§35. THE COMPLETE PARTIAL-TONE DIAGRAM
§35. The Complete Partial-Tone Diagram
Up until now, we have constructed our familiar partial-tone diagram by simply placing, under the overtone series
1/1 c 2/1 c 3/1 g 4/1 c 5/1 e ...
further overtone series with initial ratios that are reciprocal to this series, i.e. following the simple aliquot series:
1/1 c |
2/1 c′ |
3/1 g′ |
4/1 c′′ |
5/1 e′′ |
→ |
1/2 c, |
. |
. |
. |
. |
→ |
1/3 f,, |
. |
. |
. |
. |
→ |
1/4 c,, |
. |
. |
. |
. |
→ |
1/5 as,,, |
. |
. |
. |
. |
→ |
↓ |
|
|
|
|
|
Figure 298
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:
↑ |
|
|
|
|
|
3/1 g′ |
6/1 g′′ |
9/1 d′′′ |
. |
. |
→ |
2/1 c′ |
4/1 c′′ |
6/1 g′′ |
. |
. |
→ |
1/1 c |
2/1 c′ |
3/1 g′ |
. |
. |
→ |
1/2 c, |
2/2 c |
3/1 g |
. |
. |
→ |
1/3 f,, |
2/3 f, |
3/3 c |
. |
. |
→ |
↓ |
|
|
|
|
|
Figure 299
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 1/4 PE
(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.

Figure 300
The axis cross (Fig. 300) contains the reciprocal partial-tone series
in its vertical and horizontal arms, intersecting at the generator-tone
1/1 c. 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 36/1 d; the ratios of the lower left sector d are quantitatively all smaller than 1 (< 1) and descend to the corner ratio 1/36 bˇ,,,,,,-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. Ektypics
§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:
Units: 1 4 9 16 25 36 ...
Time: 1 2 3 4 5 6 ...
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:

Figure 301
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.

Figure 302
As one can see, the temporal element of the seconds of falling is equal to the equidistance of the vibration-numbers 1/1 2/1 3/1
... 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.
§35.2. Value-forms
§35.2a
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 1/1 c, we move upwards in equal steps of the primary major perception (to 6/1 g),
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 36/1 d. We are already well aware of this vitality through the direct relation along the diagonal of the 2nd power to 1/1 c. However, this backward glance to 1/1 c
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 6/1 g. At this step, however, the “falling” tendency becomes autonomous; it now turns into a minor perception narrowing down to 6/6 c,
and reinforces itself here through the reference switch to the left,
crossing over once again into the major world, as far as the ratio 1/6 f,,,.
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 1/36 bˇ,,,,,,. 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 1/6 f,,, following the same perception backwards in equidistant steps, and from there turning around in F-major up to 6/6 c, where a further turning through a “calm” minor impulse allows us to reach the ratio 6/1 g 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.
§35.2b
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:

Figure 305

Figure 303

Figure 304
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”.
§35b. Bibliography
H. Kayser: Grundriß, 100-102, 122, 225-227; for a.1: Abhandlungen, 46, 47.