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

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Physics of the Piano Piano Tuners Guild, June 5, 2000 Charles E. Hyde-Wright, Ph.D. Associate Professor of Physics Old Dominion University Norfolk VA chyde@odu.edu Physics of the Piano Oscillations &amp; Sound Vibrations of a String Travelling waves &amp; Reflections Standing Waves Harmonics

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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@odu.edu

Piano Tuners Guild

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

Piano Tuners Guild

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 doesn’t 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).

Piano Tuners Guild

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

Piano Tuners Guild

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.

Piano Tuners Guild

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

Piano Tuners Guild

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, ...

Piano Tuners Guild

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 3:2.
• 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.

Piano Tuners Guild

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.

Piano Tuners Guild

Samples of Sound Sampling
• Clarinet
• Guitar
• Piano
• Human Voice
• ...

Piano Tuners Guild

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.

Piano Tuners Guild

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 < 2 ms at ff (for middle C-264 Hz)
• Note that 2 ms = ½ period of 264 Hz oscillation

Piano Tuners Guild

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.

Piano Tuners Guild

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):
• Also complicated mechanical action.

Piano Tuners Guild

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 f1[1+(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

Piano Tuners Guild

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.

Piano Tuners Guild

Multiple Strings
• Multiple Strings store more energy--louder sound
• Strings perfectly in tune:
• Sound is loud, but decays rapidly
• Strings strongly out of tune:
• If strings are slightly out of tune
• Sound decays slowly
• Beats are slow, add richness to tone.

Piano Tuners Guild

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 alone—just louder.

Piano Tuners Guild

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.
• Cyan curve = resulting superposition of two waves
• ½ of beat period is shown. Beat period = 20*period 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

Piano Tuners Guild