EX NIHILO – Dahlgren 1
EX NIHILO: A STUDY OF CREATIVITY AND
INTERDISCIPLINARY THOUGHT-SYMMETRY IN THE ARTS
AND SCIENCES
By
DAVID F. DAHLGREN
Integrated Studies Project
submitted to Dr. Patricia Hughes-Fuller
in partial fulfillment of the requirements for the degree of
Master of Arts – Integrated Studies
Athabasca, Alberta
August, 2008
EX NIHILO – Dahlgren 2
Waterfall by M. C. Escher
EX NIHILO – Dahlgren 3
Contents Page
LIST OF ILLUSTRATIONS 4 INTRODUCTION 6 FORMS OF SIMILARITY 8 Surface Connections 9 Mechanistic or Syntagmatic Structure 9 Organic or Paradigmatic Structure 12 Melding Mechanical and Organic Structure 14 FORMS OF FEELING 16 Generative Idea 16 Traits 16 Background Control 17 Simulacrum Effect and Aura 18 The Science of Creativity 19 FORMS OF ART IN SCIENTIFIC THOUGHT 21 Interdisciplinary Concept Similarities 21 Concept Glossary 23 Art as an Aid to Communicating Concepts 27 Interdisciplinary Concept Translation 30 Literature to Science 30 Music to Science 33 Art to Science 35 Reversing the Process 38 Thought Energy 39 FORMS OF THOUGHT ENERGY 41 Zero Point Energy 41 Schools of Fish – Flocks of Birds 41 Encapsulating Aura in Language 42 Encapsulating Aura in Art Forms 50 FORMS OF INNER SPACE 53 Shapes of Sound 53 Soundscapes 54 Musical Topography 57 Drawing Inner Space 58 Exploring Inner Space 66 SUMMARY 70 REFERENCES 71 APPENDICES 78
EX NIHILO – Dahlgren 4
LIST OF ILLUSTRATIONS Page
Fig. 1 - Hofstadter’s Lettering 8 Fig. 2 - Stravinsky by Picasso 9 Fig. 3 - Symphony No. 40 in G minor by Mozart 10 Fig. 4 - Bird Pattern – Alhambra palace 10 Fig. 5 - A Tree Graph of the Creative Process 11 Fig. 6 – Rhetoric Tree Graph 12 Fig. 7 - “Hierarchic Repetition” in Beethoven, Symphony No. 5 12 Fig. 8 - Sociology Graph 12 Fig. 9 - Harmonic Series as Background Control in Harmonic Modulation 17 Fig. 10 - Reindeer, in Font-de-Gaume Cave near Dordogne, France 19 Fig. 11 - Vermeer “Tile” Paintings 22 Fig. 12 – Music Augmentation by a Factor of Two 23 Fig. 13 – Function 24 Fig. 14A - Reciprocal (a), Tonal (b) and Exact “Real” (c) Inversions in Music 26 Fig. 14B - Inverted Perspective vs Regular Perspective 26 Fig. 15 - Rocket Science Momentum Equation 26 Fig. 16 - Feynman Diagram 27 Fig. 17 - Hypercube Projection 28 Fig. 18 – Projection 28 Fig. 19 - Hawking’s Newtonian Time Model 29 Fig. 20 - Möbius Strip by Escher 29 Fig. 21 - Dali’s Crucifixion (Corpus Hypercubus) 30 Fig. 22 - Kepler’s Harmonices Mundi 33 Fig. 23 - Harmonic Series – Perspective Ratio Similarities 35 Fig. 24 – Parquet Tile Deformations 39 Fig.25 - Spectogram of “Bishop to Queen Knight Three” 43 Fig. 26 - Lexical Semantics 44 Fig. 27 - English/Arabic Words for “The House” 45 Fig. 28 - Zachary Speaking to His “Papa” in the Henderson Public Library 47 Fig. 29 - Zachary - Speech Levels in Negation 49 Fig. 30 – Visual Textures of Six Chopin Etudes 54 Fig 31A - Philips Pavilion for the 1958 Brussels World's Fair 56 Fig. 31B - Xenakis’ Mycenae – Alpha 57 Fig. 32 - The Cortex Recreates a 3D Image from a 2D Retinal Image 58 Fig 33 - Julian Hook’s Matrix 60 Fig. 34 - Tymoczko’s Matrix 61 Fig. 35 - Harmonic and Melodic Matrices 62 Fig. 36 - Galilean Space-Time Matrix 62 Fig. 37 - Poincaré Space-Time Matrix 63 Fig. 38 - Gollin Tonnetz 63 Fig. 39 - Harmonization of the First 6 Notes of Three Blind Mice 64 Fig. 40 - Assigning Chord Notes “Matrix Points” - Three Blind Mice 65 Fig. 41 - The Inner Space of the Music for Three Blind Mice 65 Fig. 42 - 3D Inner Space Recreated in 2D 66 Fig 43 – Al-Li-Cu Quasicrystal 68 Fig 44 - Crystallized Inner Space 68
EX NIHILO – Dahlgren 5
"There is nothing either good or bad, but thinking makes it so…"
(William Shakespeare – Hamlet)
“…it is a tale told by an idiot, full of sound and fury, signifying nothing.”
(William Shakespeare - Macbeth)
Note:
The mathematics discussed is kept to a layman’s level because I am not a mathematician; however, since I am a composer the discussions alluding to music theory may need further explanation and any good theory text would suffice for reference. I tried to keep this paper at an approachable level by explaining appropriate theories in the context of my discussions; an interdisciplinary learning necessity. I hope I have succeeded.
EX NIHILO – Dahlgren 6
INTRODUCTION
This paper is a semiotic study of common concepts in the liberal arts and exact
sciences. It examines thought-form similarities and cross-disciplinary concept
“translations” that exist between these two seemingly opposed thought-worlds and delves
successively deeper into different manifestations of a force many scholars agree is
common to all cultural representations - creativity. My aim is to find new
interdisciplinary passages and link them with a language of integration useful for
exploring, connecting, and understanding the interplay of all thought-forms.
Through a five-step process I consider: obvious similarities in art forms, forms of feeling
or thought energy, how the sciences have borrowed liberal arts concepts, the idea that
cultural representations act like capacitors holding the energy that shaped them in inner
spaces formed by their various nomenclatures and that these inner spaces can be mapped
and explored.
Although this paper is rather speculative in nature, I have tried to “ground” it in
accepted scholarly treatises as much as possible. For example, I am aware that Roger
Penrose’s area of expertise is not “consciousness” per se; however, I feel that sound reasoning underlies his foray into the nature of thought. Thought is, after all, common to all disciplines.
Since I am a composer, a composition component is an integral part of my final project. The work, Three Unended Waltzes, consists of three “unfinished” waltzes forming three movements within the whole. A tone row forms the basic harmonic and
melodic structure of the waltzes but I was careful to engineer a row that would sound like
late nineteenth and early twentieth century music. The row manifests itself as harmonic EX NIHILO – Dahlgren 7 material in the first waltz, formatting and structuring in the second, and melody in the third. I included a composition because linguistic codes give the impression that we think verbally; music is in direct opposition to that precept because it is non-verbal, yet communicates feeling. The stylistic similarities to waltzes by Ravel (1), Brahms (2) and
Poulenc (3) are intentional. As you listen, be reminded that music is a secret code full of multidimensional worlds. The music was recorded in the Lorne Watson Auditorium at the Brandon University School of Music; the pianist is Claudette Caron.
Click the icon to listen
Ravel
Brahms
Poulenc
EX NIHILO – Dahlgren 8
FORMS OF SIMILARITY
Forms of literature, music, and art are rife with similarities in formal structure.
Some are superficial, some deep; some are intuitive, some require analysis; however, they all seem to have common underpinnings. This chapter discusses this phenomenon in some detail and tries to discover its origins, explain why it exists, and explore how it might have ramifications beyond art forms.
My ideas on creative thought-form in literature, music, and art are based on conclusions made from what may seem rather simplistic - even naïve and superficial - observations like noticing resemblances in the feeling and form between Hofstadter’s lettering-art hobby and Picasso’s drawing of Stravinsky. (Fig.’s 1 and 2)
Fig. 1 - Hofstadter’s Lettering (Hofstadter, 1985, p. xxix)
EX NIHILO – Dahlgren 9
Fig. 2 - Stravinsky by Picasso (Edwards, 1979, p. 52)
Such surface connections may confirm deeper conceptual similarities and give rise to new insights and methods of examining yet unseen things even though they are, and have always been, in plain sight; a trait of postmodernism (Bonnycastle, 1997, pp. 231 - 241).
My investigation is a five-step process that considers: obvious similarities in art forms, forms of feeling or thought energy, how the sciences have borrowed liberal arts concepts, the idea that cultural representations act like capacitors holding the energy that shaped them in inner spaces formed by their various nomenclatures (Ogden, 1949, p. 42); and finally, that transcribing and mapping these inner spaces is both possible and necessary for furthering scientia – all knowledge. Exploring these ethereal places would be an investigation of the mercurial thing that spawned them – creative thought (Penrose, 1989, pp. 405 - 449).
Surface Connections
Mechanistic or Syntagmatic Structure
Mechanistic or syntagmatic structures are the skeletal forms that mold works of literature, music, or art. Linear in nature, they use sequencing and more complex rules of EX NIHILO – Dahlgren 10
operation like sonnet form in literature, sonata form in music, or cubist form in art, to
provide a framework to contain organic material, the ideas that fill the mechanistic
framework – discussed in detail below.
Sequencing
Lidov discusses a form of poetic sequencing Bartha called Quaternary Stanza
Structure (QSS) dealing with sets of fours used in both literature and music (Lidov, 2005, p. 31) (Fig. 7); Livio gives an example of sequencing thematic material in the first movement of Mozart’s Symphony No. 40 in G Minor (Fig. 3) as well as an example of shape sequencing in art - the repeated bird pattern on the walls of the Alhambra palace in
Granada that may have inspired Escher (Livio, 2005, pp. 16 - 18). (Fig. 4)
.
