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Musical Instruments 1 Musical Instruments 2 Introductory Question Sound can break glass. Which is most likely to break: A. A glass pane exposed to a loud, short sound A glass pane exposed to a certain loud tone A crystal glass exposed to a loud, short sound A crystal glass exposed to a certain loud tone Musical Instruments B. C. D. Musical Instruments 3 Observations about Musical Instruments They can produce different notes They must be tuned to produce the right notes They sound different, even on the same note They require power to create sound Musical Instruments 5 Musical Instruments 4 Why do strings produce specific notes? Why does a vibrating string sound like a string? Why do stringed instruments need surfaces? What is vibrating in a wind instrument? Musical Instruments 6 Question 1 4 Questions about Musical Instruments Why do strings produce specific notes? Oscillations of a Taut String A taut string has a mass that provides it with inertia a tension that provides restoring forces a stable equilibrium shape (straight line) restoring forces proportional to displacement A taut string is a harmonic oscillator It oscillates about its equilibrium shape Its pitch is independent of its amplitude (volume)! 1 Musical Instruments 7 Musical Instruments 8 A Taut String’s Pitch Stiffness of a string’s restoring forces are set by Fundamental Vibration the string’s tension the string’s curvature (or, equivalently, length) The inertial characteristics of a string are set by the string’s mass per length A string has a fundamental vibrational mode in which it vibrates as a single arc, up and down, with a velocity antinode at its center and velocity nodes at its two ends Its fundamental pitch (frequency of vibration) is proportional to its tension, inversely proportional to its length, and inversely proportional to its mass per length Musical Instruments 9 Musical Instruments 10 Question 2 Why does a vibrating string sound like a string? Overtone Vibrations A string can also vibrate as two half-strings (one extra antinode) three third-strings (two extra antinodes) etc. These higher-order vibrational modes Musical Instruments 11 Musical Instruments 12 A String’s Harmonics (Part 1) A string’s overtones are special: harmonics First overtone involves two half-strings Twice the fundamental pitch: 2nd harmonic One octave above the fundamental frequency A String’s Harmonics (Part 2) Integer overtones are called “harmonics” Bowing or plucking a string excites a mixture of fundamental and harmonic vibrations, giving the string its characteristic sound Second overtone involves three third-strings have higher pitches than the fundamental mode and are called “overtones” Three times the fundamental pitch: 3rd harmonic An octave and a fifth above the fundamental Etc. 2 Musical Instruments 13 Musical Instruments 14 Question 3 Why do stringed instruments need surfaces? Projecting Sound In air, sound consists of density fluctuations Vibrating strings barely project sound because air can’t flow around surfaces easily and is substantially compressed or rarefied Musical Instruments 16 Plucking and Bowing air flows around thin vibrating objects and is only slightly compressed or rarefied Surfaces project sound much better because Musical Instruments 15 Air has a stable equilibrium: uniform density Disturbances from uniform density make air vibrate Plucking a string transfers energy instantly Bowing a string transfers energy gradually Bow does a little work on the string every cycle Excess energy builds up gradually in the string This gradual buildup is resonant energy transfer Introductory Question (revisited) Sound can break glass. Which is most likely to break: A. A glass pane exposed to a loud, short sound A glass pane exposed to a certain loud tone A crystal glass exposed to a loud, short sound A crystal glass exposed to a certain loud tone The string will vibrate sympathetically when another object vibrates at its resonant frequency and it gradually obtains energy from that object Musical Instruments 17 B. C. D. Musical Instruments 18 Question 4 What is vibrating in a wind instrument? Oscillations of Air in a Tube Air in a tube has a mass that provides it with inertia a pressure distribution that provides restoring forces a stable equilibrium structure (uniform density) restoring forces proportional to displacement Air in a tube is a harmonic oscillator It oscillates about its equilibrium density distribution Its pitch is independent of its amplitude (volume)! 3 Musical Instruments 19 Musical Instruments 20 Fundamental Vibration Open-Open Column Air in a Tube’s Pitch Stiffness of the air’s restoring forces are set by the air’s pressure the air’s pressure gradient (or, equivalently, length) The inertial characteristics of the air are set by Air column vibrates as a single object the air’s mass per length Pressure antinode occurs at column center Pressure nodes occur at column ends Pitch (frequency of vibration) is proportional to air pressure inversely proportional to column length inversely proportional to air density Musical Instruments 21 Fundamental Vibration Open-Closed Column Air column vibrates as a single object Musical Instruments 22 Air Harmonics (Part 1) Pressure antinode occurs at closed end Pressure node occurs at open end Air column in open-closed pipe vibrates as half the column in an open-open pipe at half the frequency of an open-open pipe In open-open pipe, the overtones are at twice fundamental (two pressure antinodes) three times fundamental (three antinodes) etc. (all integer multiples or “harmonics”) In open-closed pipe, the overtones are at three times fundamental (two antinodes) five times fundamental (three antinodes) etc. (all odd integer multiples or “harmonics”) Musical Instruments 23 Musical Instruments 24 Air Harmonics (Part 2) Blowing across the column tends to excite a mixture of fundamental and harmonic vibrations Examples Organ pipes Recorders Flutes Whistles Surface Instruments Most 1-dimensional instruments Most 2- or 3- dimensional instruments can vibrate at half, third, quarter length, etc. harmonic oscillators with harmonic overtones have complicated higher-order vibrations harmonic oscillators with non-harmonic overtones Examples: drums, cymbals, bells Reeds and horns also use a vibrating air column 4 Musical Instruments 25 Drumhead Vibrations Musical Instruments 26 Summary of Musical Instrument use strings, air, etc. as harmonic oscillators pitches independent of amplitude/volume tuned by tension/pressure, length, density often have harmonic overtones project vibrations into the air as sound 5