December 13, 2017 Cantor, Einstein, Escher, Nancarrow: √2, Irrationality, Infinity, and Imagining the Boundaries of the Impossible A Research Paper Presented by Jordan Alexander Key 111 NW 15th Terrace, Apt A1 Gainesville, Florida 32603 (540) 588-2409 jordanalexanderkey@gmail.com to Dr. Silvio dos Santos Associate Professor of Musicology For PhD Composition Degree University of Florida School of Music Gainesville, Florida Fall 2017 Jordan Alexander Key Paper Presentation: December 6, 2017 Title: Cantor, Einstein, Escher, Nancarrow: √2, Irrationality, Infinity, and Imagining the Boundaries of the Impossible Featured Pieces: Conlon Nancarrow, Study No. 33 for Player Piano MC Escher, Metamorphosis II Mc Escher, Circle Limit III Select Bibliography: Bill, Max. "Die mathematische Denkweise in der Kunst unserer Zeit ( The mathematical way of thinking in the visual art of our time )." In The Visual Mind: Art and Mathematics, by Michele Emmer, trans. by Michele Emmer, 5-9. Cambridge, MA: MIT Press, 1994 Drott, Eric. "Conlon Nancarrow and the Technological Sublime." American Music. 22, no. 4 (2004): 533-63. Emmer, Michele, Doris Schattschneider, and M. C. Escher. M.C. Escher’s legacy: a centennial celebration: collection of articles coming from the M.C. Escher Centennial Conference, Rome, 1998. New York: Springer-Verlag, 2003. Gann, Kyle. The music of Conlon Nancarrow. Cambridge University Press, 2006. Schattschneider, Doris . "The Mathematical Side of M. C. Escher." Notices of the American Mathematical Society. 57, no. 6 (2010): 706-18. Thomas, Margaret Elida. Conlon Nancarrow's "Temporal Dissonance": Rhythmic and Textural Stratification in the Studies for Player Piano. Ann Arbor (MI): UMI, 1997. _____. "Nancarrow's "temporal Dissonance": Issues of Tempo Proportions, Metric Synchrony, and Rhythmic Strategies." Intégral. 14/15 (2000): 137-180. Abstract: Due to the 20th century mathematical and scientific developments of Georg Cantor, Max Karl Planck, Albert Einstein, and Werner Heisenberg, concepts once relegated to obscurity, such as irrationality, infinity, insolvability, and chaos, were brought to mainstream attention, ultimately changing the course of technological and scientific development into the 21st century. Before these seminal thinkers, concepts like numerical irrationality and infinity were considered by many to be worthless if not amoral; such attitudes can be found persisting back to the ancient Greeks under the Pythagoreans. Interestingly, the aesthetic of irrationality follows a similar historical trajectory, mostly finding relegation in peripheral movements and specific artists before the 20 th century. However, the 20th century has seen the greatest and longest persisting resurgence in mathematically irrational thought within the arts. This paper compares the visual and musical experiments in irrationality, incommensurability, and infinity in the works of MC Escher and Conlon Nancarrow during the early and mid-twentieth century, showing a correlation between contemporary mathematical and physical innovations and specific aesthetic pursuits in art and music. Key | 1 Cantor, Einstein, Escher, Nancarrow: √ , Irrationality, Infinity, and Imagining the Boundaries of the Impossible When children enter their secondary education, most are thoroughly familiar with P thago as s most famous eponymous theorem.1 While this is a useful and beautiful proof, allowing us to construct objects from the Great Pyramids to Sagrada Família, at its core broods an implication that troubled the Pythagoreans, an implication which, by many accounts, they would closely guard from the world – the existence irrational numbers. This discovery greatly disturbed the Pythagoreans because their philosophy preached that all numbers could be expressed as rational ratios of integers.2 They lived by the dictus that all is number3 and that all universal phenomenon could be explained by harmonious relations between those numbers.4 P thago as s theo e ould e the u de i i g fo e to tea apa t his s hool s e ti e os olog . Gi e sele t e a ples, P thago as s theorem is not cosmologically contradictory. Take for example a right angel with side lengths three and four; by the Pythagorean Theorem we can deduce the length of the hypotenuse is 5. 1 Namely, the square of the side length formed by connecting the two end points of a right angel (the hypotenuse) is equal to the sum of the squares of the side lengths forming that right angel, commonly expressed as a2+b2=c2, with c representing the length of the hypotenuse. It is now well known that this formulation was known by earlier civilizations, such as the Babylonians and Egyptians. See Neugebauer, The exact sciences in antiquity, 36; Teresi, Lost Discoveries, 52; Robson, "Words and Pictures," 105–120. 2 B atio al atio of i tege s, e ea that a u e is o side ed a atio al u e if that number a e e p essed as the atio of t o i tege u e s, hi h i lude all ou ti g u e s, thei egati es, a d ze o. For example, a whole number like 2 can be expressed as 2/1, and thus it is rational. Many decimal numbers can also be considered atio al; fo e a ple, . … a e e p essed as / . E e a de i al e pa sio like 0.00000648871 can be considered rational; its ratio of integer numbers is 4,379/674,864,263. 3 Fo the P thago ea s, u e ea t all those u e s p io to the discovery of irrational numbers, namely the rational numbers – i tege , hole, a d ou ti g u e s … -3, -2, - , , , , … . 4 Gullberg, Mathematics, 84 and 398. Key | 2 = , = + = + = + = = √ =√ = There is no number here which cannot be expressed as either a whole number or the ratio of two other whole numbers. However, let us consider two sides equal to one. = = + = + = + = = √ =√ √ = This cannot be simplified any further. There is no precise square root of two. We could approximate it by decimal expansion as 1.4142, but this is only an approximation. To demonstrate this approximation, let s s ua e . . (multiply the number by itself). . � . While 1.99996164 is close to two, it is not precise. In fact, there is no rational number that we can multiply by itself such that we will get 2. We can prove this through contradiction, by assuming such a rational number does exist. Thus, assume there exists a rational number (as Pythagoras might have wanted) in lowest terms such that = √ . This then implies that (squaring and multiplying both sides by b). Given that have been divisible by 2 since 2 = = = → , we can assume that (dividing both sides by 2). Then, we understand = must as Key | 3 e uali g so e othe i tege ultiplied some �. Now, we may substitute � for � = t o; let s all this u k o i tege �. Thus, back into our original premise, giving us . Note that this is now a contradiction since and = � for �2 2 = → were initially assumed to be in lowest terms; they are no longer in lowest terms since we can divide through by 2 on both sides to simplify the expression. Thus, there can be no such rational such that = √ .5 Such results were known to the Pythagoreans. However, discovery of these incommensurate numbers wrecked their cosmological ontology, leading to a mathematical impasse that would be continually ignored for centuries. One example contemporary to the Pythagoreans is Hippasus of Metapontum (fl. 5th century BC), the Pythagorean philosopher often credited with discovering irrational numbers.6 While accounts are slightly varied, Hippasus, after revealing his discovery, reportedly drowned at sea, apparently a divine punishment for divulging this impiety. If this story holds any truth, it seems reasonable to extrapolate, given no evidence for the existence of Greek pantheons, that Hippasus was executed by his fellow Pythagoreans for challenging their cosmology and undermining their school by circulating such knowledge.7 While the Islamic Middle East expanded our understanding of irrational numbers during the first millennium CE,8 Europe would not seriously encounter such concepts again until the 12th century. However, due to Medieval European Aristotelian and Platonic philosophical hegemony – both of which were heavily influenced by Pythagorean philosophy – irrationality and 5 Thanks is given to my partner, Jason Johnson, for assisting me with this classical Euclidean proof for the irrationality of the square root of two. 6 Iamblichus. The life of Pythagoras, 327. 7 Kline, Mathematical thought, 32. 8 Such as in the work of Persian mathematician, Al-Mahani (d. 874/884), and the Egyptian mathematician, A ū Kā il “hujā i Asla (c. 850 – 930). Key | 4 incommensurability, along with most proof-based mathematics and empirical sciences, were eschewed until the Enlightenment. The 17th century witnessed an explosion of mathematical inquiry. The irrational numbers pi and e Eule s u e o ed to e te stage, despite their clear incommensurability. In the 18th century, mathematicians confronted irrationality with renewed vigor; Leonhard Euler wrote the first proof of e s i atio alit i , and in 1761 Johann Heinrich Lambert proved pi cannot be rational. Subsequently in 1794, Adrien-Marie Legendre provided a proof showing the square of pi is irrational, consequently proving pi is itself irrational. Furthermore, Euler and Abraham de Moivre began theorizing i agi a field of complex numbers. u e s 9 and their combination with the reals, forming the 10 The 19th century saw the greatest proliferation of numerical genera. Liouville established the existence of transcendental numbers 11 in 1844 and 1851. Slightly later in 1873 and 1882 respectively, Charles Hermite and Ferdinand von Lindemann proved e and pi transcendental. The 19th century development of Eule s prior work in complex numbers expanded our understanding of irrationals, dividing them into algebraic12 and transcendental irrationals. Such resurgence in the academic study of irrational and transcendental numbers had not been witnessed since Euclid. The s ua e oot of a egati e u e is a i agi a u e . The e a ot e a eal s ua e oot of negative numbers since any number multiplied by itself will always equal a positive number. Thus, there is no number such that when multiplied by itself equals a negative number. The √− is the most commonly referred to imaginary u e a d is i st u e tal i fo i g hat a e k o as o ple u e s, hi h a e u e ith oth a eal and imaginary part. Complex numbers usually come in the form + �, where and are real numbers and � is the √− . While i agi a , these u e s ha e a real applications in mathematics and science today. 10 See prior footnote. 11 Transcendental numbers are never solutions to nth-degree non-zero polynomials with integer coefficients. 12 Those numbers which are solutions to nth-degree non-zero polynomial equations with integer coefficients. 