Art and Mathematics
The preceding discussion suggests several instances in which CAS might form the basis for research in applied science sometime in the future. A proper medium for presenting natural laws is the manner in which all phenomena have been expressed well before the dawn of written history: as an art form. Science may hold a treasured position in the collective opinion of humanity because its innumerable applications have done much to raise our overall standard of living, and the longevity required for enjoying its benefits. But certain intangibles, particularly self-expression and the means for delivering it, have proven to be essential to our mental and emotional evolution.
Furthermore, while mathematics―that which is usually viewed as the purest form and language of science―is the basis for the proposed visual co-expression of music, scientific study first requires something to be observed, and the art form proposed should supply ample substance for observation. This is the realm of “non-consensus reality.” It is further noted that even though science and mathematics are more consensus-based than other disciplines, at some level they, too, are based on axioms and postulates that are beyond deductive proof and that are provisionally accepted “on faith.”
The concept of mathematics-as-art has already provided classroom teachers with a means to render math in a less abstract form, through the construction of tessellations. A tessellation is a visual composition created by drawing a given shape repeatedly to cover a plane―such as a sheet of paper―without allowing gaps or overlaps to occur. The composition is then colored to suit the composer’s―the student’s―own tastes.18 Although the skill levels involved are considerably different, tessellations are in some respects a highly elementary version of the gorgeous mandalas, reverential geometric artworks with roots in Hindu and Buddhist (especially Tibetan) traditions. Mandalas are strikingly colorful depictions of the universe, and Tibetan monks fashion them with painstaking care on scrolls or consecrated ground―and in the latter
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instance, the compositions are erased at the conclusion of sacred rituals. While they are rich in devotional symbolism, and beyond the scope this presentation, there are two aspects which give them significance regarding CAS.19 Scholars of religion and culture might also make a case for other architecture and other art connections as well―for example, Islamic traditions.
First of all, mandalas begin with a dimensionless point and evolve in all directions into items of awesome beauty. This comports very nicely with one of CAS’s aspects: to add dimensions to music in order to enhance the (often casual) listeners’ enjoyment and appreciation of the composition.
Secondly, each color used to create a mandala has its own pertinent and dynamic meaning. Each begins as an element of delusion that is transformed into a seed of wisdom. What each represents is, however, less relevant to CAS than the simple matter that it has relevance. To consider that colors have meanings beyond their aesthetic values is a fundamental premise of the project.
The unfolding of sublime importance from a central point is a motif that―perhaps unexpectedly―may be found in the iconography of Eastern Orthodox and Byzantine Catholic religious traditions. One such example, glorifying the Theotokos (Virgin Mary), depicts her and the Christ child, literally surrounded by residents of Heaven and Earth―much as the planets of our solar system surround the Sun.20 This is clearly part of the mandala tradition, though in a Christian manifestation. The theology of icons in general was established by the Seventh Ecumenical Council (Nicaea II), which met in 787. The decrees of this council reveal a distinct similarity with the Mandala Tradition, in particular,
“As the sacred and life-giving cross is everywhere set up as a symbol, so also should the images of Jesus Christ, the Virgin Mary, the holy angels, as well as those of the saints and other pious and holy [persons] be embodied in the manufacture of sacred vessels, tapestries, vestments, etc., and exhibited on the walls of churches, in the homes, and in all conspicuous places, by the roadside and everywhere, to be revered by all who might see them. For the more they are contemplated, the more they move to fervent memory of their prototypes. Therefore, it is proper to accord to them a fervent and reverent adoration, not, however, the veritable worship which, according to our faith, belongs to the Divine Being alone—for the honor accorded to the image passes over to its prototype, and whoever adores the image adores in it the reality of what is there represented.”21
In all such traditions, the mandala or icon is “symbolic,” that is, it is a mysterion connecting one level of reality with another, and this connection is real―actually more real than quotidian realities. This accords with the ancient and current practices of those traditions that employ these images.
Sympathetic Vibrations
Resonance as it applies to CAS specifically concerns those frequencies that are sympathetic vibrations; these are essentially multiples of a given tone’s Hz value.22 Regarding visible light colors, also known as the visible-light spectrum from deep red to deep violet, frequencies are so great that they are expressed in terahertz, 1012 or one trillion Hz, also denoted “THz.”
