Harmonics as a Science
The Ear
The physiological basis of Harmonics as a science
is primarily in the ear. A precise description of the anatomy of the
human ear and its evolutionary origin can be found in the relevant
specialized works. In evolutionary terms, the ear as a trained organ is
a relatively late development; but not so the sense of hearing, i.e.
the living being's acoustic perceptiveness, which goes back to the
beginnings of the animal kingdom. If one considers, in addition to
this, the most recent results of experiments with ultrasound waves, one
can accept and assume that matter itself has a universal sensitivity to
acoustic stimuli. “The Greeks believed the sense of hearing was the
most 'elevated' of all, i.e. the sense though which the psyche received
the deepest, most vividly stimulating impressions.”
The ear, in its most highly developed state, is an extremely
complicated mechanism, whose individual functions have still not all
been completely explained. However, to mention only a few important
elements, it is known that the so-called “basilar membrane”-a tiny
structure of 0.8 square millimeters which transmits to the auditory
nerve the tone-vibrations that are filtered, so to speak, by the
eardrum and the oval window of the middle ear-is a technical marvel,
compared to which even our most highly sensitive acoustic membranes are
primitive hackwork. The center of the inner ear, the cochlea, has a
spiral form. Harmonics is in a position to show, for the first time in
the history of physiology, precisely why this central structure has
this spiral form. The harmonic tone-spiral, and the spatial tone-spiral
developed from it, show that the precise form
of the cochlea is in no way accidental, but was ordained by creative
forces according to tone-law and its geometric-harmonic evolutions.
More or less puzzling still for physiology is the placement of the
semicircular canals in the ear. They serve mainly as an organ of
balance, but what does balance, and therefore spatial orientation, have
to do with acoustics? Here, also, harmonics gives an explanation in the
tertium comparationis
of the spatial partial-tone coordinates, i.e. the tone-space built from
the law of the overtone series, whose three coordinate axes correspond
to the three spatial directions of the semicircular canals in the ear.
“For the Gods, the four quarters of the world are the ear,” reads an
ancient Indian text of the Satapatha Brahmana.
Nature appears to have placed the inner ear so as to be particularly
protected. It is surrounded by bones deep in our skull, making any
surgery extremely difficult. The auditory nerve is in direct, close
connection with the entire nervous system and the sphere of perception,
just as the optic nerve is connected to the brain, the sphere of
consciousness. To this is connected the fact that in general, people
with damaged hearing are far more prone to psychic disturbances than
those with damaged vision-a fact well known to every psychiatrist.
Naturally there are exceptions, and one must also distinguish between
congenital and acquired deafness. The two great exceptional cases of
the latter affliction were Beethoven and Goya.
Beethoven
Anyone who knows the life history and work of the late Beethoven can
clearly follow the great intensification of this work paralleling his
progressing deafness. Besides the typical indicators of psychic
disturbance in Beethoven's personal conduct towards other people, in
which paranoia is especially characteristic, it appears that the loss
of his sensory hearing caused the entire strength of his genius to
concentrate upon purely mental hearing, and thus conjured up visions of
a previously hidden world, reaching the highest things granted to
humanity.
Goya
Alfred Peyser writes of Goya in Von Labyrinth aus gesehen:
“If poetic fantasy deepened the shadows, in Goya's case it is evident
that the Master's psychical life was decisively influenced by the loss
of his hearing. He [like Beethoven] grabbed 'fate by the throat' and
kept creating for more than 30 years afterward, but his works from this
period predominantly reflect wrath and turmoil; we no longer see the
images reminiscent of Gobelin tapestries, depicting the cheerful life
of the Spanish people.” For the visual man Goya, the loss of hearing
had an entirely different artistic effect from what it had on the
auditory man Beethoven. For a future “philosophy of the senses,” a
comaprative analysis of these two cases would surely be productive. Of
course, it must not be forgotten that both Beethoven and Goya
originally had intact senses of hearing.
The Sensitivity of the Ear to Time Differences
The sensitivity of the healthy ear to time differences is of the utmost
importance for harmonics. Time differences are expressed in the
precision of the apperception of the most important intervals-octave,
fifth, fourth, third, whole-tone-and the number-ratios corresponding to
them. Of course, the ear is only a mediator here, and without the
prototypical forms in our psyche, the octave, fifth, etc., there would
be no holistic forms. But the ear makes us able to distinguish these
time differences most precisely, and regarding the holistic forms of
the intervals, to judge certain number-ratios spontaneously as
tone-ratios, i.e. to apperceive exact forms directly as correct or
incorrect, which we would otherwise be able to establish only
indirectly through subsequent measurement or other manipulation.
