Physics of the Piano Piano Tuners Guild, June 5, 2000

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Physics of the PianoPiano Tuners Guild, June 5,
2000

• Charles E. Hyde-Wright, Ph.D.
• Associate Professor of Physics
• Old Dominion University
• Norfolk VA
• chyde_at_odu.edu

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The Physics of Music and Musical
ReproductionAutumn 2000, Mon.Wed.
420-535pm Call 18539
ODU PHYS 332W
Topics Physical attributes of music and
sound. Acoustics of musical instruments and
concert halls. Electronic generation and
recording of sound. Neuro-physiology of sound
perception.
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Physics of the Piano

• Oscillations Sound
• Vibrations of a String
• Travelling waves Reflections
• Standing Waves
• Harmonics
• Piano acoustics
• Hammer action
• Sound Board
• Multiple Strings
• Chords, Scales Tuning

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Why does a mass on a spring oscillate?

• It is not because I push it
• The mass continues long after I let go.
• The spring is pushing on the mass.
• Why doesnt the mass just come to rest in the
  middle?
• After all, the spring(s) exert no (net) force on
  the mass when it is exactly in the middle.
• No force seems like no motion (wrong).

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Force and Motion

• The mass moves even when nothing is pushing
• The mass moves because of inertia
• Forces do not cause motionforces cause motion to
  change
• Force is proportional (mass) to the time rate of
  change of motion (acceleration)
• F ma
• A force acting to left either
• Makes the mass go faster to left or
• Makes the mass slow down as it moves to right

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Net Force causes motion to change

• Fma
• The time rate of change of velocity
  (acceleration) is proportional (mass) to the net
  applied force.
• A force acting to left either
• Makes the mass go faster to left or
• Makes the mass slow down as it moves to right

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Fma Mass on a Spring

• For a Spring, F ? -kx
• Force pushes towards middle
• Force grows with distance
• Force -kx and Force ma
• Acceleration a - (k/m)x
• Frequency (f) measures how fast an oscillation is
  changing
• Acceleration is rate of change of rate of change
  of position

• Car on freeway on-ramp
• 60 miles per hour per 10 sec
• Acceleration is equal to a mathematical constant
  times frequency squared times position.

• Frequency increases with stiffness, decreases
  with mass

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Vibrations of a String

• Each little segment of a string is like a mass on
  a spring
• The spring force is supplied by the tension in
  the string and the curvature of the wave.
• A wave (of arbitrary shape) travels on a string
  with velocity

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Travelling waves and Reflections

• Each end of the string is held rigidly.
• To the wave, the fixed point acts like a wave of
  opposite amplitude travelling in opposite
  direction.
• Rigid end of string reflects wave with opposite
  sign
• Loose end of string (or other wave--e.g. organ
  pipe) reflects wave with equal sign.

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Standing Waves

• Each point on string experiences waves reflecting
  from both ends of string.
• For a repeating wave (e.g. sinusoidal)
• Velocity wavelength times frequency v l f
• The superposition of reflecting waves creates a
  standing wave pattern, but only for wavelengths
• l 2L, L, L/2, 2L/n)
• Only allowed frequencies are f n v/(2L)
• Pitch increases with Tension, decreases with mass
  or length

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Harmonics on string

• Plot shows fundamental and next three harmonics.
• Dark purple is a weighted sum of all four curves.
• This is wave created by strumming, bowing,
  hitting at position L/4.
• Plucking at L/2 would only excite f1, f3, f5, ...

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Pitch, Timbre, Loudness

• Equal musical intervals of pitch correspond to
  equal ratios of frequency
• Two notes separated by a perfect fifth have a
  frequency ratio of 32.
• Notice that 2nd and 3rd harmonic on string are
  perfect 5th
• Timbre is largely determined by content of
  harmonics.
• Clarinet, guitar, piano, human voice have
  different harmonic content for same pitch
• Loudness is usually measured on logarithmic
  decibel (tenths of bel) scale, relative to some
  arbitrary reference intensity.
• 10 dB is a change in sound intensity of a factor
  of 10
• 20 db is a change in sound intensity of a factor
  of 100.

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Frequency analysis of sound

• The human ear and auditory cortex is an extremely
  sophisticated system for the analysis of pitch,
  timbre, and loudness.
• My computer is not too bad either.
• Microphone converts sound pressure wave into an
  electrical signal.
• Computer samples electrical signal 44,000 times
  per sec.
• The stream of numbers can be plotted as wave vs.
  time.
• Any segment of the wave can be analysed to
  extract the amplitude for each sinusoidal wave
  component.

