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Timestamp: 2019-04-22 13:17:11+00:00

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Although this book is intended, as I indicated at the beginning, to be a textbook for a science, harmonics is one of those sciences whose mentality goes back to the earliest times of the history of human thought, and whose effective constitution is identical with so-called “Pythagoreanism.” Pythagoras lived in the 6th century b.c.; thus harmonics, as a characteristic method of study based on the primal phenomenon of tone-number, has 2,500 years to look back on. Since all the specific harmonic works of Pythagoreanism are lost, preserved merely in their rudiments, or have only survived in their effects upon other domains (mathematics, astronomy, architecture, grammar, etc.)-this applies to Kepler, A. von Thimus, and V. Goldschmidt-these observations cover the entire “history of harmonics” as an autonomous science; and the author's indication of this Textbook as the first establishment of harmonics as a science, to his own great regret, cannot be disputed.
Yet the above references show that harmonics, as a characteristic way of thinking, must have existed in some form since the earliest times, that it existed at the time of the Pythagoreans, and that its effects and after-effects, albeit no longer as an autonomous science, have continually taken shape as a basis for other investigations, up to those of A. von Thimus and V. Goldschmidt.
The reader interested in history and philology may therefore desire to see further details, whereupon the author remarks, as he has said all along, that since he is neither historian nor philologist, he can only share the information that, so to speak, fell into his hands during the course of nearly 30 years of harmonic studies. May the following material inspire a professional to write a real “history of harmonics”!
The oldest relics of the akróatic mentality are found in almost all mythologies and fundamental religious principles, as well as in cosmogonies. Here we must turn our attention to a typical unification of tone (song, word, speech, hymnal utterance), number (number mysticism and symbolism), as well as corresponding image-concepts, especially of an astral nature, which, as we saw in §54, often unveil themselves in a surprising manner through harmonic analysis. In the earliest times, “astral mythology” was in a certain sense modern and of current interest. If one adds to this concept that of the “harmony of the spheres,” which was widespread throughout Greece, then one has two terms for very large domains, in which harmonic approaches can offer things of great promise in analytical and synthetic terms. Also important are all instances of harmonic numbers, which pervade almost all ancient religions and cosmological relics, such as the ancient Indian Rigveda, the Egyptian and Babylonian concept of God, the Zend-Avesta, and above all the wisdom of China.
Here the question emerges of whether, and to what extent, people in those ancient times knew the specifically harmonic technique: the monochord. One can unhesitatingly posit it in the most ancient times as an experimental instrument, due to its simplicity, despite the lack of concrete proof (which might yet be found in the examination of archaeological relics, if people only looked for it). If, like the author, one takes the view that all harmonic forms are psychical prototypes, which the monochord best “retrieves” from the depths of our subconscious into waking consciousness, then the search for this harmonic technique in ancient times would indeed be highly interesting and exceptionally important for the history of harmonics-but irrelevant for the factual existence of akróasis in those times. Even if it is accepted that tone-number investigations were employed here and there in secret schools (Pythagoras may well have brought his ideas from Egypt), we must nevertheless take these ancient akróatic relics as forms that simply correspond to very specific harmonic-psychic structures and, as long as humanity exists and has existed, continually strive toward ektypic realization.
However, it appears that people in those early times were already performing concrete tone-number investigations. Naturally, one must assume at least a perception for primitive pure-tone ratios as present a priori in humanity. In May 1936, it was demonstrated (Basel National-Zeitung May 29, 1936) that flutes found at the excavation site at the mammoth-hunting station of Unter-Wisternitz could play minor and major thirds, as well as fourths, with a surprisingly clear tone. An ancient Babylonian baked clay pipe found at Birs Nimrud has the tuning of a major triad (R. Batka: Allgemeine Geschichte der Musik, I, p. 39); and from Scandinavia we know of the “Luren,” which were mostly discovered in groups of two or three, thus showing that intervals or chords were blown on them. All this would be impossible if these early people had not already had a developed ability to perceive tones. With the further progress of culture, and the emergence of stringed instruments (cf. the Assyrian harps mentioned later), effective tone-number investigations are almost self-evident. It appears to me beyond doubt that this was the case in ancient China. In our analysis of the I-Ching diagram in §50.8, we already mentioned the works of some French Jesuits, and referred to Windischmann's Geschichte der Philosophie, in the first volume of which the ancient Chinese doctrine of music and numbers is discussed, after which there is no question that the Chinese had a precise familiarity with typical harmonic investigations. To this is added the enormous significance of the concept of “music” as the psychic and moral norm. A practical summary of the relevant passages from Chinese philosophy is given in R. Wilhelm's Chinesische Musik (China-Institut, Frankfurt am Main, 1927) as well as the article by Heinz Trefzger, “Die Musik in China,” in the journal Sinica (year XI 1936, parts 5-6).