9 Key | 5 Pivotal during this period, was the controversial work of German mathematician, Georg Cantor.13 Born in 1845, Cantor s o k represented a paradigm shift in mathematics, a coalescence of three centuries of number theory innovation forming his theories of indefinitely large but distinct transfinite numbers.14 His theories and proofs on degrees of infinity profoundly impacted nearly all physical and mathematical discoveries for the next century.15 By 1873 i alge ais he )ahle his se i al pape , Ü e ei e Eige s haft des Inbegriffes aller reellen O a Cha a te isti P ope t of All ‘eal Alge ai Nu proved rational numbers, though infinite, are countable. 16 es Cantor Furthermore, Cantor proved the real numbers (irrational and rational numbers) infinite and uncountable. Lastly, and perhaps most paradoxically, he proved the algebraic numbers are equal in infinite magnitude to integers, but the transcendental numbers, which are a subset of irrationals, are uncountable and Ca to s o k as so o t o e sial that the ift fo ed i athematics would perhaps be greater than that formed by the radical harmonic experimentations of contemporary German composer, Richard Wagner. 14 The concept and contradictions of infinity were not new to Western thought when Cantor began to consider this concept critically. Zeno of Elea (born c. 490 BCE), an ancient Grecian pre-Socratic philosopher brought infinity into Western mathematics first with his various eponymous paradoxes. In India, infinity was even given three genres in the Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE).14 See Stewart, Infinity, 117. 15 Galilei, Dialogues, 31–33. In the 17th Century, Galileo Galilei had also tried to conceptualize the infinite, only to discover his own paradoxes through realizing that one can create comparisons of different infinities. In this, his final scientific work, Galileo made seemingly paradoxical statements about the positive integers. First, he made the true observation that some numbers are squares, while others are not (4 is the square of 2; 5 is not the square of any integer); he then intuited the conclusion that all the integers must be more numerous than just the perfect squares (for example, between the numbers 1 and 20 there are 20 integers and 5 perfect squares: 1, 2, 4, 9, and 16). However, he also recognized a contradiction by viewing the problem from a different angel. He noted that for every perfect square there is exactly one positive number that is its square root. Furthermore, for every integer there is exactly one square; Consequently, one could also conclude that both sets of numbers (integers and perfect squares) are equivalent in magnitude. Galileo also noted that a circle can be understood to comprise an infinite number of points, but when a larger circle is drawn concentrically, we must recognize that this larger circle must have a larger u e of i fi ite poi ts. I the e d, he a a do ed the p o le , hi h is toda k o as Galileo s pa ado , stati g that when one considers infinity, one cannot make comparisons of less, equal, or greater. This problem was o side ed t i ial o o luded u til Ca to s o k i the th century. Come the nineteenth century, Cantor was not satisfied ith Galileo s o lusio . Ca to elie ed that this est i tio is u e essa , a d that it is possible to define meaningful comparisons of infinite sets. 16 He de o st ated this pla i g atio als i a o e-to-o e o espo de e ith atu al u e s the positive integers, excluding zero) 13 Key | 6 consequently more numerous than integers, which must be conceived as infinite. Thus, while the integers and the ratios formed from them are infinite in number, the irrational and transcendental numbers are more numerous and thus a larger infinity. Despite natural human intuition, which had blinded mathematicians and philosophers for more than two-thousand years, Cantor had the insight to prove not only certain sets, like the naturals and the rationals, are equivalent infinities, but also there must be more irrationals than rationals and more transcendentals than irrationals. Unfortunately, Ca to s 1873 paper did not burst into the world with laud and honor. Rather, it was refused for publication o e of the jou al s efe ees a d o e of Ca to s ost vehement adversary on infinity, Leopold Kronecker. In 1874, the paper was finally published after a fello a d s patheti athe ati ia s, Richard Dedekind s, intervention.17 However, the dye as ast; Leopold s sway on the German mathematical community was strong, and his continued de ou e e t of Ca to s o k se ed o l to di i ish the sig ifi a e of this dis o e fo decades to come.18 17 Bruno, Math & mathematicians, 54. Cantor's theory of transfinite numbers was so counter-intuitive, it engendered harsh resistance not only from Kronecker but many peers both within and outside of mathematics. See Dauben, "Georg Cantor and the battle for transfinite set theory," 1. Henri Poincaré referred to Ca to s p oofs a d theo ies as athe ati s "g a e disease, and Kronecker publicly denounced Cantor as a "scientific charlatan," a "renegade" and a "corrupter of youth. Some Christian theologians even saw Cantor's proofs of infinity as an infringement on the domain God and consequently blasphemous. see Dauben, "Georg Cantor and Pope Leo XIII," 86; Dauben, Georg Cantor: his mathematics, a d . E e de ades afte Ca to s death, Aust ia philosophe Lud ig Wittgenstein argued that Ca to s p oofs of i fi it e e " idde th ough a d th ough ith the pe i ious idio s…," dis issi g his o k as "utter nonsense" that is both "laughable" and unquestionably "wrong." See Rodych, "Wittgenstein's Philosophy of Mathematics." 18 Key | 7 Despite myopic resistance, Cantor's work as the first mathematician to critically understand infinity with mathematical precision would outlive him. His theories effe t as only delayed, ultimately impacting paradigm shifting work in the early 20th century. Before the wider acceptance of irrationality and infinity in the 20th century, Western cosmology was understood through a Newtonian lens: the universe was knowable, rational, fixed in all points, and able to be perfectly predicted according to the precise clockwork of its machine. 19th century mathematical work on irrationality and infinity allowed scientists at the beginning of the 20th century to question Newtonian models. The work of physicists like Max Planck, Albert Einstein, and Werner Heisenberg shifted the cosmological model to one of constant flux, a system of probabilities and uncertainties, only knowable by approximations at any moment. I i No a di alspekt u ith his se i al pu li atio , Über das Gesetz der Energieverteilung ‘ega di g the La of E e g Dist i utio s i the No al “pe t u ,19 German physicist Max Planck was one of the first scientists to shake the foundations of modern cosmology. In this paper, he described his solution to the thermodynamics, a groundbreaking result establishing ode la k body p o le in ph si s quantum theory, which would soon challenge all lassi al ph si s firmly held beliefs developed since Newton. Just a few years later in 1905, an obscure 26 your old patient office clerk, Albert Einstein, having just completed his PhD at the University of Zurich, publish a series of four papers between March and September. These papers heralded him onto the world stage as the most innovative man of the last century. In March he solved the puzzle of the photoelectric effect by modeling 19 Planck, "Über das Gesetz," 553-563. Key | 8 energy as exchangeable in discrete quantities, harkening to the prior work of Planck.20 In May, he opened the field of statistical physics and gave observable credibility to atomic physics by demonstrating Brownian motion, a phenomenon first observed by Scottish botanist Robert Brown in 1827, was empirical evidence for the existence of atoms. In June, he reconciled James Clerk Ma ell s field e uatio s fo ele t o-magnetism with the laws of classical mechanics through application of his own near-lightspeed mechanics,21 discrediting the widely held cosmological o ept of the lu i ife ous ethe . 22 In September, he discovered his famous equivalence between matter and energy, E=mc2, a d dedu ed f o this the e iste e of est e e g , the foundations of nuclear physics, and the ability of gravity to warp light.23 Despite its ability to simply explain hitherto unexplainable or difficultly modeled problems, su h as the pe ihelio of Me u , Ei stei s theo ies e e ot et ith o plete a epta e. They seriously challenged the contemporary models, and consequently demolished many eminent ph si ists life o k. The athe ati s ehi d Ei stei s o o se a le e ide e fo his theo ies e odels as sou d, ut the e as little to p edi tio s, espe iall fo his “pe ial Theo of Relativity. It would take another 14 years before his theories would gain unwavering traction. In May 1919 during a solar eclipse, Sir Arthur Eddington confirmed a prediction made by Einstein in 1911, which claimed that light from a distant star traveling to earth in a near tangent line to the “u should e e t the “u s assi e g avitational field. Soon after, these observations were published, making Einstein an international media sensation and a household name around the 20 Das, Lectures on quantum mechanics, 59. Resulting from analysis based on empirical evidence that the speed of light is independent of the motion of the observer. See Major, The quantum beat, 142. 22 See Lindsay, Foundations of physics, 330. 23 This last pape is o k o as Ei stei s “pe ial Theo of ‘elati it . 