<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 - The Proportions.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; } *.footnote_text { font-size:10pt; } *.footnote_reference { font-size:10pt; vertical-align:super; } --></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"> </span> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">§28. THE PROPORTIONS<br /> <br /> </span></div> <div align="center"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"><img width="500" height="269" src="/EZ/kayser2/kayser2/imagelibrary/Ego2.jpg" alt="" title="" /></span><br /> </div> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§28. The Proportions</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Proportion technique is a realm of “harmonic technique” that appears from the outside to be all numbers and formulae. In spite of this, it is one of the most lively domains, if one starts by understanding the concept of proportion = relationship in terms of its actual nature: an exceptional multitude of possible forms based on principles that are very simple because they are understandable, provable by anyone with basic knowledge, and not very mathematical. In this movement back and forth between numbers, the numbers change into distances and then into tones; thus every individual number, distance, or tone must always somehow agree with two, three, or more others, or else persuade these to agree with it. This process has a seductive charm for those who calculate, draw, and listen along with it; the reader should work over the examples once again and experience them. In this way, he will surely grasp their meaning himself.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In §24a, we discussed proportions while considering “harmonic division-ratios” in the harmonic pencil of rays of the equal-tone lines. Unfortunately, the study of proportion is taught perfunctorily nowadays; textbooks usually devote only a few pages to it, and one wise writer thought that 3-4 pages “might be further abbreviated without repercussions” (Tropfke: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Geschichte der Elementarmathematik</span><span lang="en-US" xml:lang="en-US">, 1902, vol. I, p. 232). We shall see about that as we orient ourselves. </span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The three most important proportions, probably originating from Babylon, were known in the school of Pythagoras:</span></p> <p style="text-align: left; margin-left: 1.6882in;" dir="ltr"><span lang="en-US" xml:lang="en-US">the arithmetic </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a - b = c - d</span></p> <p style="text-align: left; margin-left: 1.6882in;" dir="ltr"><span lang="en-US" xml:lang="en-US">the geometric </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a : b = c : d</span></p> <p style="text-align: left; margin-left: 1.6882in;" dir="ltr"><span lang="en-US" xml:lang="en-US">and the harmonic </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a : c = (a - b) : (b - c)</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">This looks very dry. What do these formulae say? First, we will try to clarify them with lines.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Arithmetic Proportion</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> For the arithmetic proportion, we must draw two lines from which we can derive another two; the remainder must be equal. For example:</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/20.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 176</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If I subtract </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> (= 3) from </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> (= 4), the result is 1. Subtracting </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">d</span><span lang="en-US" xml:lang="en-US"> (= 5) from </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> (= 6) also yields 1. This is exactly what the formula </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a - b = c - d</span><span lang="en-US" xml:lang="en-US"> indicates. Here we see that it is not the segments or their ratios that are equal, but instead their differences: that is the so-called “arithmetic” proportion. Naturally, one can perform this subtraction with any given numbers and segments; the only requirement is that the same difference results on both sides of the equation. Surplus or shortage, then, is the characteristic of this proportion, depending on whether (in the above example) I judge by the two upper or two lower lines. “Arithmetic” proportion is thus named because an arithmetic series, for example the simple whole-number series 1 2 3 4 5 ..., represents a so-called “continuous” arithmetic proportion, in which the differences between the adjacent elements are always the same (in this case, 1). The constituent “arithmetic mean” of such a series is</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> </colgroup> <tbody><tr><td style=" 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">n =</span></p> </td><td style=" 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">a + b</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">i.e. each element of the series is half the sum of its neighbors, e.g.</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> </colgroup> <tbody><tr><td style=" 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">4 =</span></p> </td><td style=" 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">3 + 5</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td></tr></tbody></table> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Now, if we examine our diagram, we immediately notice that in the frequency diagrams, all horizontal rows of the overtone type clearly form continuous arithmetic proportions. This means that every overtone, in terms of its frequency, is half the sum of its two neighboring tones. In the string-length diagram, the “arithmetic” element is in the vertical series.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Geometric Proportion</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Whereas the arithmetic proportion only has a “ratio” in the equality of differences, the geometric proportion has, so to speak, direct ratios in the segments themselves, which must also be equal. If </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">d</span><span lang="en-US" xml:lang="en-US">, then at the simplest we are looking for numbers (in this case also fractions) with the same value, e.g. </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">8</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">6</span><span lang="en-US" xml:lang="en-US"> = </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">12</span><span lang="en-US" xml:lang="en-US">, </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">20</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">5</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">, etc. In purely numeric terms, they can even be checked in this way, since the inner and outer elements of this proportion can be multiplied:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">4 : 8 = 2 : 4 8 · 2 = 4 · 4 16 = 16 etc.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> But we can go further, using a geometric proportion to exchange the elements of one string with those of another, without changing their content:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">8 : 4 = 4 : 2 4 · 4 = 8 · 2 16 = 16</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The inner value remains the same. Now when we look back to our diagram, we will notice that all the rows standing perpendicular to the generator-tone axis (in the frequency diagrams; for string length, reciprocal numbers or tone-values) form continuous geometric proportions, for example:</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> </colgroup> <tbody><tr><td style=" 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">1 </span><span lang="en-US" xml:lang="en-US">: </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">1 </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">3</span></p> </td><td style=" 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">or</span></p> </td><td style=" 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">4 </span><span lang="en-US" xml:lang="en-US">: </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">1 </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">15</span></p> </td><td style=" 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">or</span></p> </td><td style=" 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">10</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">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><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">10</span></p> </td></tr><tr><td style=" 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><span lang="en-US" xml:lang="en-US">:</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> c </span><span lang="en-US" xml:lang="en-US">:</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> f</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" 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><span lang="en-US" xml:lang="en-US">′′ : </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">des</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" 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><span lang="en-US" xml:lang="en-US">′ : </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> </td></tr><tr><td style=" 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">fifth fifth</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" 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">semitone semitone</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> </td><td style=" 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">octave octave</span></p> </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 177</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">which can, of course, also be written in the form </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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US"> : </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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> = </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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span lang="en-US" xml:lang="en-US">, etc. As one can see, the interval is always the same for geometric proportions in harmonic terms. The characteristic thing about this proportion is not, as with the arithmetic proportion, the equality of the surplus or shortage (the difference), but instead the equality of the value of both in terms of set distances (numbers, tones). Today this geometric proportion, in the form </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a </span><span lang="en-US" xml:lang="en-US">:</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US"> b = c </span><span lang="en-US" xml:lang="en-US">:</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US"> d</span><span lang="en-US" xml:lang="en-US">, is used as the prototype of proportion itself: “</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> is to </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> as </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> to </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">d</span><span lang="en-US" xml:lang="en-US">,” in which one calls the two quotients </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a </span><span lang="en-US" xml:lang="en-US">:</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US"> b</span><span lang="en-US" xml:lang="en-US"> and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c </span><span lang="en-US" xml:lang="en-US">:</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US"> d</span><span lang="en-US" xml:lang="en-US"> a “ratio” and the equalization of two ratios a “proportion.” Every arithmetic and algebra textbook covers the manipulations of which the individual elements of “proportion” are capable, especially interchangeability and the ability of the inner and outer elements to be multiplied, as already mentioned. The concept of a “continuous” or “constant” proportion exists, then, whenever the median elements are equal, as in the above examples. Thus the “median” geometric proportional of 2 numbers emerges. When </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x = x</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">, then </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</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="font-style: italic;" lang="en-US" xml:lang="en-US">ab</span><span lang="en-US" xml:lang="en-US">, or </span></p> <p style="text-align: center;" dir="ltr"><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</span><span lang="en-US" xml:lang="en-US"> = √</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">ab</span><span lang="en-US" xml:lang="en-US"> (geometric mean).</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The median geometric proportion between 2 numbers is the square root of the product of these numbers, which can be demonstrated with our diagram. If </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a </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"> and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</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">3</span><span lang="en-US" xml:lang="en-US">, then </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</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: 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"> · </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"> = 1, thus </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</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"> · </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"> = √1 = 1. In the diagram, of course, we find other geometric proportions if we merely look for the ratios of equal intervals-for example: </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">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US"> where the equality-ratio consists of fifths.