In the years following the First World War, I lived in the same apartment building as an official of Chinese nationality from Eastern Turkistan, named Burham-Bey. As we got to know each other somewhat better, one day I showed him and his Chinese secretary my monochord. Burham-Bey was reminded immediately of an old Chinese man living “near” Urumqi, who occupied himself with such things. When I asked whether and how I could get in touch with the gentleman, Burham-Bey protested almost fearfully: people in that area would receive and read only letters in Chinese, and not even these if they came from Europe. Getting a reply was absolutely unthinkable. Furthermore, the man in question lived in a “fairly” rural place, about 700 km (!) from Urumqi. This information serves only as proof that number-harmonic investigation is still alive in modern China (see a further notice in my Klang der Welt, p. 139). The fascinating book The Music of Hindostan by A.H. Fox Strangways (Oxford, 1914) shows that ancient “Pythagorean” traditions are also still alive in India.
Count Herman Kayserling writes in an extremely beautiful and lively manner in his Reisetagebuch eines Philosophen (6th ed., 1922, vol. I, p. 398 ff) of a visit to the Tagores in Calcutta, where he was truly metaphysically “impressed” one evening by authentic Indian music. But what should interest us especially in Indian music, from the harmonic viewpoint, is its sensitivity to extremely differentiated tone-steps. The factual assumptions for the analysis of these steps are given in §39 (Scales). The theoretician Pavana identifies six main scale-types and 30 secondary types, and justifies his system by the fact “that Krishna let five Ragas come out of his head, his wife Parbuti gave a sixth, whereupon Brahma saw himself entitled to create 30 more secondary tone-types.” (R. Batka: Allgemeine Geschichte der Musik, Stuttgart, n.d., I, p. 41). These ragas were ancient melody schemata which corresponded exactly to times of the year and day, and were varied creatively by the individual singers and instrumentalists. “Best known is the Ragavibhoda [i.e. scale textbook], written by Somanatha in 1609. The description of the tones in secular music is mostly given with syllables or syllable signs, and in sacred music mostly with numbers” (Batka, ibid.). If one then considers that the presumably oldest literary relic, the Rigveda, is actually a collection of songs, with nothing but the lyrics remaining for us, whose melodies had specific names with symbolic meaning and were guarded in strictest secrecy, then we can assume for sure that number-harmonic investigations were also among these secret practices.
In Babylonian-Assyrian cultural circles (Neefe: “Die Tonkunst der Babylonier und Assyreer” in Monatshefte für Musikwissenschaft XII, 1890) the harp appears as a symbol and sacred instrument. Friedrich Delitzsch, in his work Babel und Bibel (1902-1905), showed an Assyrian relief-carving in which the court “orchestra” can be seen in procession with six great harps. Now, since one of the three Babylonian classes of priests was that of the singers, and the Assyrians, culturally dependent upon Babylon, made no exception in this, and since the technical production of a harp assumes knowledge of and familiarity with string-length ratios and the corresponding number-harmonic laws, it is more than likely that the priestly castes at least had knowledge of primitive harmonic norms. To this is connected the sexagesimal number system invented in Babylonian-Assyrian cultures-which corresponds to the harmonic prototype of the “senarius”-and in fact all of Babylonian numeric and astral symbolism. “Babylonian number symbolism itself is established beyond all doubt. Reveries about the value of numbers have a significant place among the religious philosophical concepts of the Chaldaeans. Each god was designated by one of the whole numbers between 1 and 60, which corresponded to his rank in the heavenly hierarchy. A tablet from the library of Nineveh has preserved for us the list of the most important gods together with their secret numbers. It appears, in fact, that in contrast with this scale of whole numbers, which were ascribed to the gods, another scale of fractions [!] existed, which was connected with the spirits and corresponded in the same way to their current rank” (M. Cantor: Vorlesungen über Geschichte der Mathematik, 4th ed., 1922, vol. I, p. 43). Since the study of these ancient West Asian cultural circles is in full bloom today, we can expect further important harmonic discoveries.