21 Key | 9 world. The epoch turning event is famously captured in the top contemporary British newspaper, The Times, in hi h the headli e ead, ‘e olutio i “ ie e – New Theory of the Universe – Ne to ia Ideas O e th o (Figure 1)24 Figure 1: Banner from the article Revolution in Science – New Theory of the Universe – Newtonian Ideas Overthrown" in The London Times, November 7, 1919 (page 38) Overnight, the universe had metamorphosed. Not since the migration of the Sun to the center of the solar system had such a paradigm shift occurred. Just as Galileo could no longer seriously accept planets orbiting in loop-de-loops under a geocentric model during the 16th century, scientists of the early 20th century could now safely discard the archaic notion of earth floating through a crystalline solid in space and see it for the absurdity it was.25 Rather than a static universe unaffected by the things within it, the universe was now flexible space, intimately interacting and warping with mass and energy. "‘e olutio i “ ie e, . This postulation was a result of the established Theory of the Luminiferous Ether, which attempted to explain how light, as a wave, traveled through space. Since waves cannot travel without a medium, it was believed that spa e ould ot e a a uu . ‘athe , spa e ust e filled ith a fluid ethe i hi h light ould t a el. Gi e al ulated easu e e ts of light, this fluid ethe ould ha e to ha e ee a solid crystal for the theory to work. 24 25 Key | 10 The capstone in this cosmological revolution came in 1927 with a publication by another young genius, Werner Heisenberg. While in Copenhagen working on the mathematical foundations of the new field of quantum mechanics, Heisenberg produced his paper, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, content of quantum ki e ati s a d o I dete i a O the des ipti e e ha i s outli i g hat ould e o e his U e tai t p i iple. This law asse ts a fu da e tal li it o o e s a ilit to as e tai certain physical properties of a particle with precision.26 For example, the more precisely one k o s the positio of a pa ti le, the less p e isel o e e essa il k o s that pa ti le s momentum. This principle completely overturned the cosmological deterministic model outlined under classical Newtonian physics. Rather, the Heisenberg Uncertainty Principle proposed a cosmology wherein all things are only partially knowable at any point and moment. Perhaps due to the tendency of modern education to segregate systems of knowledge, understanding science isolated from history, art as antithetical to mathematics, and music as only important if one plays in the marching band, we are often unaware of the interplay among these fields a d thei p ofou d i pa t o ea h othe . We lea P thago as s theo e ithout a concept of its controversial implications. We memorize the axioms of Euclid which grew from the work of the Pythagoreans and gain no appreciation for its model of reasoning. These i di idual s influence on thinkers like Socrates, Plato, and Aristotle, and the consequent impact these thinkers had on Western thought27 steered the course of Western history until the early modern age, at which point controversial thinkers like Kepler and Galileo began to question the amalgamated 26 Heisenberg, "Über den anschaulichen Inhalt. P o a l due to A istotle s appoi t e t as the tuto of the the time, Alexander the Great. 27 ost po e ful a i the Weste o ld at Key | 11 paradigm of Platonic philosophy and Augustinian theology of the past two millennia. It would not be until the late 19th and early 20th century that this system would be utterly overturned by the momentum of four centuries of subsequent questioning and reevaluation. One cannot overstate the collective shock of the mathematical and scientific world when this revolution took hold in 1919, perhaps why Einstein remains one of the most widely recognized individuals of all time and the paragon of genius itself. It is not coincidental that concurrently all fields of critical study were experiencing pa adig shifts. The i pa t left Ca to s a d Ei stei s os ologi al ei te p etatio transformed not only the quantitative sciences but also the social sciences and humanities. While aesthetic sentimentality of the 19th century was a chimera of Rococo frill and Neo-Gothic mysticism, the universe like a cluttered Victorian parlor – an opaque knicknackatory filled with arcane ceremony – the 20th century saw a critical decluttering of ritualistic baggage, embodied by radical aesthetic shifts like Bauhaus. The unknown exposed, the transfinite secularized, and mysticism eschewed, infinity and irrationality were legitimate and integral aspects of reality. During this time, strict and purely mathematical arts gained renewed legitimacy, emancipated from the necessity of narrative or Romanticized emotion. Many artistic movements coalesced around mathematical principles.28 The human ontological orbit had widened, and art 28 Some select examples include Geometric Abstractionists Ada K. Dietz (1882–1950) with patterns based on the expansion of multivariate polynomials, Monir Farmanfarmaian, (b. 1924) exploring the infinite in mirror mosaics, and Peter Forakis, (1927–2009) pioneering abstract geometric forms in sculpture; Constructivists John Ernest (1922– 1994) and Anthony Hill (b. 1930) using group theory in self-replicating shapes; Algorists Jacobus "Koos" Verhoeff (b. 1927), using lattice configurations and fractal formations, and Helaman Ferguson (b. 1940) employing integer relation algorithms; and members of the Pattern and Decoration Movement, which included MC Escher (1898-1972) using tessellations and hyperbolic geometry, Max Bill (1908 – 1994) inspired by abstract physics concepts and the Bauhaus movement, Tony Robbin (b. 1943) exploring hyper-dimensional geometry, and Robert Longhurst (b. 1949) exploring objects of minimal and saddle surfaces. Key | 12 could now annex fruitful fields of creativity hitherto denied it. Swiss graphic artist of the early 20th century, Max Bill, encapsulates the fervor with which some artists were trekking into the mathematical unknown. To Bill, mathematics presented space not as Classically rational, or Romantically emotional, or even Gothically spiritual, ut athe as so ethi g i e pli a le. He wrote that art can now explore …the inexplicability of space - space that can stagger us by beginning on one side and ending in a completely changed aspect on the other,… the remoteness or nearness of infinity - infinity which may be found doubling back from the far horizon to present itself to us as immediately at hand; limitations without boundaries; disjunctive and disparate multiplicities constituting coherent and unified entities; identical shapes rendered wholly diverse by the merest inflection; fields of attraction that fluctuate in strength; or, again, the space in all its robust solidity; parallels that intersect; straight lines untroubled by relativity, and ellipses which form straight lines at every point of their curves…. And despite the fact that the basis of this mathematical way of thinking in art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination. 29 Thanks to the present proliferation of computer technology and the precise digital techniques it offers, visualization of complex mathematical phenomena has become not only possible but relatively easy. Half a century before this, however, artists and musicians were still boldly attempting to explore the same phenomena with nothing more than their human capabilities and the relatively crude tools available at the beginning of the 20th century. One of the first artists to venture into this new Relativistic universe, opened by Cantor and Einstein, was Dutch graphic artist, Maurits Cornelis Escher, born in 1898. As a trained graphic artist working mostly under commercial employment, making images for postage stamps and wrapping paper as well as a few portraits, landscapes, and architectural sketches, Es he s o k as al ost 29 Bill, Max. Die mathematische Denkweise. Key | 13 completely unknown or disregarded by the artistic community until the 1950s. Beginning in the 1960s, however, Escher work gained traction among an unexpected collective: physicists and mathematicians.30 This came after his initial and stylistically innovative work in impossible spaces, tessellations, and continuous transformations from the early 1940s into the 1950s. E hoi g Ma Bill s des iptio of the e a d i e pli a le a t i , MC Escher described, in retrospect in 1959, the transformation of his aesthetic aims during the middle years of his artistic maturity in the 1930s and 1940s. He wrote: I discovered that technical mastery was no longer my sole aim, for I was seized by another desire, the existence of which I had never suspected. Ideas took hold of me quite unrelated to graphic art, notions which so fascinated me that I felt driven to communicate them to others.31 Es he does ot e plai e a tl hat these ideas that took hold of hi deduce from his subsequent work on what would later be te i possi le o je ts, as e e, ut e a ed i possi le spa es a d ell as his app o i atio s of i fi it , that his new desire in the 1930s i te se ted Ma Bill s sa e i e p essi le art. The nearly four-meter long woodcut print Metamorphosis II, created between November 1939 and March 1940 serves as the figurative and literal paradigm of Es he s aestheti transformation (Figure 2). In this work we see a manipulation of space, mirroring pe fe tl Bill s o eptio of spa e that can stagge us side a d e di g i a o pletel a which 31 i go o e ha ged aspe t o the othe , as well as expressing an infinity e fou d dou li g a k f o the fa ho izo to p ese t itself to us as immediately at ha d. While Es he takes his ie e o a t a sfo 30 egi Emmer, "Mathematics and Art, Escher, Escher on Escher, 5. . atio al jou e f o the i te se tio of the Key | 14 Os i a ious otatio s of Meta o phosis a d the ei te p etatio of M as E th ough those rotations, we find that at the end of this journey the infinite variety that is has been opened to our mind nevertheless leads us back to where we started. Figure 2: MC Es he s Metamorphosis II, originally one long panel, presented here as three in succession. The fi al pa t of Es he s last le ture, never given but published in 1989,32 was dedicated to Metamorphosis II. In this lecture, he described this piece in his own words: It is a pi ture stor o sisti g of a su essi e stages of tra sfor atio s…. If a comparison with music is allowed, one might say that, up to this point, the melody was written in two-quarter measure. Now the rhythm changes: bluish elements are added to the white and black, and it turns into a three-quarter measure. He ends his description by describing the entire surfa e of the patte . 33 This ill e o e a o k as filled ea i gful o pa iso ; e ill etu ith a h th i to Es he s aestheti and it associations to rhythmic transformation in the work of contemporary, Conlon Nancarrow. 32 33 Ibid., 48. Emmer, M.C. Es her’s lega , 146. Key | 15 In 1953, Escher completed his most recognizable work today, Relativity (Figure 3), likely i spi ed the popula izatio of Ei stei s o k o ‘elati it a d g a it du i g the sa d 1940s. In Es he s lithog aph, gravity becomes a relative phenomenon, with three equally valid orthogonal sources of gravity distributed among sixteen figures. In the following year, NG de Bruin, e og izi g Es he s s ie tifi a d athe ati al pe ha t, selected Es he s o k for exhibition at the Stedelijk Museum for the 1954 International Congress of Mathematics in Amsterdam. Two meetings of art and science would occur at this exhibition, propelling Es he s t a sfo atio al art into its most mature stage of metamorphosis: the creation of impossible objects and infinite spaces. Figure 3: MC Es he s Relativity (1953) Key | 16 At this conference were the young physicist Rodger Penrose34 and his engineer father, Lionel Penrose. Upon visiting the art exhibition of the mostly unknown MC Escher, both Penroses e e i t igued Es he s use of eal o je ts within Relativity s impossible space. Inspired, the Penroses endeavored to produce their own impossibilities, though rather than create impossible spaces with possible objects, they aimed to create impossible objects within possible spaces.35 Meanwhile a few years later in 1957, mathematician HSM Coxeter, who had also been at the 1954 exhibition, contacted Escher to ask permission to use i his pape , Crystal symmetry and its generalizations," some of Es he s tessellations made in the 1930s and 1940s.36 Later that year, after publication, Coxeter sent Escher the article. What intrigued Escher most about this article, though he testified that he himself could not understand most of it, was not Coxeter s work on crystal symmetries demonstrated through Euclidian tessellatio s, ut Co ete s figu es of hyperbolic tessellations, where the repeated tiles, rather than continuing in a regular nontransformative fashion into Euclidean infinity, rapidly grew smaller and smaller towards an infinitesimal space at the edge of a circle. (See Figure 4) This image would stick with Escher for the next decade as he endeavored to visually capture infinity. 