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Sympathetic vibrations have a special relationship to any pitch created by any means, and as we shall see, resonance can be very pleasing, very irritating, or even downright devastating.23
H. Spencer Lewis built a Sympathetic Vibration Harp, which was a simple harp of 12 strings, each one tuned to one of the twelve pitches of the chromatic scale. By striking a tuning fork that vibrated at one of those twelve pitches, an investigator could identify its corresponding frequency on the harp because the string in question would audibly vibrate “in sympathy” with the fork as it passed near the string. This remarkable characteristic could furnish a harmless delight for an interested audience.
The reality and power of sympathetic resonance can be seen in its ability to destroy structures such as bridges and buildings. An example is the Tacoma Narrows Bridge, a suspension bridge, which was opened to the public on July 1, 1940; it was hailed as an example of both elegance and economy in civil engineering. It loomed 425 feet above the Tacoma Narrows, a strait which part of Puget Sound in the state of Washington. Unfortunately, its designers had failed to recognize the impact that aerodynamics―and sympathetic vibration―might have on the structure. Almost as soon as it was opened, the bridge began to undulate in strong breezes, so much so that it earned the nickname, “Galloping Gertie.” On November 7, 1940, high winds created a very low (0.2 Hz), but also very destructive pitch and violent (twenty-eight-foot) undulations, undermining the bridge’s structural soundness and culminating in “Gertie’s” total collapse.24
The Luxatone―Visual Representation of Music
In all likelihood, people began to associate color with music almost as soon as they began to produce it. Musicians and musicologists have been using “color” (to describe tonal characteristics), and “chromatic scale” (to identify the collection of half-step pitches between any tonal note and its first octave multiple), for quite a while. These terms allude to a relationship between sound and color, which is in fact a relationship between audible and visible frequencies, respectively.25
The Luxatone, a color-organ invented by H. Spencer Lewis and first demonstrated in New York City in February 1916, is described in his article bearing the same name.26 Although the Luxatone was dismantled long ago, it may be studied and understood today through records of its construction and operation. As indicated by the Rosicrucian concept of the Cosmic Keyboard and its accompanying musical keyboard, there is a direct relationship between sound and color, in which the latter is an arithmetic multiple of the former. Musical notes have special relationships with their factors and multiples. Everything on the keyboard is based on its scientifically demonstrable vibratory levels.
The number 24 plays an easily overlooked part in the development of the CAS process; just as the number 12 represents the number of half-tones or half steps in an octave, 24 denotes the number of its quarter-tones. To many traditional Western musicians this might seem to be excessive subdivision, but as we shall see, some modes contain more than 12 notes. Although ¼ tone scales are seemingly of no particular concern in Western musical composition, there may be rare occasions when their appearance is essential; moreover, these scales have an illustrious
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history all their own.27 Here the number 24 also provides a practical way to describe factors and multiples, giving that quantity added importance.
The number 24 can be divided evenly by 1, 2, 3, 4, 6, 8 and 12; these seven numbers are 24’s factors. Now 24 times 2 is 48, 24 times 3 is 72, and 24 times 4 is 96; these three are just the first of an endless quantity of numbers that are 24’s multiples.
H. Spencer Lewis’s device intercepted the frequencies from any given sound source, measured it and translated it into its corresponding color.28 The Luxatone achieved this effect by activating red, green and/or blue colored light bulbs―these hues being the primary colors of the visible-light spectrum. Regarding visible light, red and blue form magenta, blue and green form cyan, and green and red form yellow. When all three primary light colors are combined, the product is white light, appropriately called the additive process.29 (These primary colors differ slightly from those of the pigments used in visual art, where, yellow replaces green, green is produced by mixing equal parts yellow and blue, orange by equal parts of yellow and red, and violet by equal parts of red and blue. Combining the three primary pigments results in black, the absence of all color. For this reason, mixing pigments is known as the subtractive process.30
The red, green, and blue light bulbs of the Luxatone were contained in a triangular, translucent screen. The intercepted frequencies were measured and translated into the corresponding frequency according to the Keyboard. One may infer from descriptions that the fundamental, or most recognizable pitch of the emitted sound, would instantly be translated into its visible analog, although all sounds consist of a distinguishing set of tones which are part of the overtone series. When a given pitch such as C-256 is sounded by a musical instrument (or other producers of sound, such as the human voice) other pitches, or overtones, influence the timbre or tonal quality of the sound emitted.31
The pitch most easily recognized by the listener is known as the “fundamental” or “first partial.” Simply put, all sound arises from the vibrations of some part of an instrument. In accordance with the Cosmic Keyboard, Middle C is identified as 256 cycles per second (Hz); this is a mere six Hz below the regularly accepted standard.32 It is also much easier to use as an example. When an instrument sounds “C-256,” “C” arises because the instrument is vibrating at 256 Hz. The vibrations activate or “excite” air molecules at the same number of cycles per second, and these air molecules in turn excite others, causing the sound to travel outwards from its source (the instrument) in all directions. The velocity of sound through air molecules at 20oC (68oF) at sea level is 343 meters (1,125 feet) per second,33 which translates to very nearly 767 miles per hour.