Plato, in the second book of his Laws (Phaidon ed. II, p. 553), tells us that the Egyptians, regarding the arts, “fixed and consecrated [the forms and strains of virtue] ... And this practice of theirs suggests the reflection that legislation about music is not an impossible thing. But the particular enactments must be the work of God or of some God-inspired man, as in Egypt their ancient chants are said to be the composition of the goddess Isis.” Besides harps in all sizes, early flutes and trumpets already appear here. Diodorus Siculus (I, 16) writes regarding Egyptian priests: “Common speech had its first development from Hermes, and much that was not previously designated received its naming. From him comes the invention of writing in letters, and the arrangement of the worship of the gods and the sacrifice. He was the first to observe the positions of the stars and the harmonies and the nature of tones. He invented swordsmanship and taught the rhythmical movement and the training of the body to appropriate postures. The lyre he made had three strings, to signify the three seasons. For he assumed three tones, the high, low, and middle; the high corresponds to summer, the low to winter, and the middle to spring. He also taught the Greeks about expression in speech (hermeneia), hence his name Hermes. Above all, Osiris employed him as Hierogrammateus, i.e. as composer and preserver of the holy documents; people discussed everything with him, and conducted themselves in most things according to his advice.” This interesting passage shows that “Hermes” (who appears under various names such as Anubis, Thoth, etc.) symbolizes “the embodied spiritual life, and with it self-awareness, thought, teaching, and writing, the genius of the highest science and wisdom” (Creutzer: Symbolik und Mythologie, 2nd ed., 1819, vol. 1, p. 363), in contrast and completion to the “natural” Osiris; and that the ancient form of “akróasis” is realized in the very symbol of his name. From this alone, setting aside the explicit references of the ancients to the origin of Pythagoras's wisdom in ancient Egypt, one can be certain that a definite and specific harmonic technique (monochord investigations and accompanying diagrams) was pursued, but was kept strictly secret, and imparted only to initiates. Thus we can learn nothing from inscriptions, and can only support our theories upon indirect hints such as the above.
Herodotus, the “father of history,” whose reliability (apart from a few obvious fables) is increasingly acknowledged, writes in many passages in the account of his travels of Egypt that he indeed knew more about some point or other, yet was not allowed to tell it. Reading this, one has not only the impression of an unquestionable truth, but also of how the “prohibition” of imparting certain things worked in a constraining way even on a foreigner. A. von Thimus, at many point in his works, also gave proofs from classical writers that number-harmonic theorems, in particular, were deliberately kept secret (see my essay on Pythagoras in Abhandlungen for my own views on this secrecy). Compared to this esoteric harmonic knowledge that was undoubtedly present, the exoteric side of practical musical education plays a secondary role. Sources include: Ambros: Geschichte der Musik; Lauth: “Über altägyptische Musik” in Sitzungen der bayrischen Akademie 1873; and the older work of Jomard: “Mémoire sur la musique de l'antique Egypte” in Description de l'Egypte, book III, vol. I, p. 357 ff.). The idea of Memnon, the column of Memnon that resounds at sunrise, has an entirely harmonic background; likewise, above all, the ideas dating from this time regarding the relationship of tone and light (Plutarch, Symposiaca, VIII, 3)-on this see Creutzer's Symbolik und Mythologie, vol. 1, §18.
This kind of spiritual musical expression is possible only when it is concordant in all respects with the entire spiritual disposition of the people in question. In Jewry, alone among modern European nations, the living perception of akróasis of the word has been preserved; compare with this Ben Joseph's important work: “Die Struktur der jüdischen Religionsphilosophie” in Jüdisches Jahrbuch für die Schweiz, 1919-1920, p. 88 ff. As for the Arabic cultural sphere, the history of harmonics here must first direct its attention to music theory (among others, see Rosegarten: “Die moslemistischen Schriftsteller über die Theorie der Musik” in Zeitschrift für Kunde des Morgenlandes, V). Here the octave is divided into 17 tones, whereby the sharps and flats are differentiated and the diatonic scale is played with either sharps or flats alone. This, as well as the general sensitivity of the Arabs, like the Indians, to finer tone-differentiations, is still expressed in their modern music-this is shown by any of the muezzin chants sung from the minarets of the mosques, broadcast often enough on the radio. Surely a harmonic analysis could clarify this theoretically, and it would be interesting to pursue the relationship of this filigree-like music to the geometric “arabesques”; here, too, the group-theoretical forms of the “P” provide the possible foundations. Mohammed's religious ecstasies are said to have been accompanied by ideas of tones, and if he himself only permitted serious, sacred music, the enormous musical and dance industry of the later caliphs, especially in Baghdad, shows a strong preponderance of exoteric music. But a corresponding harmonic study of Arabic philosophy, mathematics, and astronomy promises, in my opinion, to be even more fruitful. Regarding the first, I already referred in §50.7 to the encyclopedia of the “Brethren of Purity,” which preserves a wealth of ancient harmonic learning. All the ideas of this sect appear to have been influenced in a Neopythagorean manner, yet somehow developed further independently. A discussion of the mathematical content of this encyclopedia-admittedly very rudimentary-appears in Cantor's Vorlesungen über Geschichte der Mathematik, 4th ed., vol. 1, p. 738 ff.-there is also a further bibliography here. Heinrich Suter's essay, Die Mathematiker und Astronomen der Araber und ihre Werke (Leipzig 1900), names various Arabic authors (no. 63, 116, 198, 303) who wrote about “music.” The encyclopedias especially (such as that of Ibn el Khatib, no. 328 of Suter's collection, of which parts still exist), and the writings of the mystically oriented thinkers, should one day be studied harmonically, as should the many tracts on proportions, in which Pythagorean legacies (cf. Nicomachus-Iamblichus) may have been preserved that are no longer extant in the Classical legacy. It is known, indeed, that after the destruction of Constantinople by the Turks, not only many Greek scholars fled, together with their books, to Arabic or Muslim turf, but also that shortly afterwards, many Arabic scholars traveled far to the west, and collected whatever they could find in libraries.