34 Roger Penrose is now a world-famous physicist and mathematicians, perhaps most famous for his work with Stephen Hawking on black holes, for which we won the prestigious Wolf Prize in 1988. 35 See The Art of the Impossible: MC Escher and Me. Directed by Clem Hitchcock. 36 Schattschneider, "The Mathematical Side of M. C. Escher," 706–718. Key | 17 Figure 4: H pe oli tili g f o H“M Co ete s pape of stal s et . 37 In the next year, 1958, the Penroses co-authored and published their results on impossible objects in the British Journal of Psychology.38 In this publication, Rodger Penrose demonstrated his Pe ose T ia gle a d Lio el his E dless “tai ase. (See Figure 5) I e og itio of Es he s inspirational impossible space, Rodger sent the article with their sketches to Escher, who soon after in 1960 and 1961 sent Roger Penrose the lithographs Ascending and Descending and Waterfall, the fi st of hi h as ased o the E dless “tai ase a d the se o d o the Pe ose T ia gle. 39 (See Figures 6 and 7) Escher, already on the track to find infinity, was reciprocally inspired by the Penroses impossible objects as designs that could express perpetuality, and consequently infinity, under concrete terms. Co ete , C stal s et a d its ge e alizatio s, –13. Penrose, "Impossible Objects," 31-33. 39 Schattschneider, "The Mathe ati al “ide of M. C. Es he , 37 38 –718. Key | 18 Figure 5: The Penrose staircase and Triangle, two impossible objects.40 Figure 6: MC Es he s Ascending and Descending (1960), based on the Penrose Staircase. 40 Penrose, "Impossible Objects," 31-33. Figure 7: MC Es he s Waterfall (1961), based on the Penrose Triangle. Key | 19 Though Es he had fou d the fou tai head fo the est of his life s o k the late s, he recognized a frustration with his developed aesthetic. In a letter to his son, George, in the 1950s, Escher elaborates: But the sad and frustrating fact remains that these da s I’ starti g to speak a la guage which is understood by very few people. It makes me feel increasingly lonely. After all, I no longer belong anywhere.41 The emotionless, perhaps cold and mathematical language with which Escher was now speaking isolated him from broader audiences.42 Of course, Escher was not alone in this experience, though he might have not known it. Not only can we see his misunderstood isolation in light of Georg Cantor s athe ati al e plo atio s i the late 19th century, but we can also find many parallels in mathematical explorations within music from this time. Earlier in the 20th century, European dodecaphonic and serial composers, such as Arnold Schoenberg, Alban Berg, and Anton Webern, confronted unpopularity and even persecution for their innovative, more-or-less mathematically oriented art. A fe “e o d Vie de ades afte “ hoe e gs ese “ hool, Wolfgang Steinecke initiated the Darmstadt International Summer Courses for New Music in 1946, at that time the greatest annual collective of mathematically oriented composers. Members of this school, central among them Luigi Nono, Pierre Boulez, Bruno Maderna, and Karlheinz Stockhausen,43 found great collective comradery, but their music 41 Escher, M.C. Escher, his life and complete graphic work, 93. This obviously would not persist. 43 But also including many other now well-known composers such as Earl Brown, John Cage, and Luciano Berio as well as Iannis Xenakis and Olivier Messiaen (mostly through their influence on Darmstadt composer, since neither Messiaen nor Xenakis even attended Darmstadt). 42 Key | 20 was in many circles poorly received, especially those circles oriented towards a more traditional harmonic and rhythmic regimen. In the United States, Charles Ives had been struggling since the end of the 19th century with artistic dejection for his experimentations pursuing rhythmically and harmonically incommensurate music. Following Ives, composers like Edgar Varese, Henry Cowell, and Ruth C a fo d “eege eated the flo e i g of the A e i a Ult a-Mode ist o e e ti usi , and just few decades later, Milton Babbitt would take mathematically structured music to new extremes with his own version of total-serialism. Despite a wider lack of appeal, Modernism, Post-Modernism, and Ultra-Modernism were well established in the academic musical community both in Europe and the United States by the 1950s. Consequently, many composers who found this music compelling sought space to experiment in academia without the financial necessity to write music for popular demand; many became university professors and formed their own studios and schools. There were, however, a few composers who found themselves, like Escher, left out of the established fold. One such composer, born in Arkansas in 1912 during this upheaval in science, art, and music, was Conlon Nancarrow. Starting at the age of 21, Nancarrow received his first formal composition training with Nicolas Slonimsky, Walter Piston, and Roger Sessions in Boston between the years of 1933 and 1936. During this time, Nancarrow began to develop his own unique desire for musical experimentation in the domain of time and rhythm.44 The extreme technical difficulty posed by his music in the early 1930s resulted in only a few satisfactory performances prior to leaving the United States to fight in the Spanish Civil War. 44 Nancarrow, "Tempus perfectum," 266. Key | 21 Pe fo e s f ust atio s with his music no Na a o s i te est i o ple h th i configurations did not wane by his return from the war in 1939. In that same year, with a pressing eed to fi d ade uate pe fo a es of his usi , Na a o dis o e ed He Co ell s t eatise New Musical Resources, which not only contained many rhythmic ideas reinforcing his own but also ideas he had yet to explore. In his book, Cowell attempted to reinterpret the paradigm of musical time, understating rhythm not merely as a temporal scaffold but as a harmonic medium. Cowell recognized pitches were simply fast periodic impulses45 and from this extrapolated the possibility of understanding harmonic ratios between frequencies as rhythmic ratios.46 This allowed him to express harmony and harmonic progression in not only the pitch domain but also the rhythmic.47 Nancarrow realized the implications for Co ell s e pa adig of h th were perhaps limitless, but the same problem still confronted him: if his music was already too difficult to perform, implementing such rhythmic structures would only make it more so. Conveniently, Cowell recognized this difficultly and offered this offhand suggestion, which would form the primary concern of Na a o s life o k: Some of the rhythms developed through the present acoustical investigation could not be played by any living performer; but these highly engrossing rhythmical complexities could easily be cut on a player-piano roll. This would give a real reason for writing music specially for player-piano, such as music written for it at present does not seem to have, because almost any of it could be played instead by two good pianists at the keyboard.48 45 For example, 440 Hertz is simple 440 beats per second. This A pitch could be transposed down various octaves (220Hz, 110Hz, 55HZ, 27.5Hz, 13.75Hz) to 13.75Hz, at which point the pitch association of the beats would be lost totally to the rhythmic domain. 46 2:1 for an octave, 3:2 for a perfect fifth, 4:3 for a perfect fourth, 5:4 for a major third, and so on. 47 For example, one could rhythmically express a major chord (ratios 6:5:4) with three musical strata moving at tempos 120, 100 and 80 beats per minute. 48 Cowell, New Musical Resources, 108. Key | 22 When in 1940 the United States government denied renewal of Na a o s passpo t due to his prior involvement in the Communist Party, he immigrated to Mexico and, finding the musical climate there equally unfavorable towards his musical experimentations with rhythm, undertook creating music for the player piano, superimposing tempi in wholly polyrhythmic pieces. The piano rolls and the clockwork mechanism of the instrument ultimately gave Nancarrow more temporal control over music than had ever previously been possible.49 Composed primarily from the 1950s through 80s – the sa e ti e as Es he e plo ed his o i possi le a t – Nancarro s innovative Studies for Player Piano are works of incredible rhythmic complexity whose hallmark is the te po al o fli t a o g si ulta eous la e s of usi , hi h he alled Temporal Dissonance. His first study for player piano was punched between 1949 and 1950. Excited about the new possibilities now open to him, this piece had more than two hundred tempo changes. Over the next fifteen years, between 1951 and 1965, Nancarrow wrote the remainder of his first thirty studies for player piano, all of which explored ever complexifying rational relationships between musical strata.50 A pivotal turn in his explorations occurred after his 20th study, when Nancarrow While Na a o s e pe i e tatio does e eed thei s, the a o plish e ts of o pose s of the th century Ars Antiqua and Ars Nova periods, particularly those involved in the Ars Subtilior, are perhaps the only Western music co pa a le to Na a o s i h th i e pe i e tatio u til the th century. Composers of this period defined tempo through an elaborate temporal notation representing mensural relationships among the established note values and their divisions. The system and its rhythmic possibilities were so well exploited that the temporal complexities of this notation eventually exceeded the bounds of practical musical performance, still challenging performers and transcribers even today. Poignant examples of this rhythmic complexity include Johannes Ci o ia s (c. 1370 - 1412) Le ray au solely, a th ee oi e p olatio a o i a atio of : : , a d )a a a da Te a o s . 