Fig. 3 - Symphony No. 40 in G minor by Mozart
Fig. 4 - Bird Pattern – Alhambra palace
Interrupted sequences use interspersed recurring elements: ABAC, etc; a story like The
Three Little Pigs closely resembles rondo form in music or a painting of telephone poles.
The interspersed “asides” or “catalytic materials” manifest as divergent events in a story, EX NIHILO – Dahlgren 11
contrasting melodies in a piece of music, or different material between the telephone
poles in art. The wolf blowing the houses down would be a recurring event; the pigs
running from one collapsed house to the other would be different events since it is a
different number of pigs each time. Such “asides” interrupt the flow of the work whereas
the recurring events, melodies, and scenes lend impetus to works of art.
Tree Graphing
The Tree Graph format is widely used in literature, music, and art as well as the exact and social sciences. Seldom conspicuously evident, it becomes clear with analysis and can draw a picture of inter-relationships between words, notes, and shapes as well as creative processes, grammar, and society. However, it is organic structure that adds life to art forms. (Fig.’s 5 – 8.)
Fig. 5 - A Tree Graph of the Creative Process (Ehrenzweig, 1967, p. 36)
EX NIHILO – Dahlgren 12
Fig. 6 – Rhetoric Tree Graph (Eco, 1979, p. 277)
Fig. 7 - “Hierarchic Repetition” in Beethoven, Symphony No. 5 (Lidov, 2005, p. 32)
Fig. 8 - Sociology Graph (Seale, 2004, p. 319)
Organic or Paradigmatic Structure
Organic or paradigmatic structure is vertical or lateral in nature. It “colors” a work by “piling up” information about the events or characters in a story or harmonizing EX NIHILO – Dahlgren 13
notes sequenced in a melody by adding notes above or below them. Stories sequence
events, but the reader likes to know why the events take place and something of why the
characters are there, otherwise there is simply a list of “stuff,” interesting for a time, but
lacking substance. Characters differ because of how the author has them behave; two notes in a melody can sound different because of how the composer uses harmony. No one would mistake one of the pigs for the wolf in The Three Little Pigs any more than
one would say the first two harmonized notes of God Save the Queen sound exactly the
same. This type of structuring has many permutations (Bonnycastle, 1996, p. 100).
Opposing Pair Energy
One aspect of organic structure is the energy, drive, or direction of an art form;
the reader must feel the work is going somewhere, otherwise interest will flag. Like
differing harmonic intervals acting on a melody, conflicting ideas add energy. The
literary theory of French structuralist, A. J. Greimas, identifies this energy source by
naming three sets of opposing pairs (Bonnycastle, 2002, p. 159; Costa, 2000, p. 4 - 22):
The “subject/object” model parallels the archetypal quest and is very similar to
the development section in sonata form - a labyrinth of bits and pieces of the original
thematic material used in different keys and permutations leading to a final rediscovery
or recapitulation.
The “sender/receiver” model of cause and effect is the very stuff of why literature
exists, the reason for the story. If the ghost of Hamlet’s father had not asked Hamlet to
avenge his death, there would be no play. In music, there are always two parts to a
phrase; without this question – answer, or antecedent – consequent “energy”, there would
be no music. EX NIHILO – Dahlgren 14
The “helper/opponent” model fits the “tonic–dominant” (I – V) movement in a simple
sonata. In Lord of the Flies, by William Golding, this helper/opponent, or tonic-dominant
energy, exists in both a literal and a figurative way. During the course of events, Ralph defers
power to Jack then at the end of the story the power shifts back to Ralph. Golding uses music symbolism to underscore the “helper/opponent” struggle in Lord of the Flies. At the beginning of the story Ralph finds an eighteen inch conch shell (Golding, 1979, p. 17) and we learn that
Jack can sing a high C# (two ledger lines above the treble staff – well within the tessitura of a boy soprano) (Golding, 1979: 23). An eighteen inch organ pipe, or conch shell, will produce an
F# (the first space in the treble staff) when it ‘speaks’; Ralph’s F# is the tonic or home key of
Jack’s C#, the “drawing force” or “dominant” in the key of F# major or minor. In a simple sonata there is always a modulation or change of key from the tonic to the dominant, but, after this change to the “dominant”, there is always a relentless pull back to the tonic again at the end.
This is what literally happens in Lord of the Flies; the combination of these forms of organic
energy is one example of thought structure similarities in literature and music. Eero Tarasti,
Professor of Musicology at the University of Helsinki and one of the world’s leading
semioticians, uses Greimas’ theory of opposing pairs to make semiotic analyses of works of
music comparing them with the visual arts and literature. His work also shows that melding
mechanical and organic art form structure can cross disciplinary boundaries and explore
scientific concepts like space-time (Tarasti, 1994, p. 26).
Melding Mechanical and Organic Structure
Combining mechanistic and organic structure is essential to invention; one cannot
exist without the other. Mechanistic form is invisible without the “light” of organic form
to create its shadow; the “light” of organic form is invisible without a mechanistic form EX NIHILO – Dahlgren 15
to project (Dahlgren, 2007, p. 11). The essay you are reading combines mechanistic and
organic forms: the three stages of creativity provide the mechanistic form: Projection,
Integration, Re-introjection. By introducing a generative idea, expanding on it, and then
summarizing it, the mechanistic form of this essay is filled with organic material of a
kind peculiar to itself; other essays by myself or others would have different organic
material shaped by the same mechanistic form. The creative process shapes any art form
and helps capture and communicate its generative energy and feeling; thus allowing art
forms to allude to something beyond themselves (Langer, 1953, pp. 247, 392).
Form has both mechanistic and organic states. Forms of literature, music, and art
can be so similar in nature that is possible to combine them; this seems to be, at least in
part, due to the mental processes that created them. There is also a kind of energy that pervades art forms and this creative thought energy seems to be capable of communicating what are often very palpable thoughts and feelings.
EX NIHILO – Dahlgren 16
FORMS OF FEELING
Creative thought takes many forms often beginning with a generative idea that
can express a specific feeling. This generative mental energy lends power and
significance to all art forms and is the essence of how audiences react to reading, hearing,
or viewing created work.
Generative Idea
Traits
Writers, composers, and artists can transport us to an old miser’s corn exchange
office in London (Dickens), an approaching thunderstorm near Bonn (Beethoven), or a
weird staircase in Amsterdam (Escher) (Berger, 1972, p. 54). A visionary can, at least in
part, transform our experience and control how we feel about a world long faded into
oblivion. A shared vision is different from merely watching the world go by, it freezes
time and superimposes space on space providing us with a glimpse into sights and sounds
from another era forged in the crucible of a mind (Dahlgren, 2004, p. 3). Generative
ideas vary; time often contracts for composers (Mozart) and spatial relationships
disappear for artists (Dali) (Penrose, 1989, p. 444; Randall, 2005, pp. 11-30; Davenport,
1960, p. 276). An initial vision can differ widely from the end-product; vague notions of
scenery become music; sounds, art (Gray, 2006, p. 4). Above all, the generative idea
must survive the creative process and be preserved by the art-form; otherwise, no
literature, music, or art can maintain and communicate the ideas shared (Sherratt, 2002, p.
127). “Energies” pervade invention (Sherratt, 2002, p. 159).
EX NIHILO – Dahlgren 17
Background Control
Background control energy manifests itself differently depending on the
circumstances but always influences art forms with pre-existing energy; one example is
Gounod’s Ave Maria. Set against Bach’s Prelude No. 1, Bach’s harmonic structure
dictates, or at least severely limits, the melody. The reverse is true for harmonization.
Notes in the melody limit the harmonization because, simply stated, the chord must
contain the melody note. In figure 9, the very first chord (marked V7) is a list of notes
often used to harmonize C, E, G or Bb in the key of F Major. The partials in a vibrating
string tuned to the bottom note, C - specifically the first seven partials - are also the
background control that influences modulation (key change) because it forms a natural dominant seventh chord (V7) that will move naturally to the flat side of the circle of fifths
modulating to the key of F Major. (Fig. 9)
Fig. 9 - Harmonic Series as Background Control in Harmonic Modulation
By going to the dominant of the dominant of C, V/V – a D7 chord - it is possible to set up a natural movement to G7, which in turn sets up a natural movement back to C. In addition, every single note is a fundamental note of an infinite harmonic series and EX NIHILO – Dahlgren 18 combining any two notes combines two harmonic series; harmony is the result of common overtones aligning (Jeans, 1961, p. 154 – 157; Bland, 1960, p. 66). For example, combining an F and a C harmonically would cause the third partial above F to align with the second partial above C - both middle C notes in the first bar of figure 9.
Growing awareness of partials may have influenced the historical and theoretical development of music. Moving upward from the fundamental note, C, in the harmonic series shown in figure 9, you get the intervals: octave, fifth, fourth, major third, two minor thirds, major second, minor second and so on to infinity (Jeans, 1961, p. 77).
Western music evolved according to that same order: unisons and octaves were exclusively used in Gregorian Chant (circa A.D. 600), fifths and fourths were used later in parallel organum (circa A.D. 1000), St. Martial organum (circa A.D. 1100) used seconds and thirds as well as their inversions - sevenths and sixths; a full range of intervals became the norm in the Renaissance (circa A.D. 1500) and subsequent eras
(Grout, 1960).