After these brief remarks and historical notes regarding a harmonics that was on the one hand doubtless present but no longer concretely handed down, on the other hand pervasive of the most varied domains and cultural circles in its prototypical forms, we now come to Pythagoreanism as the first emergence in history of harmonics as a science and philosophy par excellence. Admittedly, this claim must immediately be dampened by the restriction that here, we no longer have any of the principal works. However, we do have concrete fragments (especially Philolaus) and traditions that situate harmonics beyond question as a science dealing with tone and number. Here I must refer the reader to my essay on Pythagoras in Abhandlungen, where the Pythagorean “complex” is considered, admittedly very briefly and incompletely, but I believe correctly in the essential outline. And for the first time in the history of research, it is considered from the viewpoint from which it must be considered: not merely from the aspect of number, but from that of tone-number. The classical testimonies that the Pythagoreans used tone-number investigations as the basis of their studies are so numerous and trustworthy that one can only marvel at how almost all of modern philology regards the auditory element, when it does not neglect it completely, as a very unwelcome and complicated element, whose whimsical nature one must somehow make allowances for; there is a desire not to place the central Pythagorean concept of “harmony” on a level with such “lowly” things as monochord experiments, etc., and it receives a fantastic inflation which no longer has anything to do with the concrete and flexible sense that this concept had for the ancients. The otherwise so meritorious Boeckh, for example, published an essay, “Über die Bildung der Weltseele im Timäus” (in Daub and Creutzer's Studien, 1806, vol. 3), in which he discusses the famous-and infamous-“Timaeus scale,” obviously stemming from Pythagorean sources, and very meticulously seeks to elicit the tone-value of the relevant numbers, but obviously has not thought of the most important thing: to test these tone-numbers on the monochord and to hear them. If one does that, then connections and explanations appear immediately, as well as corrections, at which purely logical-intellectual observation alone can never arrive.
I once excitedly brought my monochord to a similarly worthy university professor, at his wish; he was positively stuffed with knowledge of sources, references, books, etc. He was working on Pythagoras's “harmony concept,” and sought to understand the fragments of Philolaus purely abstractly, on the basis of modern “idealistic” thought. In the matter of sounding tones and numbers, I explained to him fragments 5 and 6 of Diels, and referred to my work on Pythagoras, in which not only these, but a series of other important fragments, especially those on the “apeiron” and “perisson,” etc., had found obvious clarification. But it was evident that the learned man was so unmusical that he could not judge the purity of an octave at all, let alone the fifth, third, or “scale” and their psychical forms-and he did not want to know anything about my essay on Pythagoras. When he simply asked me where I got my insights from, and hoped that I would produce a long list of literary names and references, I simply indicated the monochord and my head, as well as the spirit of Pythagoreanism, based on which I believed myself to have worked and thought. He smiled pityingly: “Well, well!” I tell this story only as characteristic of the modern situation, which every harmonist will find himself confronting in regards to science! The clogging up of what has been handed down from ancient times, especially in historical things, is so great that no number of Galilean telescopes can help, and we must wait for new spiritual powers outside the “science” that will take up and continue the new ideas with enthusiasm, and without worrying about success or failure. The only book I have found after the completion of this Textbook that seeks to grasp Pythagoreanism, and above all the “Pre-Socratics” based on the nature of these thinkers themselves, is K. Joel's excellent work: Der Ursprung der Philosophie aus dem Geiste der Mystik (Jena, Diederichs, 1906), which is recommended especially to the reader.