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Harmonic Proportion</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The predominantly geometric laws of the third of the typical proportions, known as the harmonic, were already discussed amply in §24a. Its algebraic expression </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> = (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a - b</span><span lang="en-US" xml:lang="en-US">) : (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b - c</span><span lang="en-US" xml:lang="en-US">) means that in a three-element harmonic proportion (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US">,</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US"> b</span><span lang="en-US" xml:lang="en-US">, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">) the differences between the first and middle elements have the same relationship to the difference between the middle and third as the first element has to the third. If, in a three-element (continuous) harmonic proportion, (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a - x</span><span lang="en-US" xml:lang="en-US">) : (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x - b</span><span lang="en-US" xml:lang="en-US">) = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a </span><span lang="en-US" xml:lang="en-US">:</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US"> b</span><span lang="en-US" xml:lang="en-US">, then by multiplication of the outer and inner elements, we find that (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a - x</span><span lang="en-US" xml:lang="en-US">)</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> = (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x - b</span><span lang="en-US" xml:lang="en-US">)</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US">, thus </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">ab - bx = ax - ab</span><span lang="en-US" xml:lang="en-US">, so 2</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">ab</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">= ax</span><span lang="en-US" xml:lang="en-US"> + </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">bx</span><span lang="en-US" xml:lang="en-US">, therefore</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=" 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=" 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">2 a b</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">a + b</span></p> </td><td style=" 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">(harmonic mean).</span></p> </td></tr></tbody></table> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">Look once again, more closely, at the visual geometric derivation in §24a. If we look for harmonic proportion in our frequency diagram, we find it in all the vertical columns of the model of the undertone series. Each aliquot series is a continuous harmonic proportion series. The progression </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">1</span><span lang="en-US" xml:lang="en-US"> </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">2</span><span lang="en-US" xml:lang="en-US"> </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">, brought to a common denominator, is </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">6</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">6</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">6</span><span lang="en-US" xml:lang="en-US">. (</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">6</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">6</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">6</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">6</span><span lang="en-US" xml:lang="en-US">) = </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">6</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">6</span><span lang="en-US" xml:lang="en-US">, therefore </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">6</span><span lang="en-US" xml:lang="en-US"> : </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">6</span><span lang="en-US" xml:lang="en-US"> = </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">6</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">6</span><span lang="en-US" xml:lang="en-US">,</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US"> </span><span lang="en-US" xml:lang="en-US">thus the left and right sides become </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">6</span><span lang="en-US" xml:lang="en-US">, fulfilling the harmonic proportion. In a similar way, with the introduction of tone-values, the progression </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">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US"> </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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span><span lang="en-US" xml:lang="en-US"> </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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span><span lang="en-US" xml:lang="en-US">, set to a common denominator </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">12</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">12</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">12</span><span lang="en-US" xml:lang="en-US">, yields the harmonic proportion (</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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">, </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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,, </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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span><span lang="en-US" xml:lang="en-US">) =</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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span><span lang="en-US" xml:lang="en-US">, so </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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span><span lang="en-US" xml:lang="en-US"> : </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">12</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,,</span><span lang="en-US" xml:lang="en-US"> on right and left. Visually, it can be clarified thus:</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/21.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 178</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">Here, as one can see, in contrast with arithmetic proportions, it is not the differences but the ratios of the differences that are equal to the ratios of the two pairs of section differences.</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/22.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 179</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It is now important and interesting to investigate the relationships of the three proportion types to one another. Viewed harmonically, Fig. 179 makes this relationship self-explanatory. In the P-diagram one can draw horizontal, vertical, and perpendicular lines through any point on the generator-tone line to find three intersecting “continuous” arithmetic, harmonic, and geometric proportions in each case-for example, the series through the point </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">8</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">8</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">, as illustrated in Fig. 179.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The great polarity of major and minor also once again emerges here: all major series (in terms of frequency, as in our diagram) have the arithmetic proportion as a constituent element in the background, and all minor series have the harmonic, whereas the geometric proportion invades them both from one direction and overshoots in the other, holding together the coordinate structure. </span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> For further investigations of the relationships of the three proportion types, their median proportionals are most useful. We find a remarkable conjunction in the circle (Fig. 180); here we also find the solution to the problem of the geometric construction of these three “median” proportionals.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Median Proportionals</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/23.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 180</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the circle (Fig. 180), choose an arbitrary point V on the radius MW. With the same radius </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">r</span><span lang="en-US" xml:lang="en-US">, draw an arc from V that intersects the circumference at A, and continue the line AV in the opposite direction to point H. Finally, draw a perpendicular line from V to G. In every case, VA will be the arithmetic proportional, VG the geometric proportional, and VH the harmonic proportional to the segments </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> marked off by point V on the diameter UW. Proof: VA is the arithmetic median because VA = the radius </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">r</span><span lang="en-US" xml:lang="en-US">. </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> + </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> = 2</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">r</span><span lang="en-US" xml:lang="en-US">, so </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> </colgroup> <tbody><tr><td style=" 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">r = VA =</span></p> </td><td style=" 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">a + b</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td></tr></tbody></table> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">The arithmetic mean of the reciprocal values of two quantities is equal to the reciprocal value of the harmonic mean of both quantities, i.e.:</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> </colgroup> <tbody><tr><td style=" 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_</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">a</span></p> </td><td style=" 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=" 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_</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">b</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" rowspan="2"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">=</span></p> </td><td style=" 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">a + b</span></p> </td></tr><tr><td style=" 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 style="font-size: 6pt;" lang="en-US" xml:lang="en-US">-----------</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> <br /> </td><td style=" 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: 6pt;" lang="en-US" xml:lang="en-US">-------</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2ab</span></p> </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 181</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">∆VAU is similar to ∆VWH, since </span><img src="../kayser2/imagelibrary/Geometry/24.jpg" alt="" /><span lang="en-US" xml:lang="en-US"> UAH = </span><img src="../kayser2/imagelibrary/Geometry/25.jpg" alt="" /><span lang="en-US" xml:lang="en-US"> UWH and </span><img src="../kayser2/imagelibrary/Geometry/26.jpg" alt="" /><span lang="en-US" xml:lang="en-US"> AUW = </span><img src="../kayser2/imagelibrary/Geometry/27.jpg" alt="" /><span lang="en-US" xml:lang="en-US"> AHW (angles on the circumference on equal chords), therefore:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">VA : VU = VW : VH or </span></p> <p style="text-align: center;" dir="ltr"><span style="font-style: italic;" lang="en-US" xml:lang="en-US">r</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> : VH</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> </colgroup> <tbody><tr><td style=" 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">VH =</span></p> </td><td style=" 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-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">a b</span></p> <p style="text-align: center;" dir="ltr"><span style="font-style: italic;" lang="en-US" xml:lang="en-US">r</span></p> </td><td style=" 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><span style="font-style: italic;" lang="en-US" xml:lang="en-US">r</span><span lang="en-US" xml:lang="en-US"> =</span></p> </td><td style=" 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-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">a</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + </span><span style="font-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">b</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" 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">, therefore</span></p> </td></tr></tbody></table> <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> </colgroup> <tbody><tr><td style=" 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">VH = </span></p> </td><td style=" 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-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">a b</span></p> <p style="text-align: center;" dir="ltr"><span style="font-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">a</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + </span><span style="font-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">b</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td></tr></tbody></table> <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> </colgroup> <tbody><tr><td style=" 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">VH = </span></p> </td><td style=" 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">2 </span><span style="font-style: italic; text-decoration: underline;" lang="en-US" xml:lang="en-US">a b</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> </span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">a + b</span></p> </td></tr></tbody></table> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Thus VH must be the harmonic median. VG, the geometric median, results from the right triangle UGW, whose height GV, according to a well-known law of plane geometry, is equal to the square root of the product of the sections that it marks, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</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"> Since, in Fig. 180, point V can be arbitrarily chosen, it is clear that only certain numbers are suited for an “emmelic” ratio of all three median proportionals. What was said in §24a.1 regarding the musical special case of “harmonic proportion” also applies here. We will soon see how the three “medians” relate to one another algebraically and harmonically.