1350 – c. 1416) Sumite Karissimi. Such subtle rhythmic practices fell out of favor after the late 14 th and early 15th century, but we can still find some isolated examples of the aesthetic lingering in the late 15th and early 16th century. “o e e a ples i lude i o e su ate h th i ta ti i Ale a de Ag i ola s / – Ag us Dei III f o his Missa in Myn Zyn and Joha es Mitt e s d. . Osa a f o his Missa Hercules Dux Ferrarae. Such rhythmic complexities as these pieces demonstrate would no be seen again until the work of Charles Ives at the end of the 19th century, and would not proliferate until the mid-20th century. 50 Study 14 is a canon with ratio 4/5, Studies 15 and 18 are canons with ratio 3/4, Studies 17 and 19 are threepart canons with ratios 12/15/20, Study 22 is a canon with ratios of 1%/1½%/2¼%, Study 24 is a canon in ratios 14/15/16, and Study 27 is a canon in ratio 5/6/8/11. 49 Key | 23 made some custom improvements on his play piano roll hole-punching mechanism, modifying it from a ratcheted device, only allowing fixed divisions of metric units, to one with continuous movement, allowing for fluid and even more precise metric transformations. 51 While it would take some time for Nancarrow to realize the full implications of this modification, he did realize that he could explore ever increasingly complex tempo proportions. The fi st pie e o posed u de this e f eedo as Na a o s Ca o X, or Study No. 21, which explored continuous and gradual accelerandos and decelerando. By Study No. 31 Nancarrow had increased his ratio complexity to 21/24/25. However, it ould t e u til his Study No. 33, composed sometime in his late 50s in the 1960s, that Nancarrow would realize the full temporal potential of his new mechanized marvel. As Nancarrow reached his late 50s, he become more and more committed to his conviction that ti e is the last frontier of music. 52 By using his mechanical musician, Nancarrow emancipated himself from the concerns of performability, allowing himself to exploit the possibilities of musical tempo within polyphonic forms, creating a flexible, fluctuating musical space and time. However, due to his method, his ediu s usi s as h o ous h th , his o positio s formulaic e ha ized ha a te , a d his i st u e t s i o ilit a d scarcity, his music was inaccessible to most. Nancarrow, for much of his career, could only play his music in his studio unless he laboriously moved his player piano to a concert venue; he was far too unrenowned for most of his career for any venue to provide him a player piano. Nancarrow was also reticent to play his music for anyone but himself, only demonstrating his experiments when 51 Drott, "Conlon Nancarrow and the technological sublime," 542. Nancarrow, quoted in Garland, Americas, 185. See also Tho as, "Na a o 's Te po al Disso a e : Issues of Te po P opo tio s, 7. 52 Key | 24 he felt one was truly interested.53 His compositions lacked most of the qualities one would associate with traditional usi : melodiousness, harmoniousness, regularity, and emotion. Nancarrow was too interested in the sonic implications of his mathematical experiments to be concerned with emotion. Kyle Gann, perhaps the greatest Nancarrow scholar and promoter of his o ks, e alls Na a o s opi io o usi s ability to express emotion, Gann writes: … Nancarrow was one of a trio of composers, along with John Cage and Milton Babbitt, ho did ’t elie e i usi ’s a ilit to e press e otio . I re e er Conlon saying that to him music was just an interesting pattern of sounds with no emotive connotations.54 Perhaps for these reasons, Na a o s usi e t ostl u e og ized fo ost of his life, despite the extreme precision of his craft and high quality of his work. Like Cantor before him, and like Escher contemporaneously, Na a o s e pe i e tatio s i e pli a le e a t, elegated hi to the pe iphe u til his usi ithi Ma Bill s ould e ade uatel understood and appreciated in the late 20th and early 21st century. Together, Nancarrow and Escher represent the early artistic response to the revolutionary mathematical and scientific zeitgeist of the early 20th century. Both create fantastical worlds, one in light the other in sound, similar to our own, but both continually plumb the boundaries of the possi le ithi thei ediu . Es he s Relativity shows us a space wherein one is not beholden to the typical notions of gravity; the scene pulls one in three possible directions, neither more tha primar the othe . Na a o s disto tio of ti e th ough pol te po a d te po transformation provides us with a similar relativistic gravitation, each tempo independently pulling us within its own linear system, none necessarily the true center of the piece s o it. Ti e fo ea h 53 54 Koonce, "Interview." Gann, Outside the Feed a k Loop, 3. Key | 25 voice within a Nancarrow cannon is a relative experience; though the melodic material might be the same, the experience of that musical structure in time is different for each melodic observer. Ei stei s Theories of Relativity proposed in 1905 and 1915, which describe the warping of space and time and the relativistic perception of time, artistically manifest in the works of MC Escher and Conlon Nancarrow: Es he s p i a o e i a t as the e plo atio of spa e, its limits and warping capabilities, hile Na a o s as the sa e ith ti e. The t a sfo atio of time and space, approaching the bounds of infinity and irrationality, ultimately imagining that which was essentially impossible under a Newtonian Cosmology, is not only the prerogative of scientific thought at the beginning of the 20th century, but also the arts, embodied most poignantly at an early stage in the works of Escher and Nancarrow. Alistair Riddell was correct when he observed that the e s a feeli g that ti e is being a ipulated i a e ti el diffe e t a e i Na a o s usi .55 Perhaps because Western music had for so long existed under the limitations of Pythagorean and Newtonian cosmologies, usi like Co lo Na a o s had e e ee app oa hed so fervently before. When the universe is a rationally driven clock made by a rational creator, things within that universe should appropriately reflect that divine and predictable machine. However, when the universe is shown to be unpredictable and unknowable on human terms, and discovery after discovery seems to chase any rational creator further and further into the gaps of understanding,56 so far in fact that it seems rational, if not necessary, to question whether such a rational creator might even exist, the things within that universe take on a whole new, if not infinite, set of possibilities. 55 56 Na a o , "Te pus pe fe tu , 269. Popula l alled the God of the gaps. Key | 26 Composer Roger Reynolds once remarked that "it doesn't seem possible that art like [Co lo Na a o s] could exist."57 Under past paradigms, Reynolds was correct; Na a o s music, in many ways, was impossible before the 20th century, though mechanized technology did exist such that it could have physically been done.58 What was not present, however, was the cultural climate enabling such questions to be asked and their answers sought. Composers at the beginning of the 20th century, like Nancarrow, could avail themselves of such a climate. The four primary protagonists59 of this ensuing narrative – Cantor, Einstein, Escher, Nancarrow – have been herein equated not in some effort to assert trite deification of an artist o o pose a o e thei o te po a ies th ough asso iatio ith the pa ago of ge ius. 60 What is intriguing in recognizing the shared brotherhood among these four unacquainted individuals is unlocking a broader, and consequently deeper, context, into which we may place this e i e p essi le a t e e gi g i ta de ith a e olutio a a u de sta d Es he s a t i light of Na a o s of Es he s desig s. Fu the o e, e os olog . Co se ue tl , usi , a d Na a o s e usi u de the lens a u de sta d oth – their histories and resulting oeuvre – through the seemingly unassociated fields of mathematics and physics by way of their related goals: to more deeply understand a e a d i e pli a le space and time. Let us now, in light of all that has come heretofore, return to the Pythagoreans and Hippasus over two thousand years ago and recall the cosmological quandary of √ , perhaps the 57 Nancarrow, Conlon Nancarrow: Virtuoso of the Player Piano. The player piano was an outdated form of technolog du i g Na a o s life. I fa t, it had ee pate ted in 1863 by Fourneaux, though technologies like the player piano predate this machine for centuries, such as music boxes and automated carillons and pipe organs. 59 The four antagonists perhaps being Pythagoras, Leopold Kronecker, the artistic community unwilling to recognize Escher – a graphic artist – as a t ue a tist, a d all those u illi g to pe fo o liste to Na a o s usi . 60 Einstein probably never thought that highly of himself, though the analogy has persisted. 58 Key | 27 first number to be proved irrational.61 Coincidentally, it is this incommensurable value that Nancarrow first explores in his late 50s, when he began composing his self-declared most significant,62 though not most popular, work, which was the first to explore irrationality in music. This piece, more than any that had come before, would stretch the limits of the new i e p essi le a t. In his Study No. 33 for player piano Nancarrow created various tempo canons using the ratio 2:√ . Ma aestheti a d o eptual aspe ts of this i e p essi le th ough o pa iso ith Es he s si ila e pe i e tatio s usi a e u de stood ith o ti uous t a sfo atio , infinity, and incommensurability in graphic design during the same period. This will be de o st ated th ough Na a o s Study No. 33 for Player Piano. The first a d fo e ost i pli atio of Na a o s e use of i atio al atios is that the e will never be a metric convergence point in the music if precisely realized. Since, as proven earlier, there is no rational number such that √ can be expressed as some ratio of integer numbers, √ cannot be perfectly divided by 2 and 2 cannot be perfectly divided by √ . Consequently, if we have two metric systems progressing under such an irrational relationship, they will continuously move out of phase from their outset.63 Such a metric complex maximizes what Nancarrow calls te po al disso a e. Ha ke i g a k to his studies of He Co ell s New Musical Resources in his youth, Nancarrow uses this term to describe, in terms of harmony, the relationship between independent 61 Heath, A history of Greek mathematics, 155. Nancarrow, "Terraced Dynamics." 63 This is assuming that they start at some offset not the inverse of their given tempo relationship and that there will never be any alteration in their temporal relationship. 