For a new foundation of the spiritual history of Pythagoreanism, beside the standard works-Harmonikale Symbolik by Baron A. von Thimus (2 vols., Cologne 1868-1876) and the above mentioned book by K. Joel-I know of only two useful books in the German language: Erich Frank's Plato und die sogenannten Pythagoreer (Halle 1923) and Julius Stenzel: Zahl und Gestalt bei Platon und Aristoteles (Leipzig-Berlin 1924). In neither work is “harmonics” considered as an autonomous concept, and thus Thimus's work was also unknown to them. Nonetheless, Frank is one of the few scholars who views the enormous significance of music for the ancient Greeks in the correct light, and the information, references, etc. that he gives are very useful as an initial basis; his idée fixe that Pythagoras never actually existed is unimportant for our purposes. Stenzel begins more with the “dieresis of ideas,” i.e. actually with a demonstration of ancient thought with-as we may express it harmonically-the “law of harmonic quantization,” which in fact manifests from the monochord in the “P” system. As a basis for the ancient sources, the literature about it, etc., the following things are indispensable: (1) the first volume of Überweg's Geschichte der Philosophie, and (2) Diels' Fragmente der Vorsokratiker. Especially in the latter work, the reader learned in Greek will find, in the information, quotes, etc., surrounding the fragments, a wealth of harmonic treasure, which can now be summarized, since harmonics has been newly constituted, from unified viewpoints. Admittedly, as I learned from Olof Gigon: Der Ursprung der griechischen Philosophie (Basel, Benno Schwabe, 1945, p. 11), Diels' Fragmente der Vorsokratiker offers only a restricted choice of texts; a future study of Pythagoras should first assemble all the material that has been handed down to us.
If this-the rectification of true Pythagoreanism-ever happens, then we will also see that his influence reached much farther than people have yet dared to believe. Not only music and astronomy must be included in this: not only arithmetic, geometry, grammar, rhythm, and above all mythology, Neopythagoreanism, Gnosticism, architecture, etc. receive new illuminations from it; but all these domains are finally truly understandable from an inner, central synthetic viewpoint, namely that of akróasis, above all in the deeper sense as the collective cultural and spiritual attitude of an epoch. Thus it will also be shown that we must give up the comfortable humdrum of simply denying anything for which there is no explanation, all that we do not understand or that does not fit into the common philosophical-historical judgment. A model example of this is Euclid's essay “On the Division of the Canon” (for the ancients, “canon” was synonymous with “monochord”), long seen as a “falsification.” How could such a great man devote his time to such an outmoded thing as the division of the monochord! People do not understand, or do not want to understand, that at that time the monochord was a teaching tool precisely for the knowledge of the study of proportion that was so important for the ancients, completely aside from its value as the essential Pythagorean experimental instrument. We know from “legend” (!) that Pythagoras, shortly before his death, asked one of his favorite students to strike the monochord, “thus indicating,” as Aristides Quintilianus writes in his De Musica, “that the highest and final things that music treats can be grasped not so well by means of the tones heard through sensory perception, as by way of intellectual observation of the numbers” (from Thimus I, 128). Erich Frank (op. cit., p. 182) thus writes with complete justification of Euclid's work: “This Canon Division can thus be understood as a counterpart to Euclid's Elements: while the latter provides the mathematical knowledge necessary for the Platonic construction of the world's body, the laws at the basis of the construction of the world's soul are developed in the canon.” In the same breath (pp. 183-184), however, Frank explains the “scale of the [Platonic] Timaeus” as a “theory born from the start as a dead letter,” indeed as “insane number speculation” and a “musically quite impossible scale” (p. 17). Thus the same thing happened to him that happened to all commentators before him (Tannery etc.): he did not understand this “scale” at all. A. von Thimus (Harmonikale Symbolik I, 156 ff. and II, 281 ff.) gave the only adequate interpretation of the Timaeus scale thus far, from his all-embracing knowledge of ancient number-harmonics and Pythagoreanism (see the illustration of this scale in §13a and §39.2a of this book)-of course, without eliciting the slightest result for those who study it. The reason is, as with all harmonic discoveries and references, always the same: people expect recipes, but they should realize that the necessary prerequisite of all “harmonicalia” is not only a precise familiarity with Greek language, arithmetic, and music, but also an equally precise knowledge of the harmonic technique, which must be learned and studied just like the grammar of any language. The otherwise so thorough and philologically precise E. Frank writes, for example, the following (op. cit., p. 154): “The basis of our modern music is the diatonic [!] octave, where between two tones there is always [!] a whole-tone interval [!] and hence [!] this type of tone has been thus named by the Greeks [!].” Armed with such a “knowledge” of elementary matters, he then writes for pages about Greek music theory, and projects this nonsense in his head onto the “insane and musically quite impossible scale of Timaeus”! Now, since a musical historian typically understands nothing of mathematics, the philologist nothing of music or mathematics, and the “classical” historian of philosophy nothing of any of the three subjects, and none of the three know or want to know anything about the technical-harmonic fundamentals of harmonics (monochord laws), we still have the persisting calamity of a misunderstanding of precisely such problems as the Timaeus scale, ancient Greek enharmonics, and so forth-even long after their solution has been found (by Thimus).
Observed, or rather newly observed, from the point of view of akróasis, not only Pythagoreanism, the pre-Socratics, Plato, and Aristotle, but also the entire “succession” of the ancients must be subjected to a corresponding revision.