Simulacrum Effect or Aura
Simulacrum effect or aura exerts influence on art forms and gives them their raison d’être as well as a certain “truth”; it also produces a hierarchy of creative endeavor
(Baudrillard, 1981, p. 365; Sherratt, 2002, p. 155). Rembrandt’s painting, The Night
Watch, hangs in the Rijksmuseum in Amsterdam and, the first time I saw it, it took my breath away. A security guard noticed my reaction and said that it often had that effect on people. When I told him I felt I could walk into the painting he smiled and said, “Yes, one can do this.” I had much the same experience hearing Aaron Copland’s Inscape; I was “awe-struck” by the first chord and stayed in a stupor until the piece ended. There is EX NIHILO – Dahlgren 19
a magic that lies in the ability of an artist to imbue words, notes, and lines with an energy
that can communicate the power of a vision far beyond the time and space from whence it
came (Langer, 1979, p. 158). Audiences are as stunned by Mozart’s Jupiter Symphony
today as were the Viennese in 1788 (Grout, 1960, p. 465). The prehistoric painting,
Reindeer, in Font-de-Gaume Cave near Dordogne, France has communicated the artist’s thought and feeling over millennia (Langer, 1967, p. 132). (Fig. 10)
Fig. 10 - Reindeer, in Font-de-Gaume Cave Near Dordogne, France
The Science of Creativity
Douglas Hofstadter sees creativity as the single point that every discipline
revolves around; the source for all original thought and the door to infinite possibilities
(Hofstadter, 1985, p. xxix). Penrose thinks creativity is the ability to “smell out” truth in any discipline; an indispensable tool for approaching, let alone understanding, ideas like other dimensions (Penrose, 1989, p. 282). Looking beyond the way we communicate
thought - even suggesting logical flaws might exist in scientific rigor – these and other
seekers call for new ways to explore our world, new thought processes, and new EX NIHILO – Dahlgren 20
nomenclatures (Penrose, 1989, p. 413; Dahlgren, 2007, p. 18). The world of imagination
affects everything we think about and everything we do; “all philosophies, beliefs, art-
forms, edifices and machines - are visions made concrete” (Dahlgren, 2007, p. 3).
Creative thought is the one concept that unifies all disciplines, relationally connecting all
of existence (Derrida, 1968, p. 279). The lack of objectivity linked to creativity is what
draws scientists to the phenomenon since “even more than this physical determinism” in
quantum theory “was a lack of objectivity in the way that quantum theory seemed to be described” (Penrose, 1989, p. 280). But how is it described? - by symbols. And, like all art forms, mathematics is only another form of symbolic logic (Langer, 1979, p. 234). I find it interesting that mathematics transliterated into music is rather simple stuff whereas a simple melody must be expressed by complex mathematics. Since the field of mathematics called “harmonic analysis” originated as music theory, it seems that creative thought works the same in all disciplines; it is the door to scientia - all knowledge
(Darrigol, 2007, p. 343).
Expressing feeling through what Langer calls “significant form” is the purview of creative thought (Langer, 1953). A constant source of invention and advancement, creative thought may have been responsible for our growing awareness of natural phenomena which, in turn, may have controlled the advancement of civilization (Žižek,
2007, p. 107). Its power can supersede time and space, further knowledge, thrill audiences, and is a seemingly inexhaustible supply of generative power in both the lively arts and exact sciences.
EX NIHILO – Dahlgren 21
FORMS OF ART IN SCIENTIFIC THOUGHT
Creative thought both transcends and links disciplinary boundaries. The creative
thinking of a writer, composer, or artist can become scientific theory. A concept that
works in Latin, music, or art can just as easily become a mathematical equation to
formulate rocket propulsion theories as the aura of a symphonic performance.
Interdisciplinary Concept Similarities
Cavalieri’s Latin script spawned integral calculus (Colerus, 2002, p. 180);
Sanskrit texts inspired much of the research in particle acceleration (Capra, 1976, p. 15); music theorists helped to originate theories on heat and pressure as well as a branch of mathematics called Harmonic Analysis (Darrigol, 2007, p. 345). Theories in artistic
perspective have provided neurologists with insights into perception and thought (Langer,
1967, p. 55); aura has a counterpart in engineering known as Power Spectral Density
(Reinhard, 2007); Baudrillard’s idea of simulacrum has its counterpart in mathematical simulation (Wessel, 1990, p. 771). In The Golden Ratio, Mario Livio, head of the Space
Telescope Science Institute which conducts scientific programs for the Hubble Space
Telescope, delves into poetry, music, and art and discusses how these disciplines have
influenced scientific discovery; Holmes’ poem, The Chambered Nautilus, speaks of
logarithms; Leibnitz suggests music “is a secret arithmetical exercise;” Vermeer’s tiled
floors in such paintings as The Concert or Love Letter have inspired mathematicians to
analyze perspective (Livio, 2002, p. 202). (Fig. 11) EX NIHILO – Dahlgren 22
Fig. 11 - Vermeer “Tile” Paintings
Science is beginning to discover what writers, composers, and artists have known for centuries: the arts hold secret knowledge, yet untapped. Penrose said, “Understanding is, after all, what science is all about – and science is a great deal more than mere mindless computation” (Penrose, 1995, p. 266). Artistic intuition is also a “great deal more” than
“mere mindless computation;” it is a door to knowledge and a non-algorithmic passage to scientific insight (Penrose, 1989, p. 416). “I suppose there is an outside chance Von
Braun read Verne” – Terry Holt, Head Engineer, Shuttle Engine Cooling Systems -
NASA, retired. Language is the only method of non-mathematical explanation we have to communicate specific concepts (Langer, 1979, p. 144). Terms like augmentation, equation, function, and inversion are used in both the arts and sciences. Often meaning much the same thing but in a different context, perhaps these terms can provide a glossary of interdisciplinary links.
EX NIHILO – Dahlgren 23
Concept Glossary
Augmentation
In music, augmentation is the multiplication of a rhythm by a set number; augmented by a factor of two, a rhythmic motif of four eighth notes and two quarter notes
would become four quarter notes and two half notes. (Fig. 12)
Fig. 12 – Music Augmentation by a Factor of Two
In mathematics, such an operation would simply be the multiplication of a sequence by a
factor of two. Also, an augmented matrix is a number series derived (or augmented)
from a system of linear equations (Thiessen, 1999, p. 86). In literature a sequence of
events is a “story” but a story is “augmented” to plot status when these events are
expanded. E. M. Forster used this example, ‘“The king died and the queen died” is a
story. “The king died and the queen died of grief” is a plot’ (Hay/White, 2005, p. 8).
Aura
The concept of aura translates to Power Spectral Density (PSD) in fluid
mechanics. Measured in Megahertz, engineers have compared it to the power one feels
when an orchestra plays Beethoven. Aura and PSD are also medical terms used in
migraine research to measure blood-flow pressure in the brain (Reinhard, 2007).
Chance
Chance is a subject used in both music and mathematics, as aleatory composition
in music and probability in mathematics. Found poems – poems created by using non-
poetic material found by chance (like directions on a medicine bottle) and then re-written EX NIHILO – Dahlgren 24 as poetry - are an example of the use of chance in literature. The painter, Jackson
Pollock, used chance to drip, splash, and stream paint on canvas. http://simple.wikipedia.org/wiki/Jackson_Pollock.
Equation
In both Arabic and mathematics equation means much the same thing. In Arabic, the sentence, “That man is a farmer.” is called an equation rather than a statement because it balances like the equation, x2-2x+1=0 (if x = 1), since “That man” and “a farmer” are one and the same (Mace, 2007, p. 85).
Function
Langer uses the example of a chord being a function of a bass note in music
(Langer, 1979, p. 55). (Fig. 13) Known as Figured Bass this system plays an important role in music from the Baroque Period; Figured Bass was the almost exclusive domain of harpsichord continuo playing. The numbers denote intervals.
Fig. 13 – Function
Read from the bottom up they tell the harpsichordist to play a third, a fourth, and a sixth above the given bass note in the specified key in order to “realize” the chord built on the bass note, A – a second inversion of a dominant seventh (V7) in G major. In mathematics, “a function is a special kind of relation in which each x value is paired with one and only one y value” (Theissen, 2001, p. 91). This allows graphing of shapes like a EX NIHILO – Dahlgren 25
parabola where each point has a specific address: (2, 3) would mean +2 on the x axis and
+3 on the y axis. It is interesting that both systems allow the “realization” of something
on some kind of graph; a music staff or a Cartesian grid.
Invention
We speak of inventors inventing things like can openers, steam engines, or
aircraft but invention has a much more generic connotation; the result of any creative thinking. Johanne Sebastian Bach used the term to mean composing or composition and wrote two and three voiced pieces for the keyboard that he named two and three part inventions.
Inversion
In music, inversion means turning a melody either up-side down or down-side up.
The types of inversion shown here are reciprocal inversion (-3 becomes a +6), tonal, and
exact or “real” inversions. (Fig. 14A a, b, and c) Inversion also refers to different
versions of the same chord, e.g. the chord in figure 13 is in second inversion. The word
“reciprocal” is used to describe certain kinds of inversion in both music and mathematics.
(Fig. 14A) In mathematics, one may invert or reciprocate a fraction by exchanging the
numerator and the denominator. In graphing the reciprocal of a function like a mirror
image of a parabolic curve can be drawn using the reciprocal of any number, e.g. the
reciprocal of 2 is 1/2 (Theissen, 1999, p. 181). In art, inverted perspective refers to
diverging vanishing points – making it possible to see both sides of something which
would be impossible in reality (Arnheim, 1974, p. 265). (Fig. 14B) EX NIHILO – Dahlgren 26
Fig. 14A - Reciprocal (a), Tonal (b) and Exact “Real” (c) Inversions in Music
Fig. 14B - Inverted Perspective vs Regular Perspective
Modulation
Modulation means changing something from one form to another. In
telecommunications, modulated wave-forms broadcast signals in either amplitude
modulation (AM) or frequency modulation (FM). In mathematics one can modulate a
sine wave on a graph by changing variables. In music, modulation means to change key
within a work from the tonic to another related or unrelated key. (Fig. 9)
Sequencing
Mathematical sequences abound. The Fibonacci sequence (0,1, 1, 2, 3, 5, 8, 13,
etc) - in which all the numbers are the sum of the two previous numbers - and number
series exist at every level of mathematics from high school algebra to quantum physics.
