Part 5 (2/2)
Another art theorist who had great interest in the Golden Ratio at the beginning of the twentieth century was the American Jay Hambidge (18671924). In a series of articles and books, Hambidge defined two types of symmetry in cla.s.sical and modern art. One, which he called ”static symmetry,” was based on regular figures like the square and equilateral triangle, and was supposed to produce lifeless art. The other, which he dubbed ”dynamic symmetry,” had the Golden Ratio and the logarithmic spiral in leading roles. Hambidge's basic thesis was that the use of ”dynamic symmetry” in design leads to vibrant and moving art. Few today take his ideas seriously.
One of the strongest advocates for the application of the Golden Ratio to art and architecture was the famous Swiss-French architect and painter Le Corbusier (Charles-edouard Jeanneret, 18871965).
Jeanneret was born in La Chaux-de-Fonds, Switzerland, where he studied art and engraving. His father worked in the watch business as an enameler, while his mother was a pianist and music teacher who encouraged her son toward a musician's dexterity as well as more abstract pursuits. He began his studies of architecture in 1905 and eventually became one of the most influential figures in modern architecture. In the winter of 19161917, Jeanneret moved to Paris, where he met Amedee Ozenfant, who was well connected in the Parisian haut monde of artists and intellectuals. Through Ozenfant, Jeanneret met with the Cubists and was forced to grapple with their inheritance. In particular, he absorbed an interest in proportional systems and their role in aesthetics from Juan Gris. In the autumn of 1918, Jeanneret and Ozenfant exhibited together at the Galerie Thomas. More precisely, two canvases by Jeanneret were hung alongside many more paintings by Ozenfant. They called themselves ”Purists,” and ent.i.tled their catalog Apres le Cubisme Apres le Cubisme (After cubism). Purism invoked Piero della Francesca and the Platonic aesthetic theory in its a.s.sertion that ”the work of art must not be accidental, exceptional, impressionistic, inorganic, protestatory, picturesque, but on the contrary, generalized, static, expressive of the invariant.” (After cubism). Purism invoked Piero della Francesca and the Platonic aesthetic theory in its a.s.sertion that ”the work of art must not be accidental, exceptional, impressionistic, inorganic, protestatory, picturesque, but on the contrary, generalized, static, expressive of the invariant.”
Jeanneret did not take the name ”Le Corbusier” (co-opted from ancestors on his mother's side called Lecorbesier) until he was thirty-three, well installed in Paris, and confident of his future path. It was as if he wanted basically to repress his faltering first efforts and stimulate the myth that his architectural genius bloomed suddenly into full maturity.
Originally, Le Corbusier expressed rather skeptical, and even negative, views of the application of the Golden Ratio to art, warning against the ”replacement of the mysticism of the sensibility by the Golden Section.” In fact, a thorough a.n.a.lysis of Le Corbusier's architectural designs and ”Purist” paintings by Roger Herz-Fischler shows that prior to 1927, Le Corbusier never used the Golden Ratio. This situation changed dramatically following the publication of Matila Ghyka's influential book Aesthetics of Proportions in Nature and in the Arts Aesthetics of Proportions in Nature and in the Arts, and his Golden Number, Pythagorean Rites and Rhythms Golden Number, Pythagorean Rites and Rhythms (1931) only enhanced the mystical aspects of even further. Le Corbusier's fascination with (1931) only enhanced the mystical aspects of even further. Le Corbusier's fascination with Aesthetics Aesthetics and with the Golden Ratio had two origins. On one hand, it was a consequence of his interest in basic forms and structures underlying natural phenomena. On the other, coming from a family that encouraged musical education, Le Corbusier could appreciate the Pythagorean craving for a harmony achieved by number ratios. He wrote: ”More than these thirty years past, the sap of mathematics has flown through the veins of my work, both as an architect and painter; for music is always present within me.” Le Corbusier's search for a standardized proportion culminated in the introduction of a new proportional system called the ”Modulor.” and with the Golden Ratio had two origins. On one hand, it was a consequence of his interest in basic forms and structures underlying natural phenomena. On the other, coming from a family that encouraged musical education, Le Corbusier could appreciate the Pythagorean craving for a harmony achieved by number ratios. He wrote: ”More than these thirty years past, the sap of mathematics has flown through the veins of my work, both as an architect and painter; for music is always present within me.” Le Corbusier's search for a standardized proportion culminated in the introduction of a new proportional system called the ”Modulor.”
