ACOUSTICAL LAWS OF SOUNDING STRINGS.
Sound is an impression produced upon the brain through the ear by
the motion of air particles excited by an external body. In the
transmission of sound from the vibrating or “sonorous” body to the
ear it is motion that is transferred and not the substance of the air
itself. In the same way there can be no sensation of sound without
the interposition of an elastic fluid such as air or water, and the
production of sound in a vacuum is, therefore, impossible.
Sound, in short, has no objective existence. We know it simply as a
sensation, primarily caused by certain physical processes, the nature
of which is comparatively familiar to us. We are aware of all that goes
on between a sounding body and the ear, but we know nothing of the
processes whereby these physical motions are transformed until they
become, within the brain, sensations of musical sound or of noise.
While so much of mystery clouds our conception of the nature of sound,
we may take comfort in the knowledge that to penetrate the enigma is by
no means necessary. Not even the musician requires such transcendent
knowledge. To the student of musical craftsmanship it is equally
non-essential. It is well, however, to recognize the fact that as
soon as we leave the sure ground of physical investigation, we become
lost in impenetrable mystery and find ourselves face to face with the
ancient, yet ever new, questions of our origin and destination. When
we reflect upon the essentially spiritual and unearthly influence of
music, we cannot but feel that, in the making of instruments to serve
this art, we are ourselves assisting, however blindly, at a more than
Eleusinian mystery.
The ear easily distinguishes between musical and non-musical sounds.
Nor does it fail to recognize differences in relative loudness or
softness of any given musical sound. Again, the relative degree of
acuteness or gravity is distinguished, and, lastly, the quality of the
same musical note when played upon two different instruments or when
sung by two different voices is no less easily observed.
Now we have first to ask ourselves in what the difference between
musical and non-musical sounds consists. We may say that a musical
sound is produced by regularly recurring motions of the sounding body
communicated to the air; or, more technically, a musical sound may
be defined as a sound produced by periodical vibrations. This may be
proved by holding a piece of cardboard against a rapidly revolving
toothed wheel. As long as the revolutions of the wheel are performed
at a comparatively slow speed the noise produced by the impact of the
cardboard is broken and disjointed; but as the wheel is caused to
revolve with greater rapidity the noise becomes gradually continuous
and assumes a definite pitch. By increasing the speed of the wheel we
cause a higher pitched musical sound to be produced. Now, if we arrange
a second card and wheel and cause them to be set in motion together
with the first we shall find that when the two wheels are revolved
at the same speed, they produce sounds of the same pitch. Thus it is
apparent that the pitch of a musical sound depends upon the speed of
vibration, or upon the number of vibrations per second. Without going
too deeply into technicalities it may be said that similar experiments
have enabled investigators to determine the behavior of sonorous
bodies in reference to all the other conditions that pertain to them.
Thus, in the case of strings such as are used in the pianoforte, we
are in possession of facts that make it possible for us to state
accurately the pitches that will pertain to strings of given lengths,
densities and thicknesses, which are stretched at given tensions. It
is unnecessary to go into details of the precise methods employed to
demonstrate these laws, and it will be quite sufficient to quote the
laws themselves. The reader is therefore invited to note carefully that:
1. The number of vibrations of a string is inversely proportional to
the length of the string.
2. The pitch of a musical sound is proportional to the number of
vibrations per second; the greater the number of vibrations, the
higher the pitch.
3. The number of vibrations per second of a string is proportional
to the square root of its tension. That is to say, if a string is
stretched with a weight of one pound it will give forth a sound one
octave lower than the sound that it would emit if stretched with a
weight of four pounds.
4. The number of vibrations of a string varies inversely as the
thickness of the string. So that if there are two strings of the
same material and length and subjected to the same tension, and if
the diameter of the first is twice the diameter of the second, the
first will produce one-half as many vibrations as the second.
5. The number of vibrations per second of a string varies inversely as
the square root of its density. Thus, if one string has four times
the density of another, the first will produce one-half as many
vibrations as the second.