As the central workbook for the ancients and their precursors, for a future history of harmonics, A. von Thimus's Harmonikale Symbolik is above all to be valued and used. In §25.1 I have tried to characterize the value and majesty of this work in general terms. Thimus's harmonic “toolbox” is admittedly limited in intellectual-discursive terms and ignores many geometric-visual theorems that are important and indeed indispensable for an illumination of many ancient symbols. The reader will easily be able to form his own judgment after a careful comparison of Harmonikale Symbolik with this Textbook. However, Thimus presents an almost endless array of historical material which should now be worked through anew from the point of view of harmonics as an autonomous science and way of thinking.
Firstly, ancient music theory and its succession. Its kernel is the ancient Greek “enharmonics.” This was first discovered in all its richness, unveiled, and precisely examined, by A. von Thimus, and anyone who has closely studied the relevant parts of Harmonikale Symbolik, and after this hard-earned knowledge has taken up some book on Greek music theory, will agree with me that we must start again from the square one; that all other authors, no matter how worthy their work may be (Westphal, Ambros, Riemann, Abert, etc.) are either completely ignorant or work with false assumptions about this central concept of enharmonics (which makes chromatics, diatonics, rhythm, etc. understandable for the first time). Thimus devoted his life's work to this, and any future revision of this domain that ignores what he has offered is, like everything written since his work, condemned a priori to obsolescence. If these ancient fundamentals explained by Thimus are taken up anew, then new light falls upon all of music theory from the Middle Ages to modern times. It is more than merely accidentally or “coincidentally” significant that the ancient Pythagorean experimental instrument, the monochord (see Wantzloeben: Das Monochord als Instrument und als System, Halle 1911), has been used right up to modern times as a scientific and practical teaching instrument (see §1c and §1d). It is the symbol for a pervasive, living Pythagorean legacy, and I am convinced that from the numerous writings about the monochord (see Wantzloeben's bibliography) and the ancient harmonics that still sound in it, entirely new viewpoints will arise not only for the history of music theory, but also for the related practical domains such as notation, church music, the study of instruments, etc.
(from Cantor, op. cit., pp. 583-584.) However, this figure, which then follows in the manuscripts, is simply an ordinary multiplication table, as Thimus mentioned (vol. I, table II, Fig. 1 and text p. 144 ff.), and has not the least to do with “profundity.” It is to be assumed that by the mensa Pythagorea, the ancient Pythagorean “Lambdoma” was meant, and therefore the partial-tone coordinates that even Boethius did not know correctly but only from legend, and which indeed, besides their musical norms, also contain very important arithmetical and geometrical laws, as we have shown in several passages in this book: laws that go far beyond the multiplication table and deeply into number-theoretical speculations (for which such a mathematically gifted people as the ancient Greeks had a great understanding). The foundation on a harmonic basis of arithmetical, geometric, and certain number-theoretical elements (concept of the infinite, the irrational, as well as the entire study of proportion that was so important to the ancients) is beyond question for anyone who arrives by necessity via monochord experiments upon the diagrammatic notation of the “P” and insight into its number-harmonic construction, and its obvious arithmetical, geometric, and number-speculative configurations. The famous “discovery of Pythagoras” of the dependence of tone-perceptions (quality) on string lengths (quantity) in the sense of a precisely determinable numerical relationship almost pales beside this, being reduced to an admittedly important “special case” of Pythagoreanism, when we are forced to see, in this discovery, the birth of our precise scientific methods. But as I always emphasized regarding this discovery (which was certainly ancient knowledge in all high oriental cultures!), its “flip side” must be seen as at least as important for ancient culture: that in this discovery, the quantitative (string length, or simply matter) can be qualitatively-psychically evaluated (the audible tone ratios). And it was precisely this double aspect of the tertium comparationis of the “number”-that it reaches into matter on one side, into our psyche on the other side, namely into our psyche not only as intellectual-discursive logical form, but as an entire gestalt of our perception, feeling, our soul (interval, chord, scale, etc.)-which so “intoxicated” the ancients in this “discovery of Pythagoras.” This akróatic background of ancient number-thought has been completely lost to us today, and for these reasons we also find, in almost all works on the history of mathematics, a complete lack of understanding, and connected with it, a lack of interest regarding the harmonic foundations of ancient mathematics. Here almost everything must be redone and rebuilt.
The partial-tone coordinates of index 16 (PE16) with their logarithms (base 2), coordinates and tone-values, decimals and angles (frequencies).
Next, the foundations of language (word), grammar, and rhythm. The Introduction to this book discussed the akróasis of the word, speech, and the harmonic background of all spiritual utterance given a priori with it. This background was expressed in a peculiar way in the earlier times of high cultures through the singing, or reading in an emphatic voice, of the cultic hymns and songs, and it has continued in the rituals of all religions to date. The “music” here is not only an “art” like painting or architecture, which latter gives worship more of its outward symbolic consecration. The musical element here is deeply connected with the metaphysical meaning of the “word” as that form of communication that mediates the divine by auditory means.