The Modulor was supposed to provide ”a harmonic measure to the human scale, universally applicable to architecture and mechanics.” The latter quote is in fact no more than a rephrasing of Protagoras' famous saying from the fifth-century B.C. B.C. ”Man is the measure of all things.” Accordingly, in the spirit of the Vitruvian man ( ”Man is the measure of all things.” Accordingly, in the spirit of the Vitruvian man (Figure 53) and the general philosophical commitment to discover a proportion system equivalent to that of natural creation, the Modulor was based on human proportions (Figure 79).
Figure 79 A six-foot (about 183-centimeter) man, somewhat resembling the familiar logo of the ”Michelin man,” with his arm upraised (to a height of 226 cm; 75), was inserted into a square (Figure 80). The ratio of the height of the man (183 cm; 6) to the height of his navel (at the midpoint of 113 cm; 3' 8.5) was taken to be precisely in a Golden Ratio. The total height (from the feet to the raised arm) was also divided in a Golden Ratio (into 140 cm and 86 cm) at the level of the wrist of a downward-hanging arm. The two ratios (113/70) and (140/86) were further subdivided into smaller dimensions according to the Fibonacci series (each number being equal to the sum of the preceding two; Figure 81 Figure 81). In the final version of the Modulor (Figures 79 and 81), two scales of interspiraling Fibonacci dimensions were therefore introduced (the ”red and the blue series”).
Figure 80
Figure 81 Le Corbusier suggested that the Modulor would give harmonious proportions to everything, from the sizes of cabinets and door handles, to buildings and urban s.p.a.ces. In a world with an increasing need for ma.s.s production, the Modulor was supposed to provide the model for standardization. Le Corbusier's two books, Le Modulor Le Modulor (published in 1948) and (published in 1948) and Modulor II Modulor II (1955), received very serious scholarly attention from architectural circles, and they continue to feature in any discussion of proportion. Le Corbusier was very proud of the fact that he had the opportunity to present the Modulor even to Albert Einstein, in a meeting at Princeton in 1946. In describing that event he says: ”I expressed myself badly, I explained 'Modulor' badly, I got bogged down in the mora.s.s of 'cause and effect.'” Nevertheless, he received a letter from Einstein, in which the great man said this of the Modulor: ”It is a scale of proportions which makes the bad difficult and the good easy.” (1955), received very serious scholarly attention from architectural circles, and they continue to feature in any discussion of proportion. Le Corbusier was very proud of the fact that he had the opportunity to present the Modulor even to Albert Einstein, in a meeting at Princeton in 1946. In describing that event he says: ”I expressed myself badly, I explained 'Modulor' badly, I got bogged down in the mora.s.s of 'cause and effect.'” Nevertheless, he received a letter from Einstein, in which the great man said this of the Modulor: ”It is a scale of proportions which makes the bad difficult and the good easy.”
Le Corbusier translated his theory of the Modulor into practice in many of his projects. For example, in his notes for the impressive urban layout of Chandigarh, India, which included four major government buildings-a Parliament, a High Court, and two museums-we find: ”But, of course, the Modulor came in at the moment of part.i.tioning the window area. ... In the general section of the building which involves providing shelter from the sun for the offices and courts, the Modulor will bring textural unity in all places. In the design of the frontages, the Modulor (texturique) will apply its red and blue series within the s.p.a.ces already furnished by the frames.”
Figure 82 Le Corbusier was certainly not the last artist to be interested in the Golden Ratio, but most of those after him were fascinated more by the mathematical-philosophical-historical attributes of the ratio than by its presumed aesthetic properties. For example, the British abstract artist Anthony Hill used a Fibonacci series of dimensions in his 1960 ”Constructional Relief” (Figure 82). Similarly, the contemporary Israeli painter and sculptor Igael Tumarkin has deliberately included the formula for the value of in one of his paintings. in one of his paintings.
