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History <br /> </span></em></strong></div> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Architecture unifies two elements that rarely go together otherwise: outward functionality and free creative formation. This unification, seen from the outside, is connected with an apparently inescapable one-sidedness: a complete elimination of the temporal in favor of the spatial. Analogously, one might say that music is the polar opposite of the spatial, and makes the temporal exclusive. Perhaps the feeling of the <em>coincidentia oppositorum</em>, which eternally captivates aesthetics, lies in the consciousness of this diametric opposition: namely, that deep down, a unity exists between architecture and music, and that the cliché of “frozen music” that one experiences when looking at significant architecture is just an expression of this perceived coincidence.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> In the framework of this textbook, we shall now consider as far as possible the concrete backgrounds of these collective relationships, those that will appear based upon a harmonic proportion, a <em>tertium comparationis</em>, which is no longer simply architecture (space) or simply music (time); instead, by virtue of the primal phenomenon of tone-number, which unifies space and time within itself, it establishes certain simple laws, which are used as a basis by the creative process of both these arts. Regarding general viewpoints on harmonics and architecture, as well as the value and non-value of various proportions, I suggest reading my <em>Hörende Mensch</em>, pp. 257-279, <em>Klang</em>, 140-146, and <em>Grundriß</em>, 280-288 (being-proportions); these citations save me from repeating myself here.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">Albert Eichhorn <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> It is the fate of all scholars of harmonics to stumble repeatedly upon works that, even at the time of their publication, were only noticed by a few people, and since then have been completely forgotten. The most important, besides A. von Thimus’s <em>Harmonikale Symbolik</em>, are the books of Albert Eichhorn, an architect from <place w:st="on"> <state w:st="on">Berlin </state></place>(1. <em>Die Akustik großer Räume nach altgriechischer Theorie</em>, <place w:st="on"> <state w:st="on">Berlin </state></place>1888; 2. <em>Der akustische <place w:st="on"> <city w:st="on">Maßstab </city><span style="font-style: normal;">, <state w:st="on">Berlin </state></span></place><span style="font-style: normal;">1899). Eichhorn was the first to take the trouble to untangle the complicated realm of ancient spatial proportions—previously given only a rudimentary treatment in other sources—and to make this realm once again accessible to modern thought and practice.</span></em></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> In the account of the results of Eichhorn’s studies, we must restrict ourselves to what is important for us: harmonic proportion in architecture.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">Vitruvius <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Vitruvius, a Roman architectural writer living around the time of Christ, is the source from which Eichhorn gained his insights and drew his own independent conclusions. Vitruvius calls the study of harmonics, i.e. the study of numeric relationships in music, a “dark and difficult branch of science” (<em>De architectura libri X</em>, Book V, Ch. 4), especially for those not versed in the Greek language. Here he indirectly confirms that ancient Greek building-harmonics no longer existed in his time. He states, however, that an architect must be well versed in music. His works are full of precise proportional indications, and his more specialized musical-harmonic statements—admittedly taken from Aristoxenos, who did not understand the Pythagorean tone-system—are valuable enough to be studied further.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Eichhorn uses the monochord string, which he calls the “symmetron” (= <em>tonika</em>) = unit measure for architectonic and musical size. He divides the string “canonically” (according to <place w:st="on"> <city w:st="on">Euclid </city></place>’s </span><span lang="EL">κατατομη</span><span lang="EL"> </span><span lang="EL">του</span><span lang="EN-US">̃ </span><span lang="EL">κανόνος</span><span lang="EL"> </span><span lang="EN-US">= division of the canon, i.e. the monochord) as follows:</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shapetype id="_x0000_t75"> <stroke> </stroke> <formulas> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f> <f> </f></formulas> <path> </path> <lock v:ext="edit" aspectratio="t"> </lock></shapetype> <shape style="width: 266.25pt; height: 159.75pt;" id="_x0000_i1025"> <imagedata o:title="218"> </imagedata></shape> <p><img width="392" height="233" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image001.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 218 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> A further division leads to the seventh ratios that are no longer musically, i.e. psychically perceptible as “pure,” and so we also see the senary element used as a limit here. The string, as we have already learned, is divided into two segments by each division. One segment (here the right) was called the “plagal” by the ancients—from the Doric </span><span lang="EL">πλαγά</span><span lang="EL"> </span><span lang="EN-US">and Ionic </span><span lang="EL">πληγή</span><span lang="EN-US"> = striking, impact—because this is the segment that is struck or plucked, as stringed instrument players still do today; they strike this part of the string, not the other part. The other segment (here the left) was called the “authentic,” from </span><span lang="EL">α</span><span style="font-family: Tahoma;" lang="EN-US">̕</span><span lang="EL">υθέντησ</span><span lang="EN-US">, meaning something like “the actual initiator,” that which determines the actual measure of the string’s division, <sup>1</sup>/<sub>1</sub> <sup>1</sup>/<sub>2</sub> <sup>1</sup>/<sub>3</sub> etc. For Eichhorn, the difference between the study of musical and architectonic harmony lies in the fact that the ancient architects included <em>both</em> sides of the string division, the plagal and the authentic, in their proportions—whereas, as he believes, ancient musicians only used the ratio values of the plagal segments. Through summation of the above divisions, Eichhorn arrives at the pattern:</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <div align="center" class="Section1"> <table style="border-collapse: collapse;" cellspacing="0" cellpadding="0" border="0" class="MsoTableGrid"><tbody><tr><td style=" padding: 0cm 5.4pt;" valign="top"> <p class="MsoNormal"><span lang="EN-US">1 – <sup>1</sup>/<sub>6</sub> = <sup>5</sup>/<sub>6</sub> <strong>es <p> </p></strong></span></p></td><td style=" padding: 0cm 5.4pt;" valign="top"> <p class="MsoNormal"><span lang="EN-US">= minor third</span></p></td></tr><tr><td style=" padding: 0cm 5.4pt;" valign="top"> <p class="MsoNormal"><span lang="EN-US">1 – <sup>2</sup>/<sub>6</sub> = <sup>4</sup>/<sub>6</sub> = <sup>2</sup>/<sub>3</sub> <strong>g <p> </p></strong></span></p></td><td style=" padding: 0cm 5.4pt;" valign="top"> <p class="MsoNormal"><span lang="EN-US">= fifth</span></p></td></tr><tr><td style=" padding: 0cm 5.4pt;" valign="top"> <p class="MsoNormal"><span lang="EN-US">1 – <sup>3</sup>/<sub>6</sub> = <sup>3</sup>/<sub>6</sub> = <sup>1</sup>/<sub>2</sub> <strong>c</strong>′</span></p></td><td style=" padding: 0cm 5.4pt;" valign="top"> <p class="MsoNormal"><span lang="EN-US">= octave</span></p></td></tr></tbody></table></div> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 219 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US">and writes: “From the architectonic-acoustic standpoint, whose starting point is not the tones but the segments of the measuring standard belonging to them, we obtain the rule that the divisions of the measure to be used in the composition of inner spaces must correspond to the minor triad in music” (<em>Der akustische Maßstab</em>, p. 19).</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Vitruvius used these “senary” numbers and segments—which can only be understood through monochord division and therefore through harmonics—for the specification of his “eurhythmic” ratio numbers, which Eichhorn compiles in Table 41 of <em>Der akustische Maßstab</em> (reproduced as Fig. 220 of this book).