lang="en-US"> as the two great harmonic cornerstones of modern times, the exhaustive study of which should be the first and foremost duty of every harmonist. In Kepler's other works there is also a great deal of important material for harmonics, for example his commentary on the harmonics of Claudius Ptolemy, many letters to and from Kepler, and so forth.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§41.7. Bibliography</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Besides the passages cited in the text in my </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Hörende Mensch</span><span lang="en-US" xml:lang="en-US">, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Grundriß</span><span lang="en-US" xml:lang="en-US">, and articles in the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Schweizer Rundschau</span><span lang="en-US" xml:lang="en-US">: the works of astronomer W. Kaiser, and those of the gestalt theorist Christian von Ehrenfels. Regarding </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Kepler</span><span lang="en-US" xml:lang="en-US">: the great new critical collected edition (ed. M. Caspar et al., O. Beck-Verlag Munich, starting 1938), of which volumes 1, 2, 3, 4, and 6 have been released as of now (1944); and M. Caspar's translation of the work on Mars (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Neue Astronomie</span><span lang="en-US" xml:lang="en-US">, ibid. 1929), </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Mysterium Cosmographicum</span><span lang="en-US" xml:lang="en-US"> (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Das Weltgeheimnis</span><span lang="en-US" xml:lang="en-US">, Munich, Oldenburg 1936), and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US"> (Munich, Beck 1929). Moreover, Caspar's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Johannes Kepler in seinen Briefen</span><span lang="en-US" xml:lang="en-US"> (Munich, Oldenbourg 1930) and the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Biographia Kepleriana</span><span lang="en-US" xml:lang="en-US"> (Munich, Beck 1936). Other writings on Kepler and others: Apelt, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Reformation der Sternkunde</span><span lang="en-US" xml:lang="en-US"> (Jena 1852); L. Günther, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Die Mechanik des Weltalls</span><span lang="en-US" xml:lang="en-US"> (Leipzig 1909); and the standard translation (with introduction and commentary) of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US"> by W. Harburger (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Johannes Keplers Kosmische Harmonie</span><span lang="en-US" xml:lang="en-US">, Dombücherei, Inselverlag, Leipzig 1925), in which the “table of geometric-musical-astrological-astronomical correspondences” is noteworthy. </span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> E. Britt's booklet, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Tonleitern und Sternskalen</span><span lang="en-US" xml:lang="en-US">, tr. by Felix Weingartner (Leipzig 1927) is mentioned here only as a curiosity, and as an example of how one should not engage in such investigations. In harmonics we are used to somewhat more thoroughness and exactitude; above all one should never use the tempered system as a basis here, only pure-tonal values and ratios. Viktor Goldschmidt, in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Über Harmonie im Weltraum</span><span lang="en-US" xml:lang="en-US"> (Ostwalds Annalen der Naturphilosophie V, p. 51 ff.), investigates the planetary distances, among other things, by means of his “law of complication.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"><br /> </span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§45.2. Ektypics</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> One of the most important applications and results of the interval power series is “enharmonics.” With its theorem foundations, it offers a precise representation, and we will discuss it by itself in §48. Here we will only give an example for the I-powers, namely using Kepler's Third Law, whose harmonics were discussed in §41.6. The most important interval power, the octave operation, is discussed in §41.5 (analysis of the planetary distances). Kepler performed the same operation in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, e.g. Book V, Ch. 3, in which the following table appears:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">Orbital periods in Earth days:</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> </colgroup> <tbody> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" rowspan="6"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">octave</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">reduction</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">(halving)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">↓</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">(for Saturn and Jupiter)</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">Saturn</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">Jupiter</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">Mars</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">Earth</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">Venus</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">Mercury</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" rowspan="6"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">octave</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">potentiation</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">(doubling)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">↓</span></p> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">(for Venus and Mercury)</span></p> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">10759 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">12</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">5379 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">36</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">4332 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">37</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">87 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">58</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">2689 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">48</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">2166 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">19</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">224 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">42</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">175 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">56</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1344 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">54</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">1083 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">10</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">686 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">59</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">365 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">15</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">449 