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Hans Kayser's Textbook of Harmonics - Excerpts §34. Tone Spirals And Tone Curves
A Translation Society Edition
By Hans Kayser
§34. Tone-Spirals And Tone Curves
In §27, we discussed the three characteristic curves of the second degree (parabola, hyperbola, and ellipse) within the configuration of the “P”, i.e. using as a basis only the plane “P” system or its quantitative and logarithmic numbers, to which values naturally always correspond.
§34.1. Tone-Spirals on the Basis of String Lengths and Frequencies; the Decimal Tone-Spirals
We will now examine a few typical curves that are encountered in the angle (vector) diagrams of the “P”.
We already know these decimal tone-spirals from §33.3 and §33a. (I have called this type of tone-spiral “decimal” and the one based on tone-logarithms “logarithmic,” but this can easily lead to errors in terminology—see §21.) We will now recapitulate them in Figures 290 and 291, in two variations: Fig. 290 with ratios according to string lengths and Fig. 291 with ratios according to frequencies. If we create such reciprocal diagrams, we must maintain some kind of order. Here, the common element is established as the progression of tone-steps upwards within 1 octave (= the circle), going clockwise from 360° = 0°. Thus, the two spirals necessarily go in opposite (reciprocal, mirror-image) directions; their forms are exactly reversed. Not so with the geometric intervals of the tone-steps. Here, in the string length diagram, the direction of the diminution (the interval shortening) is clockwise, from left to right; in the frequency diagram, it goes counterclockwise, from right to left. We know that this diminution is the characteristic element of the law of harmonic quantization, and we find it, among other things, in dividing the monochord, where the division steps grow continually closer together as they ascend. The question now is: which type of diminution is in agreement with the diminution of the string length, if we think of the circle’s circumference as a monochord? Clearly, we must now use as a basis the diagram of the string length spiral, Fig. 290 (which emerges from the comparison with Fig. 274), where we find, for example, the note e in the second circle of index 5 at the correct place of string division (5/5c 0°, 6/5a 72°, 7/8xfis 144°, 8/5e 216°), i.e. at 216°. An instructive overview of the reciprocal relationship of the above diagrams 290 and 291 is given in Fig. 292, where we have noted the tones es, e, f, g, as, and a. The reciprocity is very noticeable here, as well as in the intervals, the corresponding angles, and their differences.
As for the mathematical character of these tone-spirals, it is based on an Archimedean spiral. Which variant we use (string length or frequency) depends, regardless of their autonomous meaning, on how we can use them for ektypic analyses. The angles of both variants are noted in the table of ratios at the end of this book; other tables show only the frequency angle, since we are mostly working with frequencies.
§34.2. Tone-Spirals on the Basis of Logarithms (the Logarithmic Tone-Spiral)
In anticipation of the next chapter, we now discuss the logarithmic tone-spiral. Comparing it with the decimal tone-spiral allows us to see its differences and peculiarities properly. Refer to Fig. 293 for the following. Since the circles, distances, and angles here correspond not to the quantitative sizes of string lengths and frequencies, but instead to the qualitative tone-values, the octave circles 1/1c2/1c¢ 4/1c¢¢ 8/1c¢¢¢ ... must be equidistant; because, indeed, we hear the octaves as tone-spaces of equal distance. The tone-angle is calculated according to the formula at the bottom left in Fig. 293. The distances between the remaining tone-circles are always between 0 and 1000 (with 3 logarithmic spaces) and can most easily be indicated with millimeter paper as an underlay, using 10 cm for each octave 0–1/1–2/1 etc. Three octaves are sufficient, and just one for the position of the angle, as for all polar diagrams. However, several octave circles have the advantage that they produce several rotations of the spiral, and thus show their physiognomy more clearly. As for the division of tones on the circular periphery of the logarithmic tone-spiral and the logarithmic polar diagram, this is oriented according to “psychical” distances, i.e. according to intervals as we hear them, not as we count them. The “perspective” element of diminution falls away here, and the eye sees the intervals distributed in the same way as the ear hears them. It is interesting that this “logarithmic tone-spiral” is actually not a logarithmic spiral, but an Archimedean one; therefore we must differentiate it from “logarithmic spirals” in the purely mathematical sense. We will discuss this further in the next section.
