by E. Donnell Blackham December 1965
Most
musical instruments produce tones whose partial tones, or overtones, are
harmonic: their frequencies are whole-number multiples of a fundamental
frequency. The piano is an exception.
Almost every musical tone, whether it is produced by a vibrating string,
a vibrating column of air or any other vibrating system, consists of a
fundamental tone and a number of the higher-pitched but generally fainter tones
known as partial tones or over tones. The complex sound produced by this
combination of separate tones has a timbre, or characteristic quality, that is
determined largely by the number of partial tones and their relative loudness. It
is timbre that enables one to distinguish between two musical tones that have
the same pitch and the same loud ness but are produced by two different musical
instruments. A pure tone-one that consists solely of the fundamental tone-is
rarely heard in music.
It is widely believed that the partial tones produced
by all musical instruments are harmonic-that their frequencies are exact
whole-number multiples of the frequency of a fundamental tone. This certainly holds true for all the woodwinds and under certain
conditions for many of the stringed instruments, including the violin. It is
only approximately true, however, in the most familiar stringed instrument: the
piano. The higher the frequency of the partial tones of any note on the piano,
the more they depart from a simple harmonic series. In our laboratory at
The physics of the piano can best be understood by
first reviewing the evolution of the modern piano and its principal components.
Archaeological evidence shows that primitive stringed instruments existed
before the begining of recorded history. The Bible refers several times to an
instrument called the psaltery that was played by plucking strings stretched
across a box or gourd, which served as a resonator. A similar instrument
existed in
Another important component of the modem piano-the
keyboard-did not arise in conjunction with a stringed instrument but with the
pipe organ. The organ of Ctesibus, perfected at
IDEAL STRING, that is, one
without any stiffness, can be made to vibrate at many different frequencies:
the fundamental frequency (a) produces a pure tone, rarely heard in music,
whereas higher-pitched partial tones, or overtones, are produced by harmonic vibrations
("b" and "c"), whose
frequencies are whole-number multiples of the
fundamental frequency.
SIMULTANEOUS
VIBRATION Fig. [d] of a string at two
or more different frequencies is normal for stringed instruments. Here the
string vibrates at the fundamental frequency and the sec. and partial frequency
("a" and "c" in upper illustration). In the piano
the stiffness of the strings causes higher partials of a complex tone to depart
from the simple harmonic series.
We do not know who first conceived the idea
of adding keys to a stringed instrument. From this obscure beginning there eventually evolved in the 15th
century the clavichord. In the early clavichords a piece of metal mounted
vertically at the end of the key acted both as a bridge for determining
the pitch of the string and as a percussive device for producing the tone [see
upper illustration on page 28]. Since one string could be used to
produce more than one tone, there were
usually more keys than strings. A strip of cloth was interlaced among the strings at one end in order to damp
the unwanted tone from the shorter part of the string.
TOP VIEW of the interior of
a modern "baby grand" piano shows the powerful construction of the
full cast-iron frame, which sustains the tremendous tension exerted by the
strings. In this particular piano the frame, which is cast in one piece, weighs
about 250 pounds and sustains an average tension of some 50,000 pounds; in a
larger concert-grand piano the frame weighs as much as 400 pounds and sustains an average tension of
60,000 pounds. The strings are made of steel wire with an ultimate tensile strength
of from 300,000 to 400,000 pounds per square inch. In order to make the bass
strings (left) vibrate slower and thus produce a lower pitch, they are
wrapped in copper or iron wire; two such wrappings are often used in the
extreme bass. In all modern pianos the bass strings are "overstrung"
in order to conserve space and to bring them more nearly over the center of the
soundboard. Starting from the treble, or right-hand, end of the keyboard there
are 60 notes with three strings each, then 18 notes with two strings each and
finally, in the extreme bass, 10 notes with only one string each. Larger pianos
have more strings but the same total number of notes: 88. Rectangular black
objects in a row near the bottom are the heads of the dampers. Parts made of
felt are in color. Strips of cloth interlaced among the strings at top damp
unwanted tones from the short parts of the strings beyond the bridge (see
illustration on next page).