An artist who transformed the Fibonacci sequence into an important ingredient of his art is the Italian Mario Merz. Merz was born in Milan in 1925, and in 1967 he joined the art movement labeled Arte Povera (Poor Art), which also included the artists Michelangelo Pisto-letto, Luciano Fabro, and Jannis Kounellis. The name of the movement (coined by the critic Germano Celant) was derived from the desire of its members to use simple, everyday life materials, in a protest against what they regarded as a dehumanized, consumer-driven society. Merz started to use the Fibonacci sequence in 1970, in a series of ”conceptual” works that include the numbers in the sequence or various spirals. Merz's desire to utilize Fibonacci numbers was based on the fact that the sequence underlies so many growth patterns of natural life. In a work from 1987 ent.i.tled ”Onda d'urto” (Shock wave), he has a long row of stacks of newspapers, with the Fibonacci numbers glowing in blue neon lights above the stacks. The work ”Fibonacci Naples” (from 1970) consists of ten photographs of factory workers, building in Fibonacci numbers from a solitary person to a group of fifty-five (the tenth Fibonacci number).
False claims about artists allegedly using the Golden Ratio continue to spring up almost like mushrooms after the rain. One of these claims deserves some special attention, since it is repeated endlessly.
The Dutch painter Piet Mondrian (18721944) is best known for his geometric, non.o.bjective style, which he called ”neoplasticism.” In particular, much of his art is characterized by compositions involving only vertical and horizontal lines, rectangles, and squares, and employing only primary colors (and sometimes black or grays) against a white background, as in ”Broadway Boogie-Woogie” (Figure 83; in The Museum of Modern Art, New York). Curved lines, three-dimensionality, and realistic representation were entirely eliminated from his work.
Figure 83 Not surprisingly, per haps, Mondrian's geometrical compositions attracted quite a bit of Golden Numberist speculation. In Mathematics Mathematics, David Bergamini admits that Mondrian himself ”was vague about the design of his paintings,” but nevertheless claims that the linear abstraction ”Place de la Concorde” incorporates overlapping Golden Rectangles. Charles Bouleau was much bolder in The Painter's Secret Geometry The Painter's Secret Geometry, a.s.serting that ”the French painters never dared to go as far into pure geometry and the strict use of the golden section as did the cold and pitiless Dutchman Piet Mondrian.” Bouleau further states that in ”Broadway Boogie-Woogie,” ”the horizontals and verticals which make up this picture are nearly all in the golden ratio.” With so many lines to choose from in this painting, it should come as no surprise that quite a few can be found at approximately the right separations. Having spent quite some time reading the more serious a.n.a.lyses of Mondrian's work and not having found any mention of the Golden Ratio there, I became quite intrigued by the question: Did Mondrian really use the Golden Ratio in his compositions or not? As a last resort I decided to turn to the the real expert-Yves-Alain Bois of Harvard University, who coauth.o.r.ed the book real expert-Yves-Alain Bois of Harvard University, who coauth.o.r.ed the book Mondrian Mondrian that accompanied the large retrospective exhibit of the artist's work in 1999. Bois's answer was quite categorical: ”As far as I know, Mondrian never used a system of proportion (if one excepts the modular grids he painted in 19181919, but there the system is deduced from the format of the paintings themselves: they are divided in 8 8 units).” Bois added: ”I also vaguely remember a remark by Mondrian mocking arithmetic computations with regard to his work.” He concluded: ”I think that the Golden Section is a complete red herring with regard to Mondrian.” that accompanied the large retrospective exhibit of the artist's work in 1999. Bois's answer was quite categorical: ”As far as I know, Mondrian never used a system of proportion (if one excepts the modular grids he painted in 19181919, but there the system is deduced from the format of the paintings themselves: they are divided in 8 8 units).” Bois added: ”I also vaguely remember a remark by Mondrian mocking arithmetic computations with regard to his work.” He concluded: ”I think that the Golden Section is a complete red herring with regard to Mondrian.”
