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Hans Kayser's Textbook of Harmonics - Excerpts §33. POLAR DIAGRAMS
A Translation Society Edition
By Hans Kayser
We now come to the circular or polar representations of the “P” (not to be confused with “polarity” as a value-form). Here there are three different elementary possibilities: 1) the simple circular form of the “P”, in which the right- or oblique-angled coordinate grid is converted into polar coordinates; 2) the division of the circle (either of the circumference or the angle at the center) according to the measure of the partial-tone ratios; and 3) the transformation of these ratios, i.e. the “P”, into vectors (angles), while simultaneously notating them as distances from the center or the generator-tone circle.
§33.1. Circular Coordinates
These consist, in principle, simply of a circular variation of the familiar “P”. This is most clearly seen from the construction of Figures 272 and 273. The sheets of polar paper commonly available have a circle divided into 16 parts; of course, one can easily
draw these polar coordinates oneself, put them on cardboard as an underlay, and perform the experiment on transparent paper laid on top. We place 1/1 at the center (Fig. 272), then 1/2, 2/1, and 2/2 on the three corresponding points of the first circle, then continue, obtaining a grid of ratios ending at the top with index 9. The two basal series, in this permutation, form a heart-shaped curve. If we start again from 1/1 at the center, but choose the ratio progression so that the overtone series 2/1 3/1 ... proceeds regularly around the radii of the circle, while the generator-tone line 2/2 3/3 ... skips alternate radii, and the undertone series 1/2 1/3 ... skips two radii, the result is the permutation in Fig. 273.
Variations of this circular model are achieved through different divisions of the circle (here it is divided into 16 parts, i.e. “octave division”).
Permutations are achieved by choosing different starting ratios, as for the square and triangular models discussed in §31. There, as here, the “variation” and “permutation” are purely geometric concerns (see the following division of a circle by harmonic ratios); these circular models become harmonically useful only through various fixations of the ratio content, i.e. through psychical evaluation. The reader should perform his own experiments in this area, likewise with regard to possible combinations that are considerably more complicated and have not yet been investigated.
§33.2. The Divisions of a Circle by “P” Ratios
If we imagine the circumference of a circle as a bent monochord string, whose length (string length) or vibration (frequency) we divide according to the measure of the whole-number series, the result is the four possibilities of Fig. 274, shown here in seven rows up to index 8.
To further clarify Fig. 274: by “subdivision” we mean the division of the circle’s circumference (= the string). “Superdivision” uses the pitch of the whole circle as a string unit (frequency unit), and adds to this unit the corresponding sectors of the relevant circle divisions.
As for the ratios that emerge from this circle division, it is best to compare the collective ratios of a group of four, such as the division in six (6) in a diagram of the “P”. If one draws the fractions of the 4 circles:
1st and 3rd circles
2nd and 4th circles
then one will see that by means of these circle divisions:
one finally arrives at all the partial-tone coordinates. Geometrically, the so-called “regular polygons” are the result. Tonally, however, the result is even richer, since from the analysis of string lengths and frequencies, we obtain the reciprocal tone-value for every ratio, which is shown by the circles under a (string lengths) and b (frequencies). The examples of notation below the circles in Fig. 274 show the tonal characteristics of each circle division. I intentionally showed extensive variations in this figure, in order to give the reader a good basis for this kind of circle analysis; we shall summarize briefly below.
We can now tackle the individual evaluation of certain circle divisions (angles, regular polygons, positions of the tones on the circle), when we examine the third and most important type of harmonic circle diagram.
§33.3. The Vector Diagram
Previously, we divided the circle’s circumference successively into 2, 3, 4, ... equal parts (arcs), and analyzed the arcs distinguished by means of the regular polygons as “tones” (= string lengths and frequencies). We will now investigate the results when we divide the round angle of 360° according to harmonic ratios.
An angle, of course, cannot make a sound, but we are entirely authorized to set the round angle of 360° equal to the string length, or the frequency unit 1/1, and to subdivide and superdivide these 360° exactly as we did the circle circumference, as the unit 1/1. We have already done this indirectly (Fig. 274), in that the regular polygons divide the round angle around the center with their axes according to the same ratios.
