Essentials of Physics- PHYS 152

Lecture 16-- Pianos

Reading: Chapter 13

 

globe_pointLet's not forget our joke

globe_pointLet's play with our Pod and think about effects...

 

 

 

grand piano action

 

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The Piano:

Our discussion of pianos will comprise three parts: a brief history, construction of modern pianos, and piano sound production.

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A brief history of the piano:

(borrowed liberally from your textbook and other sources)

Proto-pianos (or could they be proto-guitars?) were built and played by the Greeks. An instrument called a monochord was made by Pythagorus in about 582 B.C. consisting of a string stretched across a resonator box. It had a moveable bridge dividing the string into two variable-length sections. Using this Pythagorus (could he be the one of Pythagorean equation fame?) discovered that sections of strings whose lengths were related by integer multiples (one length was 2, 3, 4, etc. times the other) and played simultaneously made sounds that were pleasant heard together.

Over time the number of strings on this type of instrument generally increased and the idea was to pluck multiple strings at the same time.

Keyboards, which eventually allowed one to push a key and pluck a string, were first developed to control pipe organs in around the second century B.C.

Ow, I think I broke my clavichord! About the 11th century the hurdy gurdy and then the clavichord were developed. These comprised multiple strings (more on the clavichord) and a keyboard. Keys were pressed that placed a tangent (a narrow bar situated orthogonal to the string that acted both as a hammer and one end of the string against a string at a prescribed position along the string (and, hence, causing a string to resonate with a certain length). A cloth listing was placed under the opposite end of the string to damped any vibrations on the unused end of the string. This mechanism also quickly damped the vibrating end of the string when the key was released and the tangent dropped from the string. Johann Sebastian Bach was one famous composer who wrote music for the clavichord.

clavichord mechanism

Clavichord or Harpsichord? The harpsichord, a keyboard instrument where strings were plucked instead of struck was developed at about the same time as the clavichord. When a harpsichord key is presed a plectrum plucks a string streched between two bridges. A fairly simple mechanism (relative to the piano anyway) provides for moving the plectrum sideways once a string is plucked and allowing it to drop to its original position for the next note. As long as the key remains pressed the string will continue to vibrate (and decay). When the key is released the plectrum slips by and a damper deadens the string.

It's all in the mechanism: In 1709 Florencian harpsichord maker Bartolommeo Cristofori invented a mechanism that replaced the harpsichord's plucker with a hammer to strike a string to sound a note. This instrument, the pianoforte (that's gravicembalo col piano e forte to you!) was able to produce both loud and soft notes, unlike the harpsichord. Cristofori's mechanism was complex and involved many of the same features as today's piano mechanisms.

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Piano construction:

Cristofori's invention of a new mechanism opened up new tonal and dynamic possibilities for keyboad instruments. The stronger excitation force provided by striking strings with hammers meant that strings could be tensioned tighter and strung longer. Multiple strings for each note could be used to both increase maximum achievable loudness, to provide for striking a reduced set of these strings for longer and louder sustain of soft notes, and to generally set up tonal possibilities for driven resonance of passive strings in a set.

exploded view of piano

The modern piano comprises:

keyboard-- the 88 keys are made of wood and covered with ivory or plastic covers. Black keys corresponding to sharp and flat notes are higher, narrower and offset back from the white keys (natural notes).

action-- the mechanism based on Cristofori's that provides for hammers striking strings when keys are pushed. This is a critical bit so we'll come back to the action mechanism.

strings-- 230 of them! Notes corresponding to individual keys are produced by sets of two (lower registers) and three strings. Lower octave strings are wirewound, meaning that they comprise a solid metal core with one (or even two) longer wires wrapped in spirals around the core. Wirewinding of strings is a way to increase their mass per length (and thus decrease their frequencies of oscillation according to Mersenne's laws) while minimizing their stiffness, which can (and unfortunately still does) lead to interharmonicity, where the frequencies of upper harmonics are shifted from integer relationships to the fundamental (text book example: sixteenth overtone (17th harmonic) is half-tone higher than predicted and will noticable produce beats when played against other notes (c.f., page 341).

pin block-- mechanism for tensioning and tuning of individual strings. The pin block is attached to a....

cast-iron frame-- whose purpose is structural, the frame supports the enormous forces of 230 strings under tension (the book notes that this can approach 60,000 lbs).

soundboard-- constructed typically of clear spruce wood, its purpose is to vibrate sympathetically with strings and help send this sound to the listener. Vibrations of the latter are coupled to the soundboard via the...

UO phys demo

bridge-- a curved construction of wood and metal that serves as a displacement node for vibrating strings. All of this is placed in or on a...

case-- usually made of wood, it houses and supports the various parts of the piano.