62 Key | 28 contrapuntal lines progressing in different tempi.64 The more extreme the difference, the more the relationship between the lines is dissonant. Nancarrow considered most of his early works before Study No. 33 to be relatively consonant. While perhaps perceptually unusual for people conditioned to a musical system under the hegemony of the dyadic rational, two voices at a ratio of 4:5 are not dissonant, only representing a harmonic major third. However, voices in an irrational relationship are perhaps as dissonant as they can be,65 since they will never have a common denominator.66 Temporal common denominators form a particularly significant structural function in u h of Na a o s oeu e. K le Ga o o de o i ato s alls the esulta t o e ge e poi ts. disso a e, Na a o s o ti uall t a sfo 67 et i oi ide es eated the Under the paradigm of consonance and i g et i elatio ships i his te po a d prolation canons create progressive levels of dissonance with movement towards and arrival on o e ge e poi ts ep ese ti g to al etu a d ade e. As Margaret Thomas eloquently explains, "the momentum of many of the studies may be best understood as a progression toward the resolution of their temporal dissonance via a process of convergence to a simultaneity."68 Consequently, the closer the canonic voices are to their convergence point, the more clearly one can hear their imitative relationship. Equally, as the voices approach their maximal distance from the convergence point, their imitative relationship moves into maximal obscurity. Tho as des i ed this as the p og essio of the disso a e Na a o , "Te pus pe fe tu , . Na a o , "Te pus pe fe tu , . 66 Co elati g to ha o i p i ipal of the fu da e tal to e. 67 Drott, "Conlon Nancarrow and the technological sublime," 540. 68 Thomas, Conlon Nancarrow's, 4. 69 Ibid., 137. 64 65 69 Such structural demarcations Key | 29 e o e pa ti ula l sig ifi a t u de the i atio alit of Na a o s late studies, he ei the e might be only one such convergence point, if there is any at all. For Nancarrow to fabricate such arrival points, he must rely on arch forms in many of his irrational canons, wherein he swaps the temporal relationship at some point in the piece, so that, by a retrograde process, they return to their outset tempo and temporally converge as they began.70 Thus, in his Study No. 33 Nancarrow does not simply initiate the two canonic tempi and let them continue their de-phase ad infinitum. Rather, Nancarrow creates a series of five concatenated arch forms, wherein the voices are either initiated at some time interval such that the canon ends at the point of convergence or directed to invert their te po al atio at the a o s midpoint. Consequently, by not minimizing the number of possible convergence points to one or none and constructing points of convergence through various transformational methods, Nancarrow presents a greater level of comprehensibility than would otherwise be possible. Interestingly however, Nancarrow evades every point of convergence in his Study No. 33, arriving at the point without any sounding convergence; instead of a convergence, he begins a new canonic section. Thus, what was the point of convergence is now a new point of transformation; as much as it might be an arrival, it is also a departure. Given Na a o s Study No. 33, let us etu to Es he s Metamorphosis II, the transformations in which he coincidentally compared to metric transformations. This woodcut print functions well as a visual metaphor for the process of te po al disso a e a d 70 This should not be confused with melodic retrograde. While Nancarrow might swap the temporal relationship between the voices to that they follow their phasing process backwards to their original metric coincidence, the other musical parameters need not, and usually do not, change. Key | 30 o e ge e poi ts i Na a o s rhythmically transformational tempo canons, particularly Study No. 33. As both Gann and Thomas have demonstrated, due to the progressive transformation of imitative structure between voices due to their differing rates of unfolding, the degree to which we perceive this imitation changes throughout the piece. Take for example the third canon in Study No. 33, a notated approximated of which appears below (see Figure 8). Figure 8: App o i ati g e ditio of thi d a o f o Paul Usher, who approximates 2:√ as 7:5.71 Na a o s Study No. 33 for Player Piano. Transcription by Here, the point of imitation begins almost simultaneously, but it expands as the piece progresses. At the midpoint of the section (no shown above), due to Na a o s inversion of the tempo ratio, the point of imitation begins to contract, ultimately returning to the convergence point at the end of the section. He e, Tho as s p og essio of the disso a e is Calle de , Clifto . "Pe fo i g the I atio al: Paul Ushe s a a ge e t of Na a o s “tud No. :√ ." Conlon Nancarrow, Life and Music: Online Symposium September 27 - October 27, 2012. 71 Ca o a ifested as , Key | 31 the the point of imitation distally migrates, increasing the difficultly in perceiving canonic relationships between the voices. As e p og ess to a ds the a i al te po al disso a e e move from relative canonic simplicity to complexity. However, there is no clear point of division signaling the shift from this textural simplicity to complexity, but such a shift does take place. This shift is significant because it compels a transformation in the listener's orientation toward the work, metamorphosing from a perception of canon to free polyphony and ultimately back again. We see this sa e p o ess i Es he s Metamorphosis II, again shown below. In this work the e a e lea o e ge e poi ts as ell. Fu the pe iods of disso a t t a sfo o e, these convergence points lie between atio . At o e i sta t e see a hess oa d a d the liza ds, ut what of the between space? As we move from the chessboard to the lizards we cannot be sure what will be the result of the transformational process; we must ultimately give ourselves up to the Es he s moment of cognitive dissonance to understand not only the final convergence, but also the metamorphic revelation that is exercised when a correlation is suddenly made between a chessboard and lizards. Figure 2: MC Es he s Metamorphosis II. Key | 32 The same is true as we move from the life cycle of bees to fish. Between we might guess a transformation to black butterflies or hummingbirds, but no. The black moves from the foreground to the background and the background, once negative space upon which the bees lived, focuses i to fish. Es he s Metamorphosis II, like Na a o s te po a o s, p ese ts us ith concatenations of increasing cognitive dissonance sandwiched by convergence points of clarity, wherein we are presented with wholly transformed and seemingly unrelated ideas. However, knowing what has come before, and having traversed the fields of dissonance, we see the clever and insightful game of relations, with which Escher is playing. Consequently, we now have a better apprehension of the transformational possibilities of space, just as we might have the same app ehe sio ith ti e i Na a o s usi . Both Na a o s a d Es he s t a sfo atio al p o esses i t igui gl efle t Heisenberg s Uncertainty Principle.72 At every moment within this piece, we are never in a fixed point of reference. Rather, we are set in a world in constant fluctuation; as soon as we identify the point of relative imitation between the canonic voices in Na a o s a o , the dista e ha ges due to the continual de-phasing processing. While we might know the temporal relationship between both voices at any moment, we cannot easily say what the exact point of imitation is. Conversely, if we simply identify the point of imitation at any moment within the piece, we cannot necessarily know the original temporal relationship that created this singular point. I Es he s Metamorphosis, at every point we can identify the shapes we see, but we can only postulate (without hindsight or foreknowledge) what these shapes grew from or will transform into. Conversely, if we know the convergence points of each transformation the shapes 72 Refe to page , if ou eed a e i de of the asi p e ise of Heise e g s U e tai t P i iple. Key | 33 at every moment lose their individuality and either are subsumed into the preceding or proceeding o e ge e. I oth Na a o a d Es he , as i Heise e g s ua tu o ld, e a o l ie these works under one lens at a time, and as soon as we have chosen one lens, we lose the ability to see through the other in that instance. As we can see the implications of Heise e g s uncertain cosmology in Escher and Nancarrow, we can also trace the indirect influence of Georg Cantor through Es he s a d Na a o s explorations of the infinite and infinitesimal. Es he s o k e plo es oth the oncepts of bounded and unbounded infinities. Bounded infinities express some convergence on a point, line, or region of space that extends towards infinity. Unbounded infinities encompass a limitless region, localizing on no point, line, or space. These are demonstrated in Figures 9 and 10. Figure 9: Example of a convergent series, while this series continues into infinity, it is clear, it is collapsing collapsing to some infinitesimal point (image in public domain). Figure 10: Example of parabolic function, the graph of which grows into an indefinite infinity. As the value of x increases, the value of y increases indefinitely. Key | 34 Escher explores unbounded infinities in both two-dimensional and three-dimensional spa e. All of Es he s tessellatio s are examples of a Euclidean space with an implied extension i to i fi it at the o k s ou da ies. Gi e the o ti uous patte n of the tessellations, once one ea hes the ou da ies of Es he s p i ts, it a e assu ed that the patte , if ot arbitrarily halted, will continue indefinitely in two-di e sio al spa e see Figu e . Es he s Cubic Space Division (1953) is an example of unbounded three-dimensional space. Here, Escher uses a fading effect to halt our further observation of the infinite expansion, since he ultimately cannot draw the implied infinite space, though we can assume that such an imagined infinity it is intended (see Figure 12). Figure 11: Es he s Pegasus (No. 105) tessellation (1959). Figure 12: Es he s Cubic Space Division (1952) After his correspondence with HSM Coxeter, Escher began his exploration of tessellations under bounded infinities with non-Euclidean spaces. By 1956, Escher has fully developed infinitesimal convergence at a point in Smaller and Smaller (Figure 13). However, by 1960, Escher had mastered Co ete s h pe oli spa e i his Circle Limit series (Figure 14). In the mid-1960s, he Key | 35 extended the hyperbolic design to incorporate line limits in his Square Limit (Figure 15), thus mastering bounded infinities on points, lines, and polygonal spaces. Figure 13: Smaller & Smaller (1956) While a Figure 14: Circle Limit III (1959) Figure 15: Square Limit (1964) of Na a o s pie es are pre-designed such that the various canonic voices will at some point converge, all his i atio al canons use ratios that, by their nature, can never converge, only doing so under arbitrary transformations.73 Ho e e , Na a o s o e ge e points and his approaches towards them do represent an approximation of these bounded infinities. Just as Escher could not actually illustrate the infinitesimal or boundless infinity, and had to terminate his process at some arbitrary point, leaving his viewer to inwardly continue the process in abstraction, Nancarrow chooses terminations in his processes, his music ultimately only suggesting an infinite expansion or contraction, which is ultimately impossible to full express.74 Again, take for example the beginning of the third canon from Nanca o s Study No. 33. The egi 73 i g of the se tio i Na a o s o igi al otatio is gi e elo Figu e . He e, e Such as the arch-form process previously outlined. As hu a s a d a hi es, e a e ulti atel li ited i ou a ilit to full e p ess i fi it s all o large), a d thus these a e app oa hes to a ds a li it at i fi it . 