<div id="LayerE10" style="visibility: visible; overflow: visible; z-index:0; left:10px; top:5px; position:absolute; "><?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> <head> <!-- ======================================================= --> <!-- Created by AbiWord, a free, Open Source wordprocessor. --> <!-- For more information visit http://www.abisource.com. --> <!-- ======================================================= --> <meta http-equiv="content-type" content="text/html;charset=UTF-8" /> <title>Harmonic Theories & Diagrams - Lambdoma Applications.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> <span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Image 14</span> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Lambdoma as a Model</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Pattern developed by Albert von Thimus (1806-1878)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Model drafted and constructed by Peter Neubäcker</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">In 1868, Baron Albert von Thimus published his two-volume book </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Die harmonikale Symbolik des Alterthums</span><span lang="en-US" xml:lang="en-US">, in which he develops, among other things, the Lambdoma-he considered it to be the rediscovery of a Pythagorean scheme, but it was more likely a new construction in the Pythagorean spirit. From ancient times we only know of the Lambdoma's basic form, from which it got its name: when all whole numbers expressed as fractions, and their reciprocals, are arranged in two basal series in the form of the Greek letter </span><span lang="el-GR" xml:lang="el-GR">Λ</span><span lang="en-US" xml:lang="en-US">, this image emerges:</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Theories/20.jpg" /></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> These basal series simultaneously represent the overtone series and its reflection, the undertone series. If the space between them is filled out in terms of the overtone series for each undertone, or vice versa, the result is the complete diagram, which can be continued in both directions without limit. Image 14 is from Kayser's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Lehrbuch der Harmonik</span><span lang="en-US" xml:lang="en-US">, and goes up to index 16; the model exhibited goes up to index 32.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> This diagram exhibits various peculiarities which are highly interesting both musically and symbolically. One of these peculiarities is that all fractions that have the same tone-value are connected by a straight line, and all these lines meet at a point which is actually outside the diagram: the point 0/0. These lines are called equal-tone lines; the string in the model represents one of these equal-tone lines in each position. If the string is positioned, for example, at the value 2/3, it can be seen also to pass through the values 4/6, 6/9, 8/12, etc., which all produce a fifth. At the same time the bridge on the monochord marks off precisely these proportions, so that the ratio 2/3 is audible as a fifth (the upper part of the string represents the interval in each case). The second string represents the base tone, always remaining the same. In this manner all numeric proportions can be made audible in the model.</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 alt="" src="../kayser2/imagelibrary/Theories/21.jpg" /><span lang="en-US" xml:lang="en-US"><br /> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 15</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Spatial Logarithmic Illustration of the Lambdoma</span></p> <p style="text-align: center;" dir="ltr"><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">Displayed at the exhibition “Phenomena” in Zurich, 1984</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Photo by Dieter Trüstedt</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The structures of the Lambdoma can be further clarified by illustrating the values that exist in each field in the third dimension. For example, the values 1/1, 2/1, 3/1, and 4/1 represent the single, doubled, tripled, and quadrupled string lengths of the monochord, which can then be built in a model as column heights reaching upwards.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> But the model thus emerging does not correspond to the auditory experience, since for the ear the differences in tones become continually smaller toward the higher numbers; in other words, the hearing of the intervals takes place logarithmically. The model shown here takes these conditions into account.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The outline of this model is quadratic, like the Lambdoma itself, and the diagonals of the Lambdoma run directly through the middle of the image at a constant height. Upon these diagonals lie the values 1/1, 2/2, 3/3, etc., i.e. all values that are identical to the base tone. This line is called the “generator-tone line.” Starting from it, all tone-values go upwards and downwards as logarithmic curves.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The Swiss harmonist Rudolf Stössel studied the three-dimensional illustration of the Lambdoma from various points of view and hearing, built several models, and published the results in his paper </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Modelle zur Veranschaulichung der Ober- und Untertonreihe</span><span lang="en-US" xml:lang="en-US">. At the 1984 exhibition “Phenomena” in Zurich, an especially large version of this model was displayed, built specially for the exhibition; this model is shown here.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Theories/22.jpg" /><span lang="en-US" xml:lang="en-US"><br /> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 16</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Tones of the Lambdoma Represented as Tubular Bells</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">by Dieter and Ulrike Trüstedt</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Displayed at the exhibition “Phenomena” in Zurich, 1984</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Photo by Dieter Trüstedt</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The musician and physicist Dieter Trüstedt, from Munich, explored several possibilities of using the structures of the Lambdoma as a musical instrument. Among them was a framework in the form of the Lambdoma with index 12 × 12, upon which the intervals that corresponded to the proportions of the Lambdoma could be electronically generated through striking.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Another instrument was based on the ch'in, a Chinese stringed instrument. 