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Samples of Sound Sampling

• Clarinet
• Guitar
• Piano
• Human Voice
• ...

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Piano keys (Grand Piano)

• Key is pressed down,
• the damper is raised
• The hammer is thrown against string
• The rebounding hammer is caught by the Back Check.

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Hammer action

• Throwing the hammer against the string allows the
  hammer to exert a very large force in a short
  time.
• The force of the hammer blow is very sensitive to
  how your finger strikes the key, but the hammer
  does not linger on the string (and muffle it).
• From pianissimo (pp) to fortissimo (ff) hammer
  velocity changes by almost a factor of 100.
• Hammer contact time with strings shortens from
  4ms at pp to
• Note that 2 ms ½ period of 264 Hz oscillation

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From Strings to Sound

• A vibrating string has a very poor coupling to
  the air. To move a lot of air, the vibrations of
  the string must be transmitted to the sound
  board, via the bridge.
• The somewhat irregular shape, and the off center
  placement of the bridge, help to ensure that the
  soundboard will vibrate strongly at all
  frequencies
• Most of the mystery of violin making lies in the
  soundboard.

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Piano frame

• A unique feature of the piano, compared to
  violin, harpsichord. is the very high tension in
  the strings.
• This increases the stored energy of vibration,
  and therefore the dynamic power and range of the
  piano.
• Over 200 strings for 88 notes,each at ? 200 lb
  tension
• Total tension on frame 20 tons.
• The Piano is a modern instrument (1709, B.
  Cristofori)
• High grade steel frame.
• Also complicated mechanical action.

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Piano strings

• An ideal string (zero radius) will vibrate at
  harmonics
• fn n f1
• A real string (finite radius r) will vibrate at
  harmonics that are slightly stretched
• fn n f11(n2-1)r4k/(TL2)
• Small radius-r, strong wire (k), high tension
  (T), and long strings (L) give small
  in-harmonicity.
• For low pitch, strings are wrapped, to keep r
  small

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In-harmonicity tone color

• Perfect harmonics are not achievable--and not
  desirable. A little in-harmonicity gives
  richness to the tone, and masks slight detunings
  of different notes in a chord.
• Each octave is tuned to the 2nd harmonic of the
  octave below.

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Multiple Strings

• Multiple Strings store more energy--louder sound
• Strings perfectly in tune
• Sound is loud, but decays rapidly
• Strings strongly out of tune
• Ugly beats occur as vibrations from adjacent
  strings first add, then cancel, then add again.
• If strings are slightly out of tune
• Sound decays slowly
• Beats are slow, add richness to tone.

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Multiple Strings, Power and Decay Time

• Decay time of vibration Energy stored in string
  divided by power delivered to sound board.
• Power delivered to sound board force of string
  velocity of sound board (in response to force)
• Three strings store 3 times the kinetic energy of
  one string
• If three strings are perfectly in tune, Force is
  3 times larger, velocity is three times larger,
  power is 9 times larger, Decay time is 3/9 1/3
  as long as one string alone (Una corda pedal).
• If strings are slightly mistuned, motion is
  sometimes in phase, sometimes out of phase,
  average power of three strings is only 3 times
  greater than power of one string. Decay time of
  3 strings is SAME as decay time of one string
  alonejust louder.

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Beats from mistuned strings

• Two tones are mistuned by 10. One string makes
  10 oscillations in the time it takes the other to
  make 11 oscillations.
• Yellow curve resulting superposition of two
  waves
• ½ of beat period is shown. Beat period
  20period of individual wave.
• Acoustic power would be 4x individual wave, if
  strings were perfectly in tune. Because of
  beats, average acoustic power is 2x individual
  contribution

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Chords, Scales, and Tuning

• For the simple chords of perfect fifths and
  thirds, the harmonics of each note match the
  fundamental of the other notes in the chord.
• Equal tempered tuning was designed (by J.S. Bach
  others) to give the closest possible match
  between pitch and harmonics within chords, for
  any possible starting note.

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Conclusions

• The mathematical theory of music dates to
  antiquity.
• Music illustrates fundamental physical and
  mathematical concepts.
• Music is influenced by technology
• Modern examples
• Piano, Clarinet
• Musical Reproduction and Processing (Digital
  predates Analog!).
• Music appreciation has many layers.
• Science can add another layer