Helmholtz
Regarding sensitivity to time differences, Helmholtz wrote in his Lehre von den Tonempfindungen:
“Compared with the other nervous apparatuses, the ear has a great
superiority in this regard; it is, to an eminent degree, the organ of
small time differences, and has long been used as such by astronomers.
It is known that when two pendulums strike next to each other, the ear
can detect down to approximately 1/200 of a second whether their strikes are simultaneous or not. The eye goes wrong at 1/24 of a second, or sometimes more, when trying to decide whether two light flashes are simultaneous or not.”
The ear, like everything else in this world, also has its limits. But
anyone who has experimented on the monochord himself, and thus
established that within an octave we can easily distinguish not just 12
tones but hundreds of tones and tone-ratios as entirely separate
psychic forms, will find Helmholtz's statement to be completely
supported.
Euler
Euler, in his Tentamen novae theoriae musicae, wrote of a “perception of the order”
of tone-ratios. As regards the ear as a specialized organ for the
spontaneous apperception of certain number-ratios, and the psychic
arrangements connected a priori
with them, this ability is constantly trained and practiced by every
piano tuner and stringed instrument player-indeed by every practicing
musician and listener, and above all by every composer. The task of
harmonics is to clarify the laws behind this and interpret their
meaning. But at this place, harmonics steps out of the special case of
music alone, and expands to become a universal doctrine of akróasis, of
Weltanhörung.
Universality of Harmonic Number-Ratios
The universality of the harmonic number-ratios is initially based upon
our tone-perception, irrespective of whether our ear hears these ratios
with complete or partial precision, or of whether people have been able
to hear pure tone-ratios since the earliest times or have acquired the
ability gradually. All laws, indeed, start as ideal cases, which must
then find their verification in fact, although this verification is
only ever attained approximately. Today, for example, it would not
occur in physical optics to make the optic laws discovered over the
course of time dependent on whether the eyes
of all people at all times saw or evaluated these laws exactly as
modern physicists do. The same goes for the laws of acoustics, and of
course for those of harmonics. The universality of harmonic law is
therefore initially, like every law and norm, only a postulate, an
ideal. The task of the relevant discipline, in our case harmonics, is
to find the proof of universality in as many individual cases as
possible.
If we understand the acoustic in the familiar broadest sense, it is
evident that the ear as a receptive organ has a substantially greater
significance in modern times than before. Radio and gramophone records
alone have widened the acoustic field so enormously that we almost find
ourselves in need of an inner defense against them. Despite this, no
insightful person would want to dispense with the positive aspect of
these inventions; and from this very discrepancy emerges the
requirement of turning our renewed attention to the acoustic, the
auditory, in the broadest sense.
The task of harmonics as a science, then, is primarily to illustrate
the order of the tone-ratios. Here one must refer to a fact that may
sound somewhat unfamiliar to “physical” ears, but which is nonetheless
true: Science as yet knows of no system of tones!
System of Tones; Importance for Physics (Acoustics)
By this I do not mean a “tone-system,” a term that belongs in music
theory and has purely theoretical-musical significance with regard to
the tonal material from which music is made today as it was in earlier
times. It would be better to say that acoustics, as a division of
physics, still knows no system of tones. Physical acoustics has
examined and theoretically established many individual phenomena-such
as the dispersion of sound waves, the emergence of sounds, the law of
the overtone series (tone-color theory), Fourier's series, resonance
theory, and many others. There is, of course, also a universal theory
of vibration that applies to phenomena in the acoustic, optic, and
electromagnetic domains of physics. But an actual theory of tones,
built upon a system of tones, is not yet known to physics, and I hold
this deficiency to be one of the reasons why in most modern physics
textbooks, the chapter on “acoustics” as a separate entity has
disappeared, and the relevant individual acoustic phenomena are treated
as paradigms of the general study of waves.