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From examining Fig. 180, it is directly evident that the arithmetic mean must always be greater than the geometric, and the geometric greater than the harmonic:</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=" 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">a + b</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" 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">> √ab ></span></p> </td><td style=" 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">_2ab_</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">a + b</span></p> </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 182</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">and that the geometric mean is also the geometric mean of the arithmetic and harmonic means:</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=" 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">a + b</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" 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">: √ab = √ab :</span></p> </td><td style=" 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">_2ab_</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">a + b</span></p> </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 183</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The correctness of this proportion comes from the equality of the product of the inner and outer elements. We also come upon this remarkable symmetry for the median proportions, which we have already come to know from the continuous proportions belonging to them in the partial-tone diagram. This indication of symmetry led A. von Thimus to further investigate the behavior of the three proportion types in algebraic abstract terms. In vol. II, p. 38 of his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Symbolik</span><span lang="en-US" xml:lang="en-US">, Thimus writes: “To gain a deeper insight into the number-harmonic construction of the musical system, we wish to illustrate the process of the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Medietäten</span><span lang="en-US" xml:lang="en-US"> (= median proportionals) and third proportionals more generally, by algebraic methods. […] For this we start with the most general ideas. We place the measure of the width of the still undefined interval in the middle 'as beginning of everything,' take the same constructive primal tone of the one, α (alpha) and the other, ω (omega) from the pole (</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">̕</span><span lang="el-GR" xml:lang="el-GR">αρχαί</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">= </span><span lang="en-US" xml:lang="en-US">beginning of the up and down way </span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">̔</span><span lang="el-GR" xml:lang="el-GR">οδ</span><span lang="en-US" xml:lang="en-US">ò</span><span lang="el-GR" xml:lang="el-GR">ς</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">̕</span><span lang="el-GR" xml:lang="el-GR">άνω</span><span lang="en-US" xml:lang="en-US"> κάτω) ... the geometric median will equal the square root √αω of the product of the two outer terms. The arithmetic median is known to be half the sum of the outer term, so</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> </colgroup> <tbody><tr><td style=" 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="el-GR" xml:lang="el-GR">α + ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">(</span></p> </td><td style=" 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=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">)</span></p> </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 184</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">As a general formula for the harmonic, the expression</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> </colgroup> <tbody><tr><td style=" 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">2</span><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">αω</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α + ω</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">(</span></p> </td><td style=" 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">= α ω</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">)</span></p> </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 185</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">is found.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Thimus now proceeds as follows: first he writes the three median proportionals between the two “poles” α and ω, adding the arithmetic in parentheses as above, and replacing</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=" 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="el-GR" xml:lang="el-GR">α + ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" 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">with</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> </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 186</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">for reasons of symmetry. He notes the harmonic proportional replacement of </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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">2 α ω</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">α + ω</span></p> </td><td style=" 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">with</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0 </span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1 </span><span lang="en-US" xml:lang="en-US">ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> </td></tr></tbody></table> <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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span></p> </td><td style=" 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><td style=" 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><span lang="el-GR" xml:lang="el-GR">αω</span></p> </td><td style=" 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><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">ω</span><span lang="en-US" xml:lang="en-US"> 1st geometric medians</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="5"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">. . . . . . . . . . . . . . . . . . . . .</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" 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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">ω</span><span lang="en-US" xml:lang="en-US"> 1st arithmetic medians</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span></p> </td><td style=" 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=" 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">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0 </span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1 </span><span lang="en-US" xml:lang="en-US">ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="5"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">. . . . . . . . . . . . . . . . . . . . . . .</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">ω</span><span lang="en-US" xml:lang="en-US"> 1st harmonic medians</span></p> </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 187</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">Now he begins to proportionate according to the series. First he seeks the third arithmetic proportional </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</span><span lang="en-US" xml:lang="en-US"> as the “second arithmetic median proportional” to</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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">2 α ω</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">α + ω</span></p> </td><td style=" 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">and</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">α +</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US"> </span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </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 188</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">According to the law of arithmetic proportion, the median element</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> </colgroup> <tbody><tr><td style=" 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="el-GR" xml:lang="el-GR">α + ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span><span lang="el-GR" xml:lang="el-GR"> </span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">(</span></p> </td><td style=" 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=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">)</span></p> </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 189</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">must equal the sum of the first element</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> </colgroup> <tbody><tr><td style=" 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">2</span><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + </span><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α + ω</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">(</span></p> </td><td style=" 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">= α ω</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">)</span></p> </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 190</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">and the desired third element, divided by 2:</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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" rowspan="2"> <p style="text-align: center;" dir="ltr"><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> = ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" rowspan="2"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">=</span></p> </td><td style=" 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">2 α ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">α + ω</span></p> </td><td style=" 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></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <br /> <br /> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="2"> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </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 191</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">and therefore</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=" 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 = 2</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">(</span></p> </td><td style=" 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">α + ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">)</span></p> </td><td style=" 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=" 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">2 α ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α + ω</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">which yields:</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> </colgroup> <tbody><tr><td style=" 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">(</span><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + </span><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">ω</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">)</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> - 2 α ω</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α + ω</span></p> </td><td style=" 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=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span lang="en-US" xml:lang="en-US"> + ω</span></p> </td><td style=" 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=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> </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 192</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">If we seek the “second harmonic median proportional” </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">y</span><span lang="en-US" xml:lang="en-US"> according to the equation</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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α : y =</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> </td><td style=" 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">: ω</span></p> </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 193</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">then after calculation, we find:</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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">y =</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α ω</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> </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 194</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">In a similar way, we find the third arithmetic median proportional:</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> </colgroup> <tbody><tr><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">3</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> </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 195</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">the third harmonic median proportional:</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> </colgroup> <tbody><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α ω</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">3</span></p> </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 196</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">and so forth. The final collective result of this proportioning of the formula is:</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/28.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 197</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The brackets above the formula represent geometric proportions, those further above represent arithmetic proportions, and those below represent harmonic proportions. The collective formula obeys the law of the hyperbola, and shows the inner symmetry of the three interrelated proportion types most beautifully. If one substitutes α = 1 and ω = 2 in the formula, that being the framework-interval of the octave, the result is the series:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 ... </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">10</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">9</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">9</span><span lang="en-US" xml:lang="en-US">) </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">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">3</span><span 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">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</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">9</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"> ... 2</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 198</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If we now pursue the two halves of this series to the right and left of √2 in the “P” system, we will see that these two halves, </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"> </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">9</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">3</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">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">5</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">9</span><span lang="en-US" xml:lang="en-US"> ... each lie upon a straight line going out from </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">1</span><span lang="en-US" xml:lang="en-US">-a very remarkable condition if we seek the possible “location” of √2, and even more significant if we examine the two end-values which these lines approach, or at which they also finish in the formula, namely α = 1 and ω = 2, as “limits” of these lines. The mathematically adept reader may investigate whether this is true!</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> We can now operate a three-element continuous proportion with the third proportionals instead of with the median proportionals, to obtain new steps and values. For this proportion technique, which is often important for practical harmonic tasks, one should note the following scheme.