With this work, I offer those readers interested in a new philosophical direction (philosophia = love of wisdom, spiritual aspiration) a textbook for a field of study which, after existing in classical times under the name of Pythagoreanism, appeared only sporadically in later times, and currently has no existence in our modern worldview.

            Harmonics” comes from ̔αρμονικός (= harmonic), but its root is the verb ̔αρμόττω (= to put together, arrange). This arrangement, put together from tone and number, flowing out in a harmony of the world (cosmos), is the original concept of “harmonics” as Pythagoras and his successors understood it. The concept of “akróasis” (= listening), in opposition or complement to aisthesis (= viewing), which I first applied in my book Akróasis, initially serves simply to prevent the continual confusion of “harmonics” with the musical study of chords. But perhaps it will be better in future to use “harmonics” and “harmonic” more in the sense of individual harmonic investigations, “akróasis” and “akróatic” more in the higher sense of a general mentality pervading everything—just as the following Introduction is entitled “Akróasis,” while the book itself is called Textbook of Harmonics.

            In most modern reference books, “harmonics” is explained simply as “the study of harmony,” i.e. the study of chords in music. This definition relates to our concept of harmonics only in that the chord, as a theorem, represents one of the many special cases of general harmonics (see §40). Only in the Schweizer Lexikon have I found an entry on “harmonics” in which the concept is explained speculatively with reference to A. von Thimus and my works (E. von der Nüll’s work mentioned there has nothing to do with harmonics in our sense). Harmonics is defined there as a “metaphysical rationalism”—not a bad definition, if one knows from experience how hard it is to describe, in a few words, a discipline that is in the process of rebirth. But this definition is too narrow, and is dictated by the understandable difficulty of classifying harmonics purely in terms of the history of philosophy.

            Because harmonics, as I have attempted to reawaken and reestablish it in my thirty years of work based on Pythagorean sources (see the summary of my works at the end of this book), is not simply a philosophical speculation in the sense of a “metaphysical rationalism.” It is founded upon three pillars:

            a) Upon scientifically testable conditions. These are the harmonic theorems of tone-number, psychophysical realities that we can prove in nature and in our psyche. This book is organized according to them. In this way, harmonics is a science.

            b) Upon a progression of correspondences. But these correspondences do not occur in a series of vague analogies. Their roots are in the harmonic theorems, and through these they are verified. This way of thinking and investigating is no longer merely scientific, but takes place in a field of connections between material, psychic, and spiritual forms which, without reference to the harmonic theorem, appear to have nothing to do with one another. In this way, harmonics is a doctrine of correspondences.

            c) Upon a system of value-forms which, conceived autonomously but confirmed by theorems and made universal in their meaning through correspondences, achieve symbolic character. Even individual theorems, and yet more so the value-forms, can become harmonic symbols. Their true domain is that of metaphysical, religious, and mythological forms. In this way, harmonics is symbolism.

            If I nevertheless believed myself entitled to call harmonics a new philosophical direction, it was only in view of the renewal of the original concept of “philosophia,” as I defined it above.

            Every major harmonic work, with its tone-number descriptions, tables, formulae, and diagrams, initially gives the impression that at least a great understanding of music theory and mathematics is required to comprehend it. This is not the case. To understand the most important fundamentals of harmonics, it is sufficient to know the multiplication tables and a few basic geometric and musical concepts—these latter are explained simply in this book for those unfamiliar with them. But in any case, the reader must work along with the book, carefully drawing the diagrams with a pencil and ruler, continually working out the tables himself with their simple tones and numbers, and hearing the tones for himself on the monochord; otherwise he risks getting out of his depth with the optical and acoustic apparatus that is fundamentally important for understanding. If he makes this effort, he will soon see the apparent complication of the tables, drawings, and diagrams resolve into a few very simple facts. Besides, it is precisely this working along, rather than simply reading and following intellectually, that builds those foundations from which harmonic image-concepts can begin to come alive in the psyche.

            For the understanding of harmonic fundamentals, then, a standard secondary school education suffices. These fundamentals are, in any case, explained in the most elementary manner possible in the early chapters of this book. If the fundamentals later expand into the widest domains of knowledge, and reference is sometimes made to more complex things, this is requisite for an establishment of harmonics in those domains and is a concern of the individual disciplines in question. This “ektypics” does not itself affect the fundamentals, so the reader who lacks the relevant specialized knowledge should accept them as far as he understands them, and pass them by when he can no longer follow along. He will still find enough to encourage him to contemplation and study.

