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Hans Kayser's Textbook of Harmonics - Excerpts §34. The Complete Partial-Tone Diagram
A Translation Society Edition
By Hans Kayser
Up until now, we have constructed our familiar partial-tone diagram by simply placing, under the overtone series
further overtone series with initial ratios that are reciprocal to this series, i.e. following the simple aliquot series:
If we interpolate upwards in the same way, then obviously, if we want to stay with the pattern of the overtone series developing towards the left, we get the following progression:
If we continue the procedure to the left up to index 6, the result is Fig. 300. We call this the open or complete partial-tone plane (PE), in contrast with, or rather in completion of, the previously familiar partial-tone plane, which we can conveniently designate as 1/4PE (the quarter partial-tone plane). This is a regular continuation of the overtone series in all four directions in the flat coordinate plane. This open partial-tone diagram is especially interesting in various ways, but we will analyze it here only in terms of its main content. In regard to various important details (cadencing etc.) we will return to it later.
The axis cross (Fig. 300) contains the reciprocal partial-tone series in its vertical and horizontal arms, intersecting at the generator-tone 1/1c. This divides the coordinate field into four sectors, a b c d. Of these sectors, a and b are the same in their content, in the position of their vectors; but their location is reversed like a mirror image. In both cases, then, we have a model of the original 1/4 partial-tone plane. The sectors c and d, however, are completely different in appearance. The ratios of the upper right sector c are quantitatively all greater than 1 (> 1) and climb very steeply upwards to the peak of the corner ratio 36/1d¢¢¢¢¢; the ratios of the lower left sector d are quantitatively all smaller than 1 (< 1) and descend to the corner ratio 1/36bˇ,,,,,,—therefore, between these outermost poles of this small index of 6, there are eleven octaves of tonal space. As the numeric expression of these two corner ratios shows, these two sectors c and d are reciprocal in terms of their numbers (quotients) and tone-values; thus, the number groups of these sectors follow the same law of reciprocity as the simple linear partial-tone series. Furthermore, we see two more diagonals, drawn as dotted and dashed lines. The first (from b to a) joins together only generator-tones n/n; we call this the “generator-tone diagonal.” The entire diagram is divided into two halves by these diagonals: an upper right half with only ratios greater than 1, and a lower left half with only ratios smaller than 1. These two halves are also in exact reciprocity in terms of both number and value. The second dashed diagonal from c to d joins the ratios of greatest upwards and downwards expansion; since there are only second powers of the whole number series and its reciprocals in numeric terms here, we call it the “diagonal of 2nd powers,” or of “directional powers.” The diagram is divided by this second diagonal into two halves of identical ratios, and therefore of the same content. These halves, however, are reversed like mirror images. These two diagonals embody, geometrically, the most extreme opposites contained in the diagram: the generator-tone diagonals, the static element of the self-contained generator-tones; the diagonal of 2nd powers, the dynamic element of exceptional vitality. Later, we will further discuss the laws and norms of this complete partial-tone diagram. Seen from outside, this complete, open “P” diagram (or however one wishes to describe it) has great similarity to a 4-fold combination of our beginning diagram. But it is not a “combination model”; instead it is a simple further development of the “P” according to the laws lying immanent within it. To investigate whether, and how far, this complete P-diagram can be varied, permuted, and combined, would require far more space than this textbook allows. This is left for the reader to work out on his own initiative. In the section on “tone-space” (§37), we will construct the complete “P” spatially; the reader who enjoys drawing will be able to exercise his skills there!
§35a.1. The Law of Falling Bodies
The physical law of “free fall” states that a body will fall one unit of length in one second, 22 = 4 units in the 2nd second of falling, 32 = 9 units in the 3rd second, and so on. If one writes the numbers of the units and those of the corresponding times below them:
then one sees instantly that the upper series is perspective, and the lower series equidistant. The two dimensions in which this dialectic takes place are time (seconds) and space (units of distance). It is interesting that from the point of view of this physical law, there is an even closer relationship to our partial-tone coordinates than that of perspective and equidistance. If we observe the numeric terms in the haptic illustration of the “P” not arithmetically, i.e. not from the simple viewpoint of the number progression, but geometrically, i.e. from the viewpoint of true size, then calculating with vibration-numbers and string lengths yields the following two basal series:
Here, with the purely geometric observation of the number sizes, within the conjugated overtone and undertone series, we see the perspective and equidistant elements appear together. The dialectic mentioned above, then, appears within the number sizes, whereby the observation of the tone-values shows that the perspective side reveals a minor impulse under temporal observation and a major impulse under spatial observation. For the equidistant side, it is reversed. Thus the situation is exactly the same as for the law of gravity: space and time are in a constant perspective-equidistant relationship. Only in the acoustic domain, this relationship is mutual (reciprocal) and can reverse itself depending on whether we calculate with vibration-numbers or string lengths. In the physical domain, on the other hand, it is one-sided, since the measure of time always remains equidistant and the measure of space always remains perspective.
One can now derive this law of falling bodies directly from our completed partial-tone diagram (Fig. 300, sector c), as shown in Fig. 302.