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> a) If I am seeking, for example, the third arithmetic proportionals downwards for the partial-tone coordinates </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">10</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="font-weight: bold;" lang="en-US" xml:lang="en-US">a</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> and </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">8</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="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US">, then because the denominators must be equal for a continuous arithmetic proportion, and the numerators must be equidistant, I write:</span></p> <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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> ← </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">8</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="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> → </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">10</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="font-weight: bold;" lang="en-US" xml:lang="en-US">a</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Thus </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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> is the desired third arithmetic proportional:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">(</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">10</span><span lang="en-US" xml:lang="en-US">/- </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">8</span><span lang="en-US" xml:lang="en-US">/- </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">6</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"> b) Conversely, for a continuous harmonic proportion, the number-values must be equal and the denominators must be equidistant. For example, if I am looking for the third harmonic proportionals upwards from </span><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">10</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US"> and </span><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">9</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">, I write:</span></p> <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">10</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US"> → </span><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">9</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> → </span><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><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span></p> <p style="text-align: left;" 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><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span><span lang="en-US" xml:lang="en-US"> is the desired harmonic proportional:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">(/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">10</span><span lang="en-US" xml:lang="en-US"> - /</span><span style="vertical-align: sub;" 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><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"> c) If I seek the third geometric proportional, e.g. for 10 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> and 12 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> downwards, then I write:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">x : 10</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span lang="en-US" xml:lang="en-US"> = 10</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span lang="en-US" xml:lang="en-US"> : 12</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">x = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">100</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">12</span><span lang="en-US" xml:lang="en-US"> = </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">6</span><span lang="en-US" xml:lang="en-US"> = </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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">cis</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Thus </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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">cis</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> is the desired third geometric proportional. In pure tones (without numbers) these are equal intervals, as noted above: i.e. the interval </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g-e </span><span lang="en-US" xml:lang="en-US">is equal to the interval </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e-cis </span><span lang="en-US" xml:lang="en-US">(minor thirds). Without arithmetic calculation, then, I could continue the example, saying that the further geometric proportionals to 10 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> and </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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">cis</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US"> must be </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">ais</span><span lang="en-US" xml:lang="en-US">, since </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">cis-ais </span><span lang="en-US" xml:lang="en-US">is another minor third; and so on.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If we once again use the two poles α and ω as starting points, setting α = 1 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c </span><span lang="en-US" xml:lang="en-US">and ω = 2 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US">, then we find the corresponding third proportionals by proportioning back and forth, as follows:</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> </colgroup> <tbody><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α : ω</span></p> </td></tr></tbody></table> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 : 2</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c c</span><span lang="en-US" xml:lang="en-US">′</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 199</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">to this we add the third geometric proportion upwards and downwards:</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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">ω </span></p> </td><td style=" 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">: α : ω</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α : ω :</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">α</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="2"> <p style="text-align: left;" 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><span lang="en-US" xml:lang="en-US"> 1 2</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="2"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 2 4</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="2"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span lang="en-US" xml:lang="en-US"> c c′</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="2"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">c c′ c′′</span></p> </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 200</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The third arithmetic proportion upwards and downwards:</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> </colgroup> <tbody><tr><td style=" 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">2α - </span><span lang="el-GR" xml:lang="el-GR">ω</span><span lang="en-US" xml:lang="en-US"> : </span><span lang="el-GR" xml:lang="el-GR">α</span><span lang="en-US" xml:lang="en-US"> : </span><span lang="el-GR" xml:lang="el-GR">ω</span></p> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">0 1 2</span></p> <p style="text-align: right;" dir="ltr"><span lang="en-US" xml:lang="en-US">(</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"> = </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">∞</span><span lang="en-US" xml:lang="en-US">!) </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c c</span><span lang="en-US" xml:lang="en-US">′</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: left;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α : ω : 2ω </span><span lang="en-US" xml:lang="en-US">- </span><span lang="el-GR" xml:lang="el-GR">α</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 2 3</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c c</span><span lang="en-US" xml:lang="en-US">′ </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">′</span></p> </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 201</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The third harmonic proportion upwards and downwards:</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=" 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">_</span><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">α ω</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">_</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2ω - α</span></p> </td><td style=" 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">: α : ω</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α : ω :</span></p> </td><td style=" 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">_</span><span style="text-decoration: underline;" lang="el-GR" xml:lang="el-GR">α ω</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US">_</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">2</span><span lang="el-GR" xml:lang="el-GR">α</span><span lang="en-US" xml:lang="en-US"> - </span><span lang="el-GR" xml:lang="el-GR">ω</span></p> </td></tr><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="2"> <p style="text-align: right;" 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><span lang="en-US" xml:lang="en-US"> 1 2</span></p> <p style="text-align: right;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> c c</span><span lang="en-US" xml:lang="en-US">′</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" colspan="2"> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 2 ∞</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c c</span><span lang="en-US" xml:lang="en-US">′ (</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"> = </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">∞</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></p> </td></tr></tbody></table> <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> </colgroup> <tbody><tr><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α →</span><span lang="en-US" xml:lang="en-US"> √</span><span lang="el-GR" xml:lang="el-GR">αω</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> </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 202</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">The third harmonic proportion upwards:</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=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α →</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + ω</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">1</span></p> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">α</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span><span lang="en-US" xml:lang="en-US"> + ω</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">0</span></p> </td><td style=" padding: 0pt 2.16pt; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"> <p style="text-align: center;" dir="ltr"><span lang="el-GR" xml:lang="el-GR">ω</span></p> </td><td style=" 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="el-GR" xml:lang="el-GR">α</span><span style="text-decoration: underline; vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span style="text-decoration: underline;" lang="en-US" xml:lang="en-US"> + αω</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">3α - ω</span></p> </td></tr><tr><td style=" 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><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </td><td style=" 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><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> </td><td style=" 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><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">′</span></p> </td><td style=" 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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">′</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: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 203</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">This strictly systematic back-and-forth proportioning yields formula 204:</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/29.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 204</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If one uses octave reduction to bring all the tones in this series into the same octave, in their correct scale-arrangement, then the result is the diatonic scale:</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">ˇ </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c d es f g a </span><span lang="en-US" xml:lang="en-US">(</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">ˇ),</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">which to our perception is a </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">B</span><span lang="en-US" xml:lang="en-US">-major (</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">-minor) scale for a generator-tone of </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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">. For the ancient Greeks, however, it was a “Dorian” scale, whose characteristic is that with </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> as the generator-tone, it begins a whole-tone lower and proceeds like a </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">B</span><span lang="en-US" xml:lang="en-US">-major scale, or with </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d </span><span lang="en-US" xml:lang="en-US">as the generator-tone, it begins with </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> and proceeds like </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">C</span><span lang="en-US" xml:lang="en-US">-major. This, of course, only explains it in terms of today's very primitive perception of scales. For the ancient Greeks, and for those nowadays who truly understand “Gregorian chant,” these “church modes” are completely independent sound-forms with different psychical characteristics.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Thimus then continues by differentiating the above proportioning further, and thus comes to a new elicitation of the so-called “enharmonics” of ancient Greek tone-systems that was lost, or at least no longer used, by the later Greeks (Aristoxenos, Ptolemy, etc.): an exquisitely differentiated ratio-construction with deep speculative and symbolic significance.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The reader who, armed with a beginner's background in algebra, calculates the above formulae, works through the proportions, and devotes himself to further related exercises, will certainly appreciate the deeper understanding of the laws of harmonic series-construction thus made possible, despite the apparent schoolbook-like circumstances. In a textbook, these things cannot be ignored. And those for whom such numbers and formulae are an expression of actual relationships, rather than just something external, will share the author's astonishment and wonder at the phenomenon of proportion itself, which is revealed in its innermost nature in the above examples. Especially in the last formula, the inner laws are perceived with redoubled effect. All other readers, though, who follow the suggestion in the preface and skip over the above pages, will be rewarded by the following ektypic excursions.