<div id="LayerE10" style="visibility: visible; overflow: visible; z-index:0; left:10px; top:5px; position:absolute; "><?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> <head> <!-- ======================================================= --> <!-- Created by AbiWord, a free, Open Source wordprocessor. --> <!-- For more information visit http://www.abisource.com. --> <!-- ======================================================= --> <meta http-equiv="content-type" content="text/html;charset=UTF-8" /> <title>Harmonics & Music Theory - Interference.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; } *.header { } *.caption { font-size:10pt; font-weight:bold; margin-bottom:6pt; margin-top:6pt; } --></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">§10. INTERFERENCE</span> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§10. Interference</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Interference is generally understood to mean a reciprocal action of concurring waves. If wave peaks meet, there is a strengthening; if wave troughs meet, there is a weakening; if peak and trough (at equal height or depth) come together, they cancel each other out. This phenomenon applies in the same way to waves in water, sound, and light (electricity), thus it must be inherent in matter.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> We can distinguish three different types of interference: 1) the meeting of simple vibrations in proportion to the first whole numbers; 2) the meeting of vibrations of equal phase; and 3) the meeting of vibrations of nearly equal phase.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> No. 1 is used in physics textbooks to explain the origin of a musical sound. If A, B, and C are different partial vibrations of a string (see Fig. 23), then the sum of these three “tones” is the “sound” D: in the case of acoustics, one with a dominant fundamental tone and, in the background, the two first overtones.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/0.jpg" /></p> <p style="text-align: center; margin-bottom: 6pt; margin-top: 6pt;" class="caption" awml:style="Caption" dir="ltr"><span lang="en-US" xml:lang="en-US">Figure 23</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Construction of the Sine Curve; “Amplitude”; Construction of the Resultant</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> We can learn two things from Fig. 23. Firstly, the geometrically precise construction of these and all following waves or curves, called “harmonic” or “sine curves.” We divide a circle with radius equal to the width of the wave (= amplitude) into however many parts we want (here 12); we draw a horizontal line through the diameter and extend it beyond, then divide it into the same number of parts; then, by drawing vertical and horizontal parallel lines, we construct the points that delineate the curve. If we want a narrower wave (musically: a higher tone), we make the horizontal axis of the wave shorter, still dividing it into the same number of equal parts (the circle's division also remaining the same), as B and C show, compared to A (Fig. 23). If, on the other hand, we want a greater or smaller amplitude, then of course we must enlarge or reduce the radius of the circle accordingly. The precise construction of the resultant D (Fig. 23) arises from the addition of the heights of the curves above the horizontal axis and the subtraction of those below it, as the analysis of a single point on the curve (a + b + c) will show; by this method the reader can calculate all the other points on the resultant curve himself.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Helmholtz defined Fig. 24 (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Tonempfindungen</span><span lang="en-US" xml:lang="en-US">, 6</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> ed., p. 265) in the narrower sense of his time as “interference,” while he designated Fig. 25 as “wavering.” Fig. 24, the meeting of equal vibrations (equal pitches) can go through endless phase shifts, which however fall between two limiting cases (Fig. 24, from Helmholtz).</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/1.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 24</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the first limiting case, both waves proceed in exactly the same way and add up to the resultant wave form 3. In the second, both waves are displaced by a half phase and add up to resultant 4, i.e. they cancel each other out, whereas before they doubled each other. Obviously, innumerable in-between stages are possible between these two limiting cases, 3 and 4, depending on where one places the beginning of the second wave, i.e. depending on the phase shift.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The “wavering” can be visualized in the following way. The wave trains A and B (Fig. 25 from Scheminski)</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/2.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Figure 25</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">show only a small phase difference. As tones they would sound “impure” to us, thus if A were the note </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">, B would represent a slightly lower </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">c</span><span lang="en-US" xml:lang="en-US">. Now if we add A and B, as above, we get the resultant C, which, translated into tonal terms, creates a tone that sometimes sounds loud, at other times soft; thus exactly expressing what is meant by “wavering.” (See the acoustic textbooks cited in §6 for the acoustic-experimental demonstration of these three types of interference, the discussion of which would take us too far afield.) For present purposes, the resultants of various coincident vibrations should be clear from the examples just given. These resultants interest us both as phenomena in themselves and with regard to their formal realization. If we understand in the broadest sense the factuality of all realizations whose separate stages we designate as “being-values” (see §11), and assign a “tone,” i.e. an individual phase, to each being-value, then the material aspect of this tone, the tone-number of its vibration, is always in a position to cause an interference with the vibrations of other being-values. However, since “vibration” = frequency is only the material expression of a value-the tone-value-that is closely connected with it, the principle of interference extends to become a psychic value-form of universal significance.