13 strings are tuned in the progression of the undertone series; the overtones for each of these undertones are generated by striking the strings so that the flageolet tones sound, which are then electronically amplified. Thus the tones of the Lambdoma can be made audible up to index 13.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Dieter Trüstedt built a third instrument for the “Phenomena” exhibition-this is shown in Image 16. Here, the tones of the Lambdoma are represented by 12 × 12 steel tubular bells; the longest bell is over 2 meters long. The lengths of the tubular bells do not correspond to the numbers of the Lambdoma, since the vibrations of bars and bells obey different laws from those of strings.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Another form of the Lambdoma as a musical instrument was realized by Peter Neubäcker in a computer program: here the Lambdoma is shown on the screen up to a given index-the Lambdoma tone to which the player points on the screen is then sent to a synthesizer. In this way all tones of the Lambdoma are available, throughout the entire auditory domain.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The most charming thing about playing with the Lambdoma is the fact that musically related tones are always nearby, in contrast with a piano keyboard, on which the available tones are simply laid out in a row. Also, in the Lambdoma all the tones are available in their pure tuning, and thus also the intervals that are formed from the higher prime numbers, such as 7, 11, 13, etc., which cannot be achieved at all in our traditional music.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Theories/23.jpg" /><span lang="en-US" xml:lang="en-US"><br /> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 17</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Development of the Surfaces of Crystals Represented in the Lambdoma</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Above: up to 7 complication steps</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Below: up to 12 complication steps</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Conceived and drawn by Peter Neubäcker</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Image 18</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Surface Development of Crystals Represented in the Lambdoma</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Four Quadrants in the Cubic System</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Conceived and drawn by Peter Neubäcker</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">In the construction of the surfaces of crystals, laws can be established which can also be expressed in the form of musical intervals in the realm of harmonics. For example, if a coordinate axis is drawn through the simplest crystal form, the cube, one surface is parallel to an axis, the other perpendicular to it. This can be expressed in the ratios </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</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"> = 0 for the perpendicular, </span><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">0</span><span lang="en-US" xml:lang="en-US"> = ∞ for the parallel surfaces. These two surfaces are described as primary surfaces. The further appearing surfaces are arranged so that they form an angle between the two primary surfaces-thus only certain angles are possible, namely those which always mark off whole numbers from both axes. So the next possible surface would be that which forms an angle of 45 degrees to the primary surfaces; since it marks off equal parts from both axes, it is described as 1/1-other possible surfaces include 1/2, 2/3, etc. If the axes of the crystal system are now imagined as two monochord strings placed at a right angles to each other, then the crystal surfaces always mark off consonant intervals, and the more consonant the corresponding interval, the more likely the surface is to occur.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The crystallographer Victor Goldschmidt developed another law which determines the probability of crystal surfaces-he calls it the Law of Complication. Thus the following “normal series” appear:</span></p> <table style="border: 0.8pt solid ; width: 100%; border-collapse: collapse; empty-cells: show; table-layout: fixed;" rules="all" cellpadding="0" border="1"> <colgroup> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> <col></col> </colgroup> <tbody> <tr> <td style="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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">Primary surfaces: 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">.</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">.</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">.</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">.</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">.</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">.</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">.</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">B</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: right;" dir="ltr"><span style="font-family: 'Arial Black';" lang="en-US" xml:lang="en-US">N</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> = 0</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">.</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">.</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">.</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">.</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">.</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">.</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">.</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">∞ = normal series 0.</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">1</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">st</span><span lang="en-US" xml:lang="en-US"> complication: 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">.</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">.</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">.</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">C</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">.</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">.</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">.</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">B</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: right;" dir="ltr"><span style="font-family: 'Arial Black';" lang="en-US" xml:lang="en-US">N</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> = 0</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">.</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">.</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">.</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: 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">.