Sound Physics: The Piano

By: Greg Bozoian and Hannah Lim

In the1700s, Bartolomeo Cristofori diFrancesco came up with the original conceptof striking the strings with a hammer, whichproduced a much more pronounced andsustained sound.Shortly after Cristofori’s new concept, thepiano “action” was developed, which not onlykept the strings damped when not beingplayed, but also kept the hammers fromremaining on the strings after striking them.

Struck vs. Plucked

Piano to Forte- When the pianist makesvolume changes the tone becomes louder orsofter and brighter in color.Frequency Vibrations and DecayModes and Nodes encounteredThe piano's 88 keys range in frequency fromA0 at 27.5 Hz to C8 at 4186 Hz.

Polarization Coupled Oscillations

Three strings arecoupled on thesame bridgeHigher notes onlyHigher stringsmade of steelLower strings madeof steel with copperwrapped aroundthem

Symmetrical (a) All strings move in thesame directionMoves bridgeefficiently, resulting ina shorter decay timeUnsymmetrical (b) (c) Two strings pull upjust as hard as thethird string pulls downcreating a zero netforceResults from a slightlylopsided hammerhitting two only twostrings Una corda pedal

The way in which string lengths anddiameters change from note to note and fromoctave to octaveFundamental frequency of a string f₁ (1 / 2L) (T / μ) These factors only get the desired frequency,but cost, convenience, and tone quality mustalso be considered

Length Too short problems with tone quality Too long bulky and expensive piano Wide range from a few centimeters to one or twometers

TensionCannot exceed the breaking strength of the wireT (π/4)(d 2)HSteel has H 1.5 X 10 9 N/m2Smaller tension decreases ability of the string tofeed energy to the soundboard Difference in impedances Measure of how much force should be applied to awave to produce a given amount of motion For the piano, a soundboard impedance only a fewthousand times greater than the string impedance isdesired

Diameter Too small weak sound Too large not fully flexible Stiffness Higher modes are affected and destroyed by stiffness Causes inharmonicity Modest stiffness enough harmonic for tone to bepleasing and musical Excessive siffness string behaves like a rod; metallicsound Compromise Higher frequencies: decrease string length anddiameter to avoid stiffness Lower frequencies: making strings heavier ratherthan longer Triple stringing the treble to deliver adequateenergy to the soundboard

In the middle range, a tuner sets the octavesat 2:1 frequency ratiosTop octave strings (more stiff) are stretchedslightly (C7-C8) to ratios of 2.025:1Similar effect for lowest bass notes

Physical Higher note matched precisely with the secondmode of the lower note so no beats are produced Higher strings are more stiff, resulting in lower notefrequency to be slightly more than twice itsfundamental Dependent on piano Large grand piano: little stretching; inharmonicity of bassstrings is hardly any greater than the midrange Small spinet: much more stretching; relatively short, stuffbass strings are quite inharmonic Psycological Difficult pitch perception in extreme ranges Similar stretched octaves on any piano

In the key of CHand Mute Technique creates different waveUse of all PedalsFluctuating frequenciesKey changes to show difference in contour,brightness, tone and emotion.

Title: Sound Physics: The Piano