EXPLODED VIEW of the baby-grand piano depicted from
above moving parts involved in the actual striking of the string. Three on the
preceding page shows the relations of several main components (bottom center)
serve to control the dampers in the components. The keyboard (bottom
le/t) has 88 keys divided intonation. When a key is struck, the hammer sets
the strings in vibration seven and a third octaves. Each octave has eight white
keys for and, after a very short interval known as the attack time, sound is
playing the diatonic scale (whole notes) and five raised black keys translated
by means of a wooden bridge to the soundboard, from for playing the chromatic scale
(whole notes plus sharps and flats). which it is
radiated into the air. During the attack time sound is Connected to the
keyboard is the action, which includes all the also radiated to a lesser degree
from both the strings and the bridge.
Several essential characteristics of the modem piano
are inherited from the clavichord. The clavichord had metal strings, a
percussive device for setting the strings in vibration, a damping mechanism and
also an independent soundboard: the board at the bottom of the case did not
also serve as the frame for mounting the strings. Moreover,
al though the tone of the clavichord was weak, the instrument allowed for the
execution of dynamics, that is, for playing either loudly or softly.
At about the same time another fore runner of the
modern piano was in process of development. In the spinet, or virginal, longer
strings were introduced to produce a louder tone. Now the metal percussive
device of the clavichord was no longer adequate to produce vibration in the
strings. Instead the strings were set in motion by the plucking action of a
quill mounted at right angles on a "jack" at the end of the key
[see lower illustration on next page]. When the key was depressed, the jack
moved upward and the quill plucked the string. When the jack dropped back, a
piece of cloth attached to it damped the vibration of the string.
Around the beginning of the 16th century experiments
with still longer strings and larger soundboards led to the development of the
harpsichord. Al though this instrument was essentially nothing more than an
enlarged spinet, it incorporated several important innovations that have
carried over to the modern piano. The wing-shaped case of the harpsichord is
imitated by that of the grand piano. The stratagem of using more than one
string per note in order to increase volume was adopted for the harpsichord by
the middle of the 17th century. The harpsichord also had a "forte
stop," which lifted the dampers from the strings to permit sustained
tones, and a device for shifting the key board, both of which are preserved in
the modern piano.
The invention of the piano was fore cast by inherent
defects in both the clavichord and the harpsichord. Neither the spinet nor the
harpsichord was capable of offering the composer or per former an opportunity
to execute dynamics. The clavichord, on the other hand, allowed a modest range
of dynamics but could not generate a tone nearly as loud as that of the harpsichord.
Attempts to install heavier strings in order to increase the volume of either
instrument were futile; neither the metal percussive device of the clavichord
nor the quill of the harpsichord could excite a heavy string. Moreover, the
cases of these early instruments were not strong enough to sustain the in
creased tension of heavier strings.
JURY composed of both
musicians and nonmusicians was asked [ no proper photo] to distinguish
between recordings of real piano tones and synthetic ones. When the synthetic
tones were built up of harmonic partials, the
musicians on the jury were able to distinguish 90 percent of these tones from real piano tones; the
nonmusicians, 86 percent. When inharmonic partials were used, results
showed that in most cases both the musicians and the nonmusiciana were guessing; both groups identified only about
50 percent of the tones correctly. In this photo. graph two members of the jury are listening to tones in an anechoic, or echoless, chamber.
A remedy for these defects was provided by the Italian
harpsichord-maker Bartolommeo Christofori, who in 1709 built the first
hammer-action keyboard instrument. Christofori called his original instrument
the "piano=forte," mean ing that it could be played both softly and
loudly. The idea of having the strings struck by hammers was probably suggested
to him by the dulcimer, a stringed instrument played by hammers held in the
hands of the performer. Christofori recognized that his new instrument would
need a stronger case to withstand the increased tension of the heavier strings.
By 1720 an improved model of the pianoforte included an escapement device that
"threw" the free swinging hammer upward at the string and also a
back-check that regulated the hammer's downward return [see upper
illustration on page 29]. An individual damper connected to the action of
the hammer was provided for each note.
For a century and a half after Christofori s first piano appeared inventors worked to improve the new instrument, particularly its novel hammer action. Several other types of action were developed, some new and others modeled closely on Christofori s original. Pianos were built in a variety of forms: traditional wing-shaped pianos; square pianos, upright pianos and even a piano organ combination.