§34.3. The Tone-Curves of the Polar Arrangement
In sections 1 and 2 we constructed tone-spirals starting from the fixed center of a circle. If we now allow this center to “wander” regularly along the monochord string, a most remarkable curve appears, which I call the “partial-tone curve” (Harmonia Plantarum, p. 127) and which is shown in Fig. 294. Consider the entire length of the curve as a monochord string of 120 cm. For the angle, we use only the string-length angle. Halfway along the string, at the point 60 (cm), we place the angle 0° (= c), whose vector will correspond to the upper half of the string. The next tone is 16/15h (always string length ratios). We first divide half of the monochord string into 15 parts, add 1/15 below, and find the string position for 16/15h (the vector for this tone was erroneously omitted in the drawing). Its angle is found according to the scheme (x/y · 360) – 360, i.e. 16 · 360 = 5760 : 15 = 384 – 360 = 24° (16/15h). This angle, with the corresponding vector, is missing from Fig. 294, as noted. The next value is 12/11 °h. To find the place on the string, we can now divide the half of the string again, in 11 parts, and add 1/11 to it. Or else we calculate: 60 : 11 = 5.454 · 12 = 65.44 cm, and subtract this amount from the 120 cm monochord string, getting the length 54.5 cm as a result, which we measure from the bottom up, thus yielding the remainder of 54.5 cm for the tone 12/11 °h. Now we calculate the corresponding angle analogously to the above (12/11 · 360) – 360, yielding 32.7°. We add this angle (to the left or right), draw the vector (ray), and add to this the remainder of the string, measured from below (therefore always diminishing with the following ratios), from 54.5. We proceed in the same way with all the remaining tones, and so construct the partial-tone curve of Fig. 294. The tone-values can naturally be chosen arbitrarily from any partial-tone diagram; only they must be reduced to one octave first, and must be selected so that the vectors are distributed upon the curve as evenly as possible, i.e. so that this curve, as the line connecting the end-points of the vectors, can be constructed as accurately as possible. Like the circular periphery of the preceding tone-spirals, the partial-tone curve contains all possible tones, and therefore an infinite number of tones reduced by octaves.
§34.3a. The Tone-Cycloid
Instead of the shorter remainders of the string, we can also remove the longer ones. That is, we take the upper, longer segment of the monochord string in the circle, instead of the lower, shorter segment, and draw it along the vector. If we do this in the same way for all tones, the result is an even more interesting curve, which I call the “tone-cycloid”—an irregular ellipse almost circular in form (see Fig. 295). The partial-tone curve, to which it is reciprocal in terms of the monochord string, is drawn inside the cycloid for better comparison. This tone-cycloid is especially interesting in various ways. Assuming that it is indeed an ellipse, I have first constructed the ellipse from the narrowest and widest diameters of the cycloid—a process that can be found in every mathematics textbook. If one lays this regular ellipse upon the cycloid, one can see that with a few widenings and indentations, the cycloid aligns with the ellipse. In Fig. 307a the ellipse is printed on transparent paper, allowing comparison with the cycloid. If we had simply derived and established the cycloid as a curve drawn from observation data, as the expression of some natural phenomenon (e.g. the paths of the planets), without knowledge of its regular harmonic information, then obviously the ellipse would be highlighted as a mathematical relation. For the deviations of the ellipse, one would doubtless look for “disturbance factors”—to stick with the example of planetary orbits—in this case accepting gravitational effects from other planets, etc., as an explanation. However, in the cycloid we have a legitimate clarification of these irregularities in the harmonic emergence of the curve itself.
