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.”[1] 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.[2] 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.



            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.



            Alfred Peyser writes of Goya in Von Labyrinth aus gesehen:[3] “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.



            Regarding sensitivity to time differences, Helmholtz wrote in his Lehre von den Tonempfindungen:[4] “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, in his Tentamen novae theoriae musicae,[5] 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.



            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,[6] 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,[7] and in the “harmonic division canon,”[8] 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.



            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,”[9] 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”[10] 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,[11] 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 excellencephysics, 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.[12] 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.”[13]


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.


[1] Ambros: Geschichte der Musik, vol. I, 2nd ed., 1880, p. 323.

[2] Aus Brahmanas und Upanishaden, tr. by A. Hillebrant, Jena 1942, p. 22.

[3] Zürich 1942, p. 201.

[4] 6th ed., Braunschweig 1913, p. 289.

[5] Petersburg 1739, p. 6.

[6] For example, Grundriß der Harmonielehre, by Basel Conservatory Director W. Müller von Kulm (Amerbach-Verlag, Basel 1948), which appeared after the completion of this Textbook, and the course “The Pythagorean Tradition,” organized by Ernst Levy, Professor of Music at the University of Chicago in winter 1948-49.

[7] See my harmonic study, Die Form der Geige, aus dem Gesetz der Töne gedeutet, Occidentverlag, Zürich, 1947.

[8] See Ein harmonikaler Teilungskanon, Zürich, 1946.

[9] In Abhandlungen zur Ektypik harmonikaler Wertformen, Occident-Verlag, Zürich, 1938 (1946).

[10] See §13a and §19.

[11] Die Welt der Formen, 2nd ed., Munich 1930.

[12] Ernst Hoffmann: “Nikolaus von Kues” in Neue deutsche Biographie, vol. I, Berlin, 1935, p. 255.

[13] Ibid., p. 256.