All of this intricate history does leave us with a puzzling question. Short of intellectual curiosity, for what reason would so many artists even consider employing the Golden Ratio in their works? Does this ratio, as manifested for example in the Golden Rectangle, truly contain some intrinsic, aesthetically superior qualities? The attempts to answer this question alone resulted in a mult.i.tude of psychological experiments and a vast literature.
THE SENSES DELIGHT IN THINGS DULY PROPORTIONED.
With the words in the t.i.tle of this section, Italian scholastic philosopher St. Thomas Aquinas (ca. 12251274) attempted to capture a fundamental relations.h.i.+p between beauty and mathematics. Humans seem to react with a sense of pleasure to ”forms” that possess certain symmetries or obey certain geometrical rules.
In our examination of the potential aesthetic value of the Golden Ratio, we will concentrate on the aesthetics of very simple, nonrepresentational forms and lines, not on complex visual materials and works of art. Furthermore, in most of the psychological experiments I shall describe, the term ”beautiful” was actually shunned. Rather, words like ”pleasing” or ”attractive” have been used. This avoids the need for a definition of ”beautiful” and builds on the fact that most people have a pretty good idea of what they like, even if they cannot quite explain why.
Numerous authors have claimed that the Golden Rectangle is the most aesthetically pleasing of all rectangles. The more modern interest in this question was largely initiated by a series of rather crankish publications by the German researcher Adolph Zeising, which started in 1854 with Neue Lehre von den Proportionen des menschlichen Korpers Neue Lehre von den Proportionen des menschlichen Korpers (The latest theory of proportions in the human body) and culminated in the publication (after Zeising's death) of a ma.s.sive book, (The latest theory of proportions in the human body) and culminated in the publication (after Zeising's death) of a ma.s.sive book, Der Goldne Schnitt Der Goldne Schnitt (The golden section), in 1884. In these works, Zeising combined his own interpretation of Pythagorean and Vitruvian ideas to argue that ”the part.i.tion of the human body, the structure of many animals which are characterized by well-developed building, the fundamental types of many forms of plants,... the harmonics of the most satisfying musical accords, and the proportionality of the most beautiful works in architecture and sculpture” are all based on the Golden Ratio. To him, therefore, the Golden Ratio offered the key to the understanding of all proportions in ”the most refined forms of nature and art.” (The golden section), in 1884. In these works, Zeising combined his own interpretation of Pythagorean and Vitruvian ideas to argue that ”the part.i.tion of the human body, the structure of many animals which are characterized by well-developed building, the fundamental types of many forms of plants,... the harmonics of the most satisfying musical accords, and the proportionality of the most beautiful works in architecture and sculpture” are all based on the Golden Ratio. To him, therefore, the Golden Ratio offered the key to the understanding of all proportions in ”the most refined forms of nature and art.”
One of the founders of modern psychology, Gustav Theodor Fechner (18011887), took it upon himself to verify Zeising's pet theory. Fechner is considered a pioneer of experimental aesthetics. In one of his early experiments, he conducted a public opinion poll in which he asked all the visitors to the Dresden Gallery to compare the beauty of two nearly identical Madonna paintings (the ”Darmstadt Madonna” and the ”Dresden Madonna”) that were exhibited together. Both paintings were attributed to the German painter Hans Holbein the Younger (14971543), but there was a suspicion that the ”Dresden Madonna” was actually a later copy. That particular experiment resulted in a total failure-out of 11,842 visitors, only 113 answered the questionnaire, and even those were mostly art critics or people who had formed previous judgments.
