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Music and Mathematics – Focusing on the Numerical Systems of the Pre-Columbian Middle America

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Music and Mathematics – Focusing on the Numerical Systems of the Pre-Columbian Middle America Hudební fakulta Janáčkovy akademie múzických umění v Brně Composition, Conducting and Opera Direction Department Subject: Music Composition and Theory of Composition Academic Year: 2011/2012 MgA. Edgar Omar Rojas Ruiz Music and Mathematics – Focusing on the Numerical Systems of the Pre-Columbian Middle America Doctoral Dissertation Supervisor: Profr. Ing. MgA. Ivo Medek PhD. Brno 2012 1 2 By the current, I declare that this Doctoral Dissertation was entirely worked out by myself, and all the information, and quotes that it might include will be properly documented. Brno, 7th September of 2012 Prohlašuji, že jsem tuto disertační práci vypracoval samostatně a uvedl v ní veškerou literaturu a další zdroje informací, ze kterých jsem čerpal. V Brně dne 7. září 2012 MgA. Edgar Omar Rojas Ruiz 3 The conception and writing of this work would not be possible without the precious time and support of the following people to whom I would like to deeply render thanks: Profr. Ing. MgA. Ivo Medek PhD. For his great assessment, guidance and remarkable knowledge; to my teachers Profr. Arnost Parsch, Dr. Dan Dlouhy, Doc. Jaroslav Stastny, Dr. H.C. María Antonieta Lozano, Dr. Víctor Rasgado, Dr. Jorge Sosa, Dr. Rodrigo Sigal, MgA. Tomás Barreiro, MgA. Enrico Chapela, Guadalupe Sotres and Alejandro Velasco for leading me with wisdom through this long but beautiful journey of music, always with full patience and dedication. Special thanks to my family, to my wife Marketa and to my friends Jitka Brizova, Rodrigo Gonzalez, Lubomir Kopkas, Jose M. Marquez, Richard Pohl, and Kostiantyn Tyshko for their patience and never letting me down in the difficult moments. 4 In the loving memory of my grandfather Antonino Rojas Vargas ( 1933 - 2011) 5 INDEX Introduction 9 1. Influential composers for the development of this research 13 2.The Mayan Civilization 23 2.1. A brief review on the Mayan History 23 2.1.1. Mayan Pre – Classical period (1000 BC. - 320 AC.) 23 2.1.2. The Mayan Classical period (320 - 987 AC.) 24 2.1.3. The Mayan Post - Classical period (1000 - 1687 AC.) 26 2.2. The social structure of the Mayan Civilization 27 2.3. The Mayan economical organization 28 2.4. The Mayan religion 29 2.5. The arts in the Mayan civilization 31 2.5.1. Architecture 31 2.5.2. Ceramics 35 2.5.3. Sculpture 36 2.5.4. System of writing 38 2.5.5. Music 39 2.5.6. Painting 40 2.5.7. Literature 41 a) “Chilam Balam” 41 b) “Popol Vuh” 48 3. The Mayan numerical system 50 3.1. Basis of organization in the Mayan numerical system 50 3.2. The numerical relations of the four cardinal points according to the Mayans 52 3.3. The Mayans and the measure of the time 54 3.3.1. Introduction to the Mayan Calendars 54 3.3.2. The basic Mayan time measures 54 3.3.3. The Mayan Sun Calendar “Haab” 55 3.3.4. The Mayan Moon Calendar “Tzolkin” 57 3.3.5. The hidden structures of the Tzolkin 64 3.3.6. The long count and the short count 69 (turns of “Katunes” according to the Chilam Balam) 3.4. Summary of the The Mayan mystical numbers 70 6 4. Possible applications of the Mayan numerical system in the Music Composition 72 4.1. The Mayan Rows 72 4.1.1. The Total Mayan Row 72 4.1.2. The Basic Mayan Row 73 4.2. Possibilities of construction of melodic devices by the use of the Mayan Row applied to the twelve tone equal temperament 75 4.2.1. Translation of the Basic Mayan Row into tones 75 4.2.2 Possibilities of extension of the Mayan Hexachord 76 4.2.2.1. Linear extension by extremes 76 4.2.2.2. Linear extension trough parallel symmetrical axis 79 4.2.2.3. Extension trough parallel symmetrical axis in two dimensions 83 4.2.2.4. Fake extension trough parallel symmetrical axis in two dimensions 88 4.3. Possibilities of construction of harmonic devices by the use of the Mayan Row an the Mayan Hexachord 92 4.3.1. The Mayan Harmonic Modules 92 4.3.1.1. The Basic Mayan Modules 93 4.3.1.2. The False Mayan Modules 99 4.3.2. The Mayan Structural Chords 104 4.3.2.1. Basic Structural Chords 104 4.3.2.2. False Structural Chords 109 4.3.3. Fractals of Harmonic Construction and Deconstruction 113 4.3.3.1. Numerical/ Harmonic Pyramids 114 4.3.3.2. Simple – Unipolar Intervalic Diamonds 123 4.3.3.3. Simple - Bipolar Intervalic Diamonds 129 4.3.3.4. Mixed – Unipolar Intervalic Diamonds 133 4.3.3.5. Possibilities of reading of the Mixed – Unipolar Intervalic Diamonds 135 a) Readings of Homogeneous Poles 136 b) Readings of Heterogeneous Poles 138 4.3.3.6. Mixed – Bipolar Intervalic