Chromoacoustics
If the fundamental or first partial of a given pitch, such as C-256, were the only pitch being sounded, one could not distinguish between the instrument in question and any other instrument sounding that pitch. But each instrument sounds additional pitches that combine to create the unique timbre of a given instrument. Any instrument sounding C-256 is also sounding at least some of the following: 512 Hz, 768 Hz, 1,024 Hz, 1,280 Hz, 1,536 Hz, 1,792 Hz, 2,048 Hz, 2,304 Hz, and 2,560 Hz.
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512 Hz is the first overtone or second partial; it is also 2 x 256 Hz. The second overtone, or third partial, 768 Hz, is simply 3 x 256 Hz. The ninth overtone, or tenth partial, is of course, 2560 Hz, or 10 x 256 Hz.34 A given instrument can emit greater or higher frequencies, and at least in theory, the overtone series is endless; in fact, the range of human hearing is 16 to 20,000Hz.35
The preceding is relevant to CAS because, in essence, “sympathetic vibrations”―discussed above―make up what is known as the “overtone series.”36 The series is essential to establishing harmony in music, and it has another, implicit role in the project. If researchers continue to track the overtones of C-256, they would discover that at a given point, the vibrations are no longer audible. Very high partials of musical notes have the same frequencies as visible light (measured in THz), even though light, unlike sound, does not require a physical medium, to propagate through space and is a transverse vibration, in contrast with the longitudinal vibrations that characterize sound. This occurs at an extremely high frequency, its 2,199,023,255,551st overtone, or 2,199,023,255,552nd partial (!).
In the visible-light spectrum, the frequencies of colors are commonly identified in ranges. The midpoint of a given range may be deemed the “purest” hue of a given color. For example, green’s Hz-value occupies the range of 520 to 610 THz.37 The mean of this range is 565 THz, or 565 trillion Hertz; remarkably, it is also 2,199,023,255,551st overtone of the fundamental. The value 256 Hz is exactly 562,949,953,421,312―or approximately 563 trillion―Hertz. This is very nearly the midpoint of the green range, so Middle C’s visible analog is extremely close to “pure” green. And just as Middle C begins the Octave Eight of the Cosmic Keyboard, green begins Octave 49.
Of course, these and all other partials decay or “fade away,” and at a more rapid rate than does the fundamental or first partial.38 CAS does not address this occurrence because the images portrayed in its presentations correspond only to the fundamental and overtone frequencies (the partials involved) at the very instance that they are produced. Thus, the objective of CAS is to take a sort of “snapshot,” of that instant, and then to preserve the visual analogs as those analogs proceed right and left of the ordinate. The issue of partials’ decay is a completely legitimate object for study, but CAS is intended to form a visual picture of the sounds that the observers hear and recall. This is a more tangible visible version of the mental sequencing that allows listeners to link together the notes that create a melody (and “melody” is here used in its broadest application) rather than simply hearing the notes individually and with no memory to connect them. This demonstrates that CAS is an art form rather than a physicist’s examination or musician’s experience of acoustical phenomena, where the decay-rates of the partials are of greater concern.
It might, however, be possible to consider the effects of the partials’ decay, using one of the devised alternative presentations for timbres if the one now proposed proves impractical. This alternate approach is briefly demonstrated in Appendix B.
As described below, the CAS presentation uses the Cartesian coordinate graph, assigning to the ordinate or y-axis each partial, with the higher frequencies’ appearing higher on the axis. The abscissa’s or x-axis’ role is to extend the observer’s memory of the overtones sounded. It allows
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each partial to remain visible for several seconds (optimally, seven seconds), providing the observer the opportunity to view the visible imagery being produced. Such nominatives as “seconds” and “Hertz,” representing, respectively, units of time and event per duration, are employed here. They are, of course, artificial constructs for recording data, and they facilitate the dissemination of information among those who understand the designations. This does not, however, suggest that these forms of measurement, of seconds and Hertz units, are absolutely indispensible. If other reliable modes of measurement were to be preferred, these could be substituted for the units employed here. Such substitutions would invalidate neither the information contained in the Cosmic Keyboard, nor the conclusions reached through relying upon it. The abscissa affords the observer extra time to allow him to experience a visual continuity of the notes which he is subconsciously stringing together to recognize as the melody.
