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 chyde@odu.edu Piano Tuners Guild

The Physics of Music and Musical Reproduction:Autumn 2000, Mon.&Wed. 4:20-5:35pm 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. 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

Force and Motion • The mass moves even when nothing is pushing • The mass moves because of inertia • Forces do not cause motion…forces 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 Piano Tuners Guild

[Net] Force causes motion to change • F=ma • 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 Piano Tuners Guild

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. F=ma & Mass on a Spring • Frequency increases with stiffness, decreases with mass 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): • High grade steel frame. • 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: • 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 • 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. • Yellow 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

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. Piano Tuners Guild

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 Piano Tuners Guild

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