            Those who are encountering harmonics for the first time in this book are advised, although this is not entirely essential, to read Akróasis, which was written after the completion of this book and published in 1946 (Basel: Verlag Benno Schwabe; authorized reprint, Gerd Hatje Verlag, Stuttgart, 1947).* There I attempted to give a brief, concentrated illustration of the study of harmonics up to this time. In the present “Textbook” it is perhaps best to read the Introduction after this Foreword, then §51, the Dialogue on Tolerance, in which an attempt is made to negotiate a universal harmonic “equilibrium.” Readers interested in specific topics, e.g. mythology and symbolism, should perhaps read §54 (Harmonic Cosmogony) after the Introduction, and then study the fundamentals from §1 onwards. Philologists and historians might prefer to start with §55 (Summary of a History of Harmonics), and then, if there appears to be “something in this,” to bite into the tough apple of specialized harmonic study. Here the Index is especially helpful, since its terms synthetically unify the same topic often discussed variously in the individual chapters, besides directly explaining some specialized terms or indicating the passages where they are explained.**

            The musically versed reader should not confuse the concept of the “study of harmony” with “harmonics.” The latter does indeed use certain elementary expressions of the former, but otherwise goes entirely its own way, and in the process must incorporate many concepts that have nothing to do with the musical study of harmony. In a further sense, music is merely a special case, albeit the most important, of the artistic side of harmonics. In addition to music, there is a harmonics of every science, a harmonics of philosophy, a harmonics of religious symbolism, etc., and all these individual harmonic domains are unified by harmonics itself as an autonomous study, containing its own justification. The following Introduction will explain what this autonomous study is believed to be, and what its character is.

            Those familiar with my earlier harmonic works will notice the greater extent and the comparatively broader foundation of religious-symbolic elements in this book. This is not the result of a change of view. If the insights I have obtained from editing the 13-volume collection Der Dom, Bücher deutscher Mystik (Inselverlag, Leipzig, 1918-25) also offer me the possibility of illuminating harmonic problems correspondingly, it still lies in the nature of harmonic symbolism itself to arrive repeatedly and often involuntarily, by means of its akróatic image-concepts, at the deepest matters of thinking, willing, and feeling. Because surprisingly simple interpretation for many symbols from the most ancient religions, mythologies, and cosmologies can be found through harmonic analysis, close attention is paid to the relevant historical sources. It is not my intent, in the sections in question, to put forward a new foundation of religion, or even to pontificate about matters of religious belief; each reader can draw his own conclusions from the content of these sections.

            In paging through this book, some readers may be disconcerted by the hundreds of quotations and references from numerous domains of knowledge—in short, by the apparent accumulation of a vast quantity of stuff. In this regard, the reader is entreated not to let himself be duped. It is easy to cram knowledge without limit into one’s head, spit it out again in the form of books, and still remain a fool. The extreme diversity of the material treated by harmonics necessarily results in the vast scope of the literature quoted and discussed in it. But any scholar can see that despite the apparent vastness, there are great, often hardly forgivable shortcomings in the discussion of the specific domains in question, as well as in the bibliographies. Here I can only excuse myself by the fact that this work was written in the countryside during the war years, 1942-1944; therefore I relied mainly on my notes from earlier years and my own library. It will not be hard for a mathematician, astronomer, philosopher, literary historian, mythologist, architect, etc., to fill in what is missing from his own specialized knowledge. As for the universality to which every harmonic method of research inevitably leads, the reader is invited to read the discussion of this in §29.1.

            The reference to so many disciplines could also give the impression that it is the main intent of harmonics to meddle without authority in the domains of other sciences. Nothing could be further from the truth, because every harmonist, through his work, is above all required to exercise precision and exactitude, and only then set free to meditate; thus he must have the greatest respect for specialization, in the best sense of the word. Finally his science, harmonics, is fundamentally a discipline which must be learned and studied like any other. And if harmonics looks in every possible direction to support its views, if it uses various disciplines to support its insights, occasionally expressing conflicting viewpoints—well, every science does that, and without that there would be no science, let alone philosophy.

            From the specialized philosophical point of view, harmonics has been reproached for not caring about the “continuity” of the development of philosophy since Plato and Aristotle. It is true that the Pythagorean approach to tone-number, and the forms and laws given by it, offer modern harmonists a wealth of autonomous possibilities for research, which can stand perfectly well by themselves outside the development of philosophy up to now. If specialized philosophy has neglected the possibility of this Pythagorean approach or left its “material” evaluation (derivation of the qualitative tone-ratios from the quantitative number-ratios) to the exact sciences, that is its own problem. But to conclude from this that modern harmonics ignores, belittles, overlooks, or arrogantly looks down on the great philosophical achievements as realized in the systems of their propounding geniuses: that violates not only sound “harmonious” understanding but above all the tolerance, the respect for every effort, and the quiet listening to everything existing, to which every harmonist is accustomed and trained in the course of his work. Our motto here is “suum cuique”: to each his own! Let us till our ground as we believe to be proper; in any case we will look ungrudgingly and with great empathy upon the fields and meadows that flower and bear fruit around us!

