Musical Instruments 1 Musical Instruments Musical Instruments 2

Musical Instruments 1 Musical Instruments

Musical Instruments 2 Introductory Question n 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 B. C. D.

Musical Instruments 3 Observations about Musical Instruments They can produce different pitches n They must be tuned n They sound different, even on same pitch n Sound character is adjustable n Both require power to create sound n Can produce blended or dissonant sounds n

Musical Instruments 4 Strings as Harmonic Oscillators n A string is a harmonic oscillator Its mass gives it inertia n Its tension gives it a restoring force n It has a stable equilibrium n Restoring forces proportional to displacement n n Pitch independent of amplitude (volume)!

Musical Instruments 5 n String’s Inertia and Restoring Forces String’s restoring force stiffness set by string’s tension n string’s curvature (or, equivalently, length) n n String’s inertial characteristics set by n string’s mass per length

Musical Instruments 6 Fundamental Vibration n String vibrates as single arc, up and down velocity antinode occurs at center of string n velocity nodes occur at ends of string n This is the fundamental vibrational mode n Pitch (frequency of vibration) is n proportional to tension n inversely proportional to string length n inversely proportional to mass per length n

Musical Instruments 7 Overtone Vibrations n String can also vibrate as two half-strings (one extra antinode) n three third-strings (two extra antinodes) n etc. n These are higher-order vibrational modes n They have higher pitches n They are called “overtones” n

Musical Instruments 8 String Harmonics, Part 1 n In a string, the overtone pitches are at n twice the fundamental frequency n One octave above the fundamental frequency n Produced by two half-string vibrational mode n three times the fundamental frequency n An octave and a fifth above the fundamental n Produced by three half-string vibrational mode n etc.

Musical Instruments 9 String Harmonics, Part 2 Integer overtones are called “harmonics” n Bowing or plucking a string tends to excite a mixture of fundamental and harmonic vibrations, giving character to the sound n

Musical Instruments 10 Producing Sound n Thin objects don’t project sound well Air flows around objects n Compression and rarefaction is minimal n n Surfaces project sound much better Air can’t flow around surfaces easily n Compression and rarefaction is substantial n n Many instruments use surfaces for sound

Musical Instruments 11 Plucking and Bowing Plucking a string transfers energy instantly n Bowing a string transfers energy gradually n Rhythmic excitation at the right frequency causes sympathetic vibration n Bowing always excites string at the right frequency n The longer the string’s resonance lasts, the more effective the gradual energy transfer n

Musical Instruments 12 Introductory Question (revisited) n 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 B. C. D.

Musical Instruments 13 Air as a Harmonic Oscillator n A column of air is a harmonic oscillator Its mass gives it inertia n Pressure gives it a restoring force n It has a stable equilibrium n Restoring forces proportional to displacement n n Pitch independent of amplitude (volume)!

Musical Instruments 14 n Air’s Inertia and Restoring Forces Air’s restoring force stiffness set by pressure n pressure gradient (or, equivalently, length) n n Air’s inertial characteristics set by n air’s mass per length (essentially density)

Musical Instruments 15 Fundamental Vibration Open-Open Column n Air column vibrates as a single object Pressure antinode occurs at column center n Pressure nodes occur at column ends n n Pitch (frequency of vibration) is proportional to air pressure n inversely proportional to column length n inversely proportional to air density n

Musical Instruments 16 Fundamental Vibration Open-Closed Column n Air column vibrates as a single object Pressure antinode occurs at closed end n Pressure node occurs at open end n n Air column in open-closed pipe vibrates as half the column in an open-open pipe n at half the frequency of an open-open pipe n

Musical Instruments 17 Air Harmonics, Part 1 n In open-open pipe, the overtones are at twice fundamental (two pressure antinodes) n three times fundamental (three antinodes) n etc. (all integer multiples or “harmonics”) n n In open-closed pipe, the overtones are at three times fundamental (two antinodes) n five times fundamental (three antinodes) n etc. (all odd integer multiples or “harmonics”) n

Musical Instruments 18 Air Harmonics, Part 2 n Blowing across column tends to excite a mixture of fundamental and harmonic vibrations

Musical Instruments 19 Other Instruments n Most 1 -dimensional instruments can vibrate at half, third, quarter length, etc. n harmonic oscillators with harmonic overtones n n Most 2 - or 3 - dimensional instruments have complicated higher-order vibrations n harmonic osc. with non-harmonic overtones n n Examples: drums, cymbals, glass balls

Musical Instruments 20 Summary of Musical Instrument use strings and air as harmonic oscillators n pitches independent of amplitude/volume n tuned by tension/pressure, length, density n have harmonic overtones n project vibrations into the air as sound n

Musical Instruments 21 Figures