</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> The musical-senary content of these numbers is obvious. In Book V, Ch. 4, “die Lehre von der Harmonie,” i.e. the study of the musical tone ratios, Eichhorn discusses the strange sounding vessels (resonators) in the amphitheater (Ch. 5), then the explicit requirements (loc. cit.) for constructing these according to a musical diagram from Aristoxenos; also the condition that the architect must understand music; and many more things. I do not understand how an author such as G.F. Hartlaub (“Musik und Plastik bei den Griechen” in <em>Zeitschrift für Aesthetik und allgemeine Kunstwissenschaft</em>, Vol. 30, 1936) can drag out the old saw of the Greeks’ ignorance of triadic ratios and then, simply because we have not inherited any direct harmonic “recipes,” attempt to challenge and even outright deny the ratio designations dictated by tone. The most primitive monochord experiments yield intervals, triads, and chords; whether the Greeks used them in their practical music or only played and sang “horizontally” is of secondary significance. Only one-sided haptic blindness can hold on to such antiquated viewpoints, especially after reading Thimus and Eichhorn.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Eichhorn does not stop at this. His most valuable investigation, in my opinion, is the rediscovery of the remarkable ancient Greek theory of sound propagation, which is extensively discussed in his work, <em>Die Akustik großer Räume</em>, mainly on the basis of Porphyry’s statements in his commentary on Ptolemy’s harmonics. This is another ancient source-work, hardly noticed until now, whose mention in current musical texts must be sought out with a magnifying glass, if it can be found at all. A new edition of the text has been published—see §55—but I have currently heard nothing of its content being used.</span></p> <p class="MsoNormal"><span lang="EN-US"> Regarding this subject, I quote from my <em>Hörende Mensch</em>, p. 259: </span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="margin: 0cm 36pt 0pt 37.4pt; text-align: justify;" class="MsoNormal"><span style="font-size: 11pt;" lang="EN-US">“The result is summarily this: whereas today we imagine sound propagation somewhat like light propagation, i.e. spherically emanating from a center, the Greeks imagined two completely different types of sound propagation emanating from the same center. They imagined the sound propagating horizontally like a wave in water, and upwards and downwards from the center in the form of cylindrical shells. The two types of propagation, in which the vertical proceeds twice as fast as the horizontal, are described by the expressions </span><span style="font-size: 11pt;" lang="EL">πσάλλειν</span><span style="font-size: 11pt;" lang="EN-US"> (to speed forward) and </span><span style="font-size: 11pt;" lang="EL">κρούειν</span><span style="font-size: 11pt;" lang="EN-US"> (to press forward). In Eichhorn’s works, one can read about the more precise demonstration, which reveals that the ancient acousticians had an astonishing understanding of the physical process, as well as a definite ability to depict the process geometrically and mathematically. The results are important. And these lead the Greeks to two practical consequences which they followed strictly in the construction of their theaters. If the theater was open—an amphitheater—they calculated the angle of the slope of the sound exactly from the originating center, which usually yielded 1 : 3. This was the direction of sound that allowed someone sitting in the highest seats to hear a spoken word, or a sung or played note, just as clearly as those in the lowest rows. They went even further. To avoid loss of sound (through wind, etc.) as much as possible, they build “resonators” in the rows of seats at precisely proportioned places, i.e. larger or smaller drum-like or tub-like vessels of earth or clay, tuned precisely to certain tones, which, with the opening directed at an angle toward the stage, reflected the tone to which they were tuned. Imagine what broad acoustic knowledge must have been required for such arrangements, and especially what experiments must have been performed! It is clear that these things were constructed with the conviction that they would be extremely useful for the theater in acoustic terms. The magnificent acoustics of the ancient amphitheaters can still be witnessed today, even though the best of them are in ruins. The second practical application corresponded less to sound-direction experiments and more to the theoretical investigations into tone-development.” <p> </p></span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Eichhorn reexamined these investigations, especially in his second work (<em>Der akustische Maßstab</em>), and came to the conclusion, as remarked above, that “the measure to be used in the design of large indoor spaces must contain such divisions for its main dimensions as correspond to the construction numbers of the minor triad 5, 4, 3 in music (in terms of string or segment length)” (op. cit., 46). How it happened that the details and minutiae of the architectonic “appendages” required yet another special “module-calculation” must be studied in Eichhorn’s last work.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><em><span lang="EN-US">Table of eurhythmic ratios of the Greek temple according to Vitruvius <p> </p></span></em></p> <div class="Section1"> <table style="border-style: none; border-collapse: collapse;" cellspacing="0" cellpadding="0" border="1" class="MsoNormalTable"><tbody><tr><td style=" padding: 0cm 5.4pt; width: 201.75pt;" width="100%" valign="top"> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">Description of the ratio sizes <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 65.45pt;" width="87" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">old Doric <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 56.1pt;" width="75" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">Etruscan <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 56.1pt;" width="75" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">old Ionic <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 63.4pt;" width="85" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">true Ionic <p> </p></span></p></td></tr><tr style="height: 107.5pt;"><td style=" padding: 0cm 5.4pt; width: 201.75pt; height: 107.5pt;" width="100%" valign="top"> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">1. Width of antae: epistyle height; antae width: medius tetrans (Doric) <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">2. Triglyph or zophorus height: epistyle height: corona height <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3. Triglyph height: triglyph width; metope width: triglyph width <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4. <place w:st="on"> <city w:st="on">Corona </city></place>overhang: corona height <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5. Capital height: cyma height: abacus height <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6. Gutta cum regula width: epistyle-fillet width: gutta <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">7. Bottom column diameter: top column diameter <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">8. Middle cella width: width of side cella or ala <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">9. <place w:st="on"> <city w:st="on">Temple </city></place>length: temple width: cella length: cella width (old ratio) <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">10. Greatest width of the gable: whole roof <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">11. Cella length: C-height: C-width; bracket overhang: B-height: B-width <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">12. Bottom: middle: upper band of the epistyle; first: second: third molding <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">13. Cella length: cella width: pronaos width <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">14. Cella + pronaos length: C-width; side-wall length: side-wall width <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">15. Bottom: top column diameter: epistyle height: smooth zophorus (Hermogenes) <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">16. Total height of the dentils: tooth height: tooth width: intersection <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">17. Tooth height: tooth width: intersection width: cymatium height (old Ionic) <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">18. Cyma height: corona height: tooth height: cymatium height: ledge over dentil (Polias) <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">19. Cyma height: corona height, with incised cymatium (Ilissus) <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">20. Triglyph head height: triglyph width; triglyph gap: triglyph width <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">21. Width: height of the Etruscan round base <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">22. Frame height of corona overhang <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 65.45pt; height: 107.5pt;" width="87" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 : 1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 : 1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 : 1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">(2×3) : 1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">(3 × 2 × 1) · <sup>2</sup>/<sub>3</sub> <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 56.1pt; height: 107.5pt;" width="75" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 : 2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4:3:2:1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><sub><span style="font-size: 10pt;" lang="EN-US">└─┘</span></sub><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 : 2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><sub><span style="font-size: 10pt;" lang="EN-US">└─┘</span></sub><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : – : 2 <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 56.1pt; height: 107.5pt;" width="75" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 : 5 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">(5+3) : 4 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5:4:3:2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5:4:3:2:1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 × 4 × 3 <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 63.4pt; height: 107.5pt;" width="85" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6 :4 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">1 :1 late <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6 :5 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6 :3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6:5:4:3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6 : 5 : – : 3 : 2 : 1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 · 4 / 5 <p> </p></span></p></td></tr><tr style="height: 70.15pt;"><td style=" padding: 0cm 5.4pt; width: 201.75pt; height: 70.15pt;" width="100%" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">Results of the table: <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">Surface eurhythmy <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">Body eurhythmy <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">Composite eurhythmy <p> </p></span></p> <p class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">Interpolated values <sub>└─┘</sub> <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 65.45pt; height: 70.15pt;" width="87" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">3 : 2 : 1 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><sup><span style="font-size: 10pt;" lang="EN-US">2</span></sup><span style="font-size: 10pt;" lang="EN-US">/<sub>3</sub>, <sup>1</sup>/<sub>3</sub>, <sup>1</sup>/<sub>6</sub> <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 56.1pt; height: 70.15pt;" width="75" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">4 : 3 : 2 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">1, <sup>3</sup>/<sub>4</sub>, <sup>1</sup>/<sub>2</sub> <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 56.1pt; height: 70.15pt;" width="75" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">5 : 4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><sup><span style="font-size: 10pt;" lang="EN-US">12</span></sup><span style="font-size: 10pt;" lang="EN-US">/<sub>5</sub>, 2, 1 <p> </p></span></p></td><td style=" padding: 0cm 5.4pt; width: 63.4pt; height: 70.15pt;" width="85" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6 : (5,4,3) <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">6 : 5 : 4 : 3 <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span style="font-size: 10pt;" lang="EN-US">2, 1 <p> </p></span></p></td></tr></tbody></table></div> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">(From A. Eichhorn: <em>Der akustische Maßstab</em>, Berlin 1899, p. 41)</span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 220 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> In the results of Eichhorn’s half-century of study, there lies a wealth of concrete instructions for spatial acoustics, emerging tonally rather than physically. Are his claims valid? One can only decide through practice, i.e. investigating the actual proportions of spaces currently existing and known for their “good acoustics.” Eichhorn relates the measurements of the Berlin Singakademie, built in Schinkel’s time, perhaps the only venue in <place w:st="on"> <state w:st="on">Berlin </state></place>in which it was a joy to listen to music. Its proportions were:</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <div class="Section1"> <table style="border-collapse: collapse;" cellspacing="0" cellpadding="0" border="0" class="MsoTableGrid"><tbody><tr><td style=" padding: 0cm 5.4pt; width: 109.95pt;" width="147" valign="top" rowspan="3"> <p style="text-align: center;" align="center" class="MsoNormal"><em><span style="font-size: 11pt;" lang="EN-US"> <p> </p></span></em></p> <p style="text-align: center;" align="center" class="MsoNormal"><em><span style="font-size: 11pt;" lang="EN-US">For exact values; i.e. for 32.83 = 2 × 1 <p> </p></span></em></p></td><td style=" padding: 0cm 5.4pt; width: 109.8pt;" width="146" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">32.83 m long</span></p></td><td style=" padding: 0cm 5.4pt; width: 109.8pt;" width="146" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">12.85 m wide</span></p></td><td style=" padding: 0cm 5.4pt; width: 109.75pt;" width="146" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">9.81 m high</span></p></td></tr><tr><td style=" padding: 0cm 5.4pt; width: 109.8pt;" width="100%" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">32.83 m long</span></p></td><td style=" padding: 0cm 5.4pt; width: 109.8pt;" width="146" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">13.13 m wide</span></p></td><td style=" padding: 0cm 5.4pt; width: 109.75pt;" width="146" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">9.85 m high</span></p></td></tr><tr><td style=" padding: 0cm 5.4pt; width: 109.8pt;" width="100%" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">2 × <sup>5</sup>/<sub>5</sub> long</span></p></td><td style=" padding: 0cm 5.4pt; width: 109.8pt;" width="146" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><sup><span lang="EN-US">4</span></sup><span lang="EN-US">/<sub>5</sub> wide</span></p></td><td style=" padding: 0cm 5.4pt; width: 109.75pt;" width="146" valign="top"> <p style="text-align: center;" align="center" class="MsoNormal"><sup><span lang="EN-US">3</span></sup><span lang="EN-US">/<sub>5</sub> high</span></p></td></tr></tbody></table></div> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 221 <p> </p></span></div></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <p> </p></span></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> As one can see, these are the pure triadic proportions, only with minor differences arising from the building process.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">E. Schieß: Spatial Acoustics <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> <place w:st="on"> <country-region w:st="on">Switzerland </country-region></place>has the organ-building expert and acoustician Ernst Schiess to thank for several of its most beautiful organs, matched in their registers to the corresponding spaces, as well as for the good acoustics of the Bern Conservatory and other halls. He also revived the true, original relationship of tone and space, together with their ancient traditional rules. The great chamber of the new conservatory at Bern has the exact inner dimensions: length 22 m, width 11 m, height 7.3 m; thus, the proportions 6 : 3 : 2. The original suggestions, which would have surely produced better acoustics due to its inclusion of the proportions of a third interval, were these: length 28, width 11.2, height 8.4 m = 2 × 5 : 4 : 3 and 21.3, 12.8, 8.5 m = 5 : 3 : 2. Unfortunately, these suggestions were not followed for technical architectural reasons. One of the best small halls in <place w:st="on"> <country-region w:st="on">Switzerland </country-region></place>, the small 1890s Tonhallesaal in Zürich, has the same measurements as the Berlin Singakademie: 30 × 12 × 9 meters = 2 × 5 : 4 : 3. Despite protests, 2.5 meters were cut off from this hall, which detracted from its acoustic quality. The destroyed (i.e. its roof lowered) assembly hall in the Mädchenschulhaus at <place w:st="on"> <city w:st="on">Biel </city></place>also had the same proportions: 17.7 × 7 × 5.3 = 2 × 5 : 4 : 3. The best large concert hall in Switzerland, in Basel’s Casino, has the measure of the golden section: 35.5 × 21 × 14.5, and additions have changed it to 36 × 21.7 × 14.5 = 5 : 3 : 2. The destroyed Kammermusiksaal in <place w:st="on"> <city w:st="on">Basel </city></place>also had the golden section measure: 19.5 × 11 × 8.5; this room must have been acoustically superb. Since, as I have continually mentioned (especially in <em>Harmonia Plantarum</em>, p. 148ff.), the golden section is a third-sixth problem, these golden section rooms also have the triadic proportions in their backgrounds. Incidentally, one need only juxtapose two rectangles of which the first has the ratio of the harmonic sixth 5 : 8 (inversion of the third) and the second the golden ratio 21 : 34 (see Fig. 222), to see immediately that for the eye, there is barely any difference, whereas the ear, with the positioning of the corresponding string segments on the monochord, will perceive a definite difference: an “impurity” of the string lengths 21 : 34 in contrast with the absolute purity of the ratio 5 : 8. This is only a small selection of typical harmonic proportions with good acoustic results, which, as E. Schiess writes, can be greatly extended. The new Gewandhaussaal in <place w:st="on"> <city w:st="on">Leipzig </city></place>has the measurements: 38 × 19 × 14.3 = 8 : 4 : 3; the old one had the measurements 22.6 × 11.3 × 3.8 = 10 : 5 : 3; and so on.