Musica Mathematica: Traditions and Innovations in Contemporary Music (Methodology of Music Research)

The concept of «musica mathematica__»__ seeks to accurately examine the intersection of two seemingly radically different subject areas. From the perspective of a European perception, the definition of the science of music was a result of the Pythagorean concept of universal harmony. The Pythagoreans were the first in European culture to raise the issue of uniting music and mathematics, sound and number. In the three parts of the monograph, versatile cases of the intersection of music and mathematics are displayed, moving from philosophical and aesthetic considerations about __mathesis__ to practical studies, discussing the interaction between music and other kinds of art (architecture, painting, poetry and literature), and providing a practical research of contemporary music compositions. Cover 1 Contents 7 List of Analyzed Music Scores 11 Author’s Preface 13 Foreword: Mathesis as a Philosophy of the Beauty of Music 17 Part 1 A Retrospective of the Traditions of Musica Mathematica 23 1. The Constructive Relationships between Music and Mathematics: The Pythagorean Conception of Universal Music and its Spread in the Worldview of Later Periods 25 1.1. Music and the Theory of the Quadrivium 26 1.2. Review of the Harmony of Spheres 28 1.3. The Phenomenon of Mathesis Universalis 37 1.4. Expression of Numerical Proportions and Progressions 39 The Fibonacci Progression and the Golden Ratio Phenomenon 40 1.5. Ars Combinatoria and the Constructivism of Music 46 2. Semantic Interpretation of the Interaction between Music and Mathematics: Mystic Middle Ages and the Sacral Baroque 53 2.1. Semantics of the Kabbalah in Music 54 2.2. Symbolic Thinking and Sacral Numerology 57 2.3. Codes of the Numerical Alphabets in Music 62 3. Constructive Aspects of the Interaction between Music and other Arts 71 3.1. “Frozen Music”: Dialogues between Music and Architecture 71 3.2. Ut Pictura Musica: The Interaction between Music and Art 77 3.3. Musical Cryptography as a Common Denominator of the Sound and the Word 81 Application of the Morse Alphabet 91 4. Musica Mathematica in Practice: Aspects of Analysis 95 Part 2 The Renewal of Mathematical Techniques in Musical Compositions of the 20th and 21st Centuries 101 1. Constructive Aspects of Music Composition 105 1.1. The Implications of Numerical Proportions and Progressions in Music 105 1.1.1. The Number Proportions and Progressions of Antiquity 106 Olivier Messiaen. Quatre études de rythme (1949–1950) 108 Snieguolė Dikčiūtė. The Mystery of Seven Bridges (1991) 109 Steve Reich. Music for Pieces of Wood (1973) 110 Siegfried Thiele. Proportionen (1971) 114 John Cage. First Construction (1939) 118 1.1.2. Fibonacci and Number Sequences Derived from it 123 Derek Bourgeois. Symphony for Organ, Op. 48 (1975) 125 Steve Reich. Clapping Music (1972) 127 Karlheinz Stockhausen. Klavierstück IX (1961) 129 1.1.3. The Prime Numbers and other Mathematically Determined Sequences 131 Alain Louvier. L’Isola dei Numeri (1991) 131 Tom Johnson. Music for 88 (1988) 133 1.2. The Renewal of Polytempo, Polyrhythm, and Polymeter 139 John Adams. Short Ride in a Fast Machine (1986) 140 Conlon Nancarrow. Study for Player Piano No. 15 (premiere in 1962) 142 György Ligeti. Désordre (1985) 145 1.3. Symmetrical Algorithms and the Confrontation between Symmetries and Asymmetries 148 Morton Feldman. Crippled Symmetry (1983) 151 1.4. Transformational Elements (Combinatorics, Permutations, Rotations) 153 Olivier Messiaen. Île de feu II (1950) 154 Tom Johnson. Tango (1984) 159 Tom Johnson. The Chord Catalogue (1986) 160 2. Semantic Aspects of Music Composition 165 2.1. Cosmological Number Codes and Graphic Constructions 165 2.2. The Symbols of Magic Number Squares 170 Šarūnas Nakas. Ziqquratu (1998) 172 Dmitri Smirnov. Two Magic Squares (1971) 174 2.3. The Implications of Sacred Numbers 177 Bronius Kutavičius. The Gates of Jerusalem (1991–5) 180 Snieguolė Dikčiūtė. The Mystery of the Seven Bridges (1991) 183 2.4. Personalized Semantics: The Significance of Individual Numbers 185 George Crumb. Black Angels (1970) 187 John Cage. Ryoanji (1983) 193 Mikalojus Konstantinas Čiurlionis. Fugue in B-flat, Op. 34/VL 345 (1908–9) 198 Part 3 Innovations of Mathematical Techniques in 20th and 21st Century Music 203 1. The Mathematized Musical Graph 207 1.1. A Geometric Prototype as an Algorithm for Musical Composition 207 Jan Rokus van Roosendael. Rotations (1988) 208 John Adams. China Gates (1977) 213 1.2. L-system Formalities in Music 215 Tom Johnson. Piece No. 14 from the cycle Rational Melodies (1981) 218 Gary Lee Nelson. Summer Song (1991) 219 1.3. The Numericalization of the Musical Score 220 Morton Feldman. IXION (1958) 220 2. Implications of Modern Mathematical Theories 225 2.1. The Practice of Algorithmic Music and Computer-Generated Composition 225 2.2. Tonal Adaptations of Complicated Mathematical Processes 230 Iannis Xenakis. Nomos alpha (1965) 240 2.3. Fractal Theory Analogies in Musical Compositions 241 Šarūnas Nakas. Ziqquratu (1998) 245 Vytautas V. Jurgutis. Fractals (1999) 247 Charles Dodge. A Fractal for Wiley Hitchcock (1989) 250 Afterword 255 Bibliography 259 Supplementary Literature on the Subject of this Book 271 Index of Names 279 About the Author 287 The author displays versatile cases of the intersection of music and mathematics, moving from philosophical and aesthetic considerations about "mathesis" to practical studies, discussing the interaction between music and other arts, and providing a practical research of contemporary music compositions.