CLAVICHORD ACTION [upper
design] included one essential feature found in all modern pianos: a percussive
device for setting the strings in vibration. A piece .of metal mounted
vertically at the end of the key acted both as a bridge for determining the
pitch of the string and as a percussive device for producing the tone. Since
one string could be used to produce more than one tone, there were usually more
keys than strings. A strip of cloth was inter. laced among the strings at one
end to damp the tone from the shorter part of the string.
Among the major innovations toward the end of this
period was the full cast iron frame. Constant striving for
greater sonority had led to the use of very heavy strings, and the point was
reached where the wooden frames of the earlier pianos could no longer withstand
the tension. In 1855 the German-born American piano manufacturer Henry Steinway
brought out a grand piano with a cast-iron frame that has served as a model for
all subsequent piano frames. Although minor refinements are constantly being
introduced, there have been no fundamental changes in the design or
construction of pianos since 1855.
A part of the piano that has received a great deal of
attention from acoustical physicists is the soundboard. Some early
investigators believed the sound of the piano originated entirely in the
soundboard and not in the strings. We now know that the sound originates in the
strings; after a very short interval, called the attack time, it is translated
by means of a wooden bridge to the soundboard, from which it is radiated into
the air. During the attack time sound is also radiated to a lesser degree from
both the strings and the bridge. In the late 19th century Frederick Mathushek
and his associates proved that the quality of a piano's sound was not influenced
by the transverse, or horizontal, vibrations of the soundboard. They glued
together two soundboards so that the grain of one was at right angles to the
grain of the other, there by eliminating any transverse vibrations, and found
that the quality of the sound was not affected by this arrangement. The behavior
of the soundboard has also been analyzed theoretically by a number of eminent
physicists, including Hermann von Helmholtz, but such analyses have added
nothing to the principles of soundboard construction arrived at empirically by
the builders of the early clavichords and harpsichords.
The development of the full cast-iron frame gave the
sound of the piano much greater brilliance and power. The modem frame is cast
in one piece and carries the entire tension of the strings; in a large
concert-grand piano the frame weighs 400 pounds and is subjected to an average
tension of 60,000 pounds. In order to maintain the tension of the strings each
string is attached at the keyboard end to a separate tuning pin, which passes
down through a hole in the frame and is anchored in a strong wooden pin block. Since
the piano would go out of tune immediately if the tuning pins were to yield to
the tremendous tension of the strings, the pin block is built up of as many as
41 cross grained layers of hardwood.
The keyboard of the modern piano is constructed on
essentially the same principles that had been fully developed before the 15th
century. The standard keyboard has 88 keys divided into seven and a third
octaves, the first note in each octave having twice the frequency of the first
note in the octave below it. Each octave has eight white keys for playing the
diatonic scale (whole notes) and five raised black keys for playing the
chromatic scale (whole notes plus sharps and fiats). In all modern pianos the
white keys are not tuned exactly to the diatonic scale but rather to the
equally tempered scale, in which the octave is simply divided into 12 equal intervals.
The moving parts of the piano that are involved in the
actual striking of the string are collectively called the action [see lower
illustration on opposite page]. One contemporary piano manufacturer asserts
that the action in one of his pianos has some 7,000 separate parts. Nearly all
modern actions are versions of Christofori's original up ward-striking ones,
which took advantage of the downward force of gravity for the key's return. Some
workers have experimented with downward-striking actions, so far without
success.
Early in the history of piano-building the hammers
were small blocks of wood covered with soft leather. The inability of leather
to maintain its resiliency after many successive strikings led eventually to
the use of felt-covered hammers. If the felt is too hard and produces a harsh tone, it can be pricked with a
needle to loosen its fibers and will produce a mellower tone. If the tone is
too mellow and lacks brilliance, the felt can be filed and made harder.
A standard piano has three pedals that serve to
control the dampers. The forte, or sustaining, pedal on the right disengages a4
the dampers so that the strings are free to vibrate until the pedal is released
or the tones die away. The sostenuto pedal in the middle sustains only the
tongs that are played at the time the pedal is depressed; all the other tones
are damped normally when their respective keys are released. The
"soft" and also a back-check that regulated the hammer's downward return. An individual damper connected to the action of the hammer was provided
for each note. Christofori called his instrument the "piano-forte,"
meaning it could be played either softly or loudly.
