Music: A Mathematical OfferingSince the time of the Ancient Greeks, much has been written about the relation between mathematics and music: from harmony and number theory, to musical patterns and group theory. Benson provides a wealth of information here to enable the teacher, the student, or the interested amateur to understand, at varying levels of technicality, the real interplay between these two ancient disciplines. The story is long as well as broad and involves physics, biology, psycho acoustics, the history of science, and digital technology as well as, of course, mathematics and music. Starting with the structure of the human ear and its relationship with Fourier analysis, the story proceeds via the mathematics of musical instruments to the ideas of consonance and dissonance, and then to scales and temperaments. This is a must-have book if you want to know about the music of the spheres or digital music and many things in between. |
Contents
Introduction | 1 |
A mathematicians guide to the orchestra | 91 |
Consonance and dissonance | 144 |
the fivefold way | 161 |
More scales and temperaments | 210 |
Digital music | 245 |
Synthesis | 265 |
Symmetry in music | 312 |
Appendix A Bessel functions 361 | xiii |
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Common terms and phrases
Acoust Amer amplitude approximation beginning bells called cents century Chapter comma common Computer consists consonant constant continued convergence corresponding defined described differential discussion effect element envelope equal equal temperament equation example Exercise Figure file first follows formula Fourier transform frequency function fundamental Further reading given gives harmonic integral interval major third mathematical means meantone method minor modes modulation motion multiples Music namely notation notes obtain octave operator original partials perfect fifth periodic permutation pitch played positive Press problem Pythagorean q q q ratio recorded represents result rows sample scale Section semitones Show signal sine wave solution sound spectrum square string symmetry synthesis temperament theorem theory tone tuning twelve usually vectors vibration write zero