AN ENGINEER'S VIEW ON CELLO BOWING TECHNIQUE
by Roland V. Siemons
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ABSTRACT
Students developing their cello playing, teachers explaining how to play
cello - we are all in need of a principal understanding of how our cello
and our body work together. In this paper I try to reveal some of the physical
principles according to which a cello is being played. This is done by means
of a mechanical analysis and a number of simple experiments. Mechanical
analysis is a universal tool, the results of which are applicable to every
cellist. It is shown why relaxation of the right arm is of great importance
for obtaining a bright sound. A specific manner of right arm use to execute
bowing forces is advised, along with a method to pursue a relaxed bowing
technique.
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CONTENTS
1 INTRODUCTION 1
2 MECHANICAL ANALYSIS 2
2.1 THE FORCES EQUILIBRIUM OF BOW AND STRING 2
2.2 MANAGEMENT OF BOWING FORCES 7
2.3 SOUND GENERATION: THEORY AND EXPERIMENTS 9
3 RELAXED BOWING TECHNIQUE 11
3.1 A PROPOSED METHOD 11
3.2 SOME EXERCISES 15
4 CONCLUSIONS 16
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1 INTRODUCTION
It is often stated that relaxation is of utmost importance for proper cello
playing. Why is this true? While focusing on the right arm, this paper shows
that a primary reason for right arm relaxation is the improvement of sound
quality. The theories presented here have their roots in dedicated practice
with bowing technique during the last few years. The leading principle was
the improvement of sound quality. Since it is a challenge to understand
what one is doing, and since a better understanding may eventually result
in improved practice, theoretical analyses were made and experiments conducted.
Hopefully the physical understanding is increased by this paper. Perhaps
some misunderstandings can be taken away, and perhaps an interesting discussion
evolves. In addition to presenting theories, this paper also addresses a
way to pursue relaxed bowing.
Three principal physical questions are discussed first, in Section 2:
! Which are the forces executed on a bow?
! How may the body execute the necessary forces on a bow?
! Why is right arm relaxation so important for sound quality?
This is done by a mechanical analysis, which is a universal tool. Its results
are applicable to every cellist.
A more practical section addresses the physiology of relaxed bowing (Section
3).
2 MECHANICAL ANALYSIS
2.1 THE FORCES EQUILIBRIUM OF BOW AND STRING
To better understand the nature of the forces acting on the bow, an introduction
is given to the physics of sound production with a bowed string instrument.
We consider the uniform, i.e. unaccelerated, movement of a bow. In other
words, bow changes are ignored in this analysis.
A bowing cellist is making a string vibrate by applying a frictional force
on the string. He does this by moving his bow and by applying a force perpendicular
to the string. The strength of the perpendicular force, which depends on
the desired quantity of friction, can be expressed as:
Fperpendicular = Ffriction / c ,
in which c is a constant (the friction constant) (See Figure 1 for an illustration).
As a result of the applied forces the string is moved in the direction of
the combined perpendicular force (to the body of the cello) and the friction
force (along the bow movement: to the right hand when playing down-bow).
The string even slightly rotates, due to the fact that the friction force
does not exactly intersect with the axis of the string - but let us leave
this apart: we do not need such a refined theory.
fig below
Figure 1, The forces equilibrium acting on the string section which is touched
by the bow. (The forces are executed by the bow and by the adjacent string
parts "cut away" from the string section).
Since the bow's hair is sticky due to the applied rosin, the string does
not stay just displaced in equilibrium with the rubbing and pressing bow.
It starts to vibrate parallel to the resulting forces - not exactly in the
bowing direction, but also a bit perpendicular to the bow towards the body
of the cello. The bow, in turn, is subjected to reactive forces executed
by the string on the bow. Therefore the perpendicular force executed by
the bow on the string is not constant in time, but rather changing according
to the same vibration pattern as the string. The same applies to the friction
force executed by the bow on the string. If the string's vibrations would
be described by a perfect sinus, which is only an approximation, then the
size of the two forces can be expressed as:
Fperpendicular = A + B sin(< B t)
Ffriction = c [A + B sin(< B t)] ,
in which A and B are constants, < denotes the frequency and t represents
time. In reality, a sound shows more than a single frequency (<). The
above two formulas should therefore be a bit more complicated, but it serves
our purpose to show that there exists a basic value for the described forces
executed by the bow, around which there is a certain fluctuation which varies
with the same pattern as the frequencies of the sound produced. The basic
forces are A and c A, respectively for the perpendicular force and the friction
force. The fluctuations are B sin(< B t) and c B sin(< B t), respectively.
The fluctuations are extremely small relative to the basic forces (A and
c A) applied. This is confirmed by the observation that the displacement
of the bow's hair seems to be constant rather than oscillating with the
pitch of the tone produced. It is oscillating though - but our eyes cannot
see it.