Between cycloid and corresponding ellipse, however, even closer relationships exist. The monochord axis is divided by the cycloid into 3 equal “octaves”: the 2 octaves of the monochord itself and an additional octave of the segment lengthened below from the monochord. This “octave” appears again in the ellipse as the distances of the two focal points F and F1 from the two axis points B and A of the ellipse curve. The angles at which the major (A B) and minor (C D) ellipse axes intersect the monochord line at E and G are both 45°, and with the center of the ellipse, S, an isosceles right triangle SEG is constructed, whose height E G (at the tone f) divides in half. I hardly believe that these relationships could be merely coincidental. Remarkable above all is the vertical position of the monochord line that produces the cycloid, and the center S of the corresponding ellipse, apparently existing in complete isolation. Assuming that we can see the prototype for the paths of the planets in this harmonic cycloid—a figure that the ancient harmonists must have known of, considering the Greeks’ great talent for geometric constructions, even if they intentionally kept their other important harmonic theorems secret—then from the Pythagorean viewpoint, the center S of the ellipse must be given the name of the secret Pythagorean “antihelion” or “central sun”—a concept with which no one has been able to do anything up to this point, and which emerges inevitably and obviously from the harmonic cycloid and its ellipse. If we pursue this astronomical-symbolic ektypic further, then we come to further important “sphere-harmonic” realizations regarding the Pythagorean octave that played such a great role in ancient times. Here, inside the cycloid, we see this as a generating element. Modern literature always mentions the “scale” and the “7 planets” as the two fundamental concepts of the ancient harmony of the spheres. This is understandable on the basis of the existing exoteric ancient sources, whose writers had no knowledge of the true esoteric backgrounds. But if we return to
ancient Pythagorean thought and begin to study in Pythagorean terms, it is everywhere apparent that Pythagoreanism was a very different and exquisitely harmonically ramified domain of thought and observation, which absolutely did not bow to such primitive idols as people today imagine. Thus it is evident to me that the generating space for the ancient harmony of the spheres was not the “scale” that occupies the octave space, but instead the “octave” itself, and that the corresponding important diatonic steps and their tone-values and vectors (circle-spheres) were chosen within this space, so as to arrive at a comparison and an interpretation of the bodies visible in the skies. Now we see, in Fig. 295, that there are significantly more than seven important tone-values within the octave. In future harmonic study, however, this tone-cycloid with its ellipse is not only valuable for historical analyses, especially those arising out of the harmony of the spheres, but far more so for a prototypical interpretation of the planetary orbits. To develop each planet’s harmonic vector, its distance from the sun, and its characteristic ellipse from the tone-cycloid would require a specialized and intensive harmonic undertaking on the part of a learned astronomer, for which the above can only serve as an encouragement. If this were to succeed, a complete union between modern astronomy and Pythagoreanism would be achieved—something that Kepler attempted in his Weltengeheimnis, partially realized in his Third Law, and in which he believed with every fiber of his being during his lifetime.
§34.3b. The Primordial Leaf
If we now take the angles that we have previously set on one side of the monochord (here the left, but one can also use the right side, obtaining a mirror image of the same figures) for the partial-tone curve and the cycloid, and place them symmetrically on the middle of the axis, then the result is the tone-curve of the “primordial leaf”—a description that will be retained here for simplicity’s sake, since it has already been shown in Harmonia Plantarum (p. 125) as the harmonic prototype of the leaf in general. Here again we can construct two different figures, depending on whether we use the longer or shorter—“plagal” or “authentic” (see §29.1)—remainders on the vectors of the respective tone-locations on the monochord string. Both figures are shown together in Fig. 297. For reasons of exactitude, the inner figure of the primordial leaf is printed separately, and its development is described (Fig. 296).
The first small square at the top contains the tone-values and angles of the partial-tone plane of index 7, based on string lengths, which have a reciprocal relationship to the vibration numbers (frequencies). The second square at the top gives the corresponding string lengths, reduced by octaves—calculated for a monochord 120 cm long—and the remainders of these string lengths. For example: 1/3 of the string length will produce the duodecimal (2nd upper fifth) g¢, and 2/3 of the string length produces a tone an octave lower, thus the 1st upper fifth g. 1/2 of the circumference of the circle, i.e. the string bent into a circle (360°), yields 120°, 2/3, since we reduce all the tones within one octave, i.e. all g-values are on the vector 120°. The same goes for the position of tones on the monochord, if I wish to bring them all into one octave. 1/3g¢ is 40 cm of the 120-cm monochord string, and the remainder of the string is 80 cm long. 2/3g is 2 × 40 = 80, the octave below g¢; but since we want to bring all tones into an octave of 1-1/2 (0-60 cm string length or 0°-360° of the circle’s circumference), all g-values remain at a string length of 40 cm and a remainder of 80 cm. Fig. 296 is now easy to construct. Draw 40 units from the bottom up on a middle axis of 60 units to fix the tone g. Set the corresponding angle 120° symmetrically on this point, and set the lengths of the two angle legs equal to the corresponding string length—likewise 40 units. One proceeds analogously with all tones, thus getting the primordial leaf as the line connecting all the endpoints of the angle legs obtained. The greater the partial-tone coordinate index one uses, i.e. the more one fills out the octave with tones, the more precisely the primordial leaf can be constructed. Its form, however, always remains the same. But this means nothing other than that the primordial leaf is a form-expression of the very nature of tones—a discovery that deepens and confirms the fundamental ideas of Goethe’s morphology of plants in a completely new way.
