<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 & Mathematics - Number Distribution.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 19</span></div> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Model: The Distribution of All Rational Numbers</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">or Musical Intervals on the Monochord</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 order to investigate how all fractions, or rational numbers, are distributed, one can divide segments of a unit of length-here the monochord string-progressively by all the fractions, and mark the place of the division with a line. For example, one could begin with the number 2 and mark half of the length, then </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">3</span><span lang="en-US" xml:lang="en-US"> and </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">3</span><span lang="en-US" xml:lang="en-US">, then </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">4</span><span lang="en-US" xml:lang="en-US">, </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">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">4</span><span lang="en-US" xml:lang="en-US">, then </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">5</span><span lang="en-US" xml:lang="en-US">, </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">, </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">, </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">, and so forth. The result is represented by the drawing, placed over the monochord-here all fractions are plotted up to index 100. For clarification, the lines are of different lengths: the smaller the number, the longer the line that indicates the ratio produced by it.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In this process a very interesting structure is revealed: the smaller the number from which the interval in question is constructed, the more space is left to it from the following ratios-so the degree of consonance of the interval can be determined directly from the width of the gaps that appear in the illustration.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The string itself also conforms to this pattern in its vibrations: the wider the in-between space in the drawing, the more pronounced is the vibration point at this location, responsible for producing the respective overtone. This can be tested by moving the rubber disc back and forth on the string and plucking the string as near as possible to one of the two nuts; the rubber disc stops the string from vibrating at the place in question, and thus sets a vibration point so that the corresponding overtone will sound. Thus one can establish that there are some overtones at almost every point, which sound with greater or lesser strength-but coming near to one of the large gaps in the drawing, one enters the “catchment area” of a small number, and the corresponding stronger tone overwhelms the others, which are manifested through higher numbers. One could say that the numbers have an “attractive power,” which is stronger when the number is smaller. Besides the musical properties, a plethora of other applications lie hidden in this illustration-Rudolf Stössel examined these structures more closely-with concepts appearing such as symmetry, isolation, reproduction, the rule of addition, the product law, and connections with the Fibonacci Sequence and its relatives, i.e. the Golden Section, the Lambdoma, and crystals.</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="/EZ/kayser2/kayser2/imagelibrary/Mathematics/0.jpg" alt="" /><span lang="en-US" xml:lang="en-US"></span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 20</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Distribution of All Rational Numbers</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">or Musical Intervals in Circular Form</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">Not only can the arrangement of all fractions or musical intervals be illustrated in quadratic form, as with the model of the Lambdoma (Image 14)-a circular arrangement is also possible. For this, the entire circle is defined as the unit; the fractions are marked off as fractional parts of the whole round angle. Fractions with the same numerator are on the same circle-each circle therefore corresponds to a perpendicular line in the quadratic Lambdoma. Up to index 8, this arrangement emerges:<br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="500" height="491" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/1.jpg" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Tones with the same value, i.e. fractions that can be cancelled down, are always on the same line, which leads outwards radially. These “equal-tone lines” are also plotted here. Image 20 shows only these equal-tone lines, each drawn from their origin outwards-here for all fractions up to index 100.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> While the values with the same numerator-corresponding to the undertone series-always lie upon concentric circles, the values with the same denominator form spirals, corresponding to the overtone series, and all finally run parallel to the generator-tone line.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Here the same phenomenon can be observed as was seen in the positioning of the fractions on the monochord (Image 19): the fractions are divided unequally in favor of the smaller numbers or the more consonant intervals. This can be thought of as a hierarchy in the world of numbers, or an “aristocracy of small numbers.”</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US"><img width="500" height="710" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/2.jpg" /><br /> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 21</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Distribution of All Rational Numbers</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Above: Image 20 reflected inwards</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Below: magnification of the inner space</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">The round Lambdoma illustration actually only contains half of the values of the quadratic Lambdoma, namely all true fractions in the domain between zero and one. Their reciprocals, which would correspond in the model in Image 19 to a lengthening of the unit string, are only present latently: if the circles in the round illustration are understood as “bent strings,” then all values above one can be represented in this way. The circumference of the second circle is exactly twice that of the unit circle, the third three times as long, and so on. Thus, for example, the equal-tone line 1/3 marks off the length 4/3 on the fourth circle, the length 5/3 on the fifth, etc. In this representation, the true fractions are realized by means of the angles, and their reciprocal values through the circle arcs.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Image 21 shows another way to illustrate the reciprocal values in the round Lambdoma: through reflection on the unit circle. These images must be imagined as tiny circles at the center of Image 20: All rays going out from Image 20 go inwards here, and their origin corresponds, in each case, to the reciprocal value of the point on the outside. The lower image shows a large magnification of the upper one, and moreover the fractions here are not represented by rays (equal-tone lines) but only by points; all reducible fractions are thereby discarded. An infinite number of values go inwards-these images can be seen as a symbol of the potential endlessness in the spiritual domain, which is expressed as true unending multitude after transcending the unit circle in Image 20.</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">22</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">(No image here)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Three-Dimensional Model of the Distribution of Fractions</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Corresponds to the Scheme of Image 20</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Model conceived and built by Peter Neubäcker</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> <img width="500" height="750" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/3.jpg" /><br /> <br /> Image 23</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Distribution of the Prime Numbers up to 30000</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Above: From the viewpoint of the number 210 (2 × 3 × 5 × 7)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Below: From the viewpoint of the number 211 (prime number)</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">This and the two following images relate to the distribution of the prime numbers. The first question to be asked here is, what does a problem of number theory such as the distribution of prime numbers have to do with harmonics in the narrower sense?