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p><img style="margin: 0px;" width="500" height="321" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image003.jpg" alt="" title="" /></p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US"> <shape style="width: 431.25pt; height: 276.75pt;" id="_x0000_i1026"> <imagedata o:title="222"></imagedata></shape></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 222<a name="_ftnref1" href="#_ftn1" title=""><span class="MsoFootnoteReference">*</span></a> <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> We have good reason to dwell on these things; today they are a hot topic. With half of <place w:st="on">Europe </place>destroyed, a great amount of building will soon begin. The evil of bad acoustics is widespread in halls, and even worse in most radio station studios, with their insane sound-dampening methods, by which one hears the scratching of the bow or the incidental noises of the larynx more than the tone or voice itself. Anyone who has experienced this will wonder why a reactionary routine, afflicted with only physical sound-measuring apparatuses, is continually allowed to be a haptic nuisance, and why people do not return to the norms dictated by tone-laws themselves, which have been followed for thousands of years. Naturally, this would also require an inner change, namely an additional factor of awe at the mysterious connection of tone and space with simple functionality and usefulness, while not by any means neglecting the latter.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">Vitruvius’s Resonators <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Here I will mention another interesting discovery made in 1944 by Ernst Schieß while building the new organ in the city <placetype w:st="on">church </placetype>of <placename w:st="on">Biel </placename>, <place w:st="on"> <country-region w:st="on">Switzerland </country-region></place>. High up in the wall, holes were discovered, behind which tone vessels of various dimensions were built into the wall. These various dimensions, and their regular division, could lead to no other conclusion than that these were the famous and often doubted <em>resonators of Vitruvius</em>, i.e. sound amplifiers, tuned to certain tones. Since a few of these vessels had already been removed in rebuilding, Herr Schieß was able to photograph them, and has very kindly allowed me to reproduce the images (see Fig. 223 and 224). </span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 321pt; height: 327pt;" id="_x0000_i1027"> <imagedata o:title="223"></imagedata></shape> <p><img style="margin: 0px;" width="500" height="509" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image005.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 223 <p> </p></span></div></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <p> </p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 323.25pt; height: 156pt;" id="_x0000_i1028"> <imagedata o:title="224"></imagedata></shape> <p><img style="margin: 0px;" width="500" height="241" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image007.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 224 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> This church in <place w:st="on"> <city w:st="on">Biel </city></place>was built in 1451, and as the head architect of the restoration indicated, various architectural evidences show that these resonators must have been built into it originally. Since all publications of Vitruvius’s works are of a later date, there must have already been a tradition of this Vitruvian resonator-theory in Gothic buildings, which allows corresponding conclusions to be drawn about the level of acoustic sensitivity. This city church in <place w:st="on"> <city w:st="on">Biel </city></place>has exceptionally good acoustics, and despite its obvious irregularities, its inner proportions give a very harmonic impression. </span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> The main problem Eichhorn considered was that of proportioning the inner space so as to obtain unobjectionable acoustics. Here he stumbled upon the ancient Greek harmonic proportions, which, as we have seen, are expressed within the “senary” we know so well, i.e. within a selection-principle (pure chords of ratios 1-6, etc.) anchored in the “P” system. For the ancient architects, these senary ratios dictated forms absolutely. If the reader of this book observes and tests, together with his acquired harmonic knowledge, the above “eurhythmic ratio table” of the Greek temples according to Vitruvius and their senary ratio numbers (next to which there are no tone-numbers!), he will have no doubt as to the harmonic background of these numbers, and will feel just as Vitruvius presumably felt: after ascertaining the “rightness,” “tuning,” “eurhythmics” of these numbers, adding in the tone-values is no longer necessary, since these ratios already “sound” right.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> This is at the heart of people’s misunderstanding of harmonic norms in general, or their lack of desire to learn about them. Since concrete statements with numbers <em>and</em> tones are rarely if ever found in old sources, people believe that the “musical” element in all these things is at best a superfluous decoration, if not simply a pipe-dream. In ancient times, numeric demonstration with monochord lengths was widespread (Nicomachus, Iamblichus, Euclid), and was used to test proportional ratios.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">The Difficulties of Harmonic Study <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> If an art-historian today were to take Vitruvius’s pronouncements seriously: “An architect must understand music” (I ,1) or must understand “the study of harmonics” (V, 4), then, since he would likely have no practical understanding of music theory, he would go to his musical colleagues and ask them for information. He would receive the reply: “Yes, my friend, I don’t understand much of this myself, that is physical acoustics, so you should go ask a physicist!” After going to the physicist, and telling him painstakingly all about monochord experiments, enharmonics, Greek tone-systems, resonators in old amphitheaters, and so forth, he would be told that physics no longer deals with such old-fashioned things, and if the science of music knows nothing about this either, he should go to see an ancient philologist, one who has worked with Plato or the pre-Socratics. Resigned, our art-historian would go to visit the philologist, but would find himself in a very learned yet completely unmusical house, in which the harmonic elements of ancient <place w:st="on"> <country-region w:st="on">Greece </country-region></place>are like a book with seven seals, and which he therefore completely “dismisses.” At this point, the art-historian would go home and note something like the following, on paper or in his head (quoted from Lichtenberg): “Harmonics ought to have no influence upon architecture; however, it (maybe, perhaps, possibly) does have one.” With this he would have reached an honorable conclusion—but the original question has not been answered.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> This example applies in general to the difficulties encountered in all domains of harmonic study. These difficulties, however, do not only trouble specialists. Often enough, they defeat almost anyone who studies and works with harmonics by himself. The requirements for such work appear too vast, the knowledge needed too great, for a “common mortal” to attempt such studies with a clear conscience. As a principal reason, it is always suggested that a universal knowledge of anything is impossible nowadays; Leibniz and Goethe still believed it was possible, but today even a somewhat satisfying overview is unattainable.