74 Key | 36 see that Nancarrow does, in fact, not begin the voices precisely together. Since he does not begin the voice coincidentally, there is no point, even under his arch forms, at which the voices will truly coincide. The voices can either expand apart infinitely or contract to an infinitesimal imitative distance. Since his player piano is naturally limited in the micro precisions that it can make, there is a limit to which Nancarrow can approach such an infinitesimal difference. Consequently, when we approach the convergence point at the end of this section, we can hear the collapsing imitative interval, but in abstraction, if this process were never halted, we would never reach an actual o e ge e poi t. O e the oi es get lose e ough, o as lose as the a get, Na a o stops the process and begins a new section (see Figure 17). Figure 16: The opening of the third canon i Na a o s Study No. 33 for Player Piano Figure 17: The closing of the third canon in Na a o s Study No. 33 for Player Piano Just as Escher uses points and regions of convergence, which are approached gradually to imply movement towards infinitesimality, so too Nancarrow uses very close ratios to simulate a gradual approach to infinitesimally close imitative relationships. Key | 37 Na a o s a d Es he s approaches to infinity are aptly mirrored in mathematical concepts involved in expressing irrational numbers as infinitely expanding and infinitesimally converging continued fractions. First let us consider how we might endeavors to express the √ by some other conceivable rational means. We have shown how it is impossible to represent it as a rational number and, consequently impossible to fully express it as a decimal expansion. There is, however, another, quite Escherian, way by which we can express an irrational number – a si ple continued fraction. A simple continued fraction for a positive number is defined as a sequence of numbers [ ; , , ….] = + + + where b is part of the number greater than one and each the number. The ��ℎ iterate is evaluated by considering substituting this into � − �ℎ +⋯ are fractional iterates converging to + � and then recursively back iterate. This process requires n steps and clearly terminates for any finite �. It can be shown that the simple continued fraction converges geometrically while a decimal expansion converges only additively thus the nth iterate of a continued fraction is, in general, much closer to the number it represents than the decimal expansion taken to the ��ℎ position. Rational numbers also always have a finite simple continued fraction. To demonstrate, we can approximate √ as a continued fraction: Let us begin with the most basic approximation of √ . Key | 38 √ ≅ This is obviously a gross approximation, but we can refine it with a fractional component. √ ≅ + = + . = . We are getting closer; what if we add a fractional component to our previous fractional component? √ ≅ + + = Now, in a recursive process we will continue adding = . to the smallest part of our simple continued fraction. √ ≅ √ ≅ √ ≅ + + + + + + + + + + = + = . = = = . = . … … … + This process can continue forever, allowing us to approach √ as closely as we deem necessary. As we continually add smaller and small parts to our continued fraction, approaching some Key | 39 infinitesimal limit of convergence, we can express, in a repeating pattern, what as a decimal expansion lacks periodicity. √ = + + + + + +⋱ Like Es he s Circle Limit III or Square Limit approach the shape of a circle or square at infinity, but conceptually never reach that limiting shape, we can express the infinitesimal irrationality of √ through this ever-shrinking iterative process, approaching this number by predictable pattern but never fully arriving at our point of convergence. In practical use, we must at some point, as both Escher and Nancarrow do, arbitrarily chose our terminating point based on our needs or computational limits. Escher himself addressed the issue of visualizing objects or concepts which are wholly i e pli a le. Es he ote, The esult of the st uggle et ee the thought a d the a ilit to express it, between dream and reality, is seldom more than a o p o ise o a app o i atio . This shared concern with compromise forms a 75 ea i gful elatio ship et ee Es he s isual app o i atio s of i fi it , Na a o s converging approximations of irrationals, and the mathematical concept of continued fractions, used to express periodically the seeming chaos of irrationality. All are approximations, approached through continuous gradation, of irrationality at an infinitesimal point – one visual, one sonic, and one mathematical. 75 Escher, M.C. Escher, his life and complete graphic work, 71. Key | 40 As we draw near our conclusion, let us return to Ma Bill s idea of i e pli a le a t e ause it so aptl add esses the o e s of Es he s a d Na a o s o k, as ell as the o k of Cantor and Einstein. Each of these men was intimately concerned with fathoming that which is ultimately impossible to wholly fathom. How does one quantify the infinite, demonstrate the warping of hyper-dimensionality, sonify the rhythmic incommensurability of 2 : √ , or visualize impossible objects? Ultimately, each achieved their aim, albeit some necessary approximations and abstractions along the way. How did they manage this? Each protagonist was faced with something impossible under a Pythagorean or Newtonian cosmology, yet each was able to discover a new paradigm, under which they could realize their inexplicable curiosities and see the truth within them. Due to the work of mathematicians and physicists like Cantor and Einstein, the impossible, the irrational, and the infinite were placed in closer reach than ever before. Most of Na a o s o ks are considered impossible for any human to play. However, Nancarrow made, through the player piano, impossible performances possible ith a eal performer. Given the possibilities granted to him through his mechanized musician, Nancarrow was then able to make essentially impossible musical processes possible through close approximations and suggested continuation of those processes. We can understand the aesthetic implications of his new artistic paradigm given Cantor s and Einstein s i o atio s. Na a o s music is intrinsically concerned with the warping of time and rhythmic space, breaking the bounds of a fixed musical universe. The experience of time for each voice in a Nancarrow tempo canon is wholly relative; there is rarely a concrete metrical reference, and once Nancarrow reached his irrational studies, any Key | 41 sense of absolute temporal reference is made wholly unattainable. In pieces where Nancarrow uses continuous tempo transformations, time becomes not only relative for each participating voice, but also warped by the gravity of his mathematical processes, continually fluctuating and t a sfo i g u til the pie e s te i us. Furthermore, Nancarrow warped the sonic space of the piano through constructing music that, while acoustically played on a classical instrument, is wholly new and essentially impossible for a human to perform, both physically and mentally. In the same way, Escher explored the boundaries of the impossible in his own art, propelled by the possibilities open to him through mathematics and physics. He could construct impossible three-dimensional objects within the possible spaces of his two-dimensionally represented worlds. He not only implied infinity beyond the bounds of his prints, but also construct them within strict limits. Escher warped space and drew our attention to the uncertainty and relativity of perception. Through his continuous transformational process, he imagined, like Nancarrow, a universe that is never fixed and only partially predictable, one in which the gravity of his process inexorably draws us into the infinite possibilities of his design. In Max Bill s definition of a mathematical approach to the arts, he addresses the implications of our preceding analogies. He states, It must not be supposed that an art based on the principles of mathematics… is in any sense the same thing as [mathematics]. Indeed, [art] employs virtually none of the resources i pli it i the ter pure athe ati s. The art in question can, perhaps, best be defined as the building up of significant patterns from the ever-changing relations, rhythms, and proportions of abstract forms, each one of which… is tantamount to a law unto itself. As such, it presents some analogy to mathematics itself where every fresh advance had its i a ulate o eptio i the rai of o e or other of the great pio eers.76 76 Bill, Die mathematische Denkweise. Key | 42 While Nancarrow s music and Escher s art are not the mathematics and physical principles with which they are dialoging, they stand as poignant analogies. By recognizing such analogies, we can more clearly understand both sides of this dialogue. We can understand the complex music of Nancarrow through the subtle yet transparent art of Escher. Equally we may understand the deep complexity of Escher s elegant and seemingly simple graphic designs through their apt juxtaposition to abstract physical and mathematical concepts. Furthermore, we can understand these abstract mathematical concepts more lucidly through their visualization and sonification in Escher s and Nancarrow s art. Most importantly, in recognizing such a dialogue as this even exists, we can more deeply appreciate the historical context within which these arts and sciences arose, realizing a twothousand-year cultural odyssey that unites these great thinkers despite their seeming dissociation. Bibliography Primary Sources: Art Works (listed in chronological order): Escher, Maurits Cornelis. Metamorphosis II. 1939-1940. ____. Cubic Space Division. 