June 5, 2000 Piano Tuners Guild 1 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

June 5, 2000 Piano Tuners Guild 2 Physics of the Piano zOscillations & Sound zVibrations of a String yTravelling waves & Reflections yStanding Waves yHarmonics zPiano acoustics yHammer action ySound Board yMultiple Strings zChords, Scales & Tuning

June 5, 2000 Piano Tuners Guild 3 Why does a mass on a spring oscillate? zIt is not because I push it yThe mass continues long after I let go. zThe spring is pushing on the mass. yWhy doesn’t the mass just come to rest in the middle? xAfter all, the spring(s) exert no (net) force on the mass when it is exactly in the middle. xNo force seems like no motion (wrong).

June 5, 2000 Piano Tuners Guild 4 Vibrations of a String zEach little segment of a string is like a mass on a spring zThe spring force is supplied by the tension in the string and the curvature of the wave. zA wave (of arbitrary shape) travels on a string with velocity

June 5, 2000 Piano Tuners Guild 5 Travelling waves and Reflections zEach end of the string is held rigidly. yTo the wave, the fixed point acts like a wave of opposite amplitude travelling in opposite direction. zRigid end of string reflects wave with opposite sign zLoose end of string (or other wave--e.g. organ pipe) reflects wave with equal sign.

June 5, 2000 Piano Tuners Guild 6 Standing Waves zEach point on string experiences waves reflecting from both ends of string. zFor a repeating wave (e.g. sinusoidal)  Velocity = wavelength times frequency:v =  f zThe superposition of reflecting waves creates a standing wave pattern, but only for wavelengths = 2L, L, L/2, … = 2L/n) zOnly allowed frequencies are f = n v/(2L) yPitch increases with Tension, decreases with mass or length

June 5, 2000 Piano Tuners Guild 7 Harmonics on string zPlot shows fundamental and next three harmonics. zDark purple is a weighted sum of all four curves. yThis is wave created by strumming, bowing, hitting at position L/4. yPlucking at L/2 would only excite f1, f3, f5,...

June 5, 2000 Piano Tuners Guild 8 Pitch, Timbre, & Loudness zEqual musical intervals of pitch correspond to equal ratios of frequency: yTwo notes separated by a perfect fifth have a frequency ratio of 3:2. yNotice that 2nd and 3rd harmonic on string are perfect 5th zTimbre is largely determined by content of harmonics. yClarinet, guitar, piano, human voice have different harmonic content for same pitch zLoudness is usually measured on logarithmic decibel (tenths of bel) scale, relative to some arbitrary reference intensity. y10 dB is a change in sound intensity of a factor of 10 y20 db is a change in sound intensity of a factor of 100.

June 5, 2000 Piano Tuners Guild 9 Frequency analysis of sound zThe human ear and auditory cortex is an extremely sophisticated system for the analysis of pitch, timbre, and loudness. zMy computer is not too bad either. yMicrophone converts sound pressure wave into an electrical signal. yComputer samples electrical signal 44,000 times per sec. yThe stream of numbers can be plotted as wave vs. time. yAny segment of the wave can be analysed to extract the amplitude for each sinusoidal wave component.

June 5, 2000 Piano Tuners Guild 10 Samples of Sound Sampling zClarinet zGuitar zPiano zHuman Voice z...

June 5, 2000 Piano Tuners Guild 11 Piano keys (Grand Piano) zKey is pressed down, ythe damper is raised yThe hammer is thrown against string yThe rebounding hammer is caught by the Back Check.

June 5, 2000 Piano Tuners Guild 12 Hammer action zThrowing the hammer against the string allows the hammer to exert a very large force in a short time. zThe 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). zFrom pianissimo (pp) to fortissimo (ff) hammer velocity changes by almost a factor of 100. yHammer contact time with strings shortens from 4ms at pp to < 2 ms at ff (for middle C-264 Hz) yNote that 2 ms = ½ period of 264 Hz oscillation

June 5, 2000 Piano Tuners Guild 13 From Strings to Sound zA 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. zThe somewhat irregular shape, and the off center placement of the bridge, help to ensure that the soundboard will vibrate strongly at all frequencies zMost of the mystery of violin making lies in the soundboard.

June 5, 2000 Piano Tuners Guild 14 Piano frame zA unique feature of the piano, compared to violin, harpsichord. is the very high tension in the strings. zThis 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 yTotal tension on frame > 20 tons. zThe Piano is a modern instrument (1709, B. Cristofori ): yHigh grade steel frame. yAlso complicated mechanical action.

June 5, 2000 Piano Tuners Guild 15 Piano strings zAn ideal string (zero radius) will vibrate at harmonics yf n = n f 1 zA real string (finite radius r) will vibrate at harmonics that are slightly stretched:  f n = n f 1 [1+(n 2 -1)r 4  /(TL 2 )]  Small radius-r, strong wire (  ), high tension (T), and long strings (L) give small in-harmonicity. yFor low pitch, strings are wrapped, to keep r small

June 5, 2000 Piano Tuners Guild 16 In-harmonicity & tone color zPerfect 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. zEach octave is tuned to the 2nd harmonic of the octave below.

June 5, 2000 Piano Tuners Guild 17 Multiple Strings zMultiple Strings store more energy--louder sound zStrings perfectly in tune: ySound is loud, but decays rapidly zStrings strongly out of tune: yUgly beats occur as vibrations from adjacent strings first add, then cancel, then add again. zIf strings are slightly out of tune ySound decays slowly yBeats are slow, add richness to tone.

June 5, 2000 Piano Tuners Guild 18 Multiple Strings, Power and Decay Time zDecay time of vibration = Energy stored in string divided by power delivered to sound board. yPower delivered to sound board = force of string * velocity of sound board (in response to force) zThree strings store 3 times the kinetic energy of one string yIf 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). yIf 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.

June 5, 2000 Piano Tuners Guild 19 Beats from mistuned strings zTwo tones are mistuned by 10%. One string makes 10 oscillations in the time it takes the other to make 11 oscillations. zCyan curve = resulting superposition of two waves y½ of beat period is shown. Beat period = 20*period of individual wave. yAcoustic power would be 4x individual wave, if strings were perfectly in tune. Because of beats, average acoustic power is 2x individual contribution