In addition to these valuable laws, there are certain others which have
reference to the actual musical sounds produced by strings. By means of
them we know the relative proportions of the strings that will, other
things being equal, give the various notes of the musical scale. If a
perfect musical string be stretched and excited into vibration it will
be found that an exact octave above the note that the whole string
gives out may be produced by dividing the string at its precise middle
point and causing one of the halves to vibrate. Now we have already
noted that the number of vibrations of a string is proportional to its
length, and it is therefore obvious that the halves of the given string
each have double the number of vibrations of the whole, and that,
consequently, the octave to a note is produced by either twice or half
the number of vibrations that suffice to produce the given note.
Carrying the experiment further, we may, by dividing the given string
at other points on its surface, obtain all the other notes of the
musical scale. It will not be necessary to repeat the explanation in
each case, and the reader will have no difficulty in comprehending
the following table, which gives the relative string length required
to produce the eight notes of the diatonic scale of C major, taking
the length of the complete string that gives the keynote as 1, and
considering all other pertinent conditions to remain equal:
C D E F G A B C
1 8/9 4/5 3/4 2/3 3/5 8/15 1/2
Keynote 2d 3d 4th 5th 6th 7th Octave
At first sight it might appear that the above data ought to give us
all necessary information in regard to the phenomena of vibrating
strings. Undoubtedly, the difficulties that surround the pianoforte
designer would have little power to cause worry if there were nothing
more to learn. Our troubles, however, are but just now beginning, and
the difficulties that still exist are greater than any that we have yet
investigated. These difficulties have their origin in the _nature_ of
the sounds that are emitted by musical strings.
While we have been investigating the relative vibration speeds and
pitches that pertain to the strings under various conditions, we
have not as yet paid attention to any other difficulties that might
have their origin under entirely different circumstances. There are,
however, certain highly important phenomena which are determined
by the nature of the strings themselves, irrespective of all other
conditions. These phenomena affect the constitution of such sounds as
any musical string may produce. Sounds produced through the agency
of musical strings are not and cannot be simple sounds. And this
peculiarity arises from the fact that such strings in common with most
other agencies for the production of musical sounds are incapable of
performing perfectly simple vibrations. If a string vibrated as a
whole uniformly and all the time, its motions might be compared to the
rhythmic swing of a pendulum, and the sounds that it emitted would be
absolutely simple and absolutely pure. The fact, however, is that this
never occurs. No string ever vibrates as a whole without simultaneously
vibrating in segments, which are aliquot parts of the whole. These
segments, when thus vibrating, give out the sounds that pertain to them
according to their relative lengths; while the vibration of the whole
length of the string, at the same time, causes the production of the
sound proper to it, which is called the “prime” or “fundamental” tone.
The sounds produced by the simultaneously vibrating segments are called
“partial tones” or “upper partials.” In the case of sounding strings,
such as we are now investigating, the partials follow each other in
arithmetical progression and are produced by the vibrating of segments
the proportions of which may be expressed by the harmonic series 1,
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and so on ad infinitum. Now,
if we examine this series we shall see that the lower of the partial
tones that are represented by the various fractions must bear distinct
harmonic relations to the fundamental tone. It will simultaneously be
observed, however, that as the series is continued, the fractional
quantities become uniformly smaller, and the difference between any
pair of them (for the same reason) is smaller as the position of the
given pair is more remote from unity. Naturally, this means that the
partial tones represented by such fractional quantities are separated
by continually decreasing intervals. If the process is carried far
enough, the time comes when the interval of separation is less than a
semitone. Clearly, then, partial tones in this condition can bear no
proper harmonic relation to the fundamental tone. They are, in fact,
dissonant.