If we direct our viewpoint completely toward the division of angles, something new occurs: we have transformed the tone-values, previously indicated by segments or frequencies, into directions (vectors), because every angle is nothing other than the indication of a certain direction.
But here we must be careful, and must first proceed completely independently of the previous circle divisions. If we divide the round angle of 360° successively by 2, 3, 4, 5, and 6, then the result is Fig. 277. If, in contrast, we go downwards from ray (vector) 0° = 360°, the result is the octave 1/2 c¢ 180°, then 1/3 g¢ 120°, etc. (tone-values according to string lengths). It can easily be seen that all aliquot ratios end up in the upper half of the circle, whereby, according to the law of harmonic quantization and “perspective,” they approach the degree of zero ever more closely, but never reach it. Although we divide the round angle 360° according to harmonic ratios, resulting in corresponding vectors, this method does not get us much further; because we have only obtained the form of the “diminution” of the familiar partial-tone series, converted into vector form, as each string division line shows us. Besides, the equilibrium in the circle is very unequally divided. We therefore must search for another way.
For this, there are two considerations. Firstly, since the partial-tones develop “above” and “below” 1/1, it would be advantageous to set 1/1 not as the center, but as the circle itself, so that the overtone and undertone ratios (as circles) have their space outside and inside the unit circle. This has nothing to do with the tones as vectors in themselves. However, as we will soon see, their connection with the vectors creates the unified structure of the “tone-spiral,” which illustrates the synthesis of vector (angle, direction) and size (distance from the unit circle). The second consideration is the necessity of creating a distribution of the tone-vectors within the round angle of 360°; they are distributed as evenly as possible throughout the entire angle space. For this, we use a very simple means: the octave transposition that we already know of, i.e. the projection of all partial-tones in the space of 360°, which is regarded as an octave. For this it is first necessary to reduce the partial-tones to this “angle-octave,” which is done through the following operation, which we elucidate with the ratios 3/2 g and 4/3 f:
As one can see in the right-hand example, we must first bring the ratio 2/3 f into the octave of 1/1 c, above 4/3 f, and do the same for all ratios smaller than 1/1. The process is then very simple. We set:
1 : tone-number = round angle : x
and solve for x, the angle of the tone-number. As one can also see, the octave position of the ratio in question does not matter, in contrast to Figures 274 and 277; all c-octaves 1/2 c, 1/1 c 2/1 c¢ ... are on the 360° ray, all g-octaves 3/4 g 3/2 g 3/1 g¢ ... are on the 180° ray, and so on. To compensate for this lack of octave indication, and to incorporate it in the graphic image, we express these octaves as regular distances outside and inside the unit circle 1/1 c. From this, it follows that each tone-angle also has a tone-circle. Now, we want to continue the construction of this fundamental diagram, for which the frequency numbers and values serve. If we set the tone c as the unit 1/1, then its sphere is a circle with radius 1. The radius signifies the size of the tone. Its circumference is the location for all 1/1 c-values. The radius could go in any direction out from the center; therefore we must set it arbitrarily by choosing some radius (Fig. 279).
From there on, all further vectors are determined. Here, then, two completely different “dimensions” meet to become a unified idea: a dimension of size (distance from the center ○) and one of direction (radius). The adjacent ratios are 1/2 c, and 2/1 c¢. We do not need to find their directions, since those must obviously be identical with that of the 1/1 c-value. The radii will have to be respectively half and double the size of the radius of the 1/1 circle. The result is Fig. 280. If we construct this diagram according to the indexes of the “P”, i.e. with the ratio pattern:
then in index 2, the result is a doubling of the first c-value (1/1 and 2/2), which is best indicated by drawing the generator-tone circle with double thickness, triple thickness, etc. (Figures 280, 282, and 284).
Index 3 brings about two new values besides 3/3 c, whose angles we have already calculated above: 2/3 f 120° and 3/2 g 180°. First we divide the radius 0-1/1 c into three parts, set the compass to 2/3, and draw the 2/3 f, circle. With 3/2 of the 1/1 radius, we draw the 3/2 g circle. Then, moving clockwise, we take the angles 120° f and 180° g, and beginning from the circles in question, draw the f and g rays (see Fig. 282, in which the 2/1 c¢ and 1/2 c, circles are not drawn). It is clear that if we continue in the same way and draw the octave circles with the following ratios, we will soon have a comprehensive and substantial diagram. The reader should not neglect to draw at least one such diagram for himself, as far as his compass will allow.