Fechner's first experiments with rectangles were performed in the 1860s, and the results were published in the 1870s and eventually summarized in his 1876 book, Vorschule der Aesthetik Vorschule der Aesthetik (Introduction to aesthetics). He rebelled against a top-down approach to aesthetics, which starts with the formulation of abstract principles of beauty, and rather advocated the development of experimental aesthetics from the bottom up. The experiment was quite simple: Ten rectangles were placed in front of a subject who was asked to select the most pleasing one and the least pleasing one. The rectangles varied in their length-to-width ratios from a square (a ratio of 1.00) to an elongated rectangle (a ratio of 2.5). Three of the rectangles were more elongated than the Golden Rectangle, and six were closer to a square. According to Fechner's own description of the experimental setting, subjects often waited and wavered, rejecting one rectangle after another. Meanwhile the experimenter would explain that they should carefully select the most pleasing, harmonic, and elegant rectangle. In Fechner's experiment, 76 percent of all choices centered on the three rectangles having the ratios 1.75, 1.62, and 1.50, with the peak at the Golden Rectangle (1.62). All other rectangles received less than 10 percent of the choices each. (Introduction to aesthetics). He rebelled against a top-down approach to aesthetics, which starts with the formulation of abstract principles of beauty, and rather advocated the development of experimental aesthetics from the bottom up. The experiment was quite simple: Ten rectangles were placed in front of a subject who was asked to select the most pleasing one and the least pleasing one. The rectangles varied in their length-to-width ratios from a square (a ratio of 1.00) to an elongated rectangle (a ratio of 2.5). Three of the rectangles were more elongated than the Golden Rectangle, and six were closer to a square. According to Fechner's own description of the experimental setting, subjects often waited and wavered, rejecting one rectangle after another. Meanwhile the experimenter would explain that they should carefully select the most pleasing, harmonic, and elegant rectangle. In Fechner's experiment, 76 percent of all choices centered on the three rectangles having the ratios 1.75, 1.62, and 1.50, with the peak at the Golden Rectangle (1.62). All other rectangles received less than 10 percent of the choices each.
Fechner's motivation for studying the subject was not free of prejudice. He himself admitted that the inspiration for the research came to him when he ”saw the vision of a unified world of thought, spirit and matter, linked together by the mystery of numbers.” While n.o.body accuses Fechner of altering the results, some speculate that he may have subconsciously produced circ.u.mstances that would favor his desired outcome. In fact, Fechner's unpublished papers reveal that he conducted similar experiments with ellipses, and having failed to discover any preference for the Golden Ratio, he did not publish the results.
Fechner further measured the dimensions of thousands of printed books, picture frames in galleries, windows, and other rectangularly shaped objects. His results were quite interesting, and often amusing. For example, he found that German playing cards tended to be somewhat more elongated than the Golden Rectangle, while French playing cards were less so. On the other hand, he found the average height-to-width ratio of forty novels from the public library to be near . Paintings (the area inside the frame) were actually found to be ”significantly shorter” than a Golden Rectangle. Fechner made the following (politically incorrect by today's standards) observation about window shapes: ”Only the window shapes of the houses of peasants seem often to be square, which is consistent with the fact that people with a lower level of education prefer this form more than people with a higher education.” Fechner further claimed that the point at which the transverse piece crosses the upright post in graveyard crosses divides the post, on the average, in a Golden Ratio.
Many researchers repeated similar experiments over the twentieth century, with varying results. Overly eager Golden Ratio enthusiasts usually report only those experiments that seem to support the idea of an aesthetic preference for the Golden Rectangle. However, more careful researchers point out the very crude nature and methodological defects of many of these experiments. Some found that the results depended, for example, on whether the rectangles were positioned with their long side horizontally or vertically, on the size and color of the rectangles, on the age of the subjects, on cultural differences, and especially on the experimental method used. In an article published in 1965, American psychologists L. A. Stone and L. G. Collins suggested that the preference for the Golden Rectangle indicated by some of the experiments was related to the area of the human visual field. These researchers found that an ”average rectangle” of rectangles drawn within and around the binocular visual field of a variety of subjects has a length-to-width ratio of about 1.5, not too far from the Golden Ratio. Subsequent experiments, however, did not confirm Stone and Collins s speculation. In an experiment conducted in 1966 by H. R. Schiffman of Rutgers University, subjects were asked to ”draw the most aesthetically pleasing rectangle” that they could on a sheet of paper. After completion, they were instructed to orient the figure either horizontally or vertically (with respect to the long side) in the most pleasing position. While Schiffman found an overwhelming preference for a horizontal orientation, consistent with the shape of the visual field, the average ratio of length to width was about 1.9-far from both the Golden Ratio and the visual field's ”average rectangle.”