Diamonds 142 4.3.3.7. Possibilities of reading of the Mixed – Bipolar Intervalic Diamonds 144 a) Readings of Homogeneous Poles 145 b) Readings of Heterogeneous Poles 145 7 4.4. Possibilities of design of rhythmical devices by the use of the Mayan Row and the Mayan Hexachord 146 4.4.1. Translation procedures 146 4.4.2. Translation of the Mayan Row and its extensions into rhythmical devices 151 4.4.3. Translation of the numerical structure of the Extended Mayan Hexachord to rhythmical devices 169 4.4.3.1. Translation of the numerical structure of the Extended Mayan Hexachord by linear extremes 169 4.4.3.2. Translation of the numerical structure of the extended Mayan Hexachord through parallel symmetrical axis. 173 4.4.4. Translation of the Mayan Structural Chords into rhythmical devices 176 4.4.4.1. Translation of the Basic Structural Chords (Basic Rhythmic - Structural Patterns) 176 4.4.4.2. Translation of the False Structural Chords (False Rhythmic - Structural Patterns) 188 5. The Multistructural possibilities of the Tzolkin 198 5.1. Translation of the Tzolkin into notes 198 5.1.1. Direct translation 198 5.1.2. Indirect translation 199 5.2. Substitution of Numerical Values of the Tzolkin by Harmonic Devices 212 5.3. Applications of the Hidden Structures of the Tzolkin to the Music Composition 221 Conclusions 232 Summary 234 Appendix 235 Attachment 1 236 Attachment 2 258 Attachment 3 301 Attachment 4 320 Sources 339 8 INTRODUCTION Mexico - Tenochtitlan, 1521 After almost two years of a bloody war between the Mexicas (Aztecs) and the Spanish conquerors, the great city of Tenochtitlan, capital of the Aztec Empire, was turned into something so similar to a hell on the earth; even when the Aztec warriors were able to keep to the Spaniards away from the city during one year after the biggest defeat that the peninsular army suffered in the „battle of the sad night“, the inhabitants of Tenochtitlan had to face different enemies which were not less deathly than the foreign invaders. The hunger, the lack of drinking water and an aggressive epidemic of smallpox that was killing thousands of Aztecs; in the anonymous manuscripts of Tlatelolco (Mexico), the defeated survivors of this war describe this situation as follows: „The last days of the siege of Tenochtitlan“ And all of this happened with us, We lived it, We admired it. With this sorrowful and sad luck, we found ourselves distressed. In the roads lie broken darts, the hairs are spread. Our houses are destroyed and their walls are painted on red The worms were emerging from the streets and squares, and the walls are full of rests of human brain, the water was red, like painted and when we drank it, the sensation was like drinking saltpeter water.1 The Spanish troops took advantage of this situation to organize a definitive attack to Tenochtitlan, where they were joint by an army of 100,000 indigenous allies, coming mostly from tribes under the Aztec military domain. Cuahutemoc, the last Aztec emperor, was captured by the army of Hernán Cortés in 1521, the indigenous ruler asked to the Spanish captain to get a dignifying death, nevertheless, Cortés used his condition of emperor to force to the Aztecs to rebuilt the destroyed empire and after four years of constant humiliation and brutal torture, Cuahutemoc was murdered by the Spanish conquerors. In the following years to the fall of the great Tenochtitlan, the Spaniards were able to quickly expand their domain in the continent, mainly in the territories what today are the center and the south of the American continent; among these conquered lands were some of the territories included in the so called “Mayan Area” (South of Mexico, Guatemala, El Salvador and Honduras), where one of the most advanced civilizations of the the continent was located: The Mayans, whose last city to fall under the Spanish control was Tayasal in the year 1697. This was how two of the greatest civilizations of the world history came to their end; as the swards of the Spanish soldiers and the rage of the unknown diseases for the indians finished with the lives of so many brave Aztec and Mayan warriors, a new god would travel from Europe to the new lands, to spread his conception of love in the hands of his priests, the soldiers of faith who changed the 1 LEÓN PORTILLA Miguel: “ La visión de los vencidos”; Universidad Nacional Autónoma de México, Biblioteca del Estuiante Universitario, Coordinación de Humanidades, Programa Editorial. 23rd Edition 2003. Printed in Mexico City. ISBN 968 – 36 – 8202 – 2. P. 160 – 161. 9 scarify stones for crosses and the pagan idols for saints, the ones who burnt the „evil“ indigenous codices to replace them with prayers.
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