To represent dB volume, one can use a frequency analyzer approach. Frequency analyzers have been in use for some time, as H. Spencer Lewis demonstrated with his Luxatone. In 1938, Dr. Carl E. Seashore published Psychology of Music, and employed the Henrici Harmonic Analyzer to “dissect” (so to speak) timbres into their partials of the overtone series.39 Because we tend, in written musical notation as well awith temperatures, to present increases in quantities as increases in elevation, the overtone seriwould be depicted as follows: varying levels of decibel vwould correspond with varying intensities of each partial’s brilliance. The partials of a givetimbre, or of given timbres, woulissue from what is known inanalytical geometry as the “ordinate” or “y-axis,” displayexclusively positive values, andtravel both leftwards and rightwards (parallel to the “abscissa” or “x-axis”) from ordinate points desig
g
F
As noted above, the abscissa affords the observer extra time (about seven seconds) to allow the observer to experience a visual continuity of the notes which he/she is subconsciously stringing together to recognize as the melody. Although the best I can do at present is to sketch the effect,metaphorically providing you with a stick figure where a detailed portrait should be, this m
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still be a helpful starting point. Note that Sketches 3 and 4 in Figure 1 prova woefully primitive illustration of the field; for 3, the vertical (y) axis woulalmost “sl
Although the fundamental pitches might be very different from those depicted in the Cosmic Keyboard, the conversion process would be constanFor the octave of middle C, or C-4, a.k.a., the Cosmic Keyboard’s Octave Eight, the corresponding visible frequency (or color) would be the frequency of the fundamental’s 2,199,023,255,551st overtone (virtually, “pure” green). The visual presentation would appear simultaneously with the
th
But here a problem seems to arise: this process applies only to the scale in which the beginning, or Tonic, pitch is Middle C. After all, the specvisible light is confined to the range of 384 to 769 THz, or trillion Hertz―approximately one octave of vibrations. What about the many overtones of C-256 below the visible light spectrum? What is more, what about the overtones for other fre
a
Fortunately, both the Cosmic Keyboard and Prof. Michelson’s Light Wavprovide solutions. [Figure 2, right] In the former, the color series repeats itself (Octaves 48 through 50). In Light Wavecolor series, created by the refraction of ligh
o
A primary goal of CAS, particularly as an art form, is to addvisible dimension to an exclusively audible discipline. The position of the visible analog to each of the overtones wouaccurately depicted by its position on the ordinate, and its decibel value or volume would be denoted by the corresponding intensity or brightness. Otherwise, however, the hues presented would be identical for the several audible pitches of C: 16 Hz, 32 Hz, 64 Hz, 128 Hz, 256 Hz, 512 Hz, 1024 Hz, 2048 Hz, 4096 Hz, 8192 Hz, and 16384 Hz would brepresented by the an
5
Some hints on other ways to present color music can be inferred from illustrations by Professor Albert A. Michelson.40 First of all, he specifically identifies, and illustrates, Lord Kelvin’s torsion-driven wave-mo[Fig 3 from Michelson, left 41]. This model demonstrates the light wave’s imthree-dimensional nature, and was developed by Kelvin sixty years before Albe
E
Extra-Scientific Approaches to Investigations
While modern math and physics are most useful in these investigations, as with many human endeavors, there are other approaches as well. As we have seen, science deals in the verifiable (and thus falsifiable). However, there are other dimensions to explorations of these questions. In the realm of the extra-scientific practice of numerology, some interesting parallels are found.
