June 5, 2000 Piano Tuners Guild 1

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

June 5, 2000 Piano Tuners Guild 2

Physics of the Piano

Oscillations & Sound Vibrations of a String

Travelling waves & ReflectionsStanding WavesHarmonics

Piano acousticsHammer actionSound BoardMultiple 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 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).

June 5, 2000 Piano Tuners Guild 4

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

LM

T

/v

June 5, 2000 Piano Tuners Guild 5

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 signLoose 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 = f

The superposition of reflecting waves creates a standing wave pattern, but only for wavelengths = 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 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.

June 5, 2000 Piano Tuners Guild 9

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

ClarinetGuitarPianoHuman 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 < 2 ms at ff (for middle C-264 Hz) 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.

June 5, 2000 Piano Tuners Guild 14

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.

June 5, 2000 Piano Tuners Guild 15

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)r4/(TL2)] Small radius-r, strong wire (), 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 soundStrings 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 alone—just 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. 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 contributionBeats

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10

Time