The psychologist Michael G.o.dkewitsch of the University of Toronto cast even greater doubts about the notion of the Golden Rectangle being the most pleasing rectangle. G.o.dkewitsch first pointed out the important fact that average group preferences may not reflect at all the most preferred rectangle for each individual. Often something that is most preferred on the average is not chosen first by anyone. For example, the brand of chocolate that everybody rates second best may on the average be ranked as the best, but n.o.body will ever buy it! Consequently, first choices provide a more meaningful measure of preference than mean preference rankings. G.o.dkewitsch further noted that if preference for the Golden Ratio is indeed universal and genuine, then it should receive the largest number of first choices, irrespective of which other rectangles the subjects are presented with.
G.o.dkewitsch published in 1974 the results of a study that involved twenty-seven rectangles with length-to-width ratios in three ranges. In one range the Golden Rectangle was next to the most elongated rectangle, in one it was in the middle, and in the third it was next to the shortest rectangle. The results of the experiment showed, according to G.o.dkewitsch, that the preference for the Golden Rectangle was an artifact of its position in the range of rectangles being presented and of the fact that mean preference rankings (rather than first choices) were used in the earlier experiments. G.o.dkewitsch concluded that ”the basic question whether there is or is not, in the Western world, a reliable verbally expressed aesthetic aesthetic preference for a particular ratio between length and width of rectangular shapes can probably be answered negatively. Aesthetic theory has hardly any rationale left to regard the Golden Section as a decisive factor in formal visual beauty.” preference for a particular ratio between length and width of rectangular shapes can probably be answered negatively. Aesthetic theory has hardly any rationale left to regard the Golden Section as a decisive factor in formal visual beauty.”
Not all agree with G.o.dkewitsch's conclusions. British psychologist Chris McMa.n.u.s published in 1980 the results of a careful study that used the method of paired comparisons, whereby a judgment is made for each pair of rectangles. This method is considered to be superior to other experimental techniques, since there is good evidence that ranking tends to be a process of successive paired comparisons. McMa.n.u.s concluded that ”there is moderately good evidence for the phenomenon which Fechner championed, even though Fechner's own method for its demonstration is, at best, highly suspect owing to methodological artifacts.” McMa.n.u.s admitted, however, that ”whether the Golden Section per se per se is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear.” is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear.”
Figure 84 You can test yourself (or your friends) on the question of which rectangle you prefer best. Figure 84 Figure 84 shows a collection of forty-eight rectangles, all having the same height, but with their widths ranging from 0.4 to 2.5 times their height. University of Maine mathematician George Markowsky used this collection in his own informal experiments. Did you pick the Golden Rectangle as your first choice? (It is the fifth from the left in the fourth row.) shows a collection of forty-eight rectangles, all having the same height, but with their widths ranging from 0.4 to 2.5 times their height. University of Maine mathematician George Markowsky used this collection in his own informal experiments. Did you pick the Golden Rectangle as your first choice? (It is the fifth from the left in the fourth row.) GOLDEN MUSIC.
Every string quartet and symphony orchestra today still uses Pythagoras' discovery of whole-number relations.h.i.+ps among the different musical tones. Furthermore, in the ancient Greek curriculum and up to medieval times, music was considered a part of mathematics, and musicians concentrated their efforts on the understanding of the mathematical basis of tones. The concept of the ”music of the spheres” represented a glorious synthesis of music and mathematics, and in the imaginations of philosophers and musicians, it wove the entire cosmos into one grand design that could be perceived only by the gifted few. In the words of the great Roman orator and philosopher Cicero (ca. 10643 B.C. B.C.): ”The ears of mortals are filled with this sound, but they are unable to hear it.... You might as well try to stare directly at the Sun, whose rays are much too strong for your eyes.” Only in the twelfth century did music break away from adherence to mathematical prescriptions and formulae. However, even as late as the eighteenth century, the German rationalist philosopher Gottfried Wilhelm Leibnitz (16461716) wrote: ”Music is a secret arithmetical exercise and the person who indulges in it does not realize that he is manipulating numbers.” Around the same time, the great German composer Johann Sebastian Bach (16851750) had a fascination for the kinds of games that can be played with musical notes and numbers. For example, he encrypted his signature in some of his compositions via musical codes. In the old German musical notation, B stood for B-flat and H stood for B-natural, so Bach could spell out his name in musical notes: B-flat, A, C, B-natural. Another encryption Bach used was based on Gematria.