Here harmonics enters the arena of scientific research. In 1868, A. von
Thimus, on the basis of his rediscovery of the ancient Pythagorean
Lambdoma, made this system of tones accessible again for the first time
in the modern era. Thimus examined this “Tabula Pythagorica”-presumably
the “abacus” of the ancients-mainly with regard to its algebraic and
tonal laws; for the moment, we will not discuss the symbolic
interpretations he drew from them. But this “Lambdoma,” which I call
“partial-tone coordinates” and which the reader will find developed in
§20 of this book, is nothing other than a strict group-theoretical
continuation of the overtone series according to its own inherent law.
Remarkably, this discovery, so exceptionally valuable for acoustics and
tone psychology at the time it was made (1868), went completely
unnoticed by the specialists of the field. I would have expected
Helmholtz, for example, to have adjusted the later editions of his Lehre von den Tonempfindungen (1st
ed. 1862) to Thimus's rediscovery of the Pythagorean tone-system-which
is indeed based upon the linear overtone series-all the more so since
Helmholtz had a great understanding of Pythagoras and the pure-tonal
ratios. Surely he never acquired A. von Thimus's Harmonikale Symbolik,
or else, as an outspoken anti-metaphysicist, he was frightened away by
the title and remaining content of the book. But why are we speaking of
times past? Today, 80 years later, the situation has not changed:
neither the tables in Thimus's works nor those in mine (beginning 1932)
have attracted even the slightest attention from physicists, although
they could have made this discovery mostly on their own, and thus been
able to build the “tone-system of acoustics” on a new, secure
foundation.
Comparison
To clarify the progress of the harmonic system of tones in contrast
with the physical-acoustic attempts to bring order to the tone
phenomena, I can offer the following examples. Every reader of this
book will already know something about atoms and chemical elements.
They are the basic components from which matter is made. Today about 90
of these elements are known. Hydrogen, with the symbol H, is the
lightest element, followed by helium as the second lightest, and so on.
Earlier, the series of atomic weights was arranged linearly, just like
the overtone series. Chemists knew that certain elements would combine
with each other in certain ways, just as musicians knew that certain
tones in the overtone series could be combined into intervals and
chords. But there was no insight into the law governing the arrangement
of all the atomic weights among themselves. Things changed completely,
however, when the physicists Meyer and Mendeleyev had the idea of
arranging these atomic weights in groups. They broke up the series at
certain points and placed some sections beneath others. The result is
now known as the “periodic table of the elements.” This arrangement
provided a surprisingly deep insight into the laws governing all the
elements and their relationships with one another. A few spaces were
empty at the time (as dictated by the arrangement), and the
characteristics of the as yet undiscovered elements could be predicted
almost exactly. And it would not be wrong to give credit to the
discovery of this periodic system for the enormous progress
subsequently made by chemistry and atomic theorists. This periodic
table of elements, however, is obviously a purely mental arrangement of
a natural phenomenon, in this case that of the atomic weight series.
Nobody found this system outdoors, buried in a mineral mine, or
indoors, in a laboratory beaker. Although its implications and
indications are hugely important for practical chemistry, in reality it
does not exist at all!
Harmonists did something very similar 3,000 years ago. Through simple
monochord experiments, they must have learned very early on of the
linear partial-tone series with its simple number law. And one day, an
ingenious mind came upon the idea of interpreting this partial-tone
series that he had found on the monochord, which is indeed identical
with the overtone series. He calculated, from to this partial-tone
series (1 1/2 1/3 1/4 etc.), new partial-tone series, using the
individual fractional values to start new series. Then he placed some
beneath others, and thus the “periodic table of tones,” or the
“Lambdoma” as the ancients called it, or the “partial-tone coordinates”
as I call it, was discovered. I recommend that the reader consider this
comparison of the periodic table of elements with our harmonic system
of partial-tone coordinates in the same way that all comparisons should
be considered: as a parallel, incomplete but touching upon the
innermost core, between a well-known phenomenon (periodic table of the
elements) and an as yet unknown phenomenon (partial-tone coordinates).
Thimus, as has been remarked, found only the first, one might almost
say the most primitive, laws of the Lambdoma. All the more astonishing
is what he was able to do with his few formulae in terms of an
interpretation of various symbols from the traditions of ancient
wisdom. The actual fruitfulness of the Lambdoma begins with its
redevelopment, through the tone-number groups and the partial-tone
coordinates up to their construction in tone-space, which last is
investigated and illustrated, for the first time, in §37 of this book.