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">Universality <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Since this objection concerns both the author and the reader, allow me to answer it. “Universality” in the all-embracing sense, as these opponents believe, has never existed. It can easily be shown that Plato, Aristotle, Leibniz, Goethe, etc. were unaware of, or at least did not mention, many important things and events of their times. Universality in the deepest sense is not omniscience, but <em>knowledge</em> and <em>experience </em>of the <em>essential </em>in the science of the current time. Admittedly, the great minds such as Leibniz and Goethe possessed this “universality” in the highest measure. I am convinced that this actual universality can be attained again and again, up to a certain level, by anyone who is inwardly open-minded and receptive. I will give a concrete example. The domain of modern “results and problems of natural science” appears incomprehensible. But anyone who takes the trouble to thoroughly study B. Bavink’s magnificent book of the same name, <em>Ergebnisse und Probleme der Naturwissenschaft</em>, for a couple of hours daily for a three months, will understand what is essential. The only requirement is a standard high-school education, or simply a lively interest in the problems themselves. The same goes for other realms. We have such “universally” oriented books for almost all disciplines. I write this to appease my readers, from long personal experience and from experience with such open-minded people themselves, whom I have admittedly only rarely found in “expert circles” but much more among laypeople, for whom “knowledge” does not merely mean specialization, knowing much or everything, but means an approach to the essential, an orientation in it—which is precisely the purpose for which we have been placed on this earth as <em>humans</em>.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">Harmonic “Keys” <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> We now return to harmonic architecture. The search for proportional laws has constantly led and misled scholars to propose certain isolated geometric and mathematical forms as “keys” to the solution of the mysteries of form, and to emphasize them as measures for everything, not only architectonic formation. If we now investigate these keys from a harmonic standpoint, we will almost always find harmonic data as a background. Simply reducing the golden section (see the above diagram, Fig. 222) to a third-sixth problem reduces all the voluminous literature about the golden section to a special case within harmonics. The Pythagorean triangle was applied in ancient times not only as an angle designation but also with regard to its segment proportions (for example, the 3 : 4 : 5 triangle in the King’s Chamber of Cheops’s pyramid, consecrated to Osiris, Horus, and Isis; and others). We saw this above, with its triad 3 <strong>g</strong> 4 <strong>c</strong> 5 <strong>e</strong> (by frequency; in string length, the minor triad 3 <strong>f</strong> 4 <strong>c</strong> 5 <strong>as</strong> is produced), as the actual formative source of the chromatic scale. It played a significant role in proportions up to the Middle Ages. In “quadrature”—i.e. the division of a circle into four or eight, according to which, for example, the cathedral at <place w:st="on"> <city w:st="on">Aachen </city></place>was built—there are predominantly octave ratios. “The Achtort emerges from the square, on which is superimposed a second square of equal size, rotated as the term dictates. But if the square is inscribed with a smaller one, so that its corners are at the midpoints of the sides of the larger square, and if this pattern is continued, then the sides of the squares aligned with the largest square have the ratio 1 : 2 : 4 (octaves), but the sides of the tilted squares have the ratio √2 : 2√2 : 4√2” (Th. Fischer, op. cit., pp. 12-13). Eichhorn refers to this (<em>Der akustische Maßstab</em>, p. 80), calling this proportion the “optical” or <sup>π</sup>/<sub>4</sub> measure of the Strasburg masons’ lodge. In “triangulature,” division in three parts, and the triangles used in certain circle divisions, we see fifths predominantly as the constituting element. Th. Fischer (op. cit., p. 14ff.) gives the example of notes from the beginning of Beethoven’s Ninth Symphony as a musical counterpart, and writes: “Does anyone who hears the massive effect of descending fifths and fourths at the beginning of the Ninth Symphony feel it lessened because he knows that fifths and fourths, with their ratio of <sup>2</sup>/<sub>3</sub> and <sup>3</sup>/<sub>4</sub> together building the octave <sup>1</sup>/<sub>2</sub>, are the structural axis of all of music? How far these things are to be assigned to the conscious or the subconscious is a question for later; here it only serves to outline the subject to be discussed today, and while it is clearly an attempt to establish the tonic chord of the relationships, as Goethe says, perhaps it is also to penetrate, with all due respect, into the secrets that, according to him, can only be felt. Triangulature, quadrature, circle geometry, and other things that I will discuss later are fundamentally pure technical-rational processes; I am certain that harmonies emerge outside these yet from them, and appeal to the psyche. That is the subject at hand.”</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> The cathedral at Cologne, the work of Gerhard von Riles, was built with the measure of 25 or 50, according to Sulpice-Boisserée: the width of the nave = 50 Roman feet, the entire width of the five-nave church is 3 × 50 = 150 feet, its length is 9 × 50 = 450 feet. The transept had a ratio of 5 : 9 to the total length of the church, known as “German symmetry,” or “the highest and most elegant stonemason’s gauge of the triangle.” The choir was as high as the church was wide. The entire length had a ratio of 5 : 2 to the height, and the lower story has the same ratio to the ridge of the small roofs, likewise the upper story of the roof’s ridge to the cross of the pointed gables on the windows. </span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Prof. Castle identified 37 as the fundamental number of St. Stephen’s Cathedral in <place w:st="on"> <city w:st="on">Vienna </city></place>. The width of the long nave is 2 · 3 · 37 = 222 feet, the length of the church is 3 · 3 · 37 = 333 feet, the height of the tower is 4 · 3 · 37 = 444 feet, the ratio of total width to total length is 2 : 3, and the height of the central nave is 2/3 · 3 · 37 = 74 feet. The prime number 37, multiplied by 3 and its multiples, yields the products 111, 222, 333, 444, and so on. It is no wonder that Castle imagined the draftsman to be a profound theologian and mathematician for whom the number 37 had a great symbolic value: written in Roman numerals (XXXVII) it combines the triple cross X, thus the Trinity, with the mystical seven of the seven days of creation. As one can see, a “key number” was adopted here, which, like the meter, foot, etc., does not always need to be of harmonic nature. This key number plays the fundamental role in connection with typical harmonic ratios such as 3, 9, 5, <sup>2</sup>/<sub>3</sub>, etc.: some important symbolic or natural base-measure (= the true mason’s gauge), from which, by means of harmonic proportions, the whole multitude of the remaining forms radiates like a fan down to the smallest details.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">Ernst Mössel <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Ernst Mössel’s “circle geometry” is a grandiose undertaking, with substantial material documentation, to trace all visual art back to the regular circle divisions, 4, 5, 6, 7, 8, 10, and 12, and their geometric forms. In this (especially in his main work, <em>Vom Geheimnis der Form und Urform des Seins</em>, 1938) we find a plethora of typical harmonic ratios and proportions. In the beautiful introduction to this book so full of deep ideas, in the harmonic analysis of the triangle 3 : 4 : 5 (p. 430), and in the peculiar dream-vision mentioned in his previous work, <em>Die proportionen in Antike und Mittelalter</em> (1926, p. 117ff.), Mössel knocks very forcibly on the door of harmonics—but does not open it. It makes one wonder about the basis of his investigation and about the true foundation of this circle geometry. With Mössel’s express argument that the important thing is not the number (term) but the form (geometry), one gets no further than the obvious interchangeability of number and segment, with which the ancients were familiar. <div><span style="color: red;"> <p> </p></span></div></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> The irrationality of the geometric ratios also goes without saying (see<em> Geheimnis der Form</em>,<em> </em>p. 417). Geometry contains a great number of rational segments and point connections, and it can be said that all irrationalities are framed, indeed generated, by rationalities, or conversely that the irrational root-quantities are the