From Music to Mathematics
From Music to Mathematics
SKU:9781421419183
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About the Author
About the Author
Table of Content
Table of Content
<P>Preface<BR>Acknowledgments<BR>Introduction<BR>1. Rhythm<BR>1.1. Musical Notation and a Geometric Property<BR>1.1.1. Duration<BR>1.1.2. Dots<BR>1.2. Time Signatures<BR>1.2.1. Musical examples<BR>1.2.2. Rhythmic repetition<BR>1.3. Polyrhythmic Music<BR>1.3.1. The least common multiple<BR>1.3.2. Musical examples<BR>1.4. A Connection with Sanskrit Poetry<BR>References for Chapter 1<BR>2. Introduction to Music Theory<BR>2.1. Musical Notation<BR>2.1.1. The common clefs<BR>2.1.2. The piano keyboard<BR>2.2. Scales<BR>2.2.1. Chromatic scale<BR>2.2.2. Whole-tone scale<BR>2.2.3. Major scales<BR>2.2.4. Minor scales<BR>2.2.5. Why are there 12 major scales?<BR>2.3. Intervals and Chords<BR>2.3.1. Major and perfect intervals<BR>2.3.2. Minor intervals and the tritone<BR>2.3.3. Chords<BR>2.4. Tonality, Key Signatures, and the Circle of Fifths<BR>2.4.1. The critical tonic-dominant relationship<BR>2.4.2. Key signatures<BR>2.4.3. The circle of fifths<BR>2.4.4. Transposition<BR>2.4.5. The evolution of polyphony<BR>References for Chapter 2<BR>3. The Science of Sound<BR>3.1. How We Hear<BR>3.1.1. The magnificent ear-brain system<BR>3.2. Attributes of Sound<BR>3.2.1. Loudness and decibels<BR>3.2.2. Frequency<BR>3.3. Sine Waves<BR>3.3.1. The sine function<BR>3.3.2. Graphing sinusoids<BR>3.3.3. The harmonic oscillator<BR>3.4. Understanding Pitch<BR>3.4.1. Residue pitch<BR>3.4.2. A vibrating string<BR>3.4.3. The overtone series<BR>3.4.4. The starting transient<BR>3.4.5. Resonance and beats<BR>3.5. The Monochord Lab<BR>References for Chapter 3<BR>4. Tuning and Temperament<BR>4.1. The Pythagorean Scale<BR>4.1.1. Consonance and integer ratios<BR>4.1.2. The spiral of fifths<BR>4.1.3. The overtone series revisited<BR>4.2. Just Intonation<BR>4.2.1. Problems with just intonation<BR>4.2.2. Major versus minor<BR>4.3. Equal Temperament<BR>4.3.1. A conundrum and a compromise<BR>4.3.2. Rational and irrational numbers<BR>4.3.3. Cents<BR>4.4. Comparing the Three Systems<BR>4.5. Strähle's Guitar<BR>4.5.1. An ingenious construction<BR>4.5.2. Continued fractions<BR>4.5.3. On the accuracy of Strähle's method<BR>4.6. Alternative Tuning Systems<BR>4.6.1. The significance of log2(3/2)<BR>4.6.2. Meantone scales<BR>4.6.3. Other equally tempered scales<BR>References for Chapter 4<BR>5. Musical Group Theory<BR>5.1. Symmetry in Music<BR>5.1.1. Symmetric transformations<BR>5.1.2. Inversions<BR>5.1.3. Other examples<BR>5.2. The Bartók Controversy<BR>5.2.1. The Fibonacci numbers and nature<BR>5.2.2. The golden ratio<BR>5.2.3. Music for Strings, Percussion and Celesta<BR>5.3. Group Theory<BR>5.3.1. Some examples of groups<BR>5.3.2. Multiplication tables<BR>5.3.3. Symmetries of the square<BR>5.3.4. The musical subgroup of D4<BR>References for Chapter 5<BR>6. Change Ringing<BR>6.1. Basic Theory, Practice, and Examples<BR>6.1.1. Nomenclature<BR>6.1.2. Rules of an extent<BR>6.1.3. Three bells<BR>6.1.4. The number of permissible moves<BR>6.1.5. Example<BR>6.1.6. Example<BR>6.2. Group Theory Revisited<BR>6.2.1. The symmetric group Sn<BR>6.2.2. The dihedral group revisited<BR>6.2.3. Ringing the cosets<BR>6.2.4. Example<BR>References for Chapter 6<BR>7. Twelve-Tone Music<BR>7.1. Schoenberg's Twelve-Tone Method of Composition<BR>7.1.1. Notation and terminology<BR>7.1.2. The tone row matrix<BR>7.2. Schoenberg's Suite für Klavier, Op. 25<BR>7.3. Tone Row Invariance<BR>7.3.1. Using numbers instead of pitches<BR>7.3.2. Further analysis<BR>7.3.3. Tritone symmetry<BR>7.3.4. The number of distinct tone rows<BR>7.3.5. Twelve-tone music and group theory<BR>References for Chapter 7<BR>8. Mathematical Modern Music<BR>8.1. Sir Peter Maxwell Davies<BR>8.1.1. Magic squares<BR>8.1.2. Some examples<BR>8.1.3. The magic constant<BR>8.1.4. A Mirror of Whitening Light<BR>8.2. Steve Reich<BR>8.2.1. Clapping Music<BR>8.2.2. Phase shifts<BR>8.3. Xenakis<BR>8.3.1. A Greek architect<BR>8.3.2. Metastasis and the Philips Pavilion<BR>8.3.3. Pithoprakta<BR>8.4. Final Project<BR>8.4. References for Chapter 8<BR>Credits<BR>Index</P>
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