(Fig. 15)
Momentum: m1V1+m2V2+m3V3+m4V4…= 0 (V = velocity, m = mass)
Fig. 15 - Rocket Science Momentum Equation
EX NIHILO – Dahlgren 27
Translation
In mathematics translation is synonymous with transposition in music, i.e. the
movement of precise coordinates in a matrix or specified notes from one position to
another by a given number or interval (Thiessen, 1999, p. 88). In art, it is the movement
of an object from one position to another in a drawing. In language, transpose or
translate means much the same thing: “the house” in English, translates or transposes to
“la maison” in French.
Art as an Aid to Communicating Concepts
Richard P. Feynman invented quantum electrodynamics and the Feynman
Diagrams – a system of illustrating events during particle theory experiments. He felt
only art could clearly illustrate what happens in the CERN cyclotron; his diagrammatic
system is used universally by scientists (Feynman, 1985, p. 118). (Fig. 16)
Fig. 16 - Feynman Diagram
Lisa Randall’s book, Warped Passages, uses diagrams to illustrate different dimensions and how they can be seen differently from different angles – both by spatial projection
(Fig.’s 17 & 18) and inverted perspective (Fig. 15B) (Randall, 2006, p. 27). These are EX NIHILO – Dahlgren 28
not new ideas in art and are discussed in detail by both Arnheim and Ehrenzweig
(Arnheim, 1974, p. 272; Ehrenzweig, 1967, p. 110).
Fig. 17 - Hypercube Projection (Randall)
Fig. 18 – Projection (Arnheim)
In fact, the drawings used to communicate most scientific concepts are extremely simple, using only a small portion of existing artistic knowledge. Steven Hawking’s books, A
Brief History of Time and The Universe in a Nutshell are so full of artistic renditions of scientific theory that one wonders how much scientists depend on art to provide models of abstract percepts for the purpose of communicating them clearly. One of the illustrations in his book - a model of a time loop - is very similar to one of Escher’s prints, Möbius Strip (Hawking, 2001, p. 32). Did a mere artist pre-invent Hawking’s cutting-edge scientific theory from “prescientific knowledge” using sheer intuition
(Langer, 1967, p. 55)? (Fig.’s 19 & 20) EX NIHILO – Dahlgren 29
Fig. 19 - Hawking’s Newtonian Time Model
Fig. 20 - Möbius Strip by Escher
Lisa Randall touches on the idea of intuitive pre-scientific knowledge being communicated through art with her inclusion of Dali’s Crucifixion (Fig. 21) and the cubist paintings of Picasso in her book, Warped Passages. She also suggests that if two- dimensional art is a kind of “shadow” of a three-dimensional world, perhaps three- dimensional art is a shadow of a four-dimensional world (Randall, 2006, pp. 26, 27).
Detailed scientific analysis of art may become a passage to whole new concepts in scientific thought. EX NIHILO – Dahlgren 30
Fig. 21 - Dali’s Crucifixion (Corpus Hypercubus)
If such similarities in thought-energy exist, I contend that it is quite possible interdisciplinary translation can take place. Since both non-scientific and scientific symbols can encapsulate ideas, perhaps the ideas expressed in literature, music and art could be translated into the sciences and vice versa. This, in point of fact, has often taken
place.
Interdisciplinary Concept Translation
Literature to Science
Integral Calculus
The formula ∫ydx, stemmed from a statement by Cavalieri when he frustratingly wrote: “the problem of quadrature” (squaring the circle) “is solved only if we could find
summa omnium y” (the sum of all the y’s). In 1676, Leibniz made a decisive advance,
writing: “It will be useful henceforward to write Cavalieri’s summa omnium y as ∫ydx.”
This widely used formula is really only “word-spinning” and its origin is surprisingly
lacking in mathematical rigor; however, it has served the world of mathematics and
science well for centuries (Colerus, 2002, p. 180)! EX NIHILO – Dahlgren 31
Gödel’s Theorem
Gödel’s theorem - suggested to him by a literary statement – demonstrates that a mathematical axiom can be both true and improvable. The theorem demonstrates a mathematical concept derived from Epimenides’ paradox: “All Cretans are liars.” Gödel reasoned that, since a Cretan wrote it, it may well be true, even if it gives the impression of falsity. Given that, why wouldn’t the converse be true: it may well be false, even if it gives the impression of truth? This conundrum gave rise to a revolution in mathematical reasoning because it raised the spectre that any mathematical proof, although seemingly true, may well be false and vice versa. Gödel’s theorem was “unexpected and unwelcome…because it obliterated the hopes for completeness in an already known system, namely Russell and Whitehead’s Principia Mathematica” (Hofstadter’s italics)
(Hofstadter, 1985, p. 445). This new theorem showed that Principia Mathematica was
“rather limited in the types of mathematical reasoning that it actually incorporated” and
“any such precise mathematical system of axioms and rules…must contain some statements which are neither provable nor disprovable by the means allowed within the system” (Penrose, 1989, pp. 101, 102). Gödel showed that irrefutable logic doesn’t exist in any form and any logic system that both directly and indirectly depends on itself for proof will not – cannot - work. In other words, no mathematical system can use its own logic to prove or disprove its own statements. (This is the very reason I am totally against using any form of mathematical logic to compose music; it severely limits intuitive thought and is more than likely based on one or more falsehoods.) By forcing mathematicians to admit the possibility of false logic, Gödel expanded the field of mathematics. His theory proved that other forms of logic may exist outside of what was EX NIHILO – Dahlgren 32
once thought to be the only complete system of mathematical logic. He encouraged
mathematicians to seek a less mechanistic and more organic means of reasoning; the arts.
In fact, no mathematical theory can be stated in mathematical symbols alone.
Mathematics texts and theories must use language to state the thought processes involved
in the symbols. A good example is Einstein’s very neat hand-written proof for E=mc2
once displayed at the Chicago Field Museum. Predicate Calculus (based on the logic of
word-communication) and ancient Sanskrit writings are both used to explore alternate
forms of mathematical logic inherent in language and have both added much to the
thought-world of mathematics and science.
Predicate Calculus
This brief explanation is quoted from the Wikipedia article at http://en.wikipedia.org/wiki/First-order_logic:
In First Order Logic (FOL) Predicate Calculus, one can express the existence of
something ( ), as well as predicates ("functions" that are true or false) with more than
one parameter. For example, "there is someone who can be fooled every time" can be
expressed as:
Where " " means "there exists (an) x", " " means "and", and Canfool(x,y) means
"(person) x can be fooled (at time) y". One can only guess at the possibilities waiting discovery using this means of translating literary works to mathematics.
EX NIHILO – Dahlgren 33
Sanskrit Texts
Fritjof Capra, a researcher in theoretical high-energy physics at the Universities of
Paris, California (Santa Cruz and Berkeley), Stanford, and Imperial College, London, compares the relationship between modern physics and Eastern mysticism (Capra, 1976, p. 133). Such research opens doors to integrated studies in the arts and sciences because it shows that science can learn from “prescientific knowledge” a source of ideas from literature, music, and art (Langer, 1967, p. 55). The Austrian physicist Erwin
Schrodinger - of Schrodinger’s Cat fame - “devised the wave equation every quantum system must obey” and represented “quantum stuff as a waveform.” Vedic thought so profoundly influenced him that he kept copies of the Sanskrit texts of the Bhagavad Gita and Upanishads by his bed (http://inannareturns.com/articles/schrodinger.htm).
Music to Science
Astronomy
Johannes Kepler used music to find the planet orbit ratios in the Solar System
(Livio, 2002, p. 155). (Fig. 22)
Fig. 22 - Kepler’s Harmonices Mundi
He assigned melodic motifs to each planet and may have used the interplay of the overtones in his calculations but his method remains a mystery. One possible clue to his method of calculation is in Livio’s book, The Golden Ratio. Apparently, Kepler ascribed EX NIHILO – Dahlgren 34
certain characteristics to certain planets. Earth got the rather uncomplimentary
description of being a place of misery and famine, shortened to the solfege notes Mi and
Fa (Livio, 2002, p. 155). He then used this information and other melodic planetary
representations to calculate his Harmonices Mundi and develop his third law of planetary
movement – a list of ratios delineating the orbits of the then known planets. Whatever
method he used, his ratios are surprisingly close to the ratios now used by astronomers
(Livio, 2002, p. 147).
Quantum Physics
Music theory has often inspired musicians and scientists; Jean-Philippe Rameau
was both. In The Acoustic Origins of Harmonic Analysis, Darrigol, Research Director at
the Centre National de la Recherche Scientifique in Paris, explains how D’Alembert
adopted “Rameau’s idea of founding musical harmony on the simultaneous hearing of
harmonics from a single sonorous body.” The study of overtones inspired Fourier’s heat
theory, Helmholtz’s acoustical studies, Bernoulli’s speculation that light might move in
waves, and partial calculus. It also led to the invention of harmonic analysis – a branch
of mathematics used in the study of string theory (Darrigol, 2007, p. 345). Although the
definitions of harmonic analysis in music and mathematics might seem diametrically
opposed, Darrigol shows how acoustic theories for the emission, perception, and propagation of sound “constantly bridged musical and mathematical harmonics” throughout the history of science (Darrigol, 2007, p. 344). His article underlines my contention that science could continue to learn a great deal from a specific analysis of all art forms. In his book, Fractal Horizons, Clifford Pickover suggests that music from the
Baroque and Classical eras is full of fractal structures put there by intuition alone. This EX NIHILO – Dahlgren 35
further proves that our creative perception “source” is rich and that we need to find a way
for science to glean even more from this new frontier (Pickover, 1996, p. 213).