We will now further analyze the forces equilibrium acting on the bow. To
get a better understanding we will disregard the oscillations of the forces
just shown above. The issue will come back in Section 2.3. The forces applied
by a cellist's hand needed for creating a tone from a bowed string can be
distinguished into pure forces and rotational forces (also called moments
or torques). These forces are illustrated in Figure 2.
fig below
Figure 2, Forces equilibrium occurring with down-bow movement.
We have seen that, if a tone is produced, a friction force is executed by
the string on the bow in the direction opposite to the bow movement. The
strength of this force is equal to a lateral force effected to the frog
by the right hand. For a continuous tone produced by a down-bow or up-bow
movement, the size and direction of this frictional force are basically
constant in time, if we disregard the oscillations with the sound frequencies.
Also a perpendicular force is executed by the string on the bow. It is related
to the friction force by the friction constant c (see above). The basic
strength of this perpendicular force must also be constant in time if a
constant tone is to be produced. This force is compensated by two forces
of different origin, together creating an opposite perpendicular force of
the same strength. These forces are: Firstly, the weight of the bow and
secondly, a force induced by the right hand. The perpendicular forces are
basically constant in time and independent on bow position between tip or
frog (assuming a tone of a constant playing strength, i.e. somewhere between
pp and ff).
Note that for a uniform bow movement all forces are in equilibrium. Therefore,
the pure forces - one of which is executed by the string, another by the
right hand, another by gravity (the bow's weight) -, precisely compensate
each other. A torque executed by the hand is necessary to compensate the
counter-acting couple resulting from the distance l' between the two force
lines perpendicular to the bow. It can be directly observed by the cellist
as the rotating tension in his right fingers and wrist. It is the strongest
when playing ff at the bow tip. The size of the torque is expressed as:
T = l' Fperpendicular, by string - m Fbow weight ,
in which m is the distance between the bow's centre of gravity and the right
hand. l' is approximately equal to l, which is the distance between hand
and string (This approximation applies since the friction force is very
small). The distance l varies continually with time and therefore the size
of the torque is also subject to continual change. The difference between
down-bow and up-bow is that after a change in bowing direction, the frictional
force component executed by the string and the lateral force component executed
by the right hand have reversed to the opposite direction.
The illustrated system of forces applies to legato types of bowing in up-bow,
down-bow, staccato (after "take-off"), but not to bowing types
of strongly varying bow velocity (sautillJ, spiccato).
The forces perpendicular to the bow, are now analyzed further. The strength
of these forces varies with playing strength. Yet some characteristics applicable
to all playing strengths can be described. A projection of the bow, viewing
from frog to tip, is displayed in Figure 3. The forces equilibrium acting
on the bow in the plain of projection are indicated. The friction force
executed by the string and the lateral force executed by the hand cannot
be shown as they are pointed at right angles with the picture. The applied
torque is not displayed either. Three forces are shown:
! A force executed by the string, perpendicular to the string's axis,
! A force executed by the cellist's hand, towards the string's axis but
not exactly perpendicular to it, and
! The bow's weight.
It is shown that the force executed by the right hand can be resolved into
a downward and a vertical component. The size of the downward component
is:
Fperpendicular by hand, downward = cos(") Fperpendicular by string
- Fbow weight ,
while the size of the horizontal component is:
Fperpendicular by hand, horizontal = sin(") Fperpendicular by string
.
The angle " is different for each cellist, usually somewhere between
30o and 45o. Thus the size of the downward component is about 0.7 to 0.9
times the size of the perpendicular force executed by the string less the
bow weight. Similarly, the horizontal force is 0.5 to 0.7 times the size
of the perpendicular force executed by the string. Later on in this paper
we will look into how these forces are executed by the cellist's right arm.
fig below
Figure 3, Lateral view along a cello bow, and the forces equilibrium acting
on the bow in the plain perpendicular to the bow.
First, we determine the size of these components. As said before, the strength
of the perpendicular forces varies with playing strength. However, there
exists a maximum for these forces beyond which the bow's hair touches the
wood of the bow. This maximum strength was measured by pressing a bow down
until the bow strings touched the wood (Figure 4). A strength of about 0.7
kgf (7 N) was found. If we disregard the possibility that a cellist may
wish to overstress his bow, this is a good indication of the maximum perpendicular
forces ever executed during usual classical cello playing. Of course, it
should be taken into account that a somewhat larger force can be applied
near the frog or the bow tip, due to the larger distance between the bow's
wood and its hair and, thus, the potentially stronger reactive force executed
by the bow in those areas. It is estimated that in ordinary playing, the
perpendicular force can never get stronger than 1 kgf. Taking into account
that the bow itself weighs about 0.080 kg, it is concluded that the size
of the downward force component executed by the cellist's right hand is
always in the range of -0.08 (i.e. upward) to +0.8 kgf (downward). Likewise,
the size of the horizontal force towards the cellist's body centre is from
0 to 0.7 kgf.
A general conclusion from the previous section is that the management of
the forces necessary to produce a good quality tone is done by the right
arm by means of the controlled execution of torque, combined with the controlled
execution of vertical (downward) and horizontal (pulling) forces. The parameters
by which the executed torque varies are:
! Bow position between frog and tip, and
! Playing volume.