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§28a. Ektypics. Plato</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In his later work </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Timaeus</span><span lang="en-US" xml:lang="en-US">, Plato ascribes to Timaeus the following words on proportion: “two things cannot be rightly put together without a third; there must be some bond of union between them. And the fairest bond is that which makes the most complete fusion of itself and the things which it combines; and proportion is best adapted to effect such a union.”</span><span lang="-none-" xml:lang="-none-" class="footnote_reference">*</span><span lang="en-US" xml:lang="en-US"> And here a definition of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">geometric</span><span lang="en-US" xml:lang="en-US"> proportion is implied: “For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean-then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one.”</span><span lang="-none-" xml:lang="-none-" class="footnote_reference">*</span><span lang="-none-" xml:lang="-none-" class="footnote_reference">*</span><span lang="en-US" xml:lang="en-US"> (Example: </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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span lang="en-US" xml:lang="en-US"> : </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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">, and </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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US"> : </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">1</span><span lang="en-US" xml:lang="en-US">). The fact that Plato described the unity of the geometric proportions playing around the generator-tone line as the “fairest bond” is in itself a proof for the “harmonics” of late Platonic thought, a proof that will be further solidified in the harmonic analysis of the scale of the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Timaeus</span><span lang="en-US" xml:lang="en-US"> (§39).</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It is not claiming too much to refer to the phenomenon of ancient Greek thought and perception as a thought and perception in proportions, an ascendance of the spiritual (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">logos</span><span lang="en-US" xml:lang="en-US">) to a harmonically conditioned interplay of sensuality, in which this proportional relationship (in Greek, in fact, the word for proportion is </span><span lang="el-GR" xml:lang="el-GR">λόγος</span><span lang="en-US" xml:lang="en-US"> = Logos!) plays the role of psychically and spiritually saturated tectonics.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The actual theory of proportion was created by the Pythagoreans. Thimus found the fundamentals for his formulae described above mainly in Iamblichus's commentary on the arithmetic of Nicomachus, and Iamblichus remarked explicitly that the three proportions originated from Babylon. Cantor (I, 167) agrees with this, since arithmetic and geometric series were already known in Babylon and Egypt. For us, however, it is fundamental that the ancients' entire study of proportion was almost always illustrated with the introduction of tone-values; indeed, it was through this that the musical spirit and perception of the Greeks were first made understandable. At the center of this harmonic concept of proportion stood the so-called “harmonia perfecta maxima,” the “tetraktys”:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">6 : 8 = 9 : 12</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">which was revered as something holy. This tetraktys consists of the three proportions:</span></p> <p style="text-align: left; margin-left: 1.25in;" dir="ltr"><span lang="en-US" xml:lang="en-US">6 : 9 : 12 arithmetic proportion</span></p> <p style="text-align: left; margin-left: 1.25in;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f b</span><span lang="en-US" xml:lang="en-US">ˇ </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> <p style="text-align: left; margin-left: 1.25in;" dir="ltr"><span lang="en-US" xml:lang="en-US">6 : 8 : 12 harmonic proportion</span></p> <p style="text-align: left; margin-left: 1.25in;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f c f</span></p> <p style="text-align: left; margin-left: 1.0389in;" dir="ltr"><span lang="en-US" xml:lang="en-US">6 : 8 = 9 : 12 geometric proportion</span></p> <p style="text-align: left; margin-left: 1.0389in;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f c b</span><span lang="en-US" xml:lang="en-US">ˇ </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> <p style="text-align: center; margin-left: 1.0389in;" dir="ltr"><span lang="en-US" xml:lang="en-US">(tones by string-length)</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It produces the most important intervallic building blocks: the octave, fifth, fourth, and whole-tone.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The Greeks reached interesting conclusions from the discovery (or rediscovery) of the concept of proportion, only a few of which we can discuss here.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Irrational</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Firstly, the concept of the irrational numbers, which indeed belong in our continuous geometric proportion </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : √</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">ab</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">, in which the median element is usually “irrational,” i.e. inexpressible in rational numbers. For this reason, this proportion is known as the “geometric,” since it can only be understood and conceived geometrically. The classical example for the irrational is the diagonal of the square:</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/210.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 205</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">which, according to the Pythagorean theorem, yields the value √2, if the square's sides are all 1. E. Frank writes in his book </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Plato und die sogenannte Pythagoreer</span><span lang="en-US" xml:lang="en-US"> (1923, p. 224): “The deep impression that this discovery [the irrational numbers] made upon their time can still be seen in Plato's writings. It pointed to the existence of spatial quantities that could in no way be expressed with rational numbers. This discovery necessitated a complete revolution of mathematical thought, if the theorems of mathematics were to be absolutely universal, i.e. should apply to both rational and irrational numbers. To satisfy this requirement, the Pythagoreans created the new method of plane geometry and the new theory of proportions.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The new method of plane construction provided universal size-ratios, which today we notate algebraically-for example, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</span><span lang="en-US" xml:lang="en-US"> = √</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> or (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> + </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</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="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> + 2</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">ab</span><span lang="en-US" xml:lang="en-US"> + </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">-or geometrically, as in the side of a square or with the so-called “gnomon”; it was a type of geometric algebra. By these methods, the irrational quantities could be handled geometrically, and we find them perfected in Euclid's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Elements</span><span lang="en-US" xml:lang="en-US">. The theory of proportions follows directly from this attempt to conquer the irrational numbers. The irrational number √2 is the “median geometric proportional” between 1 and 2. E. Frank (op. cit. 226) writes further: “The irrational quantity √2 can be brought into an exact mathematic relationship with the whole numbers 1 and 2 by means of this so-called geometric proportion. This proportion (in general: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> or </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">d</span><span lang="en-US" xml:lang="en-US">) is however called “geometric” because the median proportional (√2) can only be found in it through geometric construction; it cannot be found or represented arithmetically through any kind of rational number, nor harmonically as the string-length of a harmonic tone. Through the concept of proportion, then, the relationship of quantities is made understandable to us irrespective of whether they are rational or irrational. On the basis of this (geometric) concept of proportion, the Pythagoreans then reconstructed all of mathematics. This study of proportion was primarily created by Hippasus and Archytas, and brought to conclusion by Archytas's student Euxodus.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Despite indications in the sources, E. Frank made the old mistake of viewing only the harmonic proportion as musically derivable. The ancients analyzed both the arithmetic and the geometric proportions harmonically, as we saw above; the latter, in fact, does not always need an irrational median or medians. In spite of this, it is indeed the irrational quantity √αω that, to a certain extent, controls and holds together the three harmonic proportions as beginning, end, and middle, and as a secret generating “ineffable” center.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> This conclusion, first made possible by harmonic analysis, contradicts the usual opinion of historians wishing to find, in the discovery of irrational numbers, an upheaval of the ancient worldview based only on rational numbers. The exact opposite is the case: the harmonic inclusion of the irrational numbers in the “cosmos” of the being-values arranged between the two infinities (</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">∞</span><span lang="en-US" xml:lang="en-US">-</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">1</span><span lang="en-US" xml:lang="en-US">-</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">∞</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">) strengthens ancient wisdom and perception in the presence of a universal harmony, precisely in that regulating power that grants the irrational numbers their value-emphasized place in the order of things. Of course, the discovery and teaching of proportional methods also had its practical significance. “For the Greeks, the inestimable value of proportions lay in the fact that they presented, with their metamorphoses, a substitute for our equations. This important application explains the extensive treatment given to proportions by the Greek mathematician Euclid (ca. 300 BC), and also by Arabic and Medieval scholars. For a long time in the Middle Ages, and almost up to modern times, when letter notation was finally incorporated into equations, people continued to write the results in the form of proportions, which took the place of our modern closed formulae. Only in modern times has the space allotted to them been severely reduced” (Tropfke: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Geschichte der Elementar-Mathematik</span><span lang="en-US" xml:lang="en-US">, 1902, I, p. 232).</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Fundamentally, every fraction is still a proportion, writes Julius Stenzel (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Zahl und Gestalt bei Platon und Aristoteles</span><span lang="en-US" xml:lang="en-US">, 1924, p. 36):</span></p> <p style="text-align: justify; margin-right: 0.5in; margin-left: 0.5in;" dir="ltr"><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">“The segmentation of important problems of fractional calculation through the so easily aesthetically elicited proportion-</span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">harmonia, logos</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">!-works in a similar way; for the Greeks, </span><span style="font-size: 11pt; vertical-align: super;" lang="en-US" xml:lang="en-US">4</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"> was not a fraction, but the abstract relationship between two relative quantities. We know from Euclid that the Greeks indicated numbers with proportions of segments, but precisely because of this, the pure relation, the indifference to absolute size, must have been a particularly lively problem for them. Once again one can say: in all this lie the presuppositions for Plato's study of the large-small, the </span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">hyle</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US"> [matter] of the extensive, that must occur to the Logos formerly indifferent to size in order for itself to 'come into being,' hence giving it reality and determination in the full sense. The necessary meeting of these two principles-a grasp of the relationship of </span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">logos</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US"> to </span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">physis</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US"> that is actually transcendent, i.e. based on the absolutely necessary mutual relationship of both-remains the core problem of philosophy, for Aristotle as well as for Plato.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">Stenzel writes further in his magnificent work (p. 93): “The manifest sensualization of the spiritual is the theme of all late Platonic philosophy.” Furthermore (p. 125): </span></p> <p style="text-align: justify; margin-right: 0.5in; margin-left: 0.5in;" dir="ltr"><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">“The late Platonic theories do not reveal the pallor of thought; on the contrary: they are the strongest confirmation of Fichte's thesis that the Greeks achieved an exquisite refinement of sensuality much earlier than that of abstract thought. One can say further that the value of their philosophy for all times, especially for today, lies in the fact they were spared from the unavoidable fate of all intellectuality, the atrophy of the sense of sight, or worse, substituting it with unexplained individual intention, because this prominence of sensuality finally led to a symmetry of all spiritual energies, whose purest expression will always remain Plato's and Aristotle's.