CHRISTOFORI ACTION, invented by Bartolommeo
Christofori in the early 18th century,
was the first hammer action and the prototype of all modern piano actions. It
included an escapement de vice that "threw" the free-swinging hammer
upward at the string
MODERN PIANO ACTION [lower design of fig.] is modeled
closely on Christofori's original
upward-striking actions, which took advantage of the downward force of gravity
for the key's return. Unlike the early hammers, which were small blocks of wood
covered with soft leather, the modern hammer is covered with felt. If the felt
is too hard and produces a harsh tone, it can be pricked with a needle to
loosen its fibers and will produce a mellower tone. If the tone is too mellow
and lacks brilliance, the felt can be filed and made harder.
pedal on the left shifts the entire action so that the hammers strike fewer
than the usual number of strings, decreasing the loudness of the instrument.
the most interesting part of the piano from the standpoint of the
acoustical physicist is of course the strings. The strings used in pianos today
are made of steel wire with an ultimate tensile strength of from 300,000 to
400,000 pounds per square inch. Additional weight is needed to make the bass
strings vibrate slower and so generate sounds of lower pitch; this is provided
by wrapping the steel wire with wire of copper or iron. Two such wrappings are
often used in the extreme bass.
INHARMONICITY of a real
piano tone is evident in this graph, based on data obtained from an electronic
analysis of the partial tones of the lowest note on the piano keyboard (an A).
The partials of the real piano tone (solid line) become increasingly
sharper-that is, higher in frequency-compared with the partials of a pure harmonic tone
(broken line).
The vibration of a string that is attached securely at
both ends is caused by a restoring force-a force that seeks to return the string
to its original position after it has been displaced from that position. In a
string that lacks stiff ness the partial tones set up under the influence of
the restoring force will be harmonic. In the piano the stiffness of the strings
affects the restoring force to such a degree that some of the partials
generated are not harmonic. This effect was known to
Lord Rayleigh, who took it into account in formulating his classic equations
for vibrating strings in the late 19th century. Many other investigators have
since worked on the problem; our current effort is a continuation of the same
line of inquiry.
Part of our program includes a series of tests in
which a jury composed of both musicians and nonmusicians is asked to
distinguish between recordings of real piano tones and synthetic ones. The
synthetic tones are made by a bank of 100 audio-frequency oscillators that can
be tuned to cover a range of from 50 to 15,000 cycles per second. Fine tuning
is achieved by means of an attenuator connected to each oscillator circuit; the
attenuator covers a range of 50 decibels, 10 decibels being a tenfold in crease
or decrease in the intensity of sound. With this apparatus it is possible to
build up synthetic tones that represent a wide variety of partial-tone combinations.
Real piano tones can be closely imitated by tuning a separate oscillator to the
precise frequency and intensity associated with each partial tone of the real
tone. The complex synthetic tone thus generated can then be fed into an
"attack and decay" amplifier in order to give it the attack-and-decay
characteristics found in the real piano tone.
In our early tests the synthetic tones were
arbitrarily built up of harmonic partials. The musicians on the jury were able
to distinguish 90 percent of these tones from real piano tones; the non
musicians, 86 percent. In later tests synthetic tones built up of inharmonic
partials were used. Results from these tests showed that in most cases both the
musicians and the nonmusicians were guessing; both groups identified only about
50 percent of the tones correctly.
Whenever a synthetic tone and a real tone were judged
to be identical, we could give a description of the quality of the real tone
based on our knowledge of the quality of the synthetic tone.
Recorded piano tones can also be analyzed directly by
means of a conventional audio-frequency analyzer that is adjusted to pass only
a narrow band of frequencies (about four cycles per second). The analyzer is
set at different frequencies until it registers a maximum response for the
particular partial being analyzed. A pure tone from one of the oscillators is
then sent through the analyzer, and its frequency is adjusted until it gives a
maximum response at the same setting as that of the real partial. An electronic
counter is used to measure the frequency of the oscillator tone to an accuracy
of within about a tenth of 1 percent. This frequency is assumed to be the
frequency of the real partial being analyzed.