As one can see, the temporal element of the seconds of falling is equal to the equidistance of the vibration-numbers 1/12/13/1 ... while the spatial element of the intervals of falling is congruent with the perspective of the vibration-numbers of the diagonal of 2nd powers.
However, regarding this example of free fall, we are interested in something fundamental that gives rise to a deeper observation.
It is known and acknowledged that this Galilean law of falling bodies is a preliminary step on the way to Newton’s law of gravity. In it lies the first quantitative-dynamic law, which precisely defines a process of movement, in contrast to or in completion of the first (alleged) quantitative-static law of the precise Pythagorean tracing of a perception (tone ratio) to a quantitative numeric relationship. According to common opinion, Galileo found his law of falling bodies through of experimental observations, and it is given in almost all textbooks as the paradigm of the so-called inductive method of modern science. In contrast to this, Hugo Dingler (Der Zusammenbruch der Wissenschaft und der Primat der Philosophie, 2nd ed., 1931, p. 125 ff.) tells us persuasively that without previous prototypical ideas, i.e. without the image-concepts already existing a priori in his psyche, Galileo would not have been able to discover his law of falling bodies—the evidence for this condition comes from a letter from Galileo’s student Toricelli (op. cit., p. 196) found by H. Wieleitner. To prove this psychic a priori quality of all great discoveries is now the main task of Dingler’s work, and likewise his resulting thesis that the decadence and “breakdown” of modern science is due to the loss of the creative image-concept, to the one-sided relocation of all scientific knowledge to induction, and to the mere questioning of experiments—the inevitable evil of which is the leveling and uniformity, the vapidity and worthlessness of the modern mode of scientific thought. Dingler explains this a priori existence of creative law concepts by means of a “happy arrangement of the empirical concepts,” thus with a certain minimum intellectual measure of energy, a looking inward, “driven by its unconscious rhythm.”
We can agree with all this from our harmonic viewpoint, and can follow along with Dingler up to this point. But now the deciding question emerges: what is the “happy arrangement of the empirical concepts” and the driving of “unconscious rhythm”—how should we explain these very general and noncommittal terms?
Here, harmonics can offer further assistance. Let us consider that the idea of expansion and contraction held by Newton and Jakob Böhme (see §19b) is present in nuce in the primary reciprocal partial-tone series; further, consider the image-concepts of the most varied disciplines, the religious symbols, etc. in our partial-tone diagram, and added to this the presence of Galileo’s law of falling bodies in a sector of the open “P” diagram; and above all, let us remain aware that all the harmonic tone-developments correspond to inner forms of our psyches, since in the end they can be controlled by means of psychical criteria—then we will grasp how it is possible that laws of nature are present within us as psychic image-concepts prior to their empirical discovery. The “happy arrangement of empirical concepts” can thus be traced to a psychical tectonics whose forms we are able to elicit in the harmonic prototypes (theorems and value-forms) in a scientifically exact and unobjectionable way. But since the law of harmonic tone-development is also manifested in nature, outside of humans, in the overtone series, on which the harmonic partial-tone diagrams are built, an explanation is given vice versa for how the leap of the psychical into the natural is possible, and how one should imagine that psychical prototypes can once again be discovered in natural phenomena. Here the Kantian problem of synthetic apperception obtains a hitherto unknown solution.
But yet another point appears to me to have considerable significance in the harmonic analysis of the law of falling bodies: the element of perspective and equidistance, expressed in spatial length and in time. As initially remarked above, in quantitative-geometric observation, space and time are reciprocal to each other in two different forms: a uniform, equidistant form and a perspectively shortening form. One might well say that the “perspective” of the spatial lengths of the law of falling bodies does not shorten, but lengthens, and therefore is not “extroverted” but “introverted” (see §19a.2). But what is important here is the element of perspective in itself. Given the reciprocal correspondence of the harmonic concept of time-space upon the background of a psychical major-minor world, which is again aligned perspectively and equidistantly, the meeting of time and space in the law of falling bodies can give rise to meaningful results under the same formal auspices of perspective and equidistance that occurred with those of harmonic space-time (see §7, §16.2).
To summarize: through the Galilean law of falling bodies, whose harmonics we have shown here, and through Kepler’s laws, the third of which has prototypical harmonic ideas and analyses to thank for its existence—as the most important preliminary steps for Newton’s law of gravitation, whose inner nature of expansion and contraction agrees with fundamental harmonic concepts in any case—we see the law of gravitation, which governs almost all exact sciences, appearing on an unequivocal harmonic background. This law has thereby found a psychical anchoring; it is no longer an abstraction that does not affect us inwardly, but instead the expression of a psychical structure of the universe.