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> All being-values have specific interference relationships to one another with regard to both their material and their psychic-spiritual realizations. To determine these fixed relationships individually would be a matter of eliciting the tone-numbers of the individual being-values. Here we come to the general principle. Regarding formal realization, as already characterized by the resultants of the three previous illustrations, one should examine the following illustrations.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/3.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 26</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Fig. 26 (from F. Auerbach: </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Physik in graphischen Darstellungen</span><span lang="en-US" xml:lang="en-US">, 1912, p. 64) shows the resultants of two simple vibrations (frequency 1 : 2), each in two different amplitudes and phase differences.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/4.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 27</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Fig. 27 shows the resultants of each of two vibrations of different frequencies, joined in pairs but of equal amplitude. Arrows are drawn pointing to the places where the two waves coincide, and where the form of the resultant is also repeated.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/5.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 28</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Fig. 28 shows the waves </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">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">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">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">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">4</span><span lang="en-US" xml:lang="en-US"> with their resultant; here the amplitudes are drawn in their greatest width, i.e. as arcs of a circle. The resultants are correspondingly far-reaching. The resultant of the second order emerges automatically in drawing, tracing only the sum of the 3</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">rd</span><span lang="en-US" xml:lang="en-US"> and 2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">nd</span><span lang="en-US" xml:lang="en-US"> waves, while the resultant of the first order traces the curve of the sums of all three waves.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/6.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 29</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Fig. 29 shows the wave divisions </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">, </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">, 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">5</span><span lang="en-US" xml:lang="en-US"> with the resultant. Here, in contrast to Fig. 28, a narrow amplitude is chosen, so as to make clear the correspondingly flatter interference curve that results. The reader is invited to draw as many varied diagrams as possible of the above type on millimeter paper, following the method of Fig. 23 (addition or subtraction of the amplitudes), preferably at intervals of 1 cm, so as to get an accurate picture of the curves. The drawings of these interference curves (resultants) are pleasing in appearance, and convey the lively and manifold relationships of concurring wave domains better than simply looking at the illustrations.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> As an overall result, the inner dynamic is just as important as the outer formal tendency of interferences. Every resultant has a high point and a low point, between which rising and falling sections move with their own forms. Even for a “non-rising” collective of waves, such as in Fig. 30, this condition is undisputed.</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/7.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 30</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> This condition, which we first witness inwardly as a dynamic impulse, verve, ascent, etc., and as exhaustion, dwindling, descent, has a physical equivalent in its graphic image, i.e. in its formal realization. In physics, interference is merely a sum or difference of haptic wavelengths. But harmonics can “listen” more deeply here, and also allow perception to speak, because the concern here is a harmonic primal phenomenon, the cooperation of vibrations with their value domains.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§10a. Ektypics</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The following ektypic examples will show how the phenomenon of interference works in various applications: </span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§10a.1. Mountain Growth</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 1. When one examines these interference curves in their graphic form, their extraordinary similarity with the profiles of mountain peaks is immediately noticeable. This admittedly seems like a superficial analogy; however, it becomes considerably stronger when we consider how a mountain range comes into being. Among the various theories of mountain growth, of which hitherto not a single one has enjoyed overwhelming acceptance, we shall mention “Vulcanism,” which was especially debated during Goethe's time; and the Wegener theory, widely prevalent today, which attributes the rising of mountain ranges to the collision of two lighter continental masses floating upon the heavy fluid core of the earth. Couldn't mountain building be just as well imagined as the crash of a wave, either from the depths of the fluid core of the earth or occurring from outside, at a given place on the malleable crust of the earth? It is known that the crust of the earth, previously thought of as solid, has a similar relationship to the fluid core as an apple's skin has to the apple itself. The idea that a sharp impulse occurs, from within or without, at certain points on the earth, in consequence of which a group of waves radiates whose interference or resultant then shapes the earth's malleable crust into mountains, is thus not at all absurd and would merit geological investigation-all the more since this idea is based on unobjectionable physical principles (wave theory). Geologist should not be irritated by the fact that the harmonist is primarily interested in the tonal, i.e. psychic impulse behind the wave group; this is the concern of harmonic research. One should look back at Fig. 30 in light of this. At point A an impulse is triggered in three waves of different lengths (2, 3, and 6) and different amplitudes. These waves cause an interference in the resultant (the solid line), which would correspond to a very irregular, jagged mountain range profile, of which there are indeed many. Upon comparing this with the resultants of the previous illustrations, and with whatever illustrations the reader has drawn himself, one cannot </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">a priori</span><span lang="en-US" xml:lang="en-US"> dismiss the possibility that mountain growth has may represent a highly rich sound synthesis.