A Smoother Pebble: Mathematical Explorations by Donald C. Benson

By Donald C. Benson

Книга A Smoother Pebble: Mathematical Explorations A Smoother Pebble: Mathematical Explorations Книги Математика Автор: Donald C. Benson Год издания: 2003 Формат: pdf Издат.:Oxford collage Press Страниц: 280 Размер: 11,1 ISBN: 0195144368 Язык: Английский0 (голосов: zero) Оценка:This e-book takes a unique examine the themes of college mathematics--arithmetic, geometry, algebra, and calculus. during this walk at the mathematical beach we are hoping to discover, quoting Newton, "...a smoother pebble or a prettier shell than ordinary..." This booklet assembles a suite ofmathematical pebbles which are very important in addition to appealing.

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Example text

The ear can detect the distinctive timbre of the clarinet or the flute. A sound spectrum breaks down a complex tone into the frequency components, together with the corresponding amplitudes. A device that achieves this decomposition is called a frequency analyzer. The Helmholtz resonator, named after the German scientist Hermann von Helmholtz (182194),9 is the oldest and simplest frequency analyzer. Helmholtz found that a spherical vessel with suitable dimensions resonates at a particular frequency with very little contamination from nearby frequencies.

Using logarithmic horizontal and vertical scales, this graph shows, for each given frequency, the widths of several frequency intervals with the given frequency as midpoint: (1) the critical bandwidth; (2) the width of the interval of maximum dissonance, 25% of the critical bandwidth. (3) The width of the intervals of the fifth, major third, and major second. Note that these standard musical intervals are perceived more dissonant at lower frequencies. (b) The perception of dissonance varies as the frequency difference between two simultaneous simple tones is varied from 0 up to the critical bandwidth.

In air at 68°F, sound waves travel at a speed of 770 miles per hour. An ear or a microphone detects the changing pressure at a particular point in space. The ear transmits this information to the brain. The electrical output from a microphone can be used to activate a loudspeaker, but it might also generate a graph of pressure versus time. The simplest musical sound is called a simple tone or a sine tone. 2. The horizontal coordinate is time, and the vertical coordinate is pressure, which varies about an equilibrium steady-state value.

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A Smoother Pebble: Mathematical Explorations by Donald C. Benson
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