Numerology pertains to something more than either is sometimes assumed in academic settings. As is well known, the study of music, as both an art and the object of scientific deliberation, was of tremendous interest to a most remarkable―and mysterious―figure of history: Pythagoras. Often considered the father of mathematics and of music, he was born ca. 575 BCE on Samos, a Mediterranean island near the coastline of Greece. So much of his biography is cloaked in mystery because he seems to have preferred it that way: his studies appear to have brought him to the Mystery Schools of Egypt and Babylon, and at one point he became associated with the Magi of Persia, who were learned individuals probably best known for their presence (as the “Wise Men” or Magi) in the story of Jesus’ birth in the Christian scriptures.43
Pythagoras organized his own initiatic mystery school, now known as the Pythagoreans, which helps explain why so little is actually known about his broader philosophy and teachings. Incidentally, these include the belief in soul transmigration―in reincarnation. But more relevantly here, it could be argued that while he revealed the science that underlies music, he also beheld and treasured the art of mathematics. It is tempting to provide at least a partial list of the Pythagoreans’ contributions to mathematics, but even a well-edited collection would begin to obscure the topic of this presentation. Suffice it to say that geometry, or “Earth measurement” was introduced ca. 300 BCE and is primarily attributable to Euclid;44 trigonometry, or “triangle measurement,” was introduced ca. 150 BCE and is primarily attributable to Hipparchus of Bithynia.45 The Pythagoreans were doubtlessly familiar with the mathematical rules of both disciplines, and yet they predated both by more than two centuries. The Pythagoreans also introduced the world to irrational number, very real values that cannot be expressed as the quotient of two integers (“fractions” to most of us), and without which neither geometry nor trigonometry could operate (or even exist).
The Pythagoreans’ considerable achievements can be generalized this way: 1) they discovered mathematical principles inherent in nature, and 2) they went on to apply those principles to art. Pythagoras is credited with having employed the arithmetic of acoustical physics to introduce octaves to the art of music. This is noteworthy by itself, but the Pythagoreans went on to uncover simple yet profound geometric laws in the natural world that continue today to exert their influence in the realm of self-expression. Among the topics they contemplated at great lengths were astrology and numerology. It might be helpful to keep in mind that even science, which insists upon following time-tested, reliable study methods, is itself a work in progress; and while the absence of a given experience might justifiably disqualify it from being considered in the rigid discipline of scientists, the absence of an experience is not synonymous with the experience of an absence. The Pythagoreans were well-versed in the elements of astronomy and chemistry, as well as mathematics, and they were apparently motivated to seek out paradigm-altering evidence.
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Thus even today, as scientists are studying synesthetic phenomena, they are far from concluding what a synesthete’s limitations might be. In fact, one recognized synesthete has already noted that to her, numbers not only have corresponding chromatic or color values, but also impress her with traits of personality.46
Further historical and cultural researches will be able to assess the scientific value of these extra-scientific data which come to us from other approaches to knowledge.
Relationships between Visible and Audible Frequencies
Returning to the Cosmic Musical Keyboard, the link between audible and visible frequencies begins to present itself at 192 Hz. That frequency or vibratory rate is identified as the note of G, five half steps―known among musicians as a perfect fifth―above C4 or Middle C.47 Although Middle C can range from 256 to 280 Hz, and the Cosmic Keyboard identifies 256 Hz as the most relevant pitch, while the commonly accepted Hz-value among American musicians appears to be 261.626, or 262, Hz .48
One can also note that the color of deep red is identified as the note’s “visible analog,” so to speak. The frequency of deep red is G-192’s 2,199,023,255,551st overtone. (NOTE: The relationship can be applied to what American musicians generally accept as the frequency for the key of G; on the scale below Middle C, that frequency is conventionally recognized again, by American musicians, to be 196 Hz, rather than 192 Hz, as noted in the Cosmic Keyboard.)
The range of human hearing is usually recognized as beginning at 16 Hz and ending at about 20,000, or 20 Kilohertz (KHz), while the visible light spectrum is expressed as factors of 100 Terahertz (THz, or 1012 Hertz).49 Using the pitch of G-196 as an example, there is no apparent connection between that pitch and the frequency of 493 THz. In terms of octaves, visible light begins in the 48th Octave of the Cosmic Keyboard, 41 octaves above G-196; G-196’s analog in Octave 48 is 431,008,558,088,192 Hz, or about 431 THz, which is unquestionably deep red in the visible-light spectrum. It is, of course, also G-196’s 2,199,023,255,551st overtone, or 2,199,023,255,552nd partial. The accepted pitch for G directly below Middle C (192 or 196 Hz) is not an issue; neither is the fact that the visible-light “scale,” so to speak, begins at G rather than C. The salient point is that every possible audible pitch has an arithmetic analog or counterpart in the visible light spectrum.
Next Steps
Understanding the preceding is quite important, and yet it brings us only to the starting point for our journey to translate music―and all audible frequencies―into a visible art form. One could develop a variety of visual presentations based upon the arithmetic link between audible and visible frequencies, and consider the simplest type, although it might not be at all simple with respect to its foundation. Those whose interests include visual art, music, and computer programming are not only invited to participate in contemplating the optimal type of