</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">.</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">∞ = normal series 1.</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">nd</span><span lang="en-US" xml:lang="en-US"> complication: 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">.</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">D</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">.</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">C</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">.</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">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">.</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">B</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: right;" dir="ltr"><span style="font-family: 'Arial Black';" lang="en-US" xml:lang="en-US">N</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> = 0</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">.</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="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">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">.</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: 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">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">.</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">∞ = normal series 2.</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: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">3</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">rd</span><span lang="en-US" xml:lang="en-US"> complication: 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">F</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">D</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">G</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">C</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> <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">I</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">B</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: right;" dir="ltr"><span style="font-family: 'Arial Black';" lang="en-US" xml:lang="en-US">N</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> = 0</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="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">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: 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">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 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">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: 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: center;" 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">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">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">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">∞ = normal series 3.</span></p> </td> </tr> </tbody> </table> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">etc.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The links between the series that follow in each case then emerge through the adding of the denominator and numerator of the neighboring fractions. In nature the emerging surfaces only go beyond the 3</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">rd</span><span lang="en-US" xml:lang="en-US"> normal series in rare cases. But the idea of this law of surface construction can be continued further, leading to the images shown.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The Lambdoma can be understood directly as a coordinate axis in the cubic crystal system (also for other crystal systems, when the angle of the axis is changed)-an equal-tone line going from point 0/0 to a certain tone-value is then identical to the “surface norms,” i.e. the lines perpendicular to the respective crystal surfaces. According to Goldschmidt, the “direction of the particle energy that constructs the surfaces.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The tone-values lying nearest to the origin of the Lambdoma thus represent the most commonly appearing crystal surfaces, and the ones further out represent the less common-but not arbitrarily chosen tone-values, instead those belonging to the next normal series in each case, according to Goldschmidt's Law of Complication. In the illustrations here, all tone-values or surface indices newly appearing in a normal series are joined by a line-which can be seen most plainly in the first image. These images can thus be understood as a direct illustration of Goldschmidt's Law of Complication, or better yet, as the illustration of the tendency of this law, because such high degrees of complication only appear rarely in nature.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The similarity in form to certain minerals that grow in a radiant pattern (for example antimony) is evident, especially when the lower picture of Image 17 is continued into the spatial realm. It is theoretically conceivable that a causal connection exists in the sense of crystallography, in addition to the morphological.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It is interesting to consider which variations from one degree of complication to the next emerge in the Lambdoma in this illustration. A process of inversion can be clearly seen here: precisely where the surface development at the lower degrees reaches farthest into the Lambdoma, gaps emerge at the higher degrees; so the greatest gap in the image is at 1/1-that is the direction in which the first surface emerges. For all further degrees the same phenomenon can be observed.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Following each of the newly emerging numbers and their directions in the Lambdoma, it becomes apparent that they are the numbers of the Fibonacci Sequence: the ray reaching the farthest tends to the direction of the Golden Section, the next is identical in its direction to that of the “main series of leaf position numbers” described in Image 8. But because mostly only the lower degrees of complication appear in nature, here we once again see the polarity of an ideal tendency towards the Golden Section on the one hand and the realization through the harmonic intervals on the other hand.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Theories/24.jpg" /><span lang="en-US" xml:lang="en-US"><br /> </span></p> <p style="text-align: left;" dir="ltr"><img style="width: 152.4mm;" alt="" src="../kayser2/imagelibrary/Theories/25.jpg" /></p></div></td></tr></tbody></table> </span></div> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"></span></div></div> </body> </html></div>
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
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