In my Grundriß, pp. 101-102, the reciprocal and mirror-image relationships of the complete P diagram are summarized in the “theorem of metamorphoses,” and their further significance is discussed under the value-form of the “reference switch” (pp. 225-227). The concept of the “gesture” outlined here can also be examined from the dynamic side (Fig. 300). For this, we imagine ourselves as the moving agent, as an expression of the “will,” and thus perceive something like the following. Starting from 1/1c, we move upwards in equal steps of the primary major perception (to 6/1g¢¢¢), and grasp this fifth-value as autonomous, i.e. we decide to make a “reference switch” to the right. Thereupon, taking further steps which we perceive as the dominant (G-major), we reach the highest peak and thus the utmost vitality of the step 36/1d¢¢¢¢¢. We are already well aware of this vitality through the direct relation along the diagonal of the 2nd power to 1/1c. However, this backward glance to 1/1c leads to an inner reversion and a further reference switch. Turning right once more, we pursue the mirror-image descent in a type of “retracting” perception of the same sequence (G-major) as far as 6/1g¢¢. At this step, however, the “falling” tendency becomes autonomous; it now turns into a minor perception narrowing down to 6/6c, and reinforces itself here through the reference switch to the left, crossing over once again into the major world, as far as the ratio 1/6f,,,. But things do not stop here; our perception changes, continually narrowing (becoming overshadowed, concentrating of its own volition) in a minor world (f-minor), and composes itself finally in the deepest agglomeration of the ratio 1/36bˇ,,,,,,. At this point, the diagonal of the 2nd power comes to our aid in a way; the reference switch upwards leads us, first up to 1/6f,,, following the same perception backwards in equidistant steps, and from there turning around in F-major up to 6/6c, where a further turning through a “calm” minor impulse allows us to reach the ratio 6/1g¢¢ once again.
The reader who follows this analysis precisely, and above all sympathizes with its forms and values—whereby it is left up to him to interpret things differently—will agree with me on this: Regardless of where I begin my “journey” in this diagram, I will always have to go through a world of disturbances, which is in tune with the two most important basic forms of human and voluntary psychical capability, i.e. oscillating back and forth between these two: a perception of lightening, dispersing, extroverted gestures, directed upward, outward, toward the light, and an equally strong perception of narrowing, introverted gestures downwards, inwards, towards the darkness. Along the way, our perception constantly changes between strength and weakness, between the ambivalence of a major and minor world. The human, and every being-value—a “cue-ball between Heaven and Hell”—is symbolized, if anywhere, by this harmonic diagram, if we include “good and evil” in the layman’s sense in the principle of polarity. Later (§53.4, §53.8, §54.7), we will see that we are not allowed to do that; that with this “inclusion” of the ethical in the familiar dualism we make a huge, fatal mistake; and that harmonics, with its offering of the selection principle and the disruption factor arrives at fundamentally different solutions. Major and minor, with their characteristic equidistant and perspective forms—which can metamorphose into one another with the shifting of frequencies (time) to string lengths (space)—are polarities like light and darkness, far and near, etc. (see §23), but not like “good and evil.” This “polarity,” if indeed it should be called that, arises from completely different backgrounds, and this confusion has led the layman’s philosophical treatment of ethical problems down a cul-de-sac.
The strange connection of advancing and delaying elements in the complete P-diagram, the peculiar “static dynamic” or “dynamic stasis” of its content, demands that we subject it to a formally symbolic examination. For this, we choose the axes of coordinates contained within it, which we attempt to analyze under the term of a “symbolism of the cross.” If we let this cross stand vertically and horizontally (Fig. 303), then “above” and “below” are in the same major-minor polarity as “left” and “right”: above and right in major, below and left in minor. The upper right sector c, bounded by the upper right arm of the cross, tends towards “light” and “height” and is reciprocal to the lower left sector d, which symbolizes depth and darkness. Both are centered by the dynamic of the diagonal of the 2nd powers. The upper left sector (b) and lower right sector (a) are mirror images of one another, symbolizing the symmetry of the world, and are centered by the stasis of the generator-tone line.
We find a completely different physiognomy when we position the cross of the axes on a slant (Fig. 304). Here, the world of light is obviously contained in the upper sector and the two upper halves of the left and right sector (above the generator-tone diagonal, which is level here), the world of darkness in the lower sector and the two lower halves of the right and left sector, whereby the diagonal of the 2nd powers (perpendicular here) reaches the extreme peaks of light and darkness, height and depth, attained in index 6 of this diagram. “Right and left” here have their actual significance as mirror-image equal symmetries. The reader will have noticed that these two types of cross:
are the Christian (Occidental) and the Greek Orthodox crosses, two different symbolic emblems whose fundamentally differing inner contents we can clarify with harmonic symbolism. The first is “realistic” in a certain sense (the “bad thief” to the left, the “good” to the right); the other, the “Greek” cross, expresses, in the localization of its psychic tendencies, that which it (like the first cross) is supposed to symbolize—the “Christ”—in a simpler, more spiritual way: heavenly and earthly realms are arranged in the directions of up and down, and right and left are in reconciled equality. Through this the bad thief also motions towards reconciliation. The uppermost sector (“ascended into Heaven”) sounds out in pure, intersecting major chords, the lowermost sector (“descended into Hell”) in pure minor chords.
We will return later (§40) to the attempt at a harmonic symbolism of the cross, this time as a morphological model, on the occasion of the chordal analysis of the “P”.
H. Kayser: Grundriß, 100-102, 122, 225-227; for a.1: Abhandlungen, 46, 47.
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