Time-Space Continuum
Penrose and Tarasti are both interested in musical time. For Penrose musical time
is a way to understand more about the nature of time and how time might take some sort of spatial form, visualized rather than heard (Penrose, 1998, p. 444). This idea is what inspired my “drawing” a simple musical phrase that measures space in units of time, time by distance, making it possible for the entire entity to be visualized at a glance – frozen, as it were. Perhaps this is what Mozart referred to when he said he could “seize” a musical composition “at a glance” (Penrose, 1989, p. 444). This makes the aspect of temporality in music (the micro-time aspect as Tarasti calls it) a non-entity because the entire work would be visible from any angle for as long as was necessary, much like a painting. Otherwise, music exists only in micro-time bits and depends entirely on memory for an audience to internalize a performance in its entirety (Tarasti, 1994, p. 59).
Art to Science
Harmonic Series and Perspective Ratio Similarities
Fig. 23 - Harmonic Series – Perspective Ratio Similarities EX NIHILO – Dahlgren 36
Some rudimentary experimentation with perspective shows that, if your point-of-
view remains constant, equally spaced objects of the same size will have the ratios shown
at the above left. This ratio scheme also holds for the overtones in a harmonic series.
Compared to the first, the second object or interval ratio is, 2:1; the third, 3:1; the fourth
4:1, etc. These ratios are also similar for any two sequentially spaced objects: the second
to the first is 2:1; the third to the second, 3:2, etc. It is interesting that there are obvious
ratio similarities between sound (music) and light (art) despite the fact that sound waves
are not part of the electro-magnetic spectrum. (Fig. 23)
Escher and the Riemann Surface
Sara Robinson’s article, M.C. Escher: More Mathematics than Meets the Eye, is
about a mathematical exploration of the inner space in Print Gallery, a print by Escher
(Robinson, 2002). By using the theory of elliptical curves, Hendrik Lenstra, a number
theorist who holds joint positions at the University of California, Berkeley and
Universitiet Leiden, shows the distortion of the quayside scene depicted in Escher’s Print
Gallery can be described by a complex exponential function - a Riemann surface. This
article proves creative perception is a window to knowledge because, with no personal knowledge of mathematics, Escher has inspired mathematicians like Lenstra to explore inner spaces that would not exist without his vision. In fact, during their work, scientists hired artists to draw and shade unfinished sections for the “straightened version” of
Escher’s print. Lenstra and his team realized artistic expertise and intuition could do what complex topographical mathematics and algorithmic computer-generated hard copy could not. This article is proof that such an exploration is both possible and rewarding. I EX NIHILO – Dahlgren 37 invite the reader to go to this website to experience this phenomenon personally as words cannot do it justice: http://www.msri.org/people/members/sara/articles/siamescher.pdf
Artistic Perspective as Mathematics
I have quoted the next few paragraphs on perspective mathematics from an article by Andrejs Treibergs. The main point I am making here is that what is intuitive for artists is actually extremely complex when stated in mathematical terms. The mathematics below explains the perspective of a simple line diagram of a house:
We can measure the distances between pairs of points in the usual way using the Euclidean metric. If X1 = (x1, y1, z1) and P1 = (u1, v1) and so on, then
2 2 2 1/2 dist(X1, X2) = {(x1 - x2) + (y1 - y2) + (z1 - z2) } ,
2 2 1/2 dist(P1, P2) = {(u1 - u2) + (v1 - v2) }
The projection from X to P is called a parallel projection if all sets of parallel lines in the object are mapped to parallel lines on the drawing. Such a mapping is given by an affine transformation, which is of the form
= f(X) = T + AX
where T is a fixed vector in the plane and A is a 3 x 2 constant matrix. Parallel projection has the further property that ratios are preserved. That is if X1, X2, X3 and X4 are collinear points in the object, then the ratio of distances is preserved under parallel projection
Of course denominators are assumed to be non-zero (Treibergs, 2001).
Again, I invite the reader to visit Treibergs website: http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm
EX NIHILO – Dahlgren 38
Reversing the Process
Science to Literature
I always found it interesting that Lewis Carol (AKA Charles Lutwidge Dodgson) was a mathematician and in the Wikipedia article, http://en.wikipedia.org/wiki/Alice's_Adventures_in_Wonderland, there are many references to mathematical terminology in his writing:
“In chapter 1, Down the Rabbit-Hole, in the midst of shrinking, Alice waxes philosophic concerning what final size she will end up as, perhaps "going out altogether, like a candle."; this pondering reflects the concept of a limit. In chapter 2, The Pool of Tears, Alice tries to perform multiplication but produces some odd results: "Let me see: four times five is twelve, and four times six is thirteen, and four times seven is--oh dear! I shall never get to twenty at that rate!" This explores the representation of numbers using different bases and positional numeral systems (4 x 5 = 12 in base 18 notation; 4 x 6 = 13 in base 21 notation. 4 x 7 could be 14 in base 24 notation, following the sequence). In chapter 5, "Advice from a Caterpillar", the Pigeon asserts that little girls are some kind of serpent, for both little girls and serpents eat eggs. This general concept of abstraction occurs widely in many fields of science; an example in mathematics of employing this reasoning would be in the substitution of variables. In chapter 7, "A Mad Tea-Party", the March Hare and Mad Hatter give several examples in which the semantic value of a sentence A is not the same value of the inverse of A (for example, "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!"); in logic and mathematics, this is discussing an inverse relationship. Also in chapter 7, Alice ponders what it means when the changing of seats around the circular table places them back at the beginning. This is an observation of addition on a ring of the integers modulo N.”
Science to Music
Composing music using mathematical probability theory may have begun with
Mozart who loved to write musical motives that one would put together by throwing dice, a form of aleatory music. Composers have also begun using set theory and other mathematical devices to create music - discussed in Part V.
EX NIHILO – Dahlgren 39
Science to Art
Hofstadter tells of a form of computer art known as Parquet Tile Deformations that are forms of computer generated art that he compares to music (Hofstadter, 1985, p.
202). (Fig. 24)
Fig. 24 – Parquet Tile Deformations
Thought Energy
These examples show that thought energy seems to be able to take different forms yet communicate similar concepts and that some common background energy may permeate all thought. If this is so, how does it work? I feel that thought energy is fundamental and independent of nomenclature. Thought is primarily abstract and we posit thought energy into nomenclatures for communication purposes only. The following statements on non-verbality of thought are from The Emperor’s New Mind by
Roger Penrose:
Einstein: …words…do not seem to play any role in my mechanism of thought.
Hadamard: I insist that words are totally absent from my mind when I really think…
EX NIHILO – Dahlgren 40
Schopenhauer: …thoughts die the moment they are embodied by words.
(Penrose, 1989, p. 423-424)
If, as these illustrious people suggest, non-verbality of thought exists, then non-verbal thought (or at least thought unconnected to any sort of nomenclature) is an entity in and of itself. Therefore, I contend that we transcribe thought energy using whatever sign system with which we are competent, whether writing, composing, drawing, or some scientific method to encapsulate abstractions into more concrete forms (Langer, 1967, p.
153). Thought energy exists naturally and permeates our entire consciousness; in fact, everything (Penrose, 1989, p. 433).
Science has learned much from the arts. Scientists borrow ideas from ancient
Sanskrit texts to better understand quantum physics; music theory has given rise to the theoretical mathematics used in string theory and simple line drawings help scientists visualize what happens inside the giant CERN cyclotron. The arts help to symbolize non-verbal thought energy of every sort and complexity thus making ideas workable and dreams, reality.
EX NIHILO – Dahlgren 41
FORMS OF THOUGHT ENERGY
Thought is energy. Signs and symbols encapsulate this energy to communicate
ideas with language, art forms, and science. There are numerous examples of different manifestations of thought energy; some more concrete than others.
Zero Point Energy
The physicist, David Bohm, felt there was a certain background energy
permeating the universe. He calculated that, even in the vacuum of space, this energy
could be measured at 1038 ergs per square centimetre – the energy equivalent to ten
billion tons of uranium (Pearce, 1971, p. 9). This “Zero Point Energy” (ZPE) can confine
the power of an atomic explosion giving the mushroom cloud its form and, in various
permutations, I think it forms everything. Bohm believed objective reality does not exist
and “despite its apparent solidity the universe is at heart a phantasm;” for him, ZPE was
proof that reality was “a gigantic and splendidly detailed hologram” (Bohm, 2003, p. 27).
The one thing that seems to unite all disciplinary forms is the energy that exists in the
spaces between the units; perhaps this energy is aura. Scientists researching the creative
arts are studying not only thought energy but also the primal energy of the universe
because it seems to permeate not only art forms but also has “some active effect” on
every natural thing; in fact, something akin to dark matter (Penrose, 1989, p. 409).
Taking many forms, aura is thought energy released in the presence of an audience;
without an audience it would not exist.
Schools of Fish - Flocks of Birds
While snorkeling in Mexico, I watched a school of Surgeon fish swimming
beneath me. The school often changed direction but kept the same basic shape and I EX NIHILO – Dahlgren 42 wondered if the “force” that kept the school together was the same one that shaped the flock of Starlings I had recently seen flying in formation over Bird’s Hill Park near
Winnipeg. Such questions are the purview of swarm theory and quantum biology
(Massey/Fraser, 2003). I wondered if Zero Point Energy (ZPE), swarm theory
(explaining the force that holds animals in herds, flocks, or schools), and field theory
(explaining the force that holds atoms together) could explain the shapes of words
(Emerton, 1985, p. 134; Penrose, 1989, p. 436). Perhaps the brain taps into ZPE to “fire” synapses. Whether or not this is so, thought involves complex temporal and spatial processes that begin to become apparent in linguistic research.