seedling points of rationalities by which the rational connections become “real,” a dialectic well known to the Greeks. We observed this in Thimus’s α—</span><span lang="EL">ω</span><span lang="EN-US"> formula (§28). Surely, number was an <em>a priori</em> form for the ancients; but one must experience the concept of geometry in a deep and almost unique way, and be as saturated by it as Mössel was, if one wishes to remain and be satisfied with it. The cardinal question is why <em>these</em> particular circle divisions, and not others, were so important and authoritative? Where is the domain of our psyche in which they are embedded, and where is the bridge for our intellect that leads into this domain? Mössel would answer: “The geometric forms, as archetypes, exist in our psyche as <em>a priori</em> forms; that is enough for me, I do not need or want to know more.” But this would not satisfy an architect so careful, so inwardly devoted to the whole complex of questions of proportion, as was Th. Fischer: “If all talk of ratios is to make sense, a bridge must be found between this reading off of segments and the visual impression of the space or body. And in the age of C.G. Jung, it is nothing extraordinary to locate this bridge in the subconscious. Older witnesses of no lesser status stand ready: Leibniz says of music, to take the parallel further: ‘<em>Musica est exercitium arithmeticum occultum nescientis se numerare animi</em>,’ or translated, ‘Music is a mysterious arithmetical process of the psyche, unaware that it is working with numbers.’” (Th. Fischer, op. cit., 80-81). Mössel’s arguments, attempting to discredit Vitruvius’s clear statement on the triadic proportions (3 : 5 : 8) of ancient architecture, are absurd and circular: he says that this is only a substitute for the true irrational ratios of circle division, especially those of the golden section. We saw above that the latter comes very close to the third-sixth. But precisely for this reason, the third-sixth is not a substitute for the golden section, but instead the opposite: the phantom of the “golden section,” coincidentally almost concordant with the triad 3 : 5 : 8, must be “substituted” with the psychophysical reality of the triad, indeed this purely geometric irrational proportion with its effective visual meaning only becomes rational and comprehensible when we realize that it is almost identical with a psychical form in us, namely the triad proportion. And just as our ear wants to hear this triad, and is pleased by it, so our eye wishes to <em>see </em>this triadic proportion (transposition of the auditory to the visual).</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Th. Fischer, in the sections cited above, and especially in the second lecture in his booklet, touches only on the periphery of harmonics. But his views point to the core of the problem: in the subconscious, there must be a hidden “common denominator” for number, form, and tone. Having arrived at this point, we can hope to found an architectural theory similar to music theory: a mode of thought around which all Th. Fischer’s architectonic work is known to have circled, especially in his buildings in <place w:st="on"> <city w:st="on">Stuttgart </city></place>.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> The common denominator is none other than the harmonic. If our psyche spontaneously recognizes the proportions 1 : 1, 1: 2, up to 1 : 6 in Classical, Roman, and Gothic architecture, and their variants 2 : 3, 3 : 4, etc., in the same way that it recognizes pure intervals, chords, and whole-tone steps through the ear and experiences their appearance as tones and tone ratios, then this means that the auditory and audible is only the secondary apperception of a primary form-structure existing in the depths of the subconscious, which is “synopsized” and reappears in other, visual ways on the outer surface of our optical view, via number and its conversion into segments.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Related to this, as we saw in §21a and will see in §38a.1.γ, §41.4, and §55.8, is the fruitfulness of the harmonic division canon for rational segment measurements, and thus the practical-technical applicability of geometric harmonics for the ancient temple builders. See my work on Villard in <em>Harmonikalen Studien</em>, Vol. I (Occident-Verlag, Zürich 1946), which I believe offers a proportional key with the proof of a “harmonic division canon” definitely still existing in the Middle Ages. This proportional solution, due to its many-sided nature, firstly allows considerably broader possibilities of application than the previous ones (circle geometry, golden section, </span><sup><span lang="EL">π</span></sup><span lang="EN-US">/<sub>n</sub> = triangle, etc.) and secondly allows a psychic-symbolic meaning through its tone-number backgrounds, which all other solutions have been missing.)</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> For the ancient Greeks, the value of the fully sounding pure chord and the senary proportions surrounding it, with the rare inclusion of the mystical Seven and other exceptional proportions, had a fully concrete “human” significance, besides its psychical significance. Vitruvius (IV, 1, 6, and 7) wrote: “When [the ancient Greeks] found that the human foot was a sixth part of the length of the body, they applied this to the column, and made the thickness of the foot of the shaft one sixth of the height, including the capital. Thus the Doric column began to show in buildings the ratio and the compact beauty of the male body. When they wanted to build a temple to Diana, with a new [Ionic] order, they took its form from the outline of feminine slenderness, and made the thickness one eighth of the height, giving it a more illustrious appearance.” I do not consider this a “fable based on philosophical aesthetics,” as did L. Curtius (<em>Die antike Kunst</em>, Vol. II, Ch. 1, p. 124), but an ancient tradition known to Vitruvius, which incorporated everything, even the human form, into a universal psychophysical proportion. In §38 it will be shown that the human body also reveals harmonic ratios.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">§29.2. Modern Times <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> But now we are satisfied with the dry historical tone, and want to build something ourselves!</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Firstly according to the “harmonic proportion” as we developed it in §24a.1. Since the product of the inner and outer elements in every proportion is equal, the surfaces of the corresponding rectangles are also equal.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 213.75pt; height: 45pt;" id="_x0000_i1029"> <imagedata o:title="225"></imagedata></shape> <p><img width="285" height="60" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image010.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 225 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> In the above harmonic proportion, AC : CB = AD : DB (2 : 1 – 6 : 3) and therefore AD · BC = AC · DB. Since the product of two segments yields a rectangle geometrically, there are two rectangles here in characteristic positions (Fig. 226). AD · BC is found in the rectangles ADst and CByz. AC · DB is found in the rectangles ACwx and BDuv.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 157.5pt; height: 155.25pt;" id="_x0000_i1030"> <imagedata o:title="226"></imagedata></shape> <p><img width="210" height="207" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image012.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 226 <p> </p></span></div></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <p> </p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 328.5pt; height: 344.25pt;" id="_x0000_i1031"> <imagedata o:title="227"></imagedata></shape> <p><img width="438" height="459" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image014.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 227 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> As one can see, the harmonics 1 <strong>c</strong> 2 <strong>c</strong> 3 <strong>f</strong> 6 <strong>f</strong> (or for 1 = <sup>6</sup>/<sub>6</sub> | <sup>1</sup>/<sub>6</sub> <strong>g</strong> <sup>2</sup>/<sub>6</sub> <strong>g</strong> <sup>3</sup>/<sub>6</sub> <strong>c</strong> <sup>6</sup>/<sub>6</sub> <strong>c</strong>) are an architectonic projection with a close inner connection. Another example from the harmonic:</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">2 <strong>c</strong> 3 <strong>f</strong> 10 <strong>as</strong> 15 <strong>des</strong></span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><span lang="EN-US"> In this harmonic proportion (see Fig. 227) the ratios are:</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">AD : AC = BD : BC</span></p> <p style="text-align: center;" align="center" class="MsoNormal"><span lang="EN-US">15 : 10 = 3 : 2</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> The corresponding rectangles AD · BC and AC · BD are drawn over the harmonically divided baseline analogous to Fig. 226. Naturally, the reader must imagine the three-dimensional (corporeal) projection for himself, or else draw it perspectively, which would be a good idea if he is working on various such projections according to the indications in §24a.1.