            The tools of the harmonist include a lead pencil, colored pencils, black ink and a few colored inks, a ruler, a square, an accurate compass, a protractor, lots of paper, especially millimeter paper and simple squared paper, and, most importantly, a monochord. As §1 shows, one can easily build this oneself, or have one built. To have this done will cost less than the average violin, which is a necessary purchase for anyone who wants to learn to play the violin.

            Harmonics is concerned with the inner development of the harmonic scholar as a self-sufficient and individually validated person. We stand facing the inevitable destiny of ever more overpowering collectivism. The demands made on the individual by a profession, his duties to society, the ever growing difficulty of quiet self-reflection amidst the din of modern times, will require strong counterweights lest unfettered depersonalization drive humanity into a universal ant-like existence. One of these counterweights can be the harmonist’s silent work for himself, without any aspiration to the outside. Just a small room, a table, a chair, and a monochord within reach: here, immersing oneself in harmonic problems, meditating upon the diagrams and tables one has drawn, upon each fine and subtle tone of the scales, chords, melodies, and rhythms—those who are called to this will become creators of a music without notes, which is sheer anachronism in contrast to the greater part of our so-called modern music! All this imparts, to those who know how to “hear,” a harmonic state of soul and spirit that will automatically affect the conduct of the entire person in his professional and exterior life. There could be harmonists in all professions, classes, countries, and peoples. They would hold no conventions, found no orders, choose no presidents, build no temples, hardly become outwardly apparent, and exchange their viewpoints and works only among themselves, in free unification. And since they would have learned to “hear,” they would also know to “speak” at the right time and place, that is, they would try to radiate the atmosphere of their mentality as far as possible. Harmonics thus understood as a self-orientation of modern people, pressured from all sides, is not a flight from reality, but an immersion in it, a listening into reality and the nature of things (see H. Augustin: “Von der Anhörung der Welt” in Schweizer Rundschau, Jan. 1947). Anyone who has worked harmonically in this way knows that a clean and pure breeze blows within “akróasis” which he can breathe freely, and that humanity, tolerance, and respect is the great three-pointed star that he gains in his work.

            This book is arranged so as to treat the realm of harmonic material successively, following the evolution of the theorems. Most things cannot initially be worked through exhaustively, and must be taken up later from different viewpoints. For example, the theorem of “enharmonics” appears in the following ways: (1) as definition, §20.2; (2) in the handling of the enharmonic scale, §39.4; (3) in the interval powers and constants, §45; and (4) as an independent section, §48. It is similar for many other theorems, concepts, etc., and for this the Index is indispensable as a summarizing thread. Especially, sections I through V of the Introduction, and §55 (Summary of a History of Harmonics), complement each other.

            The reason why the bibliographies at the end of each chapter predominantly cite my own works is that many problems were discussed much more exhaustively there than is possible in this Textbook. Also, these earlier writings are the only harmonic works accessible today, besides Kepler’s Harmonice Mundi and A. von Thimus’s Harmonikale Symbolik. Furthermore, unifying all my harmonic works thus far (see their enumeration at the end of this book) through this “retrospection” will make things easier for the reader, all the more because Vom Klang der Welt, Abhandlungen, and Grundriß do not have indexes. Obtaining my other works (especially Abhandlungen, Grundriß, Harmonia Plantarum, and Akróasis, which was written after the completion of this textbook—the remaining stock of Hörende Mensch was burned in an air attack on Leipzig) is not absolutely essential for the understanding of this textbook, but will undoubtedly make the study of harmonics easier as a whole, and will broaden and deepen its understanding.

            As a conclusion to this Foreword, I would like to thank all those who made the creation and publication of this book possible: the publisher, who took the risk of such a venture for purely idealistic reasons, the generous donor who made the printing possible, the friends who stood by me most disinterestedly in word and deed, my dear wife Clara, née Ruda, who took on the difficult and painstaking task of copying, and above all those friends and helpers who wish to remain anonymous, who made possible a year of quiet study in this island of peace, surrounded by the most terrible inferno in history.


Near Bern

at the sixth wartime Christmas 1944


Hans Kayser


(revised 1949)


* Translator’s note: Akróasis: The Theory of World Harmonics was translated into English by Robert Lilienfeld (Boston: Plowshare Press, 1970), but is now out of print.

** Translator’s note: Regrettably, the time was not available for translating and including the Index in this first edition (see also the Editor’s and Translator’s Preface).