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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 [email protected] Physics of the Piano. Oscillations & Sound Vibrations of a String Travelling waves & Reflections Standing Waves Harmonics

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Physics of the piano piano tuners guild june 5 2000
Physics of the PianoPiano Tuners Guild, June 5, 2000

Charles E. Hyde-Wright, Ph.D.

Associate Professor of Physics

Old Dominion University

Norfolk VA

[email protected]

Piano Tuners Guild

Physics of the piano
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
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
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
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
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
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
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
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
Samples of Sound Sampling

  • Clarinet

  • Guitar

  • Piano

  • Human Voice

  • ...

Piano Tuners Guild

Piano keys grand piano
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
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
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
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.

Piano Tuners Guild

Piano strings
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
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

  • 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.

Piano Tuners Guild

Multiple strings power and decay time
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
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