1952. ____. Relativity. 1953. ____. Smaller and Smaller. 1956. ____. Circle Limit III. 1959. ____. Ascending and Descending. 1960. ____. Waterfall. 1961. ____. Square Limit. 1964. Music Scores: Conlon, Nancarrow. Collected Studies for Player Piano Vol. 2: Study No. 41 for Player Piano. Santa Fe, CA: Soundings Press, 1981. _____. Collected Studies for Player Piano Vol. 3: Study No. 37 for Player Piano, Facsimile Edition. Berlin, Germany: Edition Schott, 1988. _____. Collected Studies for Player Piano Vol. 4: Study No. 3 Berlin, Germany: Edition Schott, 1988. _____. Collected Studies for Player Piano Vol. 5: Studies No. 2, 6, 7, 14, 20 ,21, 24, 26, and 33. Santa Fe, CA: Soundings Press, 1984. _____. Collected Studies for Player Piano Vol. 6: Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 19. Berlin, Germany: Edition Schott, 1988. Interviews: Koonce, Paul. "Interview with composer Paul Koonce regarding Conlon Nancarrow." Interview by author. November 27, 2017. Nancarrow, Conlon. "A KPFA interview with Conlon Nancarrow." Interview by Charles Amirkhanian. 20th-century music, March 1998, 1-3. ____. "Terraced Dynamics: Interview with Charles Amirkhanian, recorded in Mexico City, April 1977." Interview by Charles Amirkhanian. YouTube. August 23, 2016. Accessed November 4, 2017. https://www.youtube.com/watch?v=crGiIujMhN0. Nancarrow, Conlon, and Alistair M. Riddell. "Tempus perfectum: a conversation with Conlon Nancarrow". Meanjin. 47, no. 2 (1988): 266-273. Reynolds, Roger. "Conlon Nancarrow: Interviews in Mexico City and San Francisco." American Music. 2, no. 2 (1984): 1. doi:10.2307/3051655. Books: Cowell, Henry. New Musical Resources. New York: Alfred A. Knopf, 1930. Escher, M. C., Flip Bool, Bruno Ernst, and J. L. Locher. M.C. Escher, his life and complete graphic work: with a fully illustrated calatogue. New York: H.N. Abrams, 1981. Galilei, Galileo. Dialogues concerning two new sciences = Guan yu liang men xin ke xue de dui hua. Translated by Henry Crew and Alfonso De Salvio. Beijing Shi: Gao deng jiao yu chu ban she, 2016. Garland, Peter. Americas: essays on American music and culture, 1973-80. Santa Fe: Soundings Press, 1982. Iamblichus. The life of Pythagoras. Translated by Thomas Taylor. Krotona: Theosophical Publishing House, 1918. Journal Articles: Bill, Max. "Die mathematische Denkweise in der Kunst unserer Zeit ( The mathematical way of thinking in the visual art of our time )." In The Visual Mind: Art and Mathematics, by Michele Emmer, translated by Michele Emmer, 5-9. Cambridge, MA: MIT Press, 1994. Coxeter, H.S.M.. Crystal symmetry and its generalizations, A Symposium on Symmetry, Trans. Royal Society of Canada. 51, ser. 3, sec. 3 (1957), 1–13. Emmer, Michele, Doris Schattschneider, and M. C. Escher. M.C. Escher’s legacy: a centennial celebration: collection of articles coming from the M.C. Escher Centennial Conference, Rome, 1998. New York: Springer-Verlag, 2003. Heisenberg, W. "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik." Original Scientific Papers Wissenschaftliche Originalarbeiten, 1985, 478504. Penrose, L. S., and R. Penrose. "Impossible Objects: A Special Type Of Visual Illusion." British Journal of Psychology. 49, no. 1 (1958): 31-33. Planck, Max. "Über das Gesetz der Energieverteilung im Normalspektrum." Annalen der Physik. 309, no.3 (1901): 553-563. Newspaper Articles: "Revolution in Science – New Theory of the Universe – Newtonian Ideas Overthrown." The London Times, November 7, 1919. 38. Documentaries: Nancarrow, Conlon, James Greeson, and Robert Ginsburg. Conlon Nancarrow: Virtuoso of the Player Piano. [Little Rock, AR]: [publisher not identified], 2012. The Art of the Impossible: MC Escher and Me. Directed by Clem Hitchcock. Produced by Richard Bright. Performed by Roger Penrose. United Kingdom: BBC, 2015. DVD. Secondary Sources: Bruno, Leonard C., and Lawrence W. Baker. Math & mathematicians: the history of math discoveries around the world. Detroit: U.X.L, 1999. Bugallo, Helena. Selected Studies for player piano by Conlon Nancarrow: sources, working methods, and compositional strategies. Master's thesis, The State University of New York at Buffalo, 2004. Ann Arbor, Michigan: UMI Dissertation Services, 2004. Calle der, Clifto . "Perfor i g the Irratio al: Paul Usher s arra ge e t of Na arrow s “tudy No. , Ca o :√ ." Co lo Na arrow, Life and Music: Online Symposium September 27 - October 27, 2012. September 2012. Accessed December 02, 2017. http://conlonnancarrow.org/symposium/papers/callender/irrational.html. Das, Ashok. Lectures on quantum mechanics. Singapore: World Scientific, 2012. Dauben, Joseph W. Georg Cantor: his mathematics and philosophy of the infinite. Princeton, NJ: Princeton University Press, 1990. ____. "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite." Journal of the History of Ideas38, no. 1 (1977): 85-108. ____. "Georg Cantor and the battle for transfinite set theory." In Proceedings of the 9th ACMS Conference, 1-22. Westmont College, Santa Barbara, CA. New York: City University of New York, 1988. Drott, Eric. "Conlon Nancarrow and the Technological Sublime." American Music. 22, no. 4 (2004): 533-63. Emmer, Michele. "Mathematics and Art: Bill and Escher." Edited by Reza Sarhangi. 353-62. Proceedings of Bridges: Mathematical Connections in Art, Music, and Science. Southwestern College, Winfield,, Kansas: Bridges Conference, 2000. Accessed November 2, 2017. http://archive.bridgesmathart.org/2000/bridges2000-353.pdf. Fauvel, John, Raymond Flood, and Robin J. Wilson. Music and mathematics: from Pythagoras to fractals. Oxford: Oxford University Press, 2010. Fü rst-Heidtmann, Monika. 1984. "Conlon Nancarrow's "Studies for player piano": Time is the last frontier in music". Melos. 104-122 Gann, Kyle. "Conlon Nancarrow's Tempo Tornadoes," Village Voice, New York (5 October 1993), pp. 93 and 97 _____. "Downtown beats for the 1990s: Rhys Chatham, Mikel Rouse, Michael Gordon, Larry Polansky, Ben Neill". Contemporary Music Review. 10, no. 1 (1994): 3349. _____. Outside the Feed a k Loop: A Na arrow Key ote Address, Music Theory Online 20, no. 1 (March 2014): accessed November 4, 2017, http://mtosmt.org/issues/mto.14.20.1/mto.14.20.1.gann.pdf _____. The music of Conlon Nancarrow. Cambridge University Press, 2006. Garland, Peter. Americas: Essays on American Music and Culture, 1973-80. Santa Fe: Soundings Pr, 1982. Gimbel, Steven. "Flatland, Curved Space: How M.C. Escher Illustrated the History of Geometry." Edited by Reza Sarhangi. In Bridges: Mathematical Connections in Art, Music, and Science, 132. Proceedings. Southwestern College, Winfield, Kansas: Bridges Conference, 2001. Accessed November 2, 2017. http://archive.bridgesmathart.org/2001/bridges2001-123.pdf. Gullberg, Jan. Mathematics: from the birth of numbers. New York: W. W. Norton, 1997. Heath, Thomas. A history of Greek mathematics: From Thales to Euclid. Vol. 1. Oxford: Clarendon Press, 1921. Hocker, Jü rgen . Encounters with Conlon Nancarrow. Translated by Steven Lindberg. Lanham, MY: Lexington Books, 2012. _____. 2002. "My soul is in the machine: Conlon Nancarrow - composer for player piano – precursor of computer music". Music & Technology in the Twentieth Century. 84-96. Hollist, J. Taylor. "M.C. Escher's Associations with Scientists." Edited by Reza Sarhangi. 4552. Proceedings of Bridges: Mathematical Connections in Art, Music, and Science. Southwestern College, Winfield, Kansas: Bridges Conference, 2000. Accessed November 2, 2017. http://archive.bridgesmathart.org/2000/bridges2000-45.pdf. Jarvlepp, Jan. 1983. "Conlon Nancarrow's Study # 27 for Player Piano Viewed Analytically". Perspectives of New Music. 22 (1-2): 218-222. Kline, Morris. Mathematical thought from ancient to modern times. New York: Oxford University Press, 1990. Lindsay, Robert Bruce, and Henry Margenau. Foundations of physics. Woodbridge, CT: Ox Bow Press, 1981. Major, F. G. The quantum beat: principles and applications of atomic clocks. New York: Springer, 2007. Neugebauer, O. The exact sciences in antiquity. New York: Dover Publications, 1969. Povilioniene, Rima. Musica Mathematica. Traditions and Innovations in Contemporary Music. Frankfurt: Peter Lang GmbH, 2017. Rao, Nancy Yunhwa. "Cowells Sliding Tone and the American Ultramodernist Tradition." American Music23, no. 3 (2005): 281-323. Robson, Eleanor. "Words and Pictures: New Light on Plimpton 322." The American Mathematical Monthly. 109, no. 2 (2002): 105. Schattschneider, Doris . "The Mathematical Side of M. C. Escher." Notices of the American Mathematical Society. 57, no. 6 (2010): 706-18. Scrivener, Julie. "Applications of Fractal Geometry to the Player Piano Music of Conlon Nancarrow." Edited by Reza Sarhangi. 185-92. Proceedings of Bridges: Mathematical Connections in Art, Music, and Science . Southwestern College, Winfield, Kansas: Bridges Conference, 2000. Accessed November 2, 2017. http://archive.bridgesmathart.org/2000/bridges2000-185.pdf _____. "The Use of Ratios in the Player Piano Studies of Conlon Nancarrow." Edited by Reza Sarhangi. 159-66. Proceedings of Bridges: Mathematical Connections in Art, Music, and Science . Southwestern College, Winfield, Kansas: Bridges Conference, 2001. Accessed November 2, 2017. http://archive.bridgesmathart.org/2001/bridges2001-159.pdf. Stewart, Ian. Infinity: a very short introduction. Oxford, United Kingdom: Oxford University Press, 2017. Thomas, Margaret Elida. Conlon Nancarrow's "Temporal Dissonance": Rhythmic and Textural Stratification in the Studies for Player Piano. Ann Arbor (MI): UMI, 1997. _____. "Nancarrow's "temporal Dissonance": Issues of Tempo Proportions, Metric Synchrony, and Rhythmic Strategies." Intégral. 14/15 (2000): 137-180. Teresi, Dick. Lost discoveries: the ancient roots of modern science-- from the Babylonians to the Maya. New York: Simon & Schuster, 2003. Wilkes, Jonathan Wesley. Temporal friction in Conlon Nancarrows Study no. 36 for player piano. Master's thesis, 2009. University of California, Davis, 2009. Tertiary Sources: Fox, Christopher. "New Complexity." Grove Music Online. Oxford Music Online. Oxford University Press, accessed November 3, 2017, http://www.oxfordmusiconline.com/subscriber/article/grove/music/51676. Gann, Kyle. "Nancarrow, Conlon." Grove Music Online. Oxford Music Online. Oxford University Press, accessed Novenber 3, 2017, http://www.oxfordmusiconline.com/subscriber/article/grove/music/19552. _____. "Gordon, Michael." Grove Music Online. Oxford Music Online. Oxford University Press, accessed November 3, 2017, http://www.oxfordmusiconline.com/subscriber/article/grove/music/42648. Rodych, Victor . "Wittgenstein's Philosophy of Mathematics." The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University, 2007. Accessed November 27, 2017. https://plato.stanford.edu/entries/wittgenstein-mathematics/.