Music and word, seen thus, are finally the same: pronouncements from and to God. If I now refer briefly to the specifically harmonic backgrounds of ancient grammar and metrics (the study of syllables, etc.), it is on the one hand with the belief that here harmonics still has much to offer, but on the other hand with the assumption that in philosophical regards (including by the ancients) much effort has been made of which I am still unaware. As for grammar, I simply refer to Eberhard Hommel (Untersuchungen zur hebräischen Lautlehre, part I: “Der Akzent,” Leipzig 1917). I have already quoted a passage from this in §31a, and I entreat the reader to read it once more. These “threads” must be followed further by specialists. Important material related to this can also be found in Franz Dornseiff: Das Alphabet in Mystik und Magie, Leipzig 1922. In Plato's late dialogue, the Philebus, there are observations on tone, number, sound, and letters, which refer to the great age (Egypt) of these typical cross-references. A future historian of harmonics will have an easier time with the connection of harmonics to rhythm and metrics. I know of two works, (1) Aristides Quintilianus: Über die Musik (tr. and thorough commentary by Schäfke, Berlin 1937), and (2) the church father Augustine, Musik (tr. by Perl, Strasburg 1937), which contain extensive rhythmic and metric observations, together with their relationships and derivations to and from harmonic number-ratios, which can easily be developed further, both forward and backward. Moreover, these particular works have a truly Pythagorean “timbre,” i.e. an akróatic spiritual disposition pervaded by the universal significance of the musical, which in many places gives one pause and fills one with awe and astonishment at such a depth of thought-unfortunately entirely lost today.
Next, astronomy, and the astrology allied with it, as well as the harmony of the spheres and astral symbolism that emerge from both. Copernicus, in his De revolutionibus orbium coelestium (1543), explains expressly that he has taken up the heliocentric solar system of the Pythagoreans. He quotes Plutarch: “The others believe, however, that the Earth stands still, but the Pythagorean Philolaus says that it moves around the fire on the slanting circle of the ecliptic in the same direction as the sun and the moon.” “Starting from here, I [Copernicus] began to contemplate the movement of the Earth and, although the viewpoint seemed absurd, I still did it, because I knew that others before me had already been granted the freedom of conceiving arbitrary circular movements for the representation of the heavenly bodies” (from E. Frank: Plato und die sogenannten Pythagoreer, Halle 1925, p. 37). Frank presumes (p. 38) that the origin of the heliocentric idea is to be sought among the writers associated with Archytas. Archytas, like Democritus, wrote a work on harmonics that is now lost, and both appeal to Pythagoras as the ancestor of their ideas. A few fragments of Archytas's writings still remain, which O.F. Grupps published (Über die Fragmente des Archytas, Berlin 1840) and, following the skeptical custom of his time, declared that “not a single thing is to be taken as true.” This untenable viewpoint has been taken ad absurdum by A. Speiser in his translation of the beautiful fragment from the harmonics of Archytas (Klassische Stücke der Mathematik, Zürich 1925, pp. 9-11). Archytas's principles still resound in the harmonics of Ptolemy, according to E. Frank (op. cit., p. 166). Claudius Ptolemy, mainly known for his Syntaxis (the “Almagest”), wrote, among other things, a still surviving Harmonics, which, paraphrased by Porphyry in another surviving commentary (Ptolemy's study of harmonics and Porphyry's commentary were published in two excellent text-critical editions by Ingemar Düring, Göteborg 1932), was and quoted numerous times by Johannes Kepler in his Harmonice Mundi (see the entries “Ptolemäus” and “Porphyrius” in the index of Caspar's translation), and mentioned expressly by Kepler as one of his most important sources and inspirations. Kepler himself originally had the intention of commenting exhaustively on Ptolemy's Harmonics, and a brief part of this commentary is included in his Opera omnia. Due to my limited knowledge of the Greek language, I have not yet been able to study the relevant works of Ptolemy and Porphyry; surely they contain important material for the history of classical harmonics.
Little known and less investigated are the relationships of harmonic to astrology and partly also, via astrology, to alchemy. Very early on, the doctrine of aspects was already compared with intervallic consonance and dissonance. In the fourth, astrological book of his Harmonice Mundi (Chapter 5), Kepler, after a most interesting exposition on the intelligible nature of the harmonies (musical philosophy has let these valuable ideas escape it until now), discusses the actual doctrine of aspects, based on Ptolemy, Cardan, and Reinhold, tracing the musical intervals and astrological aspects to geometrical phenomena.