The construction of the outer curve of the primordial leaf (Fig. 297) emerges of itself according to what is said above and in §34a. As one can see in Fig. 297, the outer and inner curves of the primordial leaf, in contrast with the partial-tone curve and the cycloid, have a morphologically reciprocal relationship, except that the apex of the smaller inner curve points upwards, while the outer curve’s apex points downwards.
In many of my works, I have discussed the nature of the spiral extensively, and here I will only recapitulate fundamental matters and summarize the ektypic data. Mathematically, the spiral is imagined as a point P, moving at a given speed along a straight line which is continually rotating, at another given speed, around a center-point Z. Depending on the magnitude and ratio of those two speeds, the various spirals—Archimedean, logarithmic, etc.—then emerge with their various formulae. Even in this definition, one can recognize a certain paradox of the spiral: it is, so to speak, the geometric symbol of two divergent, opposing movements, a kind of frozen time-geometry, a capturing of the temporal in the spatial. As we consider it further, the concept of “speed” separates into two components: something reaching outward, in one direction, a vector, and something that holds itself in, with a circular tendency. Or one can also say that in the point moving on the spiral, two elements constituting the spiral meet: an element of direction (angle) and an element of distance (from the central point), whereby temporal turns into spatial in both cases. Thus the more or less “dynamic” behavior of all spirals is understandable. It arises from those two divergent tendencies of the linear striving forward and the circumpolar circling, the expansive and attractive.
Characteristic for the harmonic developments, then, are the spirals that result from the thought processes just described, which can also be found in both halves of the “P” diagram. Think, also, of the very significant spirals of the cochlea in our inner ears! On the basis of harmonic developments, as we have seen and will see again in §36, there are characteristic tone-spirals (taking this term generally), and the same appears as in harmonic number analysis: all tone-values have a psychical evaluation, and allow for analyses of a certain type, especially through their octave reductions (a typical harmonic operation not known to mathematics or the mathematical sciences). (See “Tonspektren” and the atom model therein.) These analyses can only be arrived at with great difficulty, if at all, by means of the familiar mathematical spirals.
The ektypics of the spiral in nature can be seen in so many examples, from almost all areas of knowledge, that we will give only a few examples here: the spiral cloud as the prototype of galaxies, spiral movements and laws in physics, the spiral of the harmonic atom model (tone-spectra) as the “motor” of the optical emission of the spectra, spirals in the morphological construction of diatoms, plants, and animals (the curves of blossoms, snails, the construction of the helix, the spiral vascular structure of plants), the idea of “spiral” developments of historical-morphological isotopes, the idea of the spiral as a universal religious symbol (P. Sarasin: Helios und Keraunos, 1924, p. 67 ff.), and many others.
People have tried, if not very often, to figure out the universal morphological significance of the spiral. But apart from the various mathematical spirals, which have no advantage over each other in terms of their formulae, all these attempts have remained mired in the mathematical concept of quantity, similarly to those attempted by means of the golden section, etc.—and from this viewpoint, no one can see why the spiral in particular should have such universal significance. However, if we trace the tectonics of this form back to certain psychical values, as we can do in harmonics, and if we see this form not only physiologically (the cochlea) anchored in the “filter” of this psychical value, but also in one of its most important modifications, namely the tone-spiral and logarithmic spiral (see §18.3b) as a morphological expression of this psychical value, then we see something completely different and much more authoritative in every regard; and we now understand that such a pronounced value-form must also have its value-formal counterparts in all of nature, and that it ties in with our spiritual and religious image-concepts.
H. Kayser: Hörende Mensch, 89-92 and Tables IV and V; Klang, 81-84; Abhandlungen: “Tonspektren,” pp.111-189 and the relevant tables; Grundriß, 120, 121, 240-253 (group-spiral); Harmonia Plantarum, 124-127, 142 ff., 152 ff., 270 ff. Harmonikale Studien II (the violin scroll).
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