</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The musical intervals are built from numbers-thus intervals that are built from the same prime numbers are closely connected with each other. The first interval-building number is two-it only ever leads to octaves. The next is three-the entire series of fifths can be built from it, which actually leads to an endless supply of tones, consisting only of fifth ratios. Until the Renaissance, this was the only tone structure used in the Western world. Then the prime number five emerged in music as the next step, leading to a more complex supply of tones, which includes harmony structures through the relationships of thirds in the sense of triads. In modern times the next prime number, seven, is also applied; but at present, its musical potentials have hardly been explored or integrated.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In studying harmonics, the entire world of numbers becomes a symbolic image for the world, in which each number plays its specific role-this role can be explored through the factorization of all numbers, which always leads to the prime numbers and thus reveals the relationships of the numbers. By listening to these numbers, one can also approach their nature more closely through perception. This is harder with the higher prime numbers, since for cultural reasons no specific musical experience is yet connected with them. But since all prime numbers manifest qualities with their own character in the world of numbers, their distribution is of interest in principle.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The distribution of the prime numbers is irregular. No mathematical law has yet been discovered with which prime numbers can be clearly predicted-probably no such law can be found-but certain structures in their distribution can be determined. Such structures are illustrated from various viewpoints in the image shown. Images 23 and 24 show the distribution of prime numbers from the “viewpoint” of individual numbers-mathematically expressed, the arrangement is the “module of a given number.”</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US"><img width="500" height="725" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/4.jpg" /><br /> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><br /> Image 24</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Distribution of Prime Numbers up to 10000</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">from the Viewpoint of the Numbers 200 to 228</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">(in increments of 0.2)</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">When all natural numbers are arranged so that they lie upon concentric circles, and the same quantity of numbers is on each circle, then the number of rays emerging corresponds to the number used as a basis. For the numbers 11 and 12, for example, these two images would be produced:<br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="500" height="186" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/5.jpg" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The prime numbers here are indicated by circles. Thus it is shown that for the number 12, prime numbers can only lie upon four of the 12 rays, since all numbers on the second ray are divisible by two, all on the third are divisible by three, all on the fourth are divisible by four, and so on-because these numbers are factors of 12. For the image of the number 11, on the other hand, prime numbers can lie on all rays (except for the 11</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> ray, on which all numbers are multiples of 11) , since 11 itself is a prime number. If the composite numbers are removed and only the prime numbers left, these images result for 11 and 12:<br /> <br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="500" height="126" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/6.jpg" /></span><br /> <span lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">From the viewpoint of the number 12, the prime numbers appear very orderly, but from the viewpoint of 11 completely disorderly. In this manner Images 23 and 24 emerge for higher numbers: the number 210, as a product of 2 × 3 × 5 × 7, puts the numbers in order to a very high degree, but the adjacent number 211 is a prime number which generates no order in itself; instead the order of the number 210 still appears here. Image 24 shows that every number has its own “face”-partly generated of itself, partly shaped by its neighbors.</span></p> <p style="text-align: center;" dir="ltr"></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><img width="500" height="707" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/7.jpg" /><br /> <br /> Image 25</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Distribution of the Prime Numbers According to the Pythagorean Polarity</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">of Square Numbers and Rectangular Numbers</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Above: numbers up to 2500</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Below: numbers up to 100000</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">The ordering principle of this image is different from those of the two previous images: for the Pythagoreans, polarity was a central concept, and they counted ten such polarities, so that concepts emerged such as limited and unlimited, odd and even, male and female, etc. The tenth of these polarities is the opposition of square and rectangular numbers.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Why precisely these two concepts should be polarities was not entirely understandable. But when these polarities are arranged in terms of all numbers, it becomes clear that this arranges the prime numbers to the highest degree. In the images shown here, all natural numbers are arranged in a spiral, so that in each revolution a square number is encountered-thus the series of rectangular numbers appears precisely on the opposite, i.e. the numbers that are the product of two consecutive numbers, such as 6 = 2 × 3, 12 = 3 × 4, 20 = 4 × 5, etc. In the upper image, all numbers up to 2500 are shown, and the prime numbers are indicated by white circles. In the lower image the numbers up to 100000 are shown; the composite numbers are omitted here, and only the prime numbers indicated by white dots.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Thus it can be seen that all prime numbers arrange themselves in curves, which approach lines parallel to the ray of the rectangular numbers. In each case such a ray represents the sum of a given number with all rectangular numbers, and can therefore be represented by the expression (n × [n + 1]) + m. For certain values of m, an especially large quantity of prime numbers is yielded, e.g. for m = 17 or even more for m = 41. The expression n × (n + 1) + 41 has already been described by Leonhard Euler as an especially fruitful prime number formula.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The numbers 17 and 41 are prime numbers themselves-so it can be said that prime numbers that generate many new prime numbers are especially “fruitful.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It is unlikely that the prime number-ordering function of this polarity was known to the Pythagoreans-it more likely emerged from the finding that the square numbers are always sums of all odd numbers and the rectangular numbers are always sums of all even numbers-but here it can be seen again that the truth is always found in testimonies from many different levels at the same time.<br /> <br /> </span></p> <div align="center"><span lang="en-US" xml:lang="en-US"><img width="500" height="734" title="" alt="" src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/8.jpg" /></span><span lang="en-US" xml:lang="en-US"></span><br /> <span lang="en-US" xml:lang="en-US"></span></div></div></td></tr></tbody></table></span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> <br /> </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
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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
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Cosmology & Cosmogony
Harmonic Cosmogony
I Ching
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Cosmic Harmony in Symbolism
Doctrine of Correspondences
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Hermann Hesse & The Glass Bead Game
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Partial Tone Diagram & Square of 9
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Harmonics in Nature
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Harmonics & Music Theory
Harmonics as a Science
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Thought as a Sense
Color & Partial-Tone Coordinates
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