Preface Author Bios Chapter 1 Introduction 1.1 Tones and notes 1.2 Scale examples 1.3 Equal temperament scales 1.4 Advantages of equal-temperament 1.5 Tone network 1.6 Chord network 1.7 Algebra of chords Chapter 2 Modes and chords 2.1 Mode 2.2 Directed chord and chord 2.3 Reduction of generalized modes 2.4 Submodes and supermodes 2.5 Complementary modes 2.6 Mode shifts and directed chord rotations 2.7 Number of chords and modes 2.8 Number of mode classes 2.9 Counting chords and modes Chapter 3 Subchords 3.1 Uniqueness of chords 3.2 Inverted modes 3.3 Relative inverted chords 3.4 Inverted chords 3.5 Structure of chords 3.6 Invariant subchords 3.7 Symmetric modes 3.8 Trichord examples 3.9 Trichords sharing two notes Chapter 4 Trichords 4.1 Chord extension [A,B,C] 4.2 Extension with a new chord 4.3 Tonnetz example 4.4 Chord extension [A,B,A] 4.5 Chord extension [A,2A,A] 4.6 Tonal cell [A,B,C] 4.6.1 Simplified diagram 4.7 Chord cell 4.8 Tonal cells [A,B,A] and [A,2A,A] 4.9 Major and minor chords in a 12-TET scale Chapter 5 Higher-dimensional chords 5.1 Higher-dimensional tone network 5.2 Modular structure 5.3 Distance on the Tonnetz 5.4 Non-degenerate Tonnetz 5.5 Generalized tonal cell 5.6 Generalized Tonnetz 5.7 Generalized chord network 5.8 Chord cell facets 5.9 Tetrachords Chapter 6 Operations on the root 6.1 Translations and inversions on directed chords 6.2 Translations on chords 6.3 Dependent translations 6.4 Prograde translations by mode intervals 6.5 Retrograde translations by mode intervals Chapter 7 Operations on the mode 7.1 Positive and negative inversions 7.2 Retrogradation and shifts 7.3 Mode intervals notation 7.4 Properties 7.5 Neighbor chords 7.6 Operating rules for transpositions 7.7 Translations by mode intervals 7.8 Relationships involving shifts and translations Chapter 8 Chord transformations 8.1 Operations on root and mode 8.2 Inversion of chords 8.3 Inversion and mirror by x 8.4 Properties 8.5 Rotations 8.6 Drifts along edges 8.7 Simple circuits 8.8 Shortcut circuits Chapter 9 Chord network 9.1 Some families of chords 9.2 Referring a chord to different cells 9.3 Co-cycles, co-cells, and congruent cells 9.4 Dependent operations on chords 9.5 Single translations 9.6 Translations towards one cell 9.7 Translations towards different cells 9.8 Honeycomb of trichords Bibliography Index
Biography
Rafael Cubarsi, mathematician and physicist by training, received his PhD from the Astronomy Department of the Universitat de Barcelona in 1988 with a dissertation on Chandrasekhar Stellar Systems. He developed and conducted research at the Universitat Aut`onoma de Barcelona and Universitat Polit`ectica de Catalunya, where he used to teach, and has more than 70 published papers in peer-reviewed journals to his credit. His research focused on the fields of Astronomy & Astrophysics and Mathematical Biology. Recently his interest is centered on Mathematical Theory of Music.