However, this artistically and visually beautiful but somewhat circuitous procedure is not necessary if all we need to do is to express as many tone-values as possible in vectors. Since, in the “P” system, all values appear at some point or other in the sector of the first octave above and below, it is best to use the space of the “P” bordered by the two equal-tone lines 0/0 2/1 c¢ and 1/2 c 0/0, at index 7, for instance (Fig. 283). If we construct a diagram from these ratios analogous to these specifications, the result is Fig. 284.
Here we attempt to indicate the individual vectors according to their “power,” likewise the “weight” of the circle. Since there are 7 c-values on the c-vector, it has 7 thin lines, and so on. As one can see, there is a fairly large differentiation in vectors and circles. Fig. 284, however, shows another new characteristic element, already mentioned above. If we connect the points where the individual vectors begin on their corresponding circles, we get a spiral. Since we are working with a decimal “P” scheme, I call this the decimal tone-spiral, as a supplement to a spiral also possible in the logarithmic scheme. Through this, we can simplify the polar diagram still further, so that we are restricted to only the angles (vectors, directions, tones) and completely ignore the corresponding tone circles. Then all we have is a circumference of arbitrary size, on which tones emerge—as in Fig. 285 and the following polar diagrams, as regards the scales and chord analysis. This simplified model is completely sufficient for many investigations.
A deep insight into our fundamental harmonic polar diagram is so important that we must examine it further, reviewing what has been said previously.
If the round angle 360°, i.e. the circle’s circumference, is continuously subdivided in a given succession, then one can produce all ratios of the overtone series by purely geometric means, without calculation. Fig. 286 attempts to clarify this. First we set the 1/1 c line (circle 1), go around the circle once (second operation), and get the octave 1/2 c¢. The third operation brings about a new value, 3/1 g¢; this is the first subdivision of 360° into 180°, which produces the g line. The 4th operation, with 4/1 c¢¢, produces no new value; its ratio is noted on the c line. The 5th operation produces the new e value; here the section of 180° must be divided into 90°, producing the e line at 90°, and so on. One can see that through regular halving for newly emerging values, i.e. through a simple geometric process, the correct tone vectors of the overtone series 1 2 3 4 are found without angle calculation. Let us try it with 11/1 °fis¢¢¢ 135° in the last circle of Fig. 286:
11/1 ºfis′′′ reduced by octaves to 11/8 ºfis
1 : 11/8 ºfis
360 : x
11/8 · 360
The position of this °fis is identical to its position in the last circle diagram on the right of Fig. 286, and precisely this comparison with the regular circle division in Fig. 274 now gives us information as to how we can construct all ratios and their angles (vectors) purely geometrically. The dyadic division of circles or angles in 2 4 8 etc. parts produces all primary overtone series beginning with the ratios 1/1 1/2 1/4 1/8 and so on. The triadic division in 3 6 12 24 ... parts produces all fifth series beginning with the ratios 1/3 ... 1/6 ... 1/12 ... etc. The pentadic division into 5 10 20 ... parts produces all third series, and so on. Here we see the “regular polygons” in a new light. Harmonically, they represent the possibility of illustrating the partial-tone coordinates geometrically in their lengths and frequencies as well as in their angles (vectors).
In the next chapter, we will return to the element of the tone-spiral.