A sample of this kind of analysis for the lowest note
on the piano keyboard (an A) is given in the illustration at the left. It is
evident that the partials of the real piano tone become sharper that is, higher
in frequency-compared with the partials of a pure harmonic tone. The 16th
partial, for example, is a semitone sharper-half a step higher than it would be
if it were harmonic. The 23rd partial is more than a whole tone sharp, the 33rd
partial is more than two tones sharp and the highest partial in the analysis,
the 49th, is 3.65 tones sharp.
In addition to the fact that the piano's i tones are
generally inharmonic, the partials of any particular note tend to vary
considerably in loudness. This variation is called the harmonic structure of
the tone, or in the case of the piano, the partial structure. One way to
analyze the partial structure of a piano tone is to measure the maximum
response of each partial as it passes through the audio frequency analyzer. This
method was used to obtain the partial structure of the four G's shown in the
illustration on the opposite page.
The foregoing method does not yield the most accurate
description of the par tial structure of a piano tone, because the structure is
continuously changing. When a piano string is struck by its hammer, its
response reaches a maxi mum an instant later. From this moment on the tone dies
away as the string gradually ceases to vibrate. Because the ear perceives the
entire tone dying away uniformly, it might seem that all the partials of the
tone die away at an equal rate. An examination of the decay curves of
individual partials proves that this is not the case [see illustration on
next page]. It is obvious from these curves that if the partial structure
of a tone were measured at any given time, it would be different from the
structure at any other time. Nonetheless, some authors still refer to a decay
rate bf a tone as so many decibels per second. In actuality the partials do not
all decay at the same rate; in some cases they may even increase in intensity
before starting to decay.
The tones used for our decay-time analyses were
recorded in an ordinary music studio. It was thought at first that the
irregular variations during decay might be related to the acoustic characteristics
of the room or the piano. Accordingly the experiment was repeated in three
different rooms: a normally reverberant studio, a very reverberant room and an
anechoic, or echoless, chamber. The irregularities in the decay curves were
present in all three rooms [see illustration on page 33].
0ne of the main advantages of our synthetic-tone
system is that it can be used to produce synthetic tones identical with one
another and with a real tone except for certain selected characteristics. For
example, a group of synthetic tones can be produced that differ only in attack
time, the time required for the loudness of the tone to reach its first maximum
after the hammer strikes the string. By presenting such a group of tones to our
jury we were able to determine that for the G just above middle C the attack
time has to be between zero and .05 second to sound like
the G on a piano. An attack time in the range of from .O5 to .12 second made
the note seem questionable, and one longer than .12 second made it sound decidedly
unlike a G struck on a piano. For lower notes the required attack time tended
to be longer; for higher notes it tended to be shorter.
Synthetic tones can also be produced that are
identical with one another and with a real tone in every respect except decay
time, the time required for the string to stop vibrating after it has reached
its maximum loudness. For an undamped G above middle C the decay time required
for the synthetic tone to sound piano-like was between two and 5.5 seconds. Again
acceptable decay times were longer for lower notes and shorter for higher
notes.
PARTIAL STRUCTURES of the
four lowest G's on the piano keyboard are presented in these four bar charts. The
partial structure of a musical tone is the variation in loudness of the partial
tones that constitute that particular tone. The partial structures of these
four notes were obtained by measuring the maximum response of each partial as
it passed through an audio-frequency analyzer that was adjusted to pass only a
narrow band of frequencies. The readings are given in relative decibel levels
with the loudest partial of each note set at zero; the other partials can then
be read as so many decibels below zero.
DECAY CURVES for nine
partial tones of the lowest C on the keyboard demonstrate that the partial
tones of a piano note do not all die away from an initial maximum at the same
rate. In some eases they may even increase in loudness before beginning to
decay. For each curve 30 measurements were made at equal intervals of .08
second each. Obviously the partial structure of a tone at any given time is
different from the structure at any other time.