Encapsulating Aura in Language
Figure 25 shows a spectrogram of someone saying the words “Bishop to Queen
Knight Three” (Hunt, 1982, p. 344). It shows the energy patterning of brain activity during speech and I think it is interesting to compare the rhythm of the spectrogram to the language rhythm as I have transcribed it. There is an almost exact match between them. EX NIHILO – Dahlgren 43
Fig. 25 - Spectogram of “Bishop to Queen Knight Three”
Lexical Semantics
In their article, How Does Consciousness Happen, Greenfield and Koch espouse somewhat similar theories on how thought becomes a conscious entity. Susan Greenfield states, “neurons across the brain synchronize into coordinated assemblies, then disband” and Cristof Koch writes, “a unique set of neurons in a particular brain region fire in a specific manner” (Greenfield/Koch, 2007, p. 76). These theories describe neurons firing in particular brain regions and, therefore, depend on spatial relations; however, these events also occur in a coordinated synchronized manner suggesting that temporal relations, like rhythm, play a large part in the process (Glezerman/Balkoski, 1999, p. 3). EX NIHILO – Dahlgren 44
Elizabeth Ahlsén depicts such space-time coordination in her discussion of Lexical
Semantics. (Ahlsén, 2006, p. 93). (Fig. 26)
Fig. 26 Lexical Semantics
It seems to me that how propriocpetive thought (high brain activity) is transliterated from firing synapses to sounds or symbols recognized as language is
presently speculative theory. However, if synapses firing somehow select phonemes,
then spoken language is thought energy captured in spaces between the sounds and
symbols used for communication (Ogden, 1949, p. 42). This may explain how concepts
are captured by commonly agreed-upon word forms. It may also explain why it is often
extremely difficult – if not impossible – to translate certain words from one language to
another: there are verb tenses in Greek that cannot be translated or even conceptualized in
English; Arabic uses equations and verbal nouns (Mace, 2007, p. 69). Spoken language
has many characteristics in common with music as well: rhythm, pitch, tempo, timbre,
dynamics, and articulation, to name a few; therefore, I think music can be useful in the
study of neurolinguistics and linguistics. Like Lidov, I contend spoken language is a form of music (Lidov, 2005). Perhaps we communicate by modulating the space-time between neuron stimulations using neutral symbols best suited to our propensities much like the programmed electrical impulses and algorithms that control artificial intelligence EX NIHILO – Dahlgren 45 or broadcast wave-forms (Hofstadter, 1985, p. 396; Penrose, 1989, p. 405). Paralleling this with Being and Nothingness makes both being and nothingness interchangeable. In language, thought energy may be transmitted in empty spaces between language neutral sound symbols called phonemes (Ahlsén, 2006, p. 113).
Phonemic Function
Phonemic function is the essence of spoken thought transmission. Spoken phonemic symbols are simply mediums of transmission; therefore, nothing is the very being of the transmission - the symbols, the being, represent, in fact, nothing (Sartre,
1981). Like Ogden, I feel that words – in fact all art forms - act like capacitors holding the energy that formed them (Ogden, 1949, p. 42). This word energy exists in the syllables and the smallest phonological units of language, phonemes, phonemic particles, and even smaller subdivisions; they all occur in time, thus have rhythm
(Fromkin/Rodman, 1983, p. 56). Phonemic combinations in different languages may differ in sound with similar meanings or similar sounds and different meanings; due to the insubstantive nature of language, this is of no consequence, since it is the energy they carry that matters (Derrida, 1968, pp 278 – 299). (Fig. 27) (Vertical letters denote “al.”)
Fig. 27 - English/Arabic Words for “The House” (Read Arabic from Right to Left) EX NIHILO – Dahlgren 46
The Arabic word, “bayt” sounds very much like the English word, “bite”. One chooses from a learned set of phonemes to make the syllables, words, phrases, and sentences to express an initially abstract non-verbal thought. If a child can do it, how hard can it be?
Language Rhythm
Some years ago, my grandson, Zachary, and I were in a library. Since I was busy
with researching a paper, I sent him on a quest for an interesting book that I hoped would occupy him while I worked. To my dismay, he returned quickly without a book; when I
asked him why, he gave the following response, ‘They got no pop-up books, Papa.’ In figure 28A, the sentence, ‘They got no pop-up books, Papa.’ shows syllabic rhythm
(note-stems down) and phonemic rhythm (note-stems up). In figure 28B, Zachary adds a phoneme, an ‘uh’ (U), to the word ‘got’ to make his point. No such phoneme exists in the word and I believe that this shows that Zachary understands phonemes and uses them for emphasis. Figure 28C shows phonemic particle rhythm for each of the phonemes
(Dahlgren, 2005). EX NIHILO – Dahlgren 47
Fig. 28 - Zachary Speaking to His “Papa” in the Henderson Public Library,
Winnipeg
These complex naturally occurring phenomena for expressing non-verbal thought in a more concrete form are impressive in themselves. But there is also a sophisticated word placement hierarchy at work based on musical beat “weight” in Zachary’s negative statement.
Negation Levels
“The introduction of alterity or negativity into the signifying unit” is a very
important aspect of Julia Kristeva’s work in linguistics (Kristeva, 1997, p. 67). My
grounded theory research report on the effects of language rhythm on negativity in EX NIHILO – Dahlgren 48 speech shows that there are inherent levels of negation (Dahlgren, 2005, p. 15). In music, note rhythm and beat “weight” are both very important. The same is true for spoken language perhaps because brain function seems to have both spatial and temporal characteristics (Ahlsén, 2006, p. 185).
Negation, and our sensitivity to it, takes place at many different levels of speech depending on the amount of emphasis we wish to employ. It constantly shifts, and I wondered why this happened. Figure 29-9A shows Level I negation which always occurs on strong beats and leaves no doubt as to non-agreement. Figure 29-9B shows Level II negation in a transcription of Zachary’s speech. Note the complexity and multiplicity of levels in this simple sentence. Levels I and II mark the 1st and 2nd divisions of the beats and sub-levels 1, 2 and 3 mark the strong, medium, and weak beats in the ¾ time signature. The negation that takes place is type II, as it occurs in the 2nd half of the first strong beat and is important because it is the first word in level II. In figure 29-9B
Zachary ‘hides’ his negation on a lower level perhaps because he hopes to promote empathy having returned without a book – contrary to my wishes. Words near the top of the diagram are generally more important than those at the bottom. Beat strength indicates word hierarchy, i.e. the words on the strong beats carry the meaning: “got no
Papa.” “They” has the lowest possible importance for Zachary because “they” are unable to comply with his wishes. Frankly, I found the level of language competence quite amazing especially when one considers that all this is done in milliseconds by a six-year- old child! By some mysterious preconscious process an intrinsic abstraction becomes an extrinsic expression; the semiotic, symbolic; genotext, phenotext (Kristeva, 1997, p. 25). EX NIHILO – Dahlgren 49
Fig. 29 - Zachary - Speech Levels in Negation
The thetic phase or mimesis captures the thought energy and transforms it by
transcription into speech. All thought probably undergoes much the same transliteration from intrinsic to extrinsic form for the purpose of communication; this is as true for a little boy talking to his “Papa” about the lack of pop-up books in a library as it is for
Mozart writing his fortieth symphony or Picasso drawing a portrait of Igor Stravinsky
(Langer, 1967, p. 153). Are the words in figure 29 the result of background control? Do the language rhythm levels underlying them control phoneme usage much like the melody in Gounod’s Ave Maria depending on a tune by Bach? Spoken language is temporal but part of its thought process seems a-temporal – even a-spatial; a seeming
“dual reality” contradiction unless one views it in the light of creative thought. If so, EX NIHILO – Dahlgren 50
what other background controls rule both linear and lateral thought? What shapes these
energies, forms their abode, and controls how they abide there?
Encapsulating Aura in Art Forms
Architecture as Space
Still considered one of the world’s foremost architectural critics, Bruno Zevi is famous for his theory of architecture as the science of creating space (Zevi, 1957). His
revolutionary idea made us look not only at the edifice that created the space but also the
space formed by it. I decided to expand this idea to include literature, music, and art. All
of these disciplines use different materials to enclose the inner spaces that house the
energy of our thoughts; novels, symphonies, and paintings act much like capacitors that
store energy in inner spaces formed by words, notes, and lines.
Art-forms as Aura Capacitors
Much like speech, art forms encapsulate an inert energy within their various
structures only released by an audience interacting with it. In literature, this energy exists
as plot in the spaces between literary devices: lists, descriptions of scenery, characters,
and events interspersed with dialogue. Plot is the energy of the story – the thing
encapsulated by words; everything and nothing; “the name of the form attached not only
to the form in an eternal connection; but something else which, not being the form, yet
never exists without it, is also entitled to be called by that name.” Plato, Phaedo
(Emerton, 1985, p. 79). Music, too, is more dependent on the spaces between the notes -
melodic or harmonic intervals - than the notes themselves. Linear intervals define
melody; lateral intervals, harmony. The nothingness - the relative distances or spaces of
silence between the notes - is the real stuff of music; “Without silence there is no music; EX NIHILO – Dahlgren 51
there is only noise.” - Beethoven. In art, line delineates the shape of an object, not the
form. The viewer can only see the form as a secondary spatial relationship resulting from
unconscious markings guided by a force or energy separating the objects known as
negative space. The apple in a still life only exists because the nothingness surrounding it
gives it shape (Arnheim, 1974, p. 96). An artist limits the negative space in order to
outline an illusion. The differences or similarities in the formal structures of literature,
music, and art may be explained in part by how their respective energies form an illusion
by filling the voids between the symbols (Langer, 1979, p. 120).