I can only refer to the broad domains of the harmony of the spheres and astral symbolism as historical domains which, with the new harmonic-analytic method achieved on the basis of this book, can be dealt with in terms of their actual inner nature. Far into the Middle Ages, there was an enormous body of literature, which for the most part still remains buried in the manuscripts of libraries, and has not yet been edited at all-the recently published catalogs of medicinal manuscripts are proof enough of this. Regarding the harmony of the spheres, Jacques Handschin published an excellent essay in the Zeitschrift für Musikwissenschaft 1926-27, “Beitrag zur Sphärenharmonie,” which can be used as a basis for further studies due to its deeply based erudition. The literature on astral symbolism is substantially larger-here I will mention only F.X. Kugler: Sternkunde und Sterndienst in Babel (1907-1924) and R. Eisler's work, already cited many times: Weltenmantel und Himmelszeit (Munich 1910), in which the interested reader will find a wealth of material and further reading.
Regarding the history of architectural harmonics, the most important things have been indicated in §29 of this book, so I can summarize quickly here. The central works here are those of Vitruvius and Eichhorn. Harmonics in painting and sculpture, as in architecture, has its background in certain proportions. There are three “primal proportions,” the “arithmetic,” the “harmonic,” and the “geometric” proportion, which, as we saw in §28, are all contained in the “P”, and are thus of harmonic nature. Thus, when one writes of the history of the harmonics of architecture and the visual arts, one must preface it with the history of proportions and of proportional technique, which was of enormous importance in classical cultures. Thimus anticipated the essentials in his “preamble” (from the commentary of Iamblichus on Nicomachus's Introduction to Arithmetic) and, in the course of his Harmonikale Symbolik, he developed the proportional technique found there most precisely and thoroughly-a very subtle method, not at all simple for us today, since we have become so “non-visual” compared with the ancients. This proportional technique should be handled and expounded in its historical metamorphosis up to the Renaissance as a fundamentally separate domain of mathematics; one will then not only see that the study of proportion has directly stimulated architecture and painting through all periods (until its downfall in modern times)-think only of the proportional studies of Leonardo da Vinci, Albrecht Dürer, as well as almost all great architects!-but one will also then be cured of the one-sidedness already mentioned in §28, such as the “golden section” etc. (which fanatics have meanwhile built into disproportionate complexes). The three primal proportions that were placed at the peak by the ancients-the arithmetic, harmonic, and geometric proportions-are harmonic by their very nature, and constitute such a wealth of formal possibilities, that in contrast, those individual proportion types such as the Golden Section, the p/n triangles, certain circle divisions, and so forth, appear sterile and poor. But these three primal proportions are concentrated in the “harmonic division canon” of the “P”, which, as I have shown in my work on Villard (Harmonikale Studien, vol. 1), must have still been known in the Gothic era as a Pythagorean legacy (see §38.1a and §41.4 of this book). The individual harmonic analyses of the individual styles must then be built upon this harmonic division canon; and Villard's problem in particular has convinced me that this canon, within the Gothic style itself, provides a method of style-analysis that will yield valuable results.
Historical information about harmonic relationships to philosophical and epistemological phenomena can be found in many chapters of this book, and regarding symbolism-especially religious and cosmological-the interested reader will find material sufficient for a start in §54. A historic view of harmonics in these domains is synonymous with the history of the great spiritual stages of harmonics. Therefore I will only mention the key terms: archaic harmonics (especially China), Pythagoreanism (fragments of Pythagoras, Aristides Quintilianus, Ptolemy), Plato's late philosophy (Timaeus scale), Augustine (De musica), various Renaissance philosophers only known to me by name, such as Marsilio Ficino, Cardan, etc., in whose works harmonicalia can presumably be found; also Robert Fludd should, despite Kepler's polemic, at least be examined in a historical-harmonic sense; then of course, above all, Kepler himself, the Harmonie universelle of Father M. Mersenne, Paris (3 vols., 1644-1647), Athanasius Kircher's Musurgia; of the more recent, especially Leibniz, Th. Fechner, J.J. Bachofen, and last and still most important for us, A. von Thimus. From these writings, details and cross-references will arise of themselves, indeed entirely new names and works will emerge, which have not yet been historically placed anywhere, and will perhaps for the first time find their homes in a “history of harmonics.” For the history of harmonic symbolism, Thimus is authoritative above all: the introduction to his work, especially, gives a wealth of names and references that are indispensable not only for symbolism, but for the historical development of the number-harmonic way of thinking in general.
Near Bern (Switzerland), November 23rd, 1944).

References: V. 
 V. 
 §54
 §50
 §39
 §18
 §50
 §13
 §39
 §25
 §1
 §1
 §31
 §29
 §28
 §28
 §38
 §41
 §54