§33a.1. Tone-Spectra and Atom Models
In my essay “Tonspektren” (in Abhandlungen), which set the relationship of the optical spectra to the laws of tone upon a new foundation, the fundamental polar diagram of the “decimal tone-spiral” is used to give an idea of how a hypostatic atom model can lead to an emission of spectra. Regarding the details, the reader interested in this subject is referred to the relevant essay, since discussing this here would take up too much space. But what he has learned in this chapter will enable him to understand two diagrams: the tone-spiral of PE 5 (Grundriß, table 19) and the acoustic atom model (in “Tonspektren,” table VIII), which are reproduced here in Figures 288 and 289, respectively. In Fig. 288, the grid (“the partial-tone plane of index 5”) shows the coordinate field to be analyzed—the partial-tone plane of index 5—according to whose the measure the angles (rays) and the circles are drawn, as well as the distances of the circles inwards and outwards from the central generator-tone circle (drawn in bold). The table (“the tone-values of PE5”) shows the tone-values arranged according to their frequency of occurrence. From this, for example, the elevenfold shading of the c ray becomes apparent on its upper end, since all the c values are added together there; likewise, this results in the “power” of the generator-tone circle, which from including the “inner” c values is 8 units “strong.” The spectrum shows the 7 spectral lines, which are the sum of the ratios of PE 5 of the atom model (reduced by octaves). This “summation,” i.e. the varying strengths of the spectral lines, for which no sufficient explanation has yet been found, can be most precisely tracked on the basis of its harmonic emergence. See “Tonspektren” for many other important elements of the finer structure of the optical spectra, which can only be explained through harmonic ideas and analyses.
The “acoustic atom model” in that essay (here, Fig. 289), whose ratios exhibit a decimal angle-spectrum of type I of the partial-tone quadrant of index 3 (see §37), will now be understandable to the reader without further description. Characteristic for this diagram, developed from the tone-cube, despite its small index of 3 and its few (5) tone-values, is its comparatively large “electron shell” (circle outside the generator-tone circle 1/1) in contrast with the small “nucleus” (circle within the generator-tone circle). Since there are over 90 “elements,” with electron paths increasing successively from the most simple element (hydrogen) as the atomic number increases, atoms with higher numbers, and therefore denser nuclei, would radiate a correspondingly great “remote influence” according to harmonic theory—an idea that might explain the puzzle of “cosmic rays,” and indeed the universal coexistence of matter despite external “empty spaces,” the hypothetical “ether,” etc., which last idea modern physics has abandoned in any case. Since harmonic prototypes otherwise consist of ideas of vibrations, i.e. waves, which certainly rest upon specific values, this would also do justice to the views of modern physics.
Here we can go a step further. Since all harmonic prototypes are figurations of values, we may imagine the human “head” as a type of harmonic sphere, in which the brain is the “nucleus” and the radiation of thoughts is the “shell.” Through formal use of the acoustic atom model, transposed onto the plane of values, it is possible at least to bring into the domain of the explainable a phenomenon whose reality should by now be undisputed, but for which we still have no acceptable scientific idea: that of the transmission of thoughts, or telepathy. However, in contrast with the atom’s “material” field of action, which is rigidly tuned to a one-time configuration of wave-spheres, humans are able to grasp their thoughts freely; harmonically speaking, they are able to freely determine the indexation and selection of their psychic radiations. Likewise, just as one can construct an image of the universal coexistence and remote effect of “matter” on the basis of the idea of the acoustic atom model, so one can, on the basis of an analogous assumption of “thought waves” over long distances, imagine a temporally “synchronized” thought transmission whose acting in unison requires a resonance in the receiver in tune with the broadcast in question. One must admit that this analogous example from the harmonic standpoint at least makes sense to the heart and the mind as a beginning for solving the enigma of physical and mental long-distance effects. Certainly much has been done to categorize the facts, and if two people halfway around the world from each other can be proven to have the same or very similar thought processes simultaneously, we are right to be astonished at this phenomenon, which is absolutely unexplainable with our current scientific methods. Precisely for this reason, we want to “know” how this can actually be explained; and here harmonics, with the polar diagram, can at least give a sufficient idea based on concepts of unprejudiced and precise research, and is all the more justified since the present theories of telepathy, etc., are vague, fantastic, or pitifully primitive.
In §44, we discuss the significance of the element of direction (vectors), which becomes independent, so to speak, in the polar diagram, but appears in all harmonic configurations.
H. Kayser: Hörende Mensch, 90, 91, and table IV; Klang, 81 and Fig. 5; Abhandlungen, 90, 149 ff., and table IV (tone-spectra), 165 ff., and table VIII (ibid.); Grundriß, 120, 121, 240ff., and table 19; Harmonia Plantarum, 142 ff., 152 ff., 270-276.
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