Toward an Exploration of Inner Space
Slavoj Žižek is a social critic and Senior Researcher of the Institute for Social
Studies in Ljubljana. The German philosophers, Schelling, Hegel, and Marx are his
primary interest but he is also working toward reinvigorating Jacques Lacan’s criticism of
psychoanalysis as it pertains to linguistics and semiotics. Žižek is a radical thinker but I
find myself agreeing with many of his ideas – especially those linking invention and aura
in his discussion of the virtual reality of ideas and their “timeless…dream-like…pseudo-
existence” (Žižek, 2007, p. 52). His book is one that has inspired me to discover and map
the inner spaces of cultural representations - the essence of this paper. If words capture energy there is even greater energy encapsulated in literature, music, and art and there may be a way of charting forms of that energy by over-stepping the traditional limits of studying form within the confines of nomenclature (Bourdieu, 1992, p. 37). In so doing, perhaps we can draw a picture of that energy as it existed in the mind’s eye before an artist shared it with the world. Maybe it is that same energy or creative perception that reconstitutes when an audience interacts with a shared vision be it a play, a musical EX NIHILO – Dahlgren 52 performance, or a painting. Reconstituting a more concrete form of any vision may, in some manner, give us the means to research thought forms existing in the inner spaces of the minds of both artist and audience or, for that matter, a six year old boy.
If different methods of communication encapsulate thought energy there should be some proof of this phenomenon. But how can one produce material evidence of such an ethereal premise? By shifting the paradigm of theoretical analysis, using symbols other than those commonly used in specific disciplines, one should be able to see literature, music, or art in a different light. Analysts must split the nucleus of the art form in order to explode the sacrosanct and limiting constraints of its specific codes in order to understand those codes more fully. Interdisciplinary exploration – using language to study mathematics or mathematics to study music – is the only way we can see the inner workings of any form and expand academe.
EX NIHILO – Dahlgren 53
FORMS OF INNER SPACE
“We are such stuff as dreams are made on.” Prospero’s epilogue to Shakespeare’s
The Tempest – one of my favorite plays because it takes place in “real time” – tells of the insubstantial nature of our world and our lives; yet, for us, there is substance to our being and our environment. But this very substance may be nothing more than the stuff of dreams because, without thinking making it so, nothing exists for us. Perhaps “reality” is all a dream, a fabrication of creative thought from the inner space of a mind - without which nothing can exist in any form.
Shapes of Sound
Hofstadter asks, “What is the secret magic of Chopin?” He also wonders how mere notational patterns can hold within them such heart-felt emotion (Hofstadter, 1989, p. 188). Figure 30 shows note patterns on the first page of six different etudes by
Chopin; one need not be a musician to see the obvious differences between them. EX NIHILO – Dahlgren 54
Fig. 30 – Visual Textures of Six Chopin Etudes
Eero Tarasti, a Finish musicologist interested in semiotics, has been working almost exclusively on the music of Chopin. Using the theories of Greimas and Peirce, he has done extensive analytical research on a number of Chopin’s piano works, most notably narrativity in Polonaise-Fantasie, Op. 61, wherein Chopin seems to not only tell a story but set the scene as well (Tarasti, 1994, p. 138).
Soundscapes
“Soundscape” is a word coined by the Canadian composer, R. Murray Schafer, to mean the sound-world we live in. For example, Mozart’s soundscape was very different from ours; it would have been relatively quiet and comprised of sounds like carriages, horses, children playing, and the conversations of people walking by on the cobblestones EX NIHILO – Dahlgren 55
outside his house in Vienna. Music also creates soundscapes. Robert Morgan’s work on
musical time and musical space explores how music exists in a space-time continuum: it
takes time to perform and the sounds created fill either the space of the concert venue or a
given three dimensional orb in the open air bounded by the distance at which the sounds
become inaudible (Morgan, 1980, pp. 527-538). Soundscapes, then, have both natural
and cultural representation; they are not only temporal and spatial, but also multi-
dimensional, entities. The Canadian composer, Monte Keene Pishny-Floyd, has many
ideas about two-dimensional shape in music; this is especially true regarding his ideas on
Schoenberg whose music, he writes, can give one the feel of a Mondrianesque still-life
(Pishny-Floyd, 2006, p. 150). He also brings language and mathematics to bear in his
rather unique analysis of Schoenberg’s Opus 25 by introducing ideas on the verb-like
quality of the Zeitmotivierungsfunktion (ZMF) as well as the mathematical permutations
of the music (Pishny-Floyd, 2006, pp. 152 - 156).
Iannis Xenakis studied music with Arthur Honegger and Olivier Messiaen, and
architecture with Le Corbusier. He pioneered the use of computers and mathematical models in the compositional process and was interested in both the two and three- dimensional aspects of music. His first composition, Metastasis (1954), originated from
architectural design theory; it is music directly and indirectly transliterated from drawings
to a musical score (Xenakis, 2001). Especially in the second movement of Metastasis,
mathematical formulae, such as the Fibonacci sequence, structure note placement.
Xenakis turned the graphic score of Metastasis into a blueprint for the Philips Pavilion
for the 1958 Brussels World's Fair. By transliterating the music into mathematical grids
– curved glissandos became hyperbolic paraboloids - he transformed the score into a EX NIHILO – Dahlgren 56
building, ultimately turning music – sound - into a three-dimensional structure. (Fig.
31A) More detail on Metastasis and its performance can be found in Appendix B and at:
http://wapedia.mobi/en/Metastasis_(Xenakis_composition).
Fig. 31A - Philips Pavilion for the 1958 Brussels World's Fair
Many of Xenakis’ scores are graphic in nature. Figure 31B shows one page of the
score for Mycenae- Alpha for mono tape to be projected onto either two or four sound
sources around the audience. It was composed on the UPIC graphic/computer system at
the Centre d’Etudes de Mathématiques et Automatique Musicales in Paris. I include this
work because of the visual quality of the score; it is music and art in a mathematically measurable entity that can be “read” by a machine – truly a mixture of the arts and sciences (Xenakis, 1987, pp. 12-15). The full score is in the Appendix. EX NIHILO – Dahlgren 57
Fig. 31B - Xenakis’ Mycenae – Alpha (Second Page of the Score)
Musical Topography
Music is a three dimensional landscape in sound (Morgan, 1980, pp. 527-538).
Claudette Caron, a concert pianist, speaks of “sculpting sound” when she performs. So it is that a kind of virtual multi-dimensional space exists in musical performance, something I feel is akin to all art-forms (Langer, 1953, p. 75). In literature, there are innumerable examples of this: the Brontes describe the Yorkshire moors, the buildings on them, and the rooms within those buildings. In music, there is the circle of fifths – keys within keys; in art, there are the many suggested spaces in a landscape and levels within the suggested perspective of a painting. Via an audience, a multiplicity of worlds can exist within the limited dimensions of a page (Bourdieu, 1992, p. 44).
Hofstadter introduces the idea of dimensions within dimensions in art. Inner space in art, music, and mathematics is one small part of the “many worlds interpretation of quantum mechanics” (Hofstadter, 1985, p. 469). Randall suggests such worlds are passages not only to the infinitely large but also the infinitely small with vectors traveling in every imaginable, or even unimaginable, direction (Randall, 2006, p. 27). Infinite EX NIHILO – Dahlgren 58 interpretations of literature, music, and art – even mathematics – are made possible through the “fractal” nature of any “dreamed-up” cultural representation because, although any “book” is frozen in a state-of-being chosen by its creator, it still has vestiges of the infinite. The multi-dimensionality of virtual space depends solely on our imagination. Further, I contend that it is possible to draw these inner spaces, to transliterate them and lift them from the page, to make them into “explorable” space.
Drawing Inner Space
My method for drawing the inner space of music stems from Art and Visual
Perception by Rudolf Arnheim. Figure 32 shows a two dimensional (2D) retina image transferred into a three dimensional (3D) object by the cortex (Arnheim, 1974, p. 260).
Fig. 32 - The Cortex Recreates a 3D Image from a 2D Retinal Image.
The retinal image is a 2D inverted mirror image of what could be a 3D object and it is up to the cortex to make the necessary mental adjustments to make the viewer aware of that fact. My idea was to use this process, to make what might be construed as a two- dimensional “image” of music into a three-dimensional form. This quest was initiated by a mental image described by Mozart, “the thing really gets to be almost completed in my head…so that thereafter I survey it in my mind at one glance, like a beautiful picture…right away all together” (wie gleich alles zusammen) (Arnheim, 1974, p. 374). EX NIHILO – Dahlgren 59
Since I also compose, I know that the thing I envision is never the same as the published
music nor is it necessarily music at all; it is more a shape that becomes transliterated into
notes. So why not reverse the process and transliterate notes into a shape?
Music Matrices
My idea was to use the notes on the score as a two-dimensional object that could be turned into a three dimensional object using a matrix invented by Dmitri Tymoczko – a form of musical set-theory. The figures shown below all represent some aspect of the
inner space of music. This concept is not new; theorists in ancient Greece, like
Pythagoras, studied the mysteries of music and this has continued via scientist-musicians
like Galileo, Fourier, and Rameau. Present day communication specialists like
Hofstadter and music theorists like Hook, Morgan, Tymoczko all allude to music as a
method of stating complex topographical mathematics like global-quotient orbifolds or a
Möbius strip (Darrigol, 2007, p. 343; Hofstadter, 1985, p. 173; Tymoczko, 2006, p. 73).