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> In observing these and similar projections, one might ask: “A factory?” Indeed, why not a factory? Once the insanity of this war is finally over, and factories are once again given over to producing things that help people rather than destroy them, nothing stands in the way of giving such buildings “human” measurements.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Let us examine this more closely. We draw a projection of the partial-tone coordinates for index 5 analogous to the model in §23b. In Fig. 128, for clarity’s sake, only the outer ratios are plotted. If one imagines a building equivalent to this projection, such as a high-rise, one must admit that there are completely new architectonic possibilities here, which are fruitful for new practical purposes—in this case, for example, an optimal exploitation of light with precise orientation of the corner <sup>1</sup>/<sub>6</sub> <strong>f</strong></span><span style="font-family: Tahoma;" lang="EN-US">,,,</span><span lang="EN-US"> towards the South, etc. If we attempt to give the continuous arithmetic and harmonic proportions an architectonic expression, then the result is something like Fig. 229 and 230. Here we would have two typical aspects of gables, as are found everywhere, especially in Nordic nations.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 203.25pt; height: 329.25pt;" id="_x0000_i1032"> <imagedata o:title="228"></imagedata></shape> <p><img width="374" height="610" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image015.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 228 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 3in; height: 201.75pt;" id="_x0000_i1033"> <imagedata o:title="229"></imagedata></shape> <p><img width="374" height="350" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image017.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 229 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> These examples, which serve merely to illustrate a conversion of harmonic proportions into optical-spatial forms, pose the core question: does the eye also have a “resonance” to certain spatial forms, especially to harmonic discontinuities?</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 218.25pt; height: 302.25pt;" id="_x0000_i1034"> <imagedata o:title="230"></imagedata></shape> <p><img width="360" height="497" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image019.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 230 <p> </p></span></div></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <p> </p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 391.5pt; height: 102.75pt;" id="_x0000_i1035"> <imagedata o:title="231"></imagedata></shape> <p><img style="margin: 0px;" width="500" height="131" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image021.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 231 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Th. Fischer, in his <em>2 Vorträgen über Proportionen</em>, gives an interesting example (Fig. 231), and writes (op. cit., p. 86):</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="margin: 0cm 36pt 0pt 37.4pt; text-align: justify;" class="MsoNormal"><span style="font-size: 11pt;" lang="EN-US">“In architecture, the ratio of height to width can be compared, as a means of expression, to the interval in music. I shall attempt to prove, without suggestive intent, that the rectangle, this most simple figure, can be inherent in an expression; so I set three rectangles in the ratio numbers of a major chord to the right of a square, and three with the same number ratios but reciprocal, corresponding to the minor chord, to the left of the square. And there is also reason to claim that a division can be made into more and less pleasant rectangle ratios, i.e. into consonances and dissonances—though admittedly more blurrily than in music! But I say “admittedly” in reference to what was said about practice, the practice of the individual, the generation, the people. In music, the peculiarity of dissonance is that it allows no pacification, instead leading onwards, towards resolution. We can observe similar things with the rectangle. Rectangles of undetermined ratios, such as <sup>8</sup>/<sub>15</sub>, <sup>6</sup>/<sub>13</sub>, <sup>9</sup>/<sub>10</sub>, or <sup>12</sup>/<sub>13</sub>, need something for their completion or division, be it only division into two rectangles of clear ratios that are in tune together, or for dissolution in a system of rectangles, where analogy plays a role. But if the rectangle of a wall surface or a spatial framework has definite ratios in itself, then one does not need to divide it, though this is not prohibited. Fifty years ago, the psychologist Fechner performed a statistical experiment, measuring the dimensions of the pictures in the great galleries; he determined average values, namely the ratio <sup>4</sup>/<sub>5</sub> for portrait format and <sup>3</sup>/<sub>4</sub> for landscape format.” <p> </p></span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 265.5pt; height: 364.5pt;" id="_x0000_i1036"> <imagedata o:title="232"></imagedata></shape> <p><img width="500" height="688" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image023.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 232 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> If we pursue these thoughts of Th. Fischer’s further, and apply them to the constant interval ratios, whose elements are indeed varied as in architecture generally, then for the octave, fifth, and third, the result is the possibilities in Fig. 232 (here, of course, only a sample).</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> From these three series of interval combinations, which can easily be imagined as the first outlines of buildings, gardens, furniture, etc., not only the great variability of the elements among themselves is important, but above all the comparison of the three series-types (octave, fifth, third) to one another. Physiognomically, seen as a whole, one could say that the median fifth-model with its variants, in contrast to the two others, has something smooth about it, whereas the first octave-model is pretentious, the last third-model almost mellifluous, dulcet. This is a matter of feeling, however, and each model will make a different impression on each viewer. The deciding factor is that these three models activate other perceptions, and that, concretely speaking, an architectonic complex based on an “octave” design will have a different character than one based on a “fifth” or “third” design. And in this last conclusion lies the sense of such a possible “normative” architecture, identical with the sense of the “true mason’s gauge” of the old cathedral architects, or the fundamental proportion of any Greek temple.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> A yet more precise indication for a future study of harmonic architectural proportion can be obtained by taking the first senary ratios—i.e. the index 6 of the partial-tone coordinates—as proportions, and drawing them analogously to the “P” diagram (Fig. 233). Here, we first notice that all forms “standing upright” above the generator-tone diagonal are reciprocal to all forms beneath the diagonal, which in this case means that they are equal in content, and different only in their position. Now, exactly as for the tones, identical forms emerge, for example the squares of the generator-tone diagonal, the rectangles <sup>3</sup>/1 <sup>6</sup>/<sub>2</sub>; <sup>3</sup>/<sub>1</sub> <sup>4</sup>/<sub>2</sub> <sup>6</sup>/<sub>3</sub>;<sub> </sub><sup>3</sup>/<sub>2</sub> <sup>6</sup>/<sub>4</sub>, which are different in their sizes but equal in their proportions. Therefore the figures that are effectively different and present within this index 6 reduce themselves collectively to twelve. If one takes out these different forms (drawn with thick lines in Fig. 233), and brings them to a common denominator—for example, the square <sup>6</sup>/<sub>6</sub> is “homologous” to the rectangle <sup>6</sup>/<sub>5</sub>, etc.—then one has an evident number (12) of <em>primary form-models</em>, by means of which an “<em>architectonic study of harmony</em>” can be built, especially if one also observes its possibilities of combination in three-dimensional space.