Another procedure is to give synthetic tones a
piano-like attack and decay but to vary the partial structure. In one test
synthetic tones were built up in such a way that the loudness of each
successive partial was a constant number of decibels less than that of the
partial just below it in frequency. For example, if the difference was two
decibels, then the second partial would be two decibels fainter than the first
partial, the third partial would be four decibels fainter than the first, and
so on. The limits of this "decibel difference" for obtaining a
piano-like tone from the G above middle C were from five to 13 decibels per partial.
In this case the acceptable range was narrower for lower notes and wider for
higher notes. Tones produced when the decibel difference was below the lower
limit were judged by the jury to sound "dead" or "hollow." Tones
above the upper limit were described as sounding "like a harpsichord"
or having "too much edge."
Synthetic tones that were built up of perfectly
harmonic partials were described by the musicians and nonmusicians alike as
lacking "warmth." Musicians generally use this term to suggest a
certain quality of musical tone. For instance, a number of violins playing the
same note at the same time produce a tone that is said to be warmer than that
produced by a single violin playing alone. This quality appears to result from
the fact that it is impossible for a number of musicians to play exactly in
tune. When two different frequencies are sounded together, "beats"
can be detected, the number of beats being equal to the difference in cycles
per second between the two tones. A difference as small as two cycles per
second between the fundamental frequencies of two tones can, however, produce
much larger differences in the upper partials. Thus the beats that occur when
two tones, each with a large number of partials, are sounded simultaneously can
be quite complex. It is such beats between tones that account for the warmth
produced by several violins or by a chord on the piano.
In the piano some additional warmth can be attributed
to the fact that most of the hammers strike more than one string at a time. If
the strings are not identically tuned, beats will occur between the high
partials produced by each string. If such beats become too prominent, of
course, the strings are declared to be out of tune.
ACOUSTICS OF ROOM in which
the tones used in the decay-time analyses were recorded were shown by the
author and his colleagues to have a negligible effect on the irregularities
present in the decay curves of different partial tones of the same note. To
obtain these curves the decay times for the first and second partials of the G
above middle C were record ed in three different
rooms: a normally reverberant studio (broken black curves), a very
reverberant room (solid black curves) and an anechoic chamber (solid
colored curves).
The quality of a piano's tone also depends on several
outside influences that are not usually considered intrinsic properties of a
vibrating string. There is the impact noise of the hammer as it strikes the
string, the mechanical noise of the damping pedals, the effect of the damper on
the end of a tone, and the noise level of all the other strings, which are free
to vibrate sympathetically when they are not damped. In early tests it became
quite evident that our juries were using these factors as clues to distinguish the
real tones from the synthetic ones.
The impact noise of the hammer is not as noticeable in
the lower register as it is in the upper. For the high strings the impact noise
is almost as loud as the tone itself. A similar noise had to be superposed on
the synthetic tones before they could be effectively used in our tests. Preference
tests were set up to see if piano tones without this noise were more acceptable
musically than tones in which the noise was present. In general the individuals
tested were satisfied with the quality of piano tones as it is, and any large
departures from this quality were disparaged. Obviously this is the result of
years of conditioning, of hearing piano tones produced by pianos. Some
composers even write with this specific quality in mind. An example can be
found in Piano Concerto No. 2 of the American composer Edward MacDowell,
in which certain passages are marked martellato, presumably to indicate
that as much hammer noise as possible should be introduced into the passage.
The mechanical action of the pedals or dampers also
makes a noise that has become part of the piano's tone. More over, there is a
distinctive effect evident when the felt on the dampers is brought into contact
with the string: the tone is not cut off cleanly but rather sounds as though it
is being swallowed. The problems involved in trying to duplicate these
"side effect" sounds can be eliminated by using piano tones that are
produced by striking a key and allowing the sound to decay naturally by holding
the key down. In this way all the other strings remain damped. The pedals are
not used and only the damper of the struck string is disengaged by the action
of the key.
0ur studies have clearly shown that a complete description of the quality of the piano's tone must contain more than partial structure, attack time and decay time. Above all, the inharmonicity of the piano's tones must not be neglected. Some believe that the tone quality of the piano could be improved merely by making the tones more harmonic. Our tests have proved that synthetic tones built of harmonic partials lack the quality of warmth that is associated with the piano as it exists today.
end