“Diatonic set theory investigates the subtle and beautiful relationship between the
12-note chromatic scale and the diatonic scales such as the C major scale, with seven
unequally-spaced notes per octave” (Hook, 2007, p. 49). Figure 33 is a map of the
harmonic path of a passage from Beethoven’s Violin Sonata, Op. 24 (“Spring”), second
movement, measures 37-54 using a neo-Riemannian Tonnetz by Julian Hook. The lines
running diagonally from left to right represent the circle of fifths; from right to left, the
chromatic scale – C = 0, through B = 11; horizontal, minor thirds; vertical, major thirds
(Hook, 2007, pp. 49-50). (This 2D graph is actually an unwrapped 3D torus; therefore, it
is a 2D representation of 3D space.) EX NIHILO – Dahlgren 60
Fig. 33 - Julian Hook’s Matrix
Dmitri Tymoczko’s Matrix, below, differs from Hook’s Matrix but both methods essentially do the same thing: turn 2D space into 3D space by combining the linear and lateral aspects of music (Tymoczko, 2006, p. 73). This 2D graph is actually an untwisted
Möbius Strip. (Fig 34) EX NIHILO – Dahlgren 61
Fig. 34 - Tymoczko’s Matrix
Each of the three Harmonic and three Melodic Matrices linked in figure 35 are a representation of Tymoczko’s Matrix. The numbers correspond to note names and these note names are spaced across the matrix as shown. The “squares” in my drawing are matrices and the notes on these matrices would be found roughly in the positions I have marked in the figure “Harmonic and Melodic Matrices”. (Fig. 35) EX NIHILO – Dahlgren 62
Fig. 35 - Harmonic and Melodic Matrices
I find it interesting that these matrices are so interdependent. If the horizontal axes
represent space and the vertical axes time, then “spatial” displacement seems necessary
for “temporal” alignment in the “harmonic matrix,” whereas “temporal” displacement seems necessary for “spatial” alignment in the “melodic matrix” almost as though time
becomes space; space, time. (The squares in the melodic matrix should be vertical.) It is interesting to compare the harmonic and melodic matrices above with the two matrices below in figures 36 and 37. Like the harmonic and melodic matrices in Figure 35, both the Galilean and Poincaré matrices are artistic representations of space-time (Penrose,
1998, p. 200). (Fig.’s 36 and 37)
Fig. 36 - Galilean Space-Time Matrix EX NIHILO – Dahlgren 63
Fig. 37 - Poincaré Space-Time Matrix
My contention is that there can be even more complex matrices, as alluded to by
Gollin in his discussion of the Tonnetz – a musical set-theory where C = 0, C# = 1…B =
11. Gollin’s representation is a purely theoretical representation of a Tonnetz; however, it shows the possibility of using a similar method for drawing the inner structure of music in three-dimensional space (Gollin, 1998, p. 202). (Fig. 38) Is this a shadow of 4D space?
Fig. 38 - Gollin Tonnetz
EX NIHILO – Dahlgren 64
3D Tonnetz Mapping
I felt that a drawing combining all aspects and representations of musical space
might be able to capture the inner space of music and perhaps the mental picture alluded
to by Mozart. Alas, my drawing is only a representation of the first three notes of Three
Blind Mice harmonized in three voices. I tried something more difficult, a minuet by
Bach, but drawing it became untenable and I (much like the fox in Aesop’s Fable)
decided that simpler was better. (Fig. 39)
Fig. 39 - Harmonization of the First 6 Notes of Three Blind Mice
A Journey to the Centre of the Mind
I harmonized the tune in three voices because four voices seemed less clearly
defined on a graph. A note about the harmony: I realize the vii6 chord has no root – the B
is in parentheses. Using Tymoczko’s matrix, the next step was to assign the chord notes specific points on the matrices. Note range determined the height of the 3D box-frames.
Please note that, although the notes look like they are at different levels their individual matrices align with notes on the staff. (Fig. 40) EX NIHILO – Dahlgren 65
Fig. 40 - Assigning Chord Note Matrix Points - Three Blind Mice
The final step was to draw the inner space that the notes outline by combing the
“boxes”. (Fig. 41)
Fig. 41 - The Inner Space of Three Blind Mice
Obviously, this is not the most stimulating picture; however, it does depict the
inner space of the notes on the page and may be a representation of the kind of “picture”
Mozart spoke of when he “saw” music in his mind’s eye. If this works one way, from the page to inner space, there is no reason to believe it could not work in reverse by transliterating the “picture” or vision into musical notation. This idea is not unlike
reversing Arnheim’s idea of a 3D object perceived in 2D on the retina and then
reconstituting itself in the cortex in 3D. Music involves a kind of 3D sight-hearing that
exists first in the cortex as 3D space, then on paper as 2D space and then, via
performance, re-envisioned by an audience as 3D space once more. (Fig. 42) EX NIHILO – Dahlgren 66
Fig. 42 - 3D Inner Space Recreated in 2D.
Recreating this shadow world by hologram technology to a degree where it could
become an entity capable of exploration may be a means of studying the inner space or
mental picture of all art forms. The most exciting aspect of this type of research is the
resulting hologram entity would be a 3D depiction of creative perception giving scientists
the capability of a leisurely exploration of a mind, thought from another time and place,
without invasive procedure. Please refer to this article on neurobiology: http://www.orgone.org/articles/ax7ignc1.htm.
Exploring Inner Space
The surfaces in figure 41 would not be flat but curved since twists and bends
would occur in order to join the points that represent notes - much like twisting one end
of a piece of cardboard while holding the other end straight. If this is the case, these
curved surfaces may be capable of being represented and studied by Reimann Surface
topography. An engineer suggested that these twists would also cause stresses
measurable by Power Spectral Density meters. Does this mean that the twists within the
inner space fabric of music manifest aura? Further, each note point in figure 41
represents not only one point in a given 3D space but also a plethora of overtone note
points stretching away to infinity. By imagining each of those infinite note points as EX NIHILO – Dahlgren 67
point zero of a 3D vector axis (x, y, z axes) with unlimited possibilities of orientation,
you begin to approach a concept very similar to the Hilbert Space theories. More than
the magic of the Möbius Strip or a musical Klein bottle exists within this inner space
thought-world (Penrose, 1989, p. 257). Even though the surfaces in figure 41 are devoid
of substance and may only exist on some energy plain, it would be interesting to explore
the possibilities of something formed from nothing but the energy of pure thought. But
this “something” may be less ethereal than it appears; these forces may well form more
material things.
In her book, The Scientific Reinterpretation of Form, Norma Emerton, a crystallographer from Oxford University, seeks to find what forces create forms in nature
(Emerton, 1984). In The Emperor’s New Mind, Roger Penrose explores tilings and quasicrystals composed of aluminum, lithium, and copper forming seemingly impossible crystal symmetry (Penrose, 1989, p. 434). (Fig. 43) Synchrotron technology produces X- rays that can permeate metallic crystals; then topographical mathematics is used to produce an image of a hypothetical diffraction pattern with the information gathered by the X-rays. The inner space formed by Three Blind Mice in figure 44 reminds me
somewhat of the crystal formations referred to by Emerton and Penrose. Therefore, the
mathematics used in synchrotron technology by crystallographers may provide a method
for studying the inner spaces of music; perhaps such a study may even shed some light on
crystallography (http://en.wikipedia.org/wiki/Crystallography). If the forces binding
crystals and music are similar, perhaps such force fields unify all art forms; perhaps
everything. EX NIHILO – Dahlgren 68
Fig. 43 – Al-Li-Cu Quasicrystal
Fig. 44 - Crystallized Inner Space
We see what we expect to see, so finding 3D shapes in music is exciting because
it is totally unexpected; it changes our world-view and opens our minds to unanticipated
possibilities. It is a journey “through the looking glass” to “The palm at the end of the
mind” and beyond – into a land of birds with dangling fire-fangled feathers. A
symphonic score may be both a picture of neurons firing and a non-algorithmic depiction
of crystallographic phenomenon. If so, music theory may explain much more than
harmonic progression. If thought is a common bond linking art and science then exploring inner space may elicit some interesting observations – perhaps even conclusions (Žižek, 2007, p. 151). If three harmonized notes in Three Blind Mice contain this much information, imagine what wonders The St. Matthew Passion must hold!
EX NIHILO – Dahlgren 69
Of Mere Being – Wallace Stevens
The palm at the end of the mind, Beyond the last thought, rises In the bronze décor,
A gold-feathered bird Sings in the palm, without human meaning, Without human feeling, a foreign song.
You know then that it is not the reason That makes us happy or unhappy. The bird sings. Its feathers shine.
The palm stands on the edge of space. The wind moves slowly in the branches. The bird’s fire-fangled feathers dangle down.
Both Shakespeare and Stevens – along with many other writers too numerous to name – have created worlds. The worlds of Prospero and the “gold-feathered bird” are as real to us as anything could be. They exist in the inner spaces of our minds along with all our memories of lived events, music heard, plays seen, food tasted, sculpture felt, and the perfume of flowers. Worlds unseen are as real as anywhere we have been. We live in the yet to be and that is what science is beginning to understand. Nothing is the very
substance of our being.
EX NIHILO – Dahlgren 70
SUMMARY
Science has already benefited from a cursory exploration of the arts; expanding such exploration can only lead to discoveries never before imagined. To integrate disciplines we must find some common ground beyond disciplinary jargon for any meaningful communication to take place; this is especially true with an amalgamation of science and the arts. Because similar concepts exist between these two thought-worlds, we must begin to create a vocabulary for interdisciplinary communication far beyond the short glossary in this paper. Inner spaces may differ in the details but what constitutes musical inner space can be re-constituted as language, flocks of birds, schools of fish, small particle physics, cosmology, even human society. In fact, all infinitely small or infinitely large groupings of anything can be re-constituted and studied as forms of inner space; one day, such a study may lead to the beginnings of a unified field theory. Artistic thought is a common link to scientific truth; after all, groupings - and the proximatey of the units within those groupings - be they musical notes or atoms, are interspersed by essentially the same thing - nothing.
EX NIHILO – Dahlgren 71
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from the World Wide Web
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EX NIHILO – Dahlgren 78
APPENDICES
A) The full score of Mycenae – Alpha by Iannis Xenakis
Page 1
Page 2
EX NIHILO – Dahlgren 79
Page 3
A performance of Mycenae – Alpha is available at the following address:
http://www.youtube.com/watch?v=yztoaNakKok&feature=related
B) Metastasis by Iannis Xenakis
A full score and performance is available at the following address:
http://www.youtube.com/watch?v=SZazYFchLRI