It is in this spatial formation of the tone-system that the most
peculiar configurations of tone-number groups appear, which mathematics
will have to deal with using new concepts such as the “geometric
discontinuum.” Acousticians and music theorists, especially, will find
a wealth of material there, whose strict logic will repeatedly lead
them back to the universal harmonic system of tones. With this system,
a new era also begins for music theory.
Importance for Music Theory
Today's music theory, just like acoustics, lacks an arrangement of
tones allowing it to derive the most important musical phenomena, such
as the whole-tone, scale, counterpoint, cadences, a legitimate
coordination of the chordal to the melodic, etc. In the harmonic system
of partial-tone coordinates, we have this arrangement. Since it can be
demonstrated on the monochord, its factual nature is beyond dispute.
Here, also, it is just as remarkable as it is incomprehensible that all
“official” music theory and musical science, with few exceptions,
have paid no attention to these things-to their own disadvantage. In my
experience, these circles are dominated by an astonishing ignorance of
the most basic acoustic and mathematical phenomena. Today books are
still appearing on harmony, tone psychology, tone character, etc.,
which either completely ignore the mathematic-acoustic data on which
fundamental elements of music rest, or handle them using tools from the
dustiest, stalest physics textbooks-without making any mention of the
results of harmonic study, which would provide the real foundation for
their work. So something is wrong here. The reasons are obvious: all
these proponents of music theory and musicology are lacking in
mathematic-acoustic education. Future musicologists and music theorists
will have to acquire this education and work with the results of
harmonic study, otherwise there is the danger that their works will
become out of date while still in manuscript, and be superseded before
they are published. A further reason for the “horror” that music theory
and music psychology have hitherto had of harmonic discoveries is the
widespread superstition that the overtone series, being a simple
natural law, is not suited for eliciting psychological laws. But who
will claim in earnest that the overtone series is only
a natural law? Are not the intervals of its primary ratios anchored in
our psyche, just as outside in nature? The intervals at the beginning
of the overtone series-the octave, fifth, fourth, major and minor
third, whole-tone-are they not spontaneously perceptible by our psyche
as correct or incorrect? Be that as it may: in the following work it
will be proven that many phenomena that were previously only
understandable “psychologically,” such as scales, counterpoint,
cadences, etc., can be derived directly from the harmonic system of
tones. Here it can also be said that music theory, by the same token as
physics, only harms itself by clinging to its aged and doctrinaire
standpoint. But with this rediscovery, reestablishment, and further
development of the harmonic tone-system, as yet unknown to modern
physics and musicology, harmonics as a science is not yet exhausted.
“Sound-Image”; Audition Visuelle
An entirely new domain opened up by harmonics is the so-called
“sound-image,” i.e. the conversion or metamorphosis of acoustic
tone-forms into optic, graphic images. As a collective term for this
domain, I have chosen the term “audition visuelle,” in conscious parallel to the concept of audition colorée (color-hearing), which plays a well-known role in synesthesia. As for audition visuelle,
it owes its possibility and emergence to the fact that every tone can
be expressed as a number, spatially (string length) or temporally
(frequency), and can thus be illustrated graphically as a line. This
applies both to the individual tone-ratios and to the tone groups and
tone curves, as well as to all polar illustrations of acoustic
phenomena and configurations. Tone curves and acoustic polar diagrams
are investigated, mostly for the first time, in my works, and are most
extensively illustrated in this book. In these “sound-images” of audition visuelle, the acoustic is thus transcribed into the visual: here we can see
the tones, not as a simple conversion of acoustic vibrations into optic
vibrations, such as is possible with electro-physical instruments, but
as a transformation of the tone-numbers and their configurations into
visual diagrams. This is something fundamentally new, different, but
still completely workable within our “scientific” way of thinking. I
have especially shown how fruitful these sound-images can
be-fundamentally, every harmonic diagram is one-in two scientific
examples of application: in the example of the “partial-tone curve,”
which turned out to be a constitutive element of the shape of the
violin body, and in the “harmonic division canon,” significant
for the history of art. The application of sound-images in the harmonic
doctrine of correspondences and symbolism is given in many places in
the following work.