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">24</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">351 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">52</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> </tr> <tr> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">672 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">27</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">541 </span><span style="font-size: 10pt; vertical-align: super;" lang="en-US" xml:lang="en-US">35</span><span style="font-size: 10pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 10pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">60</span></p> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> <td style="border: 0pt solid rgb(0, 0, 0); padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td> </tr> </tbody> </table> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 443</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> An explanation of this table and its significance in </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi </span><span lang="en-US" xml:lang="en-US">would take too long here. It represents only a stage along the path Kepler was on: the path to the proof of precise harmonies in the law of orbits. I cite it here only to show that Kepler consciously used interval powers, in this case octave operations, to find certain numeric proportions in the orbital periods of the planets.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If one is aware of this, and has pursued the many other typical harmonic analyses in </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, then one will see the mathematical result of this work by Kepler, namely his famous Third Law, through entirely different eyes. This law states that the squares of the orbital periods are proportional to the cubes of the major axes. This means, in Kepler's harmonic mode of thought, that the temporal primal intervals of the orbital periods must be raised to the second intervallic power, and the spatial primal intervals of the major axes must be raised to the third intervallic power. This means a power ratio of 2 : 3, i.e. the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">fifth</span><span lang="en-US" xml:lang="en-US">, the most important interval after the octave! And if any doubt remains about the harmonic basis of these operations, Kepler dispels it himself. In Book V, Ch. 3, Point 11, he clarifies his Third Law as follows:</span></p> <p style="text-align: justify; margin-right: 0.416in; margin-left: 0.5194in;" dir="ltr"><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">“Suppose the orbital periods of two planets are 27 and 8; their median daily movements have the ratio 8 : 27; the radii of their orbits thus have the ratio 9 : 4. The cube root of 27 is 3; that of 8 is 2. The squares of these cube roots, 3 and 2, are 9 and 4. Now suppose that the apparent movement of one planet at aphelion is 2, the other at perihelion 33 </span><span style="font-size: 11pt; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 11pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">. The median proportions between the median movements 8 and 27 and the apparent ones are 4 and 30. If the mean 4 yields 9 for the median distances of the planets, then the median movement 8 yields the aphelion distance 18, corresponding to the apparent movement 2. And if the other mean 30 also yields 4 for the median distance between the two planets, then its median movement 27 yields its perihelion distance 3 </span><span style="font-size: 11pt; vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 11pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">. I claim, then, that the aphelion distance of the one planet has a ratio of 18 : 3 </span><span style="font-size: 11pt; vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">/</span><span style="font-size: 11pt; vertical-align: sub;" lang="en-US" xml:lang="en-US">5</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US"> to the perihelion distance of the other planet. Thus, the extreme distances and the medians, i.e. the eccentricities, are automatically yielded when the </span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">harmonies</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US"> between the extreme movements of the two planets are arranged and have their orbital periods prescribed.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Here, and in many other passages, the harmonic origin of Kepler's Third Law is clearly shown. The very meritorious translator of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice Mundi</span><span lang="en-US" xml:lang="en-US">, Max Caspar, writes in his notes (p. 386) on Book V, Ch. 3, Point 8: “Kepler, in the previous section, says not a word about the physical considerations that led him to the discovery of his Third Law, as he does with the discovery of his first two laws.” This is completely correct, but not important with regard to the actual theoretical backgrounds from which the Third Law was finally obtained. These backgrounds were harmonic in nature, not physical. If one wishes to dismiss all these harmonic thought-processes as “aesthetic”-if one is not indeed babbling from the outset about simple “proof”-then one is completely overlooking the exact, numerical side of the harmonic theorems and approaches. These exempt all harmonic thought processes from the merely aesthetic, and connect them to all objective research methods; because indeed, all harmonic ratios can be measured and counted. In harmonics, then, the “aesthetic”-if that is what we wish to call the forms apperceptible via tone-perception in our psyche-is thus objectified by means of the tone-</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">numbers</span><span lang="en-US" xml:lang="en-US">. On the other hand, however, we can evaluate and perceive these numbers psychically. This is also no longer “aesthetic,” instead something much more authoritative: harmonics. It was exactly this reciprocal foundation, this intensification of the objective through the subjective and vice versa, which so impressed Kepler in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonice</span><span lang="en-US" xml:lang="en-US">, and which casts an equally impressive spell over anyone who studies harmonics.</span></p></div></td></tr></tbody></table></span></div> <p style="text-align: left;" dir="ltr"></p> </div> </body> </html></div>
History of Harmonics
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Haase - Hermann Bahr & Harmonic Thought
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Summary of History
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Magic Squares
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Parabola, Hyperbola, Ellipse
The Proportions
Tone Spirals & Tone Curves
Harmonics & Astronomy
Kepler
Haase on Plato's Harmony of The Spheres
Harmony of The Spheres
Kepler`s Platonic Solids
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Cosmic Harmony in Symbolism
Doctrine of Correspondences
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Partial Tone Diagram & Square of 9
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Harmonics in Nature
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