<div id="LayerE10" style="visibility: visible; overflow: visible; z-index:0; left:10px; top:5px; position:absolute; "><?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> <head> <!-- ======================================================= --> <!-- Created by AbiWord, a free, Open Source wordprocessor. --> <!-- For more information visit http://www.abisource.com. --> <!-- ======================================================= --> <meta content="text/html;charset=UTF-8" http-equiv="content-type" /> <title>Harmonics & Geometry - Musical Ratios.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: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> <table style="border-collapse: collapse;" width="600" cellspacing="0" cellpadding="3" bordercolor="#000000" border="0"><tbody><tr><td width="100%" valign="top"> <div> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Image 13</span></div> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Musical Number Ratios in Geometry</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">from: </span><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Wege zur Harmonik</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> by Rudolf Stössel</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Format 3 : 5 (Major Sixth)</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Just as Johannes Kepler saw the foundations of harmonics in geometry, which describes the fundamental laws of space, so Rudolf Stössel also begins with geometry in his work, and finds many ratios of small whole numbers there, which form the building blocks for musical intervals. Regarding the illustrations he writes:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “Cicero, as he told us, found an old crumbling gravestone in the undergrowth, the carving on which indicated that it was the gravestone of Archimedes.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “Figure 4 reproduces the drawing and shows the two primal geometric forms, the square and the circle in their closest relationship, and also the symmetrical form of the isosceles triangle. If this image is rotated around its axis of symmetry, three bodies emerge, a cone, a sphere, and a cylinder. Their volumes have the exact ratio 1 : 2 : 3. These primal forms correspond to a proportion of the three smallest whole numbers, to the beginning of the number series, and to mathematics generally. But they also correspond to the tones c c′ g′, i.e. the intervals of the octave and fifth, the strongest consonances, also the primal consonances, so to speak, and to the duodecimal.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The two middle rows in the illustration mostly speak for themselves-they show the many forms in which ratios of small whole numbers emerge from the combination of the primal forms of the square and circle. Regarding the lower figure, Stössel writes:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “I now wish to combine elementary forms once again, an equilateral triangle with its inscribed circle (Fig. 11). We place the compass in the top corner and draw an arc through the lower corners. This gives us a large sector. Now we construct the inner circle of the sector, which touches its radii and the arc. I will leave it to the reader to construct its center and calculate my result. In any case, the ratio of the area of the smaller circle to the greater one, and then to the sector, is 3 : 4 : 6.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “Here we already see the fifth </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">4</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">2</span><span lang="en-US" xml:lang="en-US">, the fourth </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">3</span><span lang="en-US" xml:lang="en-US">, and the octave </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">. Now we multiply the numbers by 2, getting 6 : 8 : 12, and add the small sector with the dashed arc, then we have four areas with the ratio numbers 6 8 9 12-the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonia Perfecta Maxima</span><span lang="en-US" xml:lang="en-US">.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Peter Neubäcker</span></p> <p style="text-align: left;" align="center" dir="ltr"> <img src="../kayser2/imagelibrary/Geometry/0.jpg" alt="" /></p></div></td></tr></tbody></table></span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> </span></div></div> </body> </html></div>
History of Harmonics
Ancient Civilizations
Haase - Hermann Bahr & Harmonic Thought
Haase - Leibniz & Kayser
Summary of History
Harmonics & Mathematics
Magic Squares
Number Distribution
Number Symbolism
Fractal Harmonies
Harmonics & Geometry
Musical Ratios
Parabola, Hyperbola, Ellipse
The Proportions
Tone Spirals & Tone Curves
Harmonics & Astronomy
Kepler
Haase on Plato's Harmony of The Spheres
Harmony of The Spheres
Kepler`s Platonic Solids
Planetary Harmonics
Cosmology & Cosmogony
Harmonic Cosmogony
I Ching
Pythagoreanism
Robert Fludd
Robert Fludd Images
Octaves & Spirals
Harmonics & Architecture
Harmonics & Architecture
Architectural Examples
Harmonics & Language
Harmonics & Language
Harmonics & Symbolism
Cosmic Harmony in Symbolism
Doctrine of Correspondences
Harmonics as Symbolism
Hermann Hesse & The Glass Bead Game
Gann & Financial Markets
Partial Tone Diagram & Square of 9
Polar Diagrams & Spiral Charts
Harmonics in Nature
Human Canon
Animals
Plants
Snow Crystals
Harmonics & Music Theory
Harmonics as a Science
Interference
Harmonic Theories & Diagrams
Thought as a Sense
Color & Partial-Tone Coordinates
The Lambdoma
Lambdoma Applications
The Monochord
Library & Resources
Hans Kayser Biography
Hans Kayser's Horoscope
Rudolf Haase Biography
Bibliography of Kayser's Works
Bookstore
Law of the Cosmos
Natural Architecture
The Textbook of Harmonics
News - May 2010
February 2008 Newsletter
Links
