Music: A Mathematical Offering
Cambridge University Press, 2007 - 411 Seiten
Since 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.
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A mathematicians guide to the orchestra
Consonance and dissonance
the ﬁvefold way
More scales and temperaments
Symmetry in music
Appendix A Bessel functions 361
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Acoust Amer amplitude approximation basilar membrane cents Computer Music Journal consonant continued fraction convergence corresponding cosets CSound deﬁned deﬁnition described differential equation drum element envelope equal temperament equal tempered scale example Figure ﬁlter ﬁnd ﬁnite ﬁrst ﬁve ﬁxed FM synthesis formula Fourier coefﬁcients Fourier series Fourier transform frequency ratio fundamental Further reading given gives graph harmonic instrument integral interval intonation Jn(z major third mathematical meantone scale meantone temperament minor modes modulation multiples notation notes obtain octave perfect ﬁfth periodic function permutation piano pitch class Pythagorean comma q q q quarter comma meantone reﬂection represents satisﬁes score ﬁle Section semitones signal sine wave solution sound string symmetry syntonic comma theorem theory tone equal temperament tone rows tone scale triad tuning twelve tone unison vectors vibration wave equation waveform Wendy Carlos zero