The single parameter on which the horizontal and vertical forces are dependent
is playing volume only. The principal possibility of arm relaxation for
executing these torque and forces is discussed in the following section.
2.2 MANAGEMENT OF BOWING FORCES
Body dimensions (i.c. arm weight) of cellists are reviewed in Table 1. It
is shown that the forearm usually represents sufficient mass to execute
the maximum required downward force (0.8 kgf, see above) by gravity alone.
This applies to adult men, women and children alike.
Table 1, (see web version)
Source: A.E. van Hellemond, dietitian.
fig below
Figure 4, Measurement of the maximum downward force if the string touches
the bow at the bow centre.
A further conclusion can be drawn: Since the forearm is so heavy, it needs
to be lifted continuously by the upper arm, even during the strongest ff.
A cellist must lift his arm the most strongly when playing pp. When playing
ff he does not need to press the bow downward, but he may rather relax the
muscles which are lifting his arm. Thus a stronger vertical force - albeit
upward - is to be executed when playing softly. Isn't this a paradox? Such
a paradox does not apply to the pulling horizontal force executed by the
bowing arm. If playing ff, the cellist has to pull the bow strongly horizontally
towards his body centre. If playing pp the pulling horizontal force is weaker.
2.3 SOUND GENERATION: THEORY AND EXPERIMENTS
Theory
By means of his bow, the cellist provides energy to the string. This is
done by the frictional and perpendicular forces which the bow executes on
the string. The bow is an energy transmitter. The string, in turn is supposed
to transmit the energy received from the bow to the body of the cello which
transforms it into an audible sound.
The same type of vibrations which the bow evokes in the string are adopted
by the bow. This is because the forces executed by the bow on the string
are reflected by the string into the bow. In Section 2.1 we have seen that
the string vibrates parallel to the combined friction and perpendicular
force applied by the bow. As a result the bow will respond with complementary
vibrations as indicated in Figure 5.
fig below
Figure 5, Corresponding string and bow vibrations.
Thus the bow also absorbs some of the energy of the swinging string. The
energy may even be carried further into the right arm where it may disappear
as heat (dissipation) due to internal friction. This can be demonstrated
by the following simple experiment: At the middle of the bow, the bow is
pressed as if playing ff but moved very slowly in a down-bow or up-bow movement.
The result is a loud sort of shriek. A proper sound does not develop because
the string does not transfer its energy to the body of the cello, but rather
to that of the cellist, by way of his right arm. You can feel it happen
in your shaking arm. In addition to being an energy transmitter, it appears
that the bow may also act as a brake.
However, also under ordinary circumstances, the string cannot succeed to
transfer all energy towards the body of the cello in a perfect manner either.
The induction into the bow of a vibration pattern similar to that of the
string, and thus backward energy transmission from the string into the bow
is necessary to liberate the string so that it can vibrate as freely as
possible. A further backward transmission of energy through the bow into
the right arm should however be prevented. Some experiments show how a phenomenon
called damping inside the right arm can be responsible for a decreased sound
quality. Damping is a way of energy absorption.
Experiments
Experiment 1: A lead mass of 0.5 kg (made of sheet lead) was tightly fit
to the frog (Figure 6). A G major scale was played in first position on
the G and d string. Sound quality was accurately observed and compared to
the sound quality of playing without mass attached.
fig below
Figure 6, Experiment with an inert weight attached to the frog.
Experiment 2: A mass of 0.5 kg was constructed from silicon gel and lead
grains inside a plastic bag. The mass was wrapped around the bow grip. The
bow was manipulated by holding the mass in the right hand. In theory, the
bow vibrations could be damped by the internal friction in the applied weight.
A G major scale was played in first position on the G and d string and again
sound quality accurately observed. (In two variations to this experiment
the mass was (1) tightly fit to the fingers with adhesive tape and also
(2) to the wrist).
Results: No convincing difference in sound quality could be observed between
ordinary playing - without inert mass attached - and playing with the attached
rigid mass from experiment 1. Application of the silicon based mass, whether
to the frog, the fingers or the wrist, caused a strong reduction in brightness
(less overtones could be heard). In all experiments the same total weight
was added to the arm-bow combination. Apparently the increased mass as such
was not a reason for the observed change in sound quality. Rather it was
the type of weight attached. Overtones were significantly absorbed by internal
friction inside the silicon based mass.
Interpretation and hypothesis
The mechanical theory of dynamics shows that a phenomenon called resonance
may occur if a vibrating force is applied to a mass which is balanced by
a spring and which is subjected to a damping force. The specific circumstances
under which resonance occurs are reduced damping and correspondence of the
force frequency and the natural frequency of the mass-spring system. While
mechanical engineers attempt to avoid such situations for their bridges
and other constructions in order to avoid collapse, a cello player pursues
the opposite. The more vibrations the better: a richer, brighter sound.
The experiments showed that a damping mass purposely attached to the