Here, then, we come upon a fact that has a very wide bearing. It is a
demonstrated acoustical truth that tone quality depends upon the number
and intensity of the partial tones that accompany the fundamental
during the sounding of any musical note. If, through any cause, these
high and dissonant partials are excited into undue prominence, they
may, and do, exercise a profound and maleficent influence upon the
quality of musical sounds. We shall later have occasion to confirm
the truth of this statement, and we shall learn, in the course of our
investigation, fully to appreciate its importance in the practical
problems of pianoforte design.
For the purpose of assisting the reader in the comprehension of the
above argument, the following table is given, showing the order of
succession and pitch of the partial tones of the note C (second line
below the staff in the bass clef). Taking the pitch of the octave above
middle C, for convenience of calculation, as 512 vibrations per second,
this gives us 64 for the C in question.
[Illustration:
NAME OF NOTE
ORDER OF SUCCESSION
VIBRATION PER SECOND
C1 1 64
C2 2 128
G2 3 192
C3 4 256
E3 5 320
G3 6 384
B-flat3 7 448
C4 8 512
D4 9 576
E4 10 640
F-sharp4 11 704
G4 12 768
A4 13 832
B-flat4 14 896
B-natural4 15 960
C5 16 1024
]
It should be observed that the seventh, eleventh and fourteenth
partials and their multiples cannot more than approximately be
indicated in musical notation, as they do not exactly correspond to the
notes that are written to represent them. We are obliged to be content
with an approximation to the true pitch of these partials and the
notation given here is as near as it is possible to approach.
A brief consideration of the facts thus presented will convince the
reader that a combination of any fundamental tone with its first eight
partials will produce a relatively harmonious effect. At the same time
we must observe that this harmoniousness is more and more obliterated
as the higher partials are permitted to sound simultaneously with the
others. In fact, it may be said that, although we can not and must not
eliminate the dissonant partials altogether, we should attempt to cause
the strings to vibrate with entire freedom only as far as concerns the
first eight partials, and less freely as far as concerns the others.
Now, in what manner can this desirable end be attained? To answer
this question we must first discover what pre-disposing causes, if
any, exist towards the favoring of any combination of partials at the
expense of any other.
In speaking of the automatic sub-division of a string into vibrating
segments, we omitted, at the time, to make mention of a fact which
should, however, be obvious to the reader; namely, that the various
points at which the sub-division occur are themselves motionless.
It would be more correct, perhaps, to say “apparently motionless”; for,
of course, if these dividing points or “nodes,” as they are called,
were entirely without motion, the formation of the vibrating segments
would be impossible. In most cases, however, the “amplitude” or length
of swing of the nodes when in motion is very much smaller than the
amplitude of vibration of the segments. Consequently, as the vibration
of the segments of a string is itself ordinarily invisible, the motion
of the nodes may be considered as inappreciable.
Now, these nodes exercise considerable influence upon the problems that
we are considering. For example, according to the researches of Young,
it appears that when a string is struck at any point all those partials
are obliterated that have their nodes at that point. Curiously enough,
however, it has since been found in the case of the pianoforte, that
those upper partials are not _necessarily_ eliminated that have their
nodes at the striking point. Undoubtedly, however, a properly chosen
node provides the best possible striking point, since its selection
permits the operation of its tendency to suppress those particular
partials that have their nodes at the same place.
A consideration of the phenomena already observed has caused us to
perceive that the highest partials of the compound tone produced by
a musical string do not bear precise harmonic relations to the prime
tone. As the successive sub-divisions of the string approach closer and
closer to each other, the tones thus generated are seen to be distant
by proportionately less intervals, until at length they cease to have a
close similarity to any tone of the musical scale. Consequently, as was
said before, they exercise a generally harsh and dissonant influence
upon the nature of the compound tone. We have already concluded that,
broadly speaking, we should aim to eliminate these dissonant partials
and, conversely, to favor the prominence of those which are more nearly
harmonic. The reasoning which has served to lead us to this conclusion
may profitably be carried a step further. If the highest partials are
non-harmonic, it is obvious that their presence or absence, their
prominence or the reverse, must necessarily exercise much influence
upon the actual quality of a musical sound; upon the individual color
which different generating agencies impart to the same musical note; in
effect, upon all the numerous gradations of what we are accustomed to
call harshness, hollowness or mellowness of tonal quality.