Mechanism/action/regulation:

Cristofori's mechanism has evolved into the action of the piano, a piece of complicated-looking machinery involving levers, dampers and hammers for striking a string set. Although it only addresses the action of an upright piano, which is more complicated than that of a grand piano (the difference is the orientation of the strings, etc., which are vertical for an upright piano), this page does a nice job of explaining how piano actions work.

The important tasks to be accomplished by a piano action for a given note are:

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Piano sound production:

Imagine that you were challenged with designing the different elements of a modern piano. We have already discussed one problem, that thinner, shorter strings used for high notes produce less sound, so we design our modern piano to have 3 strings for every higher note (top 68 keys) and 2 for the lower notes.

But we also want to design for a few other aspects:

Overtone content: the first design problem can be addressed empirically. We can devise an arrangement with a hammer that can be moved along the string and experiment until our ear hears what we want. We could also do this for high note triplets of strings and low note doublets.

We would find that an optimal hammer strike point is between 1/7 and 1/8 the distance of the string from the peg end.

Inharmonicity: The second problem is substantial for pianos. The textbook states that the 16th overtone for a piano string can be "off" by as much as one half-tone. If one plays two notes an octave apart (e.g., C3 and C4), the inharmonicity of the second harmonic of the lower note will cause beating against the fundamental of the higher note. To "solve" for this problem octaves are stretched, meaning that a note an octave up is tuned slighly more than double the frequency of the octave below.

Our ears are trained by hearing (well-constructed and tuned) pianos with their inharmonicities for upper harmonics. We expect this sound. Digital electric pianos that try to mimic the sound of standard pianos need provide overtone content that anticipates some inharmonicity.

Aging: Piano hammers are covered with felt. The purpose of a felt is to spread out the impact of the hammer strike-- remember that our clavichord employed sharp hammers which served as displacement antinodes after striking a string. We don't want this to happen for our piano.

One common problem with pianos is that the felts dry out and harden. This serves to emphasize higher harmonics relative to low harmonics, giving a piano a tinny 'honky tonk' sound.

Sustain: Well, sometimes we want it, right? Piano notes are characterized by a sharp attack with rapid early decay, followed by a long, slow decay over time. Term project mystery instrument number 2 was a piano and its waveform and spectra looked like:

piano waveform

 

Term project mystery So how do we design our piano for attack and sustain? There are two fundamental ways of doing this: polarization and coupled oscillation (or sympathetic resonance).

Polarization happens because of slight imperfections in the piano mechanism in contact with played strings. Although the hammer is meant to impart only vertical motion in the string of a grand piano, eventually horizontal motion also occurs. Horizontal motion is less damped over time because its motion is less coupled to the piano bridge and, ergo, soundboard.

UO phys demo

Coupled oscillations are another, more complicated story. In response to this phenomenon, piano tuners will actually tune different strings in a doublet or triplet (per note) to slightly different frequencies!

Strings that are part of a doublet or triplet are usually excited differently upon hammer strike. Althought they will initially be in phase, their amplitudes may differ at first because of imperfections in the hammer head, etc. At some point in time one string may be almost stopped while another is still oscillating. These strings interact via the piano bridge to cause coupled oscillation, where the vibrating string drives the nearly stopped string back into oscillation. When this happens the two strings are often out of phase (one is going up while the other goes down) so that the bridge moves minimally. This means that less string vibration is coupled as sound to the soundboard, and the note can sustain for a long time.

That's the simple explanation, but the reality is a bit more complicated (of course!). Depending upon how it couples to the bridge, one can view the end of a string as "springy," "massive" or "resistive." A springy end is like tacking the end of the string to a mass on a spring (see textbook, Figure 13-5). The motion of the string and support are in phase and, more importantly, the wavelength of the fundamental sound on the string is just a bit longer than (twice its...) length. The net effect makes for a slightly lower-pitched sound than what we might normally expect.

For a massive support, the motion of the string and support are out of phase, and the fundamental wavelength is a bit shorter than (twice the...) strings length.

Resistive loading of the string provides for a fundamental wavelength that is precisely 2 times the length of the string. The frequency is then what we expect, and neither lower nor higher.

Here's the key bit. The bridge of a piano acts either springy, or massive, or resistive, depending on the phase of the string motion relative to the bridge motion (are they going up and down together, or something different?). Thus piano tuners must accomodate this by tuning each member of a doublet or triplet slightly differently.

Let's monitor our old piano to see if we can measure this difference.

UO phys demo

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Resources:

Grand piano action explained, with animation.

 

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