Ultrasound
Recently “ultrasound waves” have become very important. These are tones
of extremely high frequency, which we cannot hear, but which have a
special effect, analogous to that of ultraviolet light waves, reaching
into the atomic building blocks of matter. They are used in medicine as
“ultrasound radiation.” In my essay “Tonspektren,”
I showed that the harmonic tone-system and its laws not only reveal a
whole series of close analogies to the laws of the optic spectra, but
also that retrospectively, some hitherto unexplained phenomena of the
optic spectra can be solved by means of the tone spectra-which are
actually only special cases of the harmonic tone-system. Obviously, the
harmonic tone-system points to a deeper, universal law for which the
“medium,” air or ether, is secondary to the norm governing it. If
today's “first trials” of ultrasound waves reach into those domains
that belong to atomic research, and if, as I showed in “Tonspektren,”
there is a close connection between optic-atomic and acoustic laws,
then future ultrasound research, if it is to have a theoretical basis,
will be forced to rely upon the harmonic tone-system. Here another wide
and fascinating field of activity is awaiting harmonics as a science.
Gestalt Mathematics
Other domains for a purely scientific handling from the harmonic
viewpoint, to which the above examples have already pointed, are
language and mathematics, among others. Regarding language, the
harmonics of the form, melody, and rhythm (poetry) of speech must be
examined, as well as writing; and as for mathematics, I see harmonic
efforts in this field crowned with the emergence of a “gestalt
mathematics” (see §4a and §36b), building the tone-number forms and
their formulae and configurations into an autonomous, self-validating
branch of mathematics. Unfortunately, my knowledge of this domain is
too limited for me to offer anything profitable. As far as I can see,
it requires an expert in group theory to establish gestalt mathematics
on a harmonic basis; the starting point will always be the overtone
series with its tone-number groups and geometric forms.
Here I believe I can skip the discussion of “harmonics as a science,” such as the “law of harmonic quantization”
and many other things. In this section VI, I wish merely to show that
harmonics has contributions to make within the “scientific” way of
thinking familiar to us today. The reader will find many other examples
in the chapters that follow. By “scientific thought,” I mean the
attitude, of research and of the researcher, that would be considered
“housetrained” within today's university disciplines.
Scientific Thought; Hermann Friedmann; “Haptics”
Regarding the relationship of akróasis as a whole to scientific
thought, one must first be clear about the heredity and origin of this
thought. From Hermann Friedmann,
we know that this thought is “haptified” in its fundamental structure,
i.e. oriented to the mode of thinking of the natural sciences,
especially as it has developed since the Renaissance. With harmonics,
we dismantle the foundations back down to these “haptics” (= perception
of the sense of touch, with its supporting pillars of measure and
number), namely to Pythagoreanism itself. Of the two original
approaches of Pythagoreanism, tone and number, only the numeric, haptic
aspect was subsequently developed as a basis for all further scientific
research. Naturally this did not exclude religion, philosophy, and art
in all their manifold forms. But the specific mode of thinking of the
natural sciences, especially as it has solidified in the last two
hundred years in the exact sciences, and above all in the haptic
science par excellence-physics,
and the technology born from it-has become such a dominant influence
over all domains of science, our entire thought structure, and almost
our entire lifestyle (civilization, comfort), that we would do well not
to close our eyes to this haptification and to see it for what it is: a
completely unnatural advancement, exceeding all human proportions, of a
very one-sided tendency, whose overpowering virulence, in the atomic
bomb, has given its first great warning signal, or-if people do not
take notice of this “human” invention-its last mene tekel.
“Mensura is the primal function of the mens! We only grasp nature where we can measure, count, or weigh”-Nikolaus Cusanus knew that. But “Cusanus discovers consciousness as that domain of the spirit (mens)
that deals with more than rational verdicts and conclusions, numbers
and measures, namely with higher things, 'ideas,' such as unity and
wholeness.”
Science and Harmonics
The relationship of harmonics to modern science can be summarized in
the simple statement: Not measure and number, but measure and value!
That is what modern civilization cannot avoid if it wishes to become a
culture once again. This does not mean putting measure in one place
(science) and value in another (religion, philosophy, art), but
revitalizing the scientific way of thinking with both of the
Pythagorean approaches, i.e. with the reintroduction of psychic
principles and norms (tone) into the currently purely haptic way of
thinking (number). Thus scientific thought will not only regain human
warmth and humane responsibility, but also the domains currently
outside of this thought, such as religion and the arts, will again be
connected with scientific thought by the symbols of the harmonic
value-forms, and thus relieved of their splendid isolation as
activities for holidays and leisure time. The possibility of this
revitalization lies in the primal phenomenon of tone-number and the
norms and laws coming from it, and harmonics is the study of the
transformation of this possibility into reality.