p288_05_w11 - PAT 102/Physics 288/489 Winter 2011 L e c tu...

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PAT 102/Physics 288/489 Winter 2011 Lecture 5 Graphs Logs Cents
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Periodic Waves and Vibrations • Waves and oscillations are characterized by – Frequency f – Period P (or sometimes T ) – Amplitude A – Wavelength (for waves) λ – Speed (for waves) v • For simple harmonic motion v = " f = / P P = 1/ f f ! Restoring force Inertia period
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Examples of Simple Harmonic Motion • Mass on a spring – Mass m and spring constant k • Simple pendulum – Length L • Vibrating stretched string – Tension T , length L and mass per unit length μ f = 1 2 ! k m f = 1 2 g L f = 1 2 L T μ Frequency is independent of the amplitude
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Galileo’s studies of Vibrations • Realized that pendulums provide a good way of measuring time – Needed for his careful studies of motion – Ideas about timing probably influenced by his father and musician Vincenzo Galilei • After age 70 and under house arrest, Galileo returns to pendulums, vibrations and music – Published results in a book: Discourses on two new sciences
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Galileo’s Idea About Consonance • Why are consonant musical intervals related to simple numerical ratios? • Galileo’s argument: – Musical tones are associated with oscillations – Consonance occurs when the oscillations form a regular repeating pattern Interval Ratio Unison (prime) 1:1 Octave 2:1 Perfect fifth 3:2 Perfect fourth 4:3 Major third 5:4
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Consonance of the Octave “Agreeable consonances are pairs of tones which strike the ear with a certain regularity. The first and most pleasing consonance is, therefore, the octave, for every pulse given to the tympanum by the lower string, the sharp string delivers two.” A 4 A 5 One cycle Two cycles The pulses are “in sync” every cycle of the A 4 oscillation and every other cycle of the A 5 oscillation 2:1 ratio
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Consonance of the Fifth “The fifth, however, is characterized by its displaced beats, that is, by the interposition of two solitary beats of the upper, and one of the lower, between each case of simultaneous pulses. Moreover, these three are separated by time intervals one-half of that which separates simultaneous pulses from pulses of the upper string. Thus the effect of the fifth is to produce a tickling of the eardrum, so that its gentleness is modified by sprightliness, giving the impression simultaneously of a gentle kiss and a bite.” A 4 Two cycles E 5 Three cycles 3:2 ratio
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The Major Triad • The major triad is a three note chord with a major third and a perfect fifth – The frequency ratios are f 2 / f 1 = 5/4 (major third) f 3 / f 1 = 3/2 (perfect fifth) – This can be expressed as f 1 : f 2 : f 3 = 4 : 5 : 6 • We can demonstrate this with three pendulums – Lengths: L 1 = 1 m L 2 = 16/25 m L 3 = 4/9 m C E G f 1 f 2 f 3 same as 6/4
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This note was uploaded on 01/29/2012 for the course PHY 288 taught by Professor Jamesliu during the Winter '11 term at University of Michigan.

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p288_05_w11 - PAT 102/Physics 288/489 Winter 2011 L e c tu...

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