</span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§10a.2. Rilke's “Primal Noise”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> 2. In Rainer Maria Rilke's Collected Works (vol. IV, 1930, pp. 285-294), there is a remarkable prose fragment entitled “primal noise” [</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Ur-Geräusch</span><span lang="en-US" xml:lang="en-US">]. Here, Rilke reports on the great effect that the invention of the gramophone had on him during his youth, and how, for him, “each of the waltzes” remained “carved figures” of this “independent sound which we set down and then outwardly preserve.” Fifteen years later, while studying anatomy during his years with Rodin in Paris, his zeal led him to acquire a human skull; often he would gaze at it, especially at night, “in the often so strangely wakeful and inviting light of the candle.” One night, he was suddenly entranced by the cracks in the cranial suture in the skull, “and then I knew what they reminded me of: one of those recorded tracks, as they are carved in a small wax cylinder with the point of a small needle!” Rilke then proposes timidly, hesitatingly, almost fearfully, the idea of placing a gramophone needle upon one of these skull sutures: “What would happen? A tone would have to emerge, a melody, a music ... feelings-but which? incredulity, awe, fear, reverence-which of all possible feelings is it that prevents me from suggesting a name for this primal sound that would then come into being...”</span></p> <p style="text-align: center;" dir="ltr"><img alt="" src="../kayser2/imagelibrary/Music/8.jpg" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 31</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Observe Fig. 31. Here are five different waves of randomly-chosen frequencies and amplitudes drawn together (shown for simplicity by zigzag lines). The interference resultant is highlighted by a bolder line. Imagine these resultants correspondingly reduced in size, and compare them with a skull suture! Would this not show the deeper foundation of interference in the harmonic value-form, just as Rilke anticipated in another analogy, in the idea that the waves and rhythms of growth form the strange graphics of the skull sutures as a bundle of psychic tone waves, and arrange them synthetically as interference curves? Besides this “Utopian” idea, however, Rilke's essays contain other ideas of importance for the harmonist: ideas already touched upon in the Introduction, which suggest, as Katharina Kippenberg remarked in her magnificent book on Rilke (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Rainer Maria Rilke</span><span lang="en-US" xml:lang="en-US">, 1938, p. 252 and note 18), that Rilke “would have been enthusiastically open” to harmonic research had he been acquainted with it. Rilke was struck (Gesammelte Werke IV, p. 291) by “how unequally and separately the modern European poet is served by these informants [the five senses], of which virtually one alone, sight, glutted by the world, consistently overpowers him; how paltry is by comparison the contribution made to him by the inattentive sense of </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">hearing</span><span lang="en-US" xml:lang="en-US">, not to mention the lack of involvement of the other senses ... The question emerges here, is the work of research really able to widen the extent of these sectors in the areas that we choose? Does the acquisition of the microscope, the telescope, and other such contraptions that extend the senses up or down, really transport them to </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">another</span><span lang="en-US" xml:lang="en-US"> level? Because most of what is gained thereby cannot be penetrated by the senses, and thus cannot be truly 'experienced.'”</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From H. Friedmann (</span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Die Welt der Formen</span><span lang="en-US" xml:lang="en-US">, 2</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">nd</span><span lang="en-US" xml:lang="en-US"> ed., 1930), we know that the one-sidedness of poetry and research, rightly deplored by Rilke, is actually attributable not to the tyranny of the sense of sight, but to the sense of touch (haptic), which especially “haptifies” the eye, i.e. demotes it to impressions of an exclusively material kind. The fact that the sense of hearing, heretofore completely neglected especially in its scientific aspect, can now build a bridge to a “lived experience,” and the reasons why this is possible, have been extensively explained in the Introduction and in many of the author's previous harmonic works. It is the purpose and goal of harmonics to point out correspondences in which only the poet's intuition still dares to see analogies (e.g. those of tonal graphics to the cranial sutures). Analogy is deepened through harmonic cognition into a psychic correspondence, precisely because it does not dispense with the material basis (the tone-number) and can be pursued with scientific impartiality. </span></p> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">§10b. Bibliography</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Besides the cited literature, the works cited in §6b will orient the reader regarding acoustics; on harmonic geology, see H. Kayser, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Tagebuch vom Binntal</span><span lang="en-US" xml:lang="en-US"> (in </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Abhandlungen</span><span lang="en-US" xml:lang="en-US">).</span></p></div></td></tr></tbody></table> </span></div> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"></span></div> <p style="text-align: left;" dir="ltr"></p> </div> </body> </html></div>
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