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From our harmonic viewpoint, we can only agree with these penetrating views. Unfortunately, Stenzel, like all previous philologists, knew only of the periphery of the harmonic backgrounds of Greek thought, and not of the center. The reader who has worked through harmonic fundamentals based on Thimus's studies and this textbook will find hardly anything more depressing than having to look over and work through the forest of often primitive commentary, laboring with insufficient acoustic tools in “dark” numeric-harmonic areas, especially on the fragments of Philolaus, the Pythagoreans in general, and the late philosophy of Plato. If one places all these ideas of the golden age of Greek philosophy upon the background of the concept of akróasis and its norms, its numerically precisely understandable and psychically perceivable laws-a procedure that is not at all arbitrary, but actually encouraged by hundreds of ancient sources-then this “refinement of sensuality,” which Stenzel very rightly detected in those ideas, will become graspable and understandable from a fully new, central position, inwardly adequate to this thought and accessible in the deeper sense to heart and mind. One then sees that Plato did not “come upon the unhappy [!] idea, in his old age, of developing the world of ideas as a number-system” (W. Windelband: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Platon</span><span lang="en-US" xml:lang="en-US">, 1905, p. 93). On the contrary, for Plato, number-harmonics was the bond joining idea and sensuality, and it was Windelband himself who came upon a very “unhappy idea.” In my essay on Pythagoras (in </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Abhandlungen</span><span lang="en-US" xml:lang="en-US">), I attempted for the first time to rectify the most important Pythagorean theorems of akróasis, which indeed place the eye and ear on an equal level with thought. But a harmonic analysis of all Greek philosophy up to around Proclus would require half a life's work; this analysis requires a precise familiarity with the fundamentals of harmonics </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">and</span><span lang="en-US" xml:lang="en-US"> an equally great knowledge of philosophical and natural history, supported by thorough language abilities. I am sure that whoever undertakes this project will achieve results that place all of the ancients' scientific, artistic, and philosophical thoughts and experiences upon a new, central foundation, not as seen by our modern thought but self-radiating from the cosmos of Greek thought.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> After this digression, we now return to our proportions and will note a few more typical examples for the ektypics of harmonic proportioning.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Cube</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The twelve sides, eight corners, and six faces of the cube form a harmonic proportion. Nicomachus and Iamblichus believed that on this basis, the ancients spoke of a </span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">̔</span><span lang="el-GR" xml:lang="el-GR">αρμωνία</span><span lang="en-US" xml:lang="en-US"> </span><span lang="el-GR" xml:lang="el-GR">γεωμετρική</span><span lang="en-US" xml:lang="en-US"> (geometric harmony); because the octave (12 : 6 = 2 : 1), the fifth (12 : 8 = 3 : 2), and the fourth (8 : 6 = 4 : 3) are all present in the cube's ratios. As we saw in §24, geometry, through the significance of harmonic proportion, is anchored in harmonics especially on its projective side. Clavius, in his commentary on Euclid, concentrates completely on a musical proportion, which emerges in a triangle M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> that bisects any triangle A B C and the corresponding segments on the connecting lines of the corner points and the bisection points (Fig. 206).</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/211.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 206</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Draw a triangle A B C of any size, then draw lines from the corners to the midpoints of the sides M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1,</span><span lang="en-US" xml:lang="en-US"> M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2,</span><span lang="en-US" xml:lang="en-US"> and M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">. Join M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1,</span><span lang="en-US" xml:lang="en-US"> M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2,</span><span lang="en-US" xml:lang="en-US"> and M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> together to form a smaller triangle, and indicate the intersection points with the first set of lines with HJK. Then set a compass, for example, to OH, and one will see that if OH is the unit 1, HA is 3 and OM is 2. The analogy also applies for the units OJ and OK. Here are the proportion numbers:</span></p> <p style="text-align: left; margin-left: 2.4674in;" dir="ltr"><span lang="en-US" xml:lang="en-US">OH = 1 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> <p style="text-align: left; margin-left: 2.4674in;" dir="ltr"><span lang="en-US" xml:lang="en-US">OM</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> = 2 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> <p style="text-align: left; margin-left: 2.4674in;" dir="ltr"><span lang="en-US" xml:lang="en-US">HA and HM</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US"> = 3 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> <p style="text-align: left; margin-left: 2.4674in;" dir="ltr"><span lang="en-US" xml:lang="en-US">OA = 4 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> <p style="text-align: left; margin-left: 2.4674in;" dir="ltr"><span lang="en-US" xml:lang="en-US">M</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">A = 6 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> These sectors of the middle line AM</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">, which also apply for the two others BM</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> and CM</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US">, set in relation to one another as string lengths, yield the ratios:</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=" 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">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><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </td><td style=" 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">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><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">′</span></p> </td><td style=" 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">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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">′</span></p> </td><td style=" 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">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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">′′</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">1</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">′′</span></p> </td></tr><tr><td style=" 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">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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> </td><td style=" 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">2</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </td><td style=" 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">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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> </td><td style=" 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">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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">′</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">2</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">′</span></p> </td></tr><tr><td style=" 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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> </td><td style=" 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">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> </td><td style=" 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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </td><td style=" 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">4</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">6</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">′</span></p> </td></tr><tr><td style=" 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">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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> </td><td style=" 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">4</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> </td><td style=" 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">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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> </td><td style=" 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">4</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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">4</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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> </td></tr><tr><td style=" 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">.</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">.</span></p> </td></tr><tr><td style=" 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">1</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,</span></p> </td><td style=" 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">2</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> </td><td style=" 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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> </td><td style=" 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">4</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,</span></p> </td><td style=" 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">.</span></p> </td><td style=" 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">6</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> </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 207</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Here I have written all possible proportions between one another, so that his example shows how the material of our partial-tone coordinates, albeit lacking thirds here, emerges from a very simple notation of various proportional possibilities, with an elementary geometric theorem well known to the ancients.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Tonally, the two fifths (inverted: fourths) emerge with their octaves from this triangle theorem, beside the primal tone and its octave, as well as the large whole-tone from the comparison of </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g </span><span lang="en-US" xml:lang="en-US">to </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f </span><span lang="en-US" xml:lang="en-US">(</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f </span><span lang="en-US" xml:lang="en-US">to </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US">). [Note: the reader who owns </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Hörende Mensch</span><span lang="en-US" xml:lang="en-US"> should be aware of an error on p. 121, line 7 from the bottom, corrected thus: a : (a + b) : (a + b + c).]</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Pythagorean Triangle</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The Pythagorean triangle, with sides 3, 4, and 5, yields a considerably richer psychic harvest.</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/212.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 208</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Since it was traditional for the Greeks to test all segments on the monochord and to set the tone-values obtained in relation to one another, we will set the segment-numbers and their squares together with the corresponding tone-values (since for a right triangle, c</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> = a</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> + b</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span 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> </colgroup> <tbody><tr><td style=" 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">3 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> </td><td style=" 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">4 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,</span></p> </td><td style=" 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">5 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,,</span></p> </td></tr><tr><td style=" 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">9 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">ˇ</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,,</span></p> </td><td style=" 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">16 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,,</span></p> </td><td style=" 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">25 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">fes</span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">,,,,,</span></p> </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 209</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">We will then place these values, reduced by octaves (without octave signatures) analogous to the above, in the relationship</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> </colgroup> <tbody><tr><td style=" 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">9</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="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> <p style="text-align: left;" 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">4</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">ˇ</span></p> <p style="text-align: left;" 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">5</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span><span lang="en-US" xml:lang="en-US">ˇ</span></p> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">16</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="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">16</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="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> <p style="text-align: left;" dir="ltr"><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">16</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="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span></p> <p style="text-align: left;" dir="ltr"><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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">ces</span><span lang="en-US" xml:lang="en-US">ˇ</span></p> <p style="text-align: left;" dir="ltr"><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">4</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">fes</span></p> <p style="text-align: left;" dir="ltr"><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">5</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span></p> <p style="text-align: left;" dir="ltr"><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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> <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">5</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">a</span></p> <p style="text-align: left;" dir="ltr"><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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> <p style="text-align: left;" dir="ltr"><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"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span></p> <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">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">es</span></p> <p