§10. INTERFERENCE
§10. Interference
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.
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.
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.

Figure 23
Construction of the Sine Curve; “Amplitude”; Construction of the Resultant
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.
Helmholtz defined Fig. 24 (Tonempfindungen, 6th
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).

Figure 24
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.
The “wavering” can be visualized in the following way. The wave trains A and B (Fig. 25 from Scheminski)

Figure 25
show only a small phase difference. As tones they would sound “impure” to us, thus if A were the note c, B would represent a slightly lower c.
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.
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.

Figure 26
Fig. 26 (from F. Auerbach: Physik in graphischen Darstellungen,
1912, p. 64) shows the resultants of two simple vibrations (frequency 1
: 2), each in two different amplitudes and phase differences.

Figure 27
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.

Figure 28
Fig. 28 shows the waves 1/1; 1/2, 1/2; 1/4, 1/4, 1/4, 1/4
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 3rd and 2nd waves, while the resultant of the first order traces the curve of the sums of all three waves.

Figure 29
Fig. 29 shows the wave divisions 1/1, 1/2, 1/3, 1/4, and 1/5
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.
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.

Figure 30
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.
§10a. Ektypics
The following ektypic examples will show how the phenomenon of interference works in various applications:
§10a.1. Mountain Growth
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 a priori dismiss the possibility that mountain growth has may represent a highly rich sound synthesis.
§10a.2. Rilke's “Primal Noise”
2. In Rainer Maria Rilke's Collected Works (vol. IV, 1930, pp.
285-294), there is a remarkable prose fragment entitled “primal noise” [Ur-Geräusch].
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...”

Figure 31
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 (Rainer Maria Rilke,
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 hearing,
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 another level? Because most of what is gained thereby cannot be penetrated by the senses, and thus cannot be truly 'experienced.'”
From H. Friedmann (Die Welt der Formen, 2nd
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.
§10b. Bibliography
Besides the cited literature, the works cited in §6b will orient the
reader regarding acoustics; on harmonic geology, see H. Kayser, Tagebuch vom Binntal (in Abhandlungen).