Taking A = 1, B = 2, C = 3, and so on, B-A-C-H = 14 and J-S-B-A-C-H = 41 (because I and J were the same letter in the German alphabet of Bach's time). In his entertaining book Bacha.n.a.lia Bacha.n.a.lia (1994), mathematician and Bach enthusiast Eric Altschuler gives numerous examples for the appearances of 14s (encoded BACH) and 41s (encoded JSBACH) in Bach's music that he believes were put there deliberately by Bach. For example, in the first fugue, the C Major Fugue, Book One of Bach's (1994), mathematician and Bach enthusiast Eric Altschuler gives numerous examples for the appearances of 14s (encoded BACH) and 41s (encoded JSBACH) in Bach's music that he believes were put there deliberately by Bach. For example, in the first fugue, the C Major Fugue, Book One of Bach's Well Tempered Clavier Well Tempered Clavier, the subject has fourteen notes. Also, of the twenty-four entries, twenty-two run all the way to completion and a twenty-third runs almost all the way to completion. Only one entry-the fourteenth-doesn't run anywhere near completion. Altschuler speculates that Bach's obsession with encrypting his signature into his compositions is similar to artists incorporating their own portraits into their paintings or Alfred Hitchc.o.c.k making a cameo appearance in each of his movies.
Given this historical relations.h.i.+p between music and numbers, it is only natural to wonder whether the Golden Ratio (and Fibonacci numbers) played any role either in the development of musical instruments or in the composition of music.
The violin is an instrument in which the Golden Ratio does feature frequently. Typically, the violin soundbox contains twelve or more arcs of curvature (which make the violin's curves) on each side. The flat arc at the base often is centered at the Golden Section point up the center line.
Some of the best-known violins were made by Antonio Stradivari (16441737) of Cremona, Italy. Original drawings (Figure 85) show that Stradivari took special care to place the ”eyes” of the f-holes geometrically, at positions determined by the Golden Ratio. Few (if any) elieve that it is the application of the Golden Ratio that gives a Stradivarius violin its superior quality. More often such elements as varnish, sealer, wood, and general craftsmans.h.i.+p are cited as the potential ”secret” ingredient. Many experts agree that the popularity of eighteenth-century violins in general stems from their adaptability for use in large concert halls. Most of these experts will also tell you that there is no ”secret” in Stradivarius violins-these are simply inimitable works of art, the sum of all the parts that make up their superb craftsmans.h.i.+p.
Figure 85
Figure 86 Another musical instrument often mentioned in relation to Fibonacci numbers is the piano. The octave on a piano keyboard consists of thirteen keys, eight white keys and five black keys (Figure 86). The five black keys themselves form one group of two keys and another of three keys. The numbers 2, 3, 5, 8, and 13 happen all to be consecutive Fibonacci numbers. The primacy of the C major scale, for example, is partly due to the fact that it is being played on the piano's white keys. However, the relations.h.i.+p between the piano keyboard and Fibonacci numbers is very probably a red herring. First, note that the chromatic scale (from C to B in the figure), which is fundamental to western music, is really composed of twelve, not thirteen, semitones. The same note, C, is played twice in the octave, to indicate the completion of the cycle. Second, and more important, the arrangement of the keys in two rows, with the sharp and flats being grouped in twos and threes in the upper row, dates back to the early fifteenth century, long before the publication of Pacioli's book and even longer before any serious understanding of Fibonacci numbers.
In the same way that Golden Numberists claim that the Golden Ratio has special aesthetic qualities in the visual arts, they also attribute to it particularly pleasing effects in music. For example, books on the Golden Ratio are quick to point out that many consider the major sixth and the minor sixth to be the most pleasing of musical intervals and that these intervals are related to the Golden Ratio. A pure musical tone is characterized by a fixed frequency (measured in the number of vibrations per second) and a fixed amplitude (which determines the instantaneous loudness). The standard tone used for tuning is A, which vibrates at 440 vibrations per second. A major sixth can be obtained from a combination of
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