style="text-align: left;" dir="ltr"><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">4</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span></p> </td><td style=" 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">* </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">9</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span></p> <p style="text-align: left;" dir="ltr"><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">16</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span></p> <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">25</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">cis</span></p> <p style="text-align: left;" dir="ltr"><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">9</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span></p> <p style="text-align: left;" dir="ltr"><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">16</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span></p> <p style="text-align: left;" dir="ltr"><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">25</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">gis</span></p> <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">9</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span></p> <p style="text-align: left;" dir="ltr"><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">16</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">as</span></p> <p style="text-align: left;" dir="ltr"><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">25</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">e</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">* </span><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">16</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">ˇ</span></p> <p style="text-align: left;" 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">25</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">fis</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">* </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">16</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">9</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">d</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">* </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">16</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">25</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">gis</span></p> <p style="text-align: left;" dir="ltr"><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">9</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">ges</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">* </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">16</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">fes</span></p> </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 210</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If we now eliminate the duplicates (*) and arrange the remaining tone-values in scales, the result is a nicely differentiated chromatic scale:</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/213.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 211</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> One can naturally explain this result by means of the permutation possibilities of the triadic ratios 2, 3, and 5, and say that any triangle would produce the same result with triadic ratio sides reduced by octaves. However, we are morphologists, and we think in terms of the form. If this scale emerges within a typical limit such as that of the Pythagorean triangle 3 : 4 : 5, that is reason enough for anyone who has even a minimal understanding of morphological laws to pay close attention. This is an optically viewable figure of a highly characteristic quality, namely a triangle, which satisfies the Pythagorean theorem with the smallest whole numbers possible; thus behind it lies an exceptionally normative configuration. The fact that calculating this yields a chromatic scale-admittedly not tempered, but with the finest enharmonic variants-should be food for thought for those harmonic hobbyists who tinker so long with some arbitrary numbers that a chromatic scale arises from them like a phoenix from the ashes, or who manage without a normative proof from the outset, which the cautious among them still do today, precisely because they do not know of such a proof. (Note: 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. 116-117 frequencies are used as a basis instead of string-length for analysis of the Pythagorean triangle; the result is naturally the same as here.)</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Newton</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Here we offer the reader an excerpt from Thimus (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Symbolik</span><span lang="en-US" xml:lang="en-US"> I, 48ff.) showing that the great Newton himself was interested in number-harmonic problems. This is what Thimus wrote:</span></p> <p style="text-align: justify; margin-right: 0.5in; margin-left: 0.5194in;" dir="ltr"><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">“The discoverer of the law of gravity directed his attention especially to the musical number-ratios of ancient times, and did not fail to connect his speculations on the relationships between the laws of art and the mathematical laws of nature with the observation of laws of musical harmony after the manner of the ancients. A young scholar, John </span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">Harington</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">, who occupied himself with subject matter relevant to this, had devised a compilation of those harmonic numbers which, on the basis of the 1</span><span style="font-size: 11pt; vertical-align: super;" lang="en-US" xml:lang="en-US">st</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US"> book of the Elements of Euclid, theorem 47, could be found in the right triangle from the measure of the sides of the square of the hypotenuse and the other two sides, if the three sides of the triangle have the ratio 3, 4, and 5. The ratios found in this manner through geometric-linear construction were as completely developed and as musically rich as those that furnished the geometric figure of the so-called Helikon contrived by Ptolemy. Ptolemy left out the thirds and sixths. The right-angle triangle also provided the numbers at the basis of these intervals in the linear measure. </span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">Harington, in his letters to Newton</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US">, had mentioned how strange it would be in principle that the ancients should have remained unaware of the ratios of the two thirds and sixths, in other words the arithmetic and harmonic divisions of the interval of a fifth, in which the pivotal point for the modern tone-system lies [we could substitute any naturally developed tone-system]. Here he also mentioned mystical Biblical numbers and emphasized how the ratios for the triple octave of the major thirds (300 : 30 = 10 : 1) and for the major sixth (50 : 30 = 5 : 3)-the inversion of the minor third-could be found in the numbers described in Genesis as the measures of the Ark: 300 · 50 and 30, expressed as string lengths. He asked Newton's opinion of this, and also stated the view that in architecture the art-forms that were pleasing to our senses might well have their deeper reasons in a definite agreement of the chosen measure with the harmonic numbers of musical intervals. Newton's reply to the young scholar's communication can be found in Hawkins: </span><span style="font-style: italic; font-size: 11pt;" lang="en-US" xml:lang="en-US">General History of the Science and Practice of Music</span><span style="font-size: 11pt;" lang="en-US" xml:lang="en-US"> (London 1776, vol. III, pp. 142-143). It was only because Newton was too busy with his multifarious other occupations that he did not pursue the subject of the communication and the cue therein and did not produce a more detailed work. He said that earlier he had occupied himself with investigations of a similar type, and would like to have the leisure in future to apply himself to the same. Meanwhile, he would continue contemplating this subject-matter; for these were serious investigations which showed how great is the simple beauty in all works of the Creator. Moreover, pleasure in the arts requires the closest possible approach to simple, concise forms, and this requirement became, from his point of view, most fulfilled in the visual arts, the more the chosen ratios approached the harmonic ratios. In architecture, arc segments present a more or less pleasing image, according to whether the emerging curves give the observer the idea of the law on which they are based. He assumes that Harington would have consulted Kepler's, Mersenne's, and the other authors' writings about the matter. Newton goes on to say that what Harington said about the inconceivability of the ancients' ignorance of the ratios of both thirds is quite right. It would be very surprising that such highly talented mathematicians should not have discovered in such number observations that if the ratio of the fifth 3 : 2 does not permit division in this form of its expression [through whole-number interpolation], however it immediately becomes possible with the exchange of the last term in the same ratio 6 : 4; now [through interpolation of the five] the ratios yield both the major third 5 : 4 and the minor third 6 : 5, in which the diatonic system first reaches completion, and without which the tone-system of the ancients must have been very incomplete. It appears amazing that those who brought their inquiries to a conclusion in such a minute way had not gone to work more precisely in the investigation of the larger intervals, whose divisions, just mentioned, are so fruitful for modern harmonics. Furthermore, he would be inclined to accept that certain universal laws laid down by the Creator would be authoritative for the pleasantness or unpleasantness of all our various sense-impressions; at least such an assumption would not in any way infringe upon the belief in the wisdom or omnipotence of God, but would stand much more in unison with the simplicity showing itself everywhere in the macrocosm. The letter is dated from May 30, 1693 and closes with the assurance that all further results of the investigations into the subject of the communication would interest Newton to a high degree. Newton, in his optics, based the exposition of his theory of the prism's graduated breaking up of the various color-rays upon a comparison of the graduations with the ratios of the Doric diatonic scale. This will be discussed later, and will be the subject of detailed disquisitions with which we intend to conclude our continuing investigations of harmonics and the study of ancient number-theory in the second part.”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Unfortunately, the “second part” mentioned by Thimus, i.e. the third volume of his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Symbolik</span><span lang="en-US" xml:lang="en-US">, was never published, and so far the search for the supposedly existing manuscript has proved fruitless. The letter from Newton to Harington speaks for itself. If such a great spirit was not afraid to take number-harmonic investigations seriously, and indeed to direct his special attention to them, then this is a challenge and incentive for us today to give a new foundation to the whole field of harmonics and to draw the corresponding conclusions.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Geometric Harmonics</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Further on in the domain of elementary geometry, there is a wealth of typical harmonic features, especially in the so-called “curious properties of the triangle,” the sections determined by them, and so forth. This is actually expressed in the word “chordal” (from </span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">̔</span><span lang="el-GR" xml:lang="el-GR">η</span><span lang="en-US" xml:lang="en-US"> </span><span lang="el-GR" xml:lang="el-GR">χορδή</span><span lang="en-US" xml:lang="en-US"> = the string), and “hypotenuse” (from </span><span style="font-family: 'Tahoma';" lang="en-US" xml:lang="en-US">̔</span><span lang="el-GR" xml:lang="el-GR">υποτείνω</span><span lang="en-US" xml:lang="en-US"> = to span, namely a string). The beginnings of the function-concept and the seedlings of analytical geometry are also based on the ancient concept of proportion, whose innermost nature the ancients knew how to derive harmonically and substantiate, as we have seen. The formula for changing a rectangle into a square, which we write today as the formula </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</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="font-style: italic;" lang="en-US" xml:lang="en-US">ab</span><span lang="en-US" xml:lang="en-US">, was expressed by the Greeks in the proportion </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x </span><span lang="en-US" xml:lang="en-US">= </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">x</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">, etc. There is more on this in the textbooks on the history of mathematics cited in the bibliography for this chapter.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">The Imaginary Number</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> i</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> It is highly interesting to note the sudden emergence of the imaginary number </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">i</span><span lang="en-US" xml:lang="en-US"> when the two division ratios of a harmonically constructed segment are set equal.