1. Musical Instruments 1 Musical Instruments

2. Musical Instruments 2 Introductory Question Sound can break glass. Which is most likely to break: Sound can break glass. Which is most likely to break: A. A glass pane exposed to a loud, short sound B. A glass pane exposed to a certain loud tone C. A crystal glass exposed to a loud, short sound D. A crystal glass exposed to a certain loud tone

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

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

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

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

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

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

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

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

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

12. Musical Instruments 12 Introductory Question (revisited) Sound can break glass. Which is most likely to break: Sound can break glass. Which is most likely to break: A. A glass pane exposed to a loud, short sound B. A glass pane exposed to a certain loud tone C. A crystal glass exposed to a loud, short sound D. A crystal glass exposed to a certain loud tone

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

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

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

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

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

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

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

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

21. Musical Instruments 21 Figures