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong><span lang="EN-US"> <shape style="width: 276pt; height: 290.25pt;" id="_x0000_i1037"> <imagedata o:title="233"></imagedata></shape> <p><img width="494" height="518" src="/EZ/kayser2/kayser2/imagelibrary/architecture/image025.jpg" alt="" title="" /></p></span></strong></p> <p style="text-align: center;" align="center" class="MsoNormal"><strong> <div><span style="font-size: 10pt;" lang="EN-US">Figure 233 <p> </p></span></div></strong></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> If we complain today that we are lacking a “style,” it is not because our time lacks a “feeling for style”—every era has its “style,” including ours, and it is not at all necessary for this style to be activated predominantly in architecture—it is because (as far as architecture goes) people simply build as they wish and as they see fit, and because the measurements of each individual building or building-complex follow completely arbitrary proportions, or at best utilitarian ones. If the architect in question happens to have great talent, or if he is an artist of the first rank, then he will accomplish brilliant things, even today. The thousands of others, though, who are well-intentioned but, due to their lack of original ideas, more or less rely on copying, badly need some steady guidance. This guide would give them not only a “personal mark” but also a certain uniformity in their plans and drafts, certain norms which would temper the hullabaloo caused by the modern disarray of styles. I can well imagine that the talented architect sets himself such norms in his designs, and I believe that he would not do badly with harmonic proportions, especially bearing in mind the things that have been shown and developed in this section, regarding both the “optics” and especially the “acoustics” of his buildings.</span></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Building is something wonderful! Do we not use the word alone as an expression for production and creativity? On one end of this concept is the taming and forming of raw material for framing space, to shelter human beings for their lives, reflections, and work. On the other end is the almost limitless possibility for artistic creation, and even, if we include the concept of “interior design,” all visual arts and crafts! To this it is imperative to add a coherent arrangement according to measure and number as the basis on which an architectonic work can begin to be realized: an imperative that has presumably made the most substantial contribution to a conscious grasp of number and geometric form.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <p class="MsoNormal"><strong><em><span lang="EN-US">§29a. Bibliography <p> </p></span></em></strong></p> <p style="text-align: justify;" class="MsoNormal"><span lang="EN-US"> Th. Fischer’s booklet is recommended as a first introduction. Besides the other sources mentioned in this section, anyone who wants to follow up on harmonic proportion is referred to the following authors: Bötticher: <em>Tektonik der Hellenen</em>, 1874; Reinhardt: <em>Gesetzlichkeit der griechischen Baukunst</em>, 1903; Röber: <em>Geometrische Grundformen antiker Tempel</em>, 1854; Schadow: <em>Polyklet</em>, 1866; Schulz: <em>Werkmaß und Zahlverhältnisse griechischer Tempel</em>, 1893; Koldewey-Puchstein: <em>Die griechischen Tempel in Unteritalien und Sizilien</em>; Weichert: <em>Typen der archaischen Architektur in Griechenland und Kleinasien</em>; M. Theurer: <em>Der griechische-dorische Peripteraltempel. Ein Beitrag zur antiken Proportionslehre</em>; Semper: <em>Kleine Schriften</em>; Thiersch: <em>Handbuch der Architektur</em>, IV, 1, 1904, on “proportions”; Sulpice-Boisserée and other more recent literature is named and discussed in Th. Fischer’s <em>2 Vorträge über Proportionen</em>, <place w:st="on"> <city w:st="on">Munich </city></place>1934, Oldenbourg-Verlag. Martin Strübin ( <place w:st="on"> <city w:st="on">Basel </city></place>) wrote an essay in the <em>Schweizerische Bauzeitung</em> (Sep. 20<sup>th</sup>, 1947): “Das Villard-Diagramm, ein Schlüssel zur Bauweise der Gotik?” in which, inspired by my <em>II. Harmonikale Studie</em> and other writings, he analyzed the cathedral at <place w:st="on"> <city w:st="on">Bern </city></place>by means of the harmonic division canon.</span></p> <p class="MsoNormal"><span lang="EN-US"> <p> </p></span></p> <div><br clear="all" /> <hr width="33%" size="1" align="left" /> <div id="ftn1"> <p style="text-align: justify;" class="MsoFootnoteText"></p> <div style="text-align: justify;" class="MsoFootnoteText"><a name="_ftn1" href="#_ftnref1" title=""><span class="MsoFootnoteReference"><span lang="EN-US">*</span></span></a><span lang="EN-US"> <em>Translator’s note: </em>In the original, Kayser marked the rectangle on the left as “die große Sext, Umkehrung der großen Terz” (“the major sixth, inversion of the major third”). We assume this was an error, since it is the minor sixth that is the inversion of the major third, while the major sixth is the inversion of the minor third. The tone-ratio <sup>5</sup>/<sub>8</sub> is closest to the minor sixth, therefore we have corrected the figure accordingly.</span></div> <div style="text-align: justify;" class="MsoFootnoteText"><span lang="EN-US"> </span></div> <div style="text-align: justify;" align="center" class="MsoFootnoteText"> <div align="center"><span lang="EN-US"><img width="500" height="595" src="/EZ/kayser2/kayser2/imagelibrary/architecture/13.jpg" alt="" title="" /></span><br /> <span lang="EN-US"></span></div> <div align="left"> <div align="center"><span lang="EN-US"><br /> Figure 253</span><br /> </div><span lang="EN-US"></span></div><span lang="EN-US"><br /> <br /> </span> <div align="left"> <div align="center"><span lang="EN-US"><img width="500" height="503" src="/EZ/kayser2/kayser2/imagelibrary/architecture/14.jpg" alt="" title="" /></span><br /> </div><span lang="EN-US"></span></div><span lang="EN-US"><br /> </span> <div align="left"> <div align="center"><span lang="EN-US">Figure 262<br /> </span><br /> </div><span lang="EN-US"></span></div><span lang="EN-US"><br /> </span> <div align="left"> <div align="center"><span lang="EN-US"><img width="461" height="437" src="/EZ/kayser2/kayser2/imagelibrary/architecture/15.jpg" alt="" title="" /></span><br /> </div><span lang="EN-US"></span></div><span lang="EN-US"><br /> </span> <div align="left"> <div align="center"><span lang="EN-US">Figure 269<br /> </span><br /> </div><span lang="EN-US"></span></div><span lang="EN-US"><br /> </span></div> <div style="text-align: justify;" align="center" class="MsoFootnoteText"> <div align="center"><img width="500" height="497" src="/EZ/kayser2/kayser2/imagelibrary/architecture/16.jpg" alt="" title="" /> <br /> </div><br /> <div align="center">Figure 270</div><br /> </div> <div style="text-align: justify;" align="center" class="MsoFootnoteText"><span lang="EN-US"><br /> </span> <div align="center"><span lang="EN-US"><img width="500" height="399" src="/EZ/kayser2/kayser2/imagelibrary/architecture/17.jpg" alt="" title="" /></span></div></div> <div style="text-align: justify;" align="center" class="MsoFootnoteText"><span lang="EN-US"> </span></div> <div style="text-align: justify;" align="center" class="MsoFootnoteText"> <div align="left"> <div align="center"><span lang="EN-US">Figure 271a</span><br /> </div><span lang="EN-US"></span></div><span lang="EN-US"><br /> </span></div></div></div></td></tr></tbody></table></span></strong></div> </body> </html></div>
History of Harmonics
Ancient Civilizations
Haase - Hermann Bahr & Harmonic Thought
Haase - Leibniz & Kayser
Summary of History
Harmonics & Mathematics
Magic Squares
Number Distribution
Number Symbolism
Fractal Harmonies
Harmonics & Geometry
Musical Ratios
Parabola, Hyperbola, Ellipse
The Proportions
Tone Spirals & Tone Curves
Harmonics & Astronomy
Kepler
Haase on Plato's Harmony of The Spheres
Harmony of The Spheres
Kepler`s Platonic Solids
Planetary Harmonics
Cosmology & Cosmogony
Harmonic Cosmogony
I Ching
Pythagoreanism
Robert Fludd
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Octaves & Spirals
Harmonics & Architecture
Harmonics & Architecture
Architectural Examples
Harmonics & Language
Harmonics & Language
Harmonics & Symbolism
Cosmic Harmony in Symbolism
Doctrine of Correspondences
Harmonics as Symbolism
Hermann Hesse & The Glass Bead Game
Gann & Financial Markets
Partial Tone Diagram & Square of 9
Polar Diagrams & Spiral Charts
Harmonics in Nature
Human Canon
Animals
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Snow Crystals
Harmonics & Music Theory
Harmonics as a Science
Interference
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Thought as a Sense
Color & Partial-Tone Coordinates
The Lambdoma
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The Monochord
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Hans Kayser's Horoscope
Rudolf Haase Biography
Bibliography of Kayser's Works
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News - May 2010
February 2008 Newsletter
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