</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Geometry/214.jpg" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 212</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> As we know, in the harmonically divided segment, AB and PQ,</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=" 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">AP</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">PB</span></p> </td><td style=" 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=" 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">AQ</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">QB</span></p> </td><td style=" 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">and</span></p> </td><td style=" 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">PA</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">AQ</span></p> </td><td style=" 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=" 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">PB</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">BQ</span></p> </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 213</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">i.e. the segment AB is divided inwards and outwards in the same ratio by PQ, likewise PQ by AB, but only each one in itself; the two ratios are not equal. This yields the numeric proof</span></p> <p style="text-align: justify;" 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">6</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">6</span><span lang="en-US" xml:lang="en-US"> = </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></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> therefore 2 and </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></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 214</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">One can now calculate one ratio with the other. If one writes</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=" 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=" 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">AP</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">PB</span></p> </td><td style=" 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">and y =</span></p> </td><td style=" 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">PA</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">AQ</span></p> </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 215</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">and then continues this with BP,</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=" 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">y =</span></p> </td><td style=" 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">PA</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">BP</span></p> </td><td style=" 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=" 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">BP</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">AQ</span></p> </td><td style=" 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=" 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">BP</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">AQ</span></p> </td><td style=" 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=" 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">_x_</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">AQ</span></p> </td><td style=" 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">(AQ - BQ - AP)</span></p> </td></tr></tbody></table> <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> <col></col> <col></col> <col></col> <col></col> </colgroup> <tbody><tr><td style=" 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=" 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: 30pt;" lang="en-US" xml:lang="en-US">(</span></p> </td><td style=" 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=" 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">BQ</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">AQ</span></p> </td><td style=" 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=" 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">AP</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">AQ</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">)</span></p> </td><td style=" 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=" 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: 30pt;" lang="en-US" xml:lang="en-US">(</span></p> </td><td style=" 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=" 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</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">x</span></p> </td><td style=" 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">- y</span></p> </td><td style=" 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: 30pt;" lang="en-US" xml:lang="en-US">)</span></p> </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 216</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Therefore, </span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">y = x - 1 - xy or y + xy = x - 1</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">from which it follows that </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=" 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">y =</span></p> </td><td style=" 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">x - 1</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">x + 1</span></p> </td><td style=" 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">or y + 1 = x - xy</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">and thus </span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"><br /> </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> </colgroup> <tbody><tr><td style=" 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=" 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 + y </span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 - y</span></p> </td></tr></tbody></table> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">If we now set the two ratios equal, i.e. set x = y, then</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=" 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=" 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">x - 1</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">x + 1</span></p> </td><td style=" 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><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US"> = - 1</span></p> </td><td style=" 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 = y = √-1 = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">i</span></p> </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 217</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From all these examples, and those in §24, it is undoubtedly evident that seen historically, the fundamentals of the “new” projective geometry, the germ of the function-concept, analytical geometry, the concept of the infinite, the irrational, and the imaginary (the number </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">i</span><span lang="en-US" xml:lang="en-US">), are all based on a common denominator of the universal concept of proportion. The fact that the Greeks did not yet need our modern formulae makes no difference regarding the actual knowledge that they obviously possessed about these things. The common view of Greek thought is that it was directed only towards the finite, the “rational”; but for us, from the harmonic standpoint, the derivation and basing of all these very “modern” concepts upon that of harmonic proportion is of exceptional importance. This shows that the concept of the ratio in itself, which indeed is most distinctly characterized in the interval, has a constructive significance in the whole structure of mathematics.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Proportion as Value-Form</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If we grasp the nature of proportion in the value-formal sense, as I tried to do in my </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Grundriß</span><span lang="en-US" xml:lang="en-US"> (p. 277ff. and 280ff.) under the value-forms of “ratio-equalization” and “being-proportion,” this meaning becomes almost universal. The reader should use his own knowledge and experience, and the parts of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Grundriß</span><span lang="en-US" xml:lang="en-US"> just mentioned, to make up for the lack of space to discuss further ektypic examples here. The “golden section,” harmonically a third-sixth problem, was specially discussed in my </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonia Plantarum</span><span lang="en-US" xml:lang="en-US">, p. 148ff. In §29.1 and §55.8 we will discuss this further.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From our human viewpoint, the concept of the “relationship” of humans to one another is nothing other than proportions of psychic individualities, whose “tones” must or should be tuned to one another so as to somehow satisfy the proportional relationship, not in a mathematical but in a value sense. Anyone who is comfortable in a circle of friends knows well enough how someone who does not belong in the circle “disturbs” it, which says nothing about the disturber as an autonomous being-value, but simply affirms the fact that the previously existing proportion is no longer in tune when a foreign element is introduced. But a thought that is new, unfamiliar, objectionable, also acts as a power in the previously existing spiritual proportion, and must either adjust to the existing elements (if it is too weak) or else, by the strength of its own value, rectify the other elements until they enter into a new proportion with it.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">J.J. Rousseau</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> J.J. Rousseau, in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Contrat Social</span><span lang="en-US" xml:lang="en-US">, Book 3, Ch. 1, imagines the relationship of ruler (president), government, and people in the form of a geometric proportion. “One can clarify the relationships set forth with the image of a continuous proportion (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">), the outer terms </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> and </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> representing the people as ruler and the people as a crowd of individuals, while the government is the geometric mean </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">. The government receives orders from the ruler and gives them to the people, and thus the state is in a correct balance. Equality must exist between the product of the government's power times itself (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">) and the product of the ruler's power over the people times the power of the individual (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> · </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US">). (From </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span lang="en-US" xml:lang="en-US"> : </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> follows </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a</span><span lang="en-US" xml:lang="en-US"> · </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US"> = </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">b</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">2</span><span lang="en-US" xml:lang="en-US">.)” (Quoted from A. Speiser: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Klassische Stücke der Mathematik</span><span lang="en-US" xml:lang="en-US">, Zürich, 1925, p. 105.)</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The domains that emerge in a new light upon the background of harmonic proportions will be discussed later. The first, “harmonics and architecture,” will be the subject of the next chapter. The proportions of the human form will be brought into connection with the “sound-image” in §38, and proportional relationships in the planetary system will appear in §41 with the discussion of the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Timaeus</span><span lang="en-US" xml:lang="en-US"> scale; further harmonic theorems must be discussed beforehand.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§28b. Bibliography</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> A. von Thimus, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Symbolik</span><span lang="en-US" xml:lang="en-US">, preface x ff., I, 118ff., 196ff., II, 32ff. H. Kayser, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Hörende Mensch</span><span lang="en-US" xml:lang="en-US">, Ch. 2; </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Klang</span><span lang="en-US" xml:lang="en-US">, 20, 21, 36, 37, 51-2, 142, 145; </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Grundriß</span><span lang="en-US" xml:lang="en-US">, 103, 155, 287. On mathematical proportions, see the familiar textbooks. For a new discussion of Greek thoughts and perceptions from the standpoint of akróasis, indispensable initial literary tools are: 1) A. von Thimus, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Symbolik</span><span lang="en-US" xml:lang="en-US">; 2) Diels' </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Fragmente der Vorsokratiker</span><span lang="en-US" xml:lang="en-US">; 3) </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Grundriß</span><span lang="en-US" xml:lang="en-US">, vol. I; 4) E. Frank, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Plato und die sogenannte Pythagoreer</span><span lang="en-US" xml:lang="en-US">, 1923; 5) J. Stenzel, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Zahl und Gestalt bei Plato und Aristoteles</span><span lang="en-US" xml:lang="en-US">, 1924; and 6) the author's fundamentals of new harmonics compiled in his works, especially his essay on Pythagoras in </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Abhandlungen</span><span lang="en-US" xml:lang="en-US">. Aristides Quintilianus's </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">De Musica</span><span lang="en-US" xml:lang="en-US">, fundamental for the entire “atmosphere” of Pythagorean musical philosophy, has been excellently translated into German by R. Schäfke (Berlin 1937). Regarding Frank, the first thing to observe is that the main value of his book is in its wealth of dates and references, and in the fundamentally correct assessment of harmonics in Greek thought. As far as I can see, even specialists reject his quirk of proving that the “so-called” (!) Pythagoreans were mythical. Stenzel's work is valuable as an excellent collection of sources; it also does justice through many subtle and deep thoughts to the assessment of the “harmonic” side of Plato's later philosophy in its numeric-spiritual aspect. But neither Frank nor Stenzel knew that this actually had a technical harmonic background, though Thimus had brought it to light long before them, and for these reasons they often labor vainly with certain things that a harmonic analysis could make very simple. But precisely for this reason, their books are more important for us harmonists; once one has studied through them, one often has the feeling that many false conclusions, incorrect interpretations, and misunderstandings in certain places can be rectified by the single concept of tone-number, not to mention the various harmonic theorems themselves, which often shed a brilliant light in the darkest places.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> But above all, the harmonically versed philologist should make a new foray into ancient mathematics, metrics, music theory, and natural philosophy. It seems to me that there is a wealth of harmonic relics here, in manuscripts or in published but little-known authors, with which history not yet known where to start, simply because no one has been able to arrange them into any “valid” science-and thus no one has been interested in them.</span></p> <p> </p> <div> <p style="text-align: left;" dir="ltr" awml:style="Footnote Text" class="footnote_text"><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US">* Plato, Timaeus, 31, tr. by Benjamin Jowett.</span></p> <p style="text-align: left;" dir="ltr" awml:style="Footnote Text" class="footnote_text"><span style="font-size: 12pt;" lang="en-US" xml:lang="en-US">** Timaeus, 31-32.</span></p> </div></div></td></tr></tbody></table></span></div></div> </body> </html></div>
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