This inevitable conclusion has been fully substantiated by the
results of experiment. The labors of Helmholtz and Koenig have
demonstrated conclusively that the quality of a musical sound depends
upon the number and intensity of the partial tones that accompany
the fundamental. Thus the mystery of the individual tone coloring
that distinguishes the voices of different musical instruments or of
different persons is transferred from the realm of psychology to that
of science. In fine, it becomes clear that if we can govern the number
of the segments into which a vibrating string divides itself, and if we
can also control the amplitude of vibration of these segments, we shall
find it possible to alter the tone quality of a musical instrument at
our pleasure.
It has already been observed that the generation of certain partial
tones is assisted or retarded as the position of the striking point
on a string is changed. It may not be out of place to note that the
various other methods of exciting a string, such as plucking, bowing,
etc., permit the production of equally variable effects as the points
at which they operate are changed. Our inquiry, however, is confined to
the pianoforte, and we shall therefore continue to limit ourselves to
the cases of pianoforte strings as struck by the usual hammers.
The matter of choosing a proper striking point was first systematically
investigated by John Broadwood, founder of the celebrated house of that
name, in the early part of the nineteenth century. Until that time the
pianoforte makers had, apparently, paid no attention to this important
problem and had been content to follow in the steps of the builders of
harpsichords and spinets. Examination of any of the instruments that
are direct ancestors of the pianoforte will show that the strings are
struck, indifferently, at any point from one-tenth to one-half of the
speaking length. The only exceptions appear to be those clavichords in
which the strings are all of the same length and in which the tangents
on the keys impinge upon the strings at different fixed points to give
the corresponding notes of the scale. Since the time of Broadwood,
however, the vast importance of correctness in this particular has come
to be recognized with more or less unanimity.
The investigations undertaken by this eminent maker convinced him that
the ideal striking point lay between one-seventh and one-ninth of the
speaking length of the string. Now, our investigations have shown us
that the most harmonious and agreeable compound tone is that which is
formed by the combination of the first eight partials. It would seem,
therefore, that one-eighth of the speaking length would be more correct
than the approximation that was arrived at by Broadwood. Theoretically,
indeed, the latter is nearer to the ideal point; is, in fact, the
ideal point. For obvious mechanical reasons, however, it is usually
impossible to hit this point with exactitude, and the approximation
suggested and used by Broadwood has been proved, by the practice of the
best makers, to offer the nearest practical solution.
We may, then, lay it down as a rule to be followed that a point as
nearly as possible midway between one-seventh and one-ninth of the
speaking length of the string should be chosen and adhered to as the
proper place where the blow of the hammer should be struck. If this
rule be faithfully followed the greatest obstacle to purity of tone is
removed and the most harmonious and agreeable combination of partials
is in a fair way to be secured. Nevertheless, it is necessary to make
an exception for the highest notes on the piano. Practical experience
has shown that one-tenth is a better striking point for the very
highest and shortest strings.
Thus we have been able to enunciate and discuss the principal laws that
govern the activities of sounding strings, particularly those of the
pianoforte. As the argument is developed, it will often appear that the
theoretical exactitude of the rules here laid down must be modified
in practice. Such a condition is always inevitable as between a body
of laws and the application thereof. It will be found, however, that
the variations to be recorded are not generally very important, and
the reader will be well advised to make the rules enunciated in this
chapter his continual leaning post and guide.
The most conspicuous difference is, perhaps, that which exists between
the theoretical and practical results of halving string lengths to
obtain octaves. In practice it is found that pianoforte strings
generally sound a little flat of the octave when divided at exactly the
middle point. But the variation is the fault of the steel wire and not
of the rule.