<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 & Architecture - Examples.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 10</span></div> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Emergence of Three Types of Architecture</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">Lehrbuch der Harmonik </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">by Hans Kayser (1891-1964)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Format 1 : 2 (Octave)</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">This image, like the right side of the preceding one, comes from a “harmonic division canon”: If one takes any rectangle and draws diagonals in it, then divides the rectangle in two through the intersection point of the diagonals and draws more diagonals in the newly created rectangles, proceeding this way at one's discretion, the intersection points always produce harmonic intervals. This division canon was discovered by Hans Kayser in medieval architecture.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Regarding Image 10, Kayser writes: “This harmonic analysis not only gives us an idea of how the ancient architects, who were doubtless familiar with this 'rational segment division,' may have proceeded in their designs. It also gives us something far more important: an inner characterization for these three styles in relation to each other. The lengthening of the monochord-the increasing of the octave space-can be observed psychically as an enlargement of the psychical configuration space. Whereas in the Egyptian aspect, the tone-lines still adhere to the 'earthly' and allow only the pyramid as a prototypical form, Romanesque architecture introduces the 'tower' and thereby allows for basilical symmetry. In the Gothic aspect, this tower is of commanding importance and pulls all other forms upwards along with it, reaching the maximal expansion, the greatest possible upsurge of the architectonic measure as a symbol for the relationship of the earthly and the human to the divine.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “Naturally, for these three types of building, one cannot measure 'after the fact' whether the pyramid's angles are exactly in tune, whether the ratio of church and roof in the Gothic aspect is exactly 'right,' and so forth. As for all 'sound-images,' it is not immediately essential to make special individual measurement analyses, but instead to produce the evolution of the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">model-</span><span lang="en-US" xml:lang="en-US">here of three architectural styles-from a unified idea.”</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: center;" dir="ltr"><img src="../kayser2/imagelibrary/architecture/0.jpg" alt="" /><span lang="en-US" xml:lang="en-US"><br /> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 11</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Proportions of the Temple of Athena at Paestum</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">Architektur und Harmonie</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> by Paul v. Naredi-Rainer</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Format 8 : 15 (Major Seventh)</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Whereas the previous image relates to ideal observations on archetypes of fundamental architectonic principles, this and the next image examine existing buildings concretely in terms of the harmonic measures at their basis. The book </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Architektur und Harmonie</span><span lang="en-US" xml:lang="en-US"> by Paul v. Naredi-Rainer contains many such investigations. Regarding this image he writes:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">“The flourishing of the Pythagorean school, approximately contemporary with the construction of the Temple of Athena, makes it seem possible that Pythagorean speculations on numbers were reflected in the Temple's architectonic expression. The numbers of the Temple's axial measurements, 40 × 96, can be derived from the numbers of the Tetraktys (1 2 3 4), which the Pythagoreans held sacred: division by 4, the measure of half the span, yields 10 as the sum and 24 as the product of the Tetraktys numbers (40 : 4 = 10, 10 = 1 + 2 + 3 + 4; 96 : 4 = 24, 24 = 1 × 2 × 3 × 4).</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “This already points to the central meaning that is attributed to the reciprocal correspondence of numbers and tones in Pythagorean thought. Hans Kayser, who revived Pythagorean harmonics in our century, performed a harmonic analysis of the Temple of Paestum, translating the rational number ratios into musical intervals. Inspired by the romantic idea of viewing architecture as 'frozen music,' Kayser perceives numbers and proportions not as 'purely intellectual quantities,' but as 'tones, intervals, and melodic types, “</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">nomoi</span><span lang="en-US" xml:lang="en-US">,” which are identical to the forms in our psyche and express them in many direct ways as mere numbers and measurements.' Amidst all the fascination emerging from this harmonic approach, the question must be asked of whether the ancient architects were as familiar with the philosophical and mathematical theories of their time as, for example, the humanistically educated architects of the Renaissance-Kayser, in any case, is convinced that 'where the architect knew nothing of harmonic proportions, he still applied them, simply because instinct and feeling led him to arrange his plans and ideas according to these proportions.'”</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/architecture/1.jpg" alt="" /><span lang="en-US" xml:lang="en-US"><br /> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 12</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Musical Intervals in the Façade of the Palazzo Rucellai</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"> Architektur und Harmonie</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> by Paul v. Naredi-Rainer</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Format 9 : 10 (Whole-Tone)</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Regarding this image, Naredi-Rainer writes: “The architectural aesthetics of the Renaissance are saturated with the Pythagorean-Platonic idea that the harmony of the cosmos is built upon musical number ratios; its first and most significant verbalization was in Alberti's tract on architecture, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">De re aedificatoria libri decem</span><span lang="en-US" xml:lang="en-US">, which was written in the mid 15</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> century but first printed in Florence in 1485. The fundamental aesthetic principle, the 'concinnitas,' is manifested according to Alberti in certain numbers and proportions, which appear most clearly in music. Thus one should obtain the entire law of the relationship from the musicians, who know these numbers best.</span></p> <div style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> “... The three-story façade of the Palazzo Rucellai, subdivided into pilasters and cornices, is now 7 axes across, but was originally conceived upon 5 axes. The measuring unit, the Florentine braccio equal to 58.3 cm, is visible in the width of the finely carved pilasters, which together with the fascia organize the wall surface into a grid-patterned arrangement and frame the 'outer panels' thus formed. However, this grid is in no way uniform, but is subtly varied through the differences in story height and axis width: the middle axis accentuated by the doorway is wider than the other axes by the ratio of a whole-tone (5 </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">5</span><span lang="en-US" xml:lang="en-US"> : 4 </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">5</span><span lang="en-US" xml:lang="en-US"> = 9 : 8). Subtracting this difference in width of </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">5</span><span lang="en-US" xml:lang="en-US"> braccia from the total width (30 </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">5</span><span lang="en-US" xml:lang="en-US"> braccia) yields the round measure of 30 braccia, whose ratio to the total height is that of a minor third (36 : 30 = 6 : 5). The complementary interval to the minor third, completing the proportion of the façade panels, is the major sixth. It determines the proportion of the central panel on the 1</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">st</span><span lang="en-US" xml:lang="en-US"> story above the ground floor, engraved with a coat of arms, the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Piano Nobile</span><span lang="en-US" xml:lang="en-US"> (9 : 5 </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">5</span><span lang="en-US" xml:lang="en-US"> = 5 : 3). The outer panels of the remaining axes have the proportions of the major seventh on the 1</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">st</span><span lang="en-US" xml:lang="en-US"> upper story, the minor seventh in the 2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">nd</span><span lang="en-US" xml:lang="en-US"> story, while the outer panels of the middle axes on the 2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">nd</span><span lang="en-US" xml:lang="en-US"> story represent the ratio of a major sixth, which is constructed in a quasi-mirror image relationship from the ratio of the height of the doorway to its width (including the doorframe). The high rectangles of the window apertures repeat the form of the outer panels; they yield the ratios of the fourth and major third.”</span></div> <p style="text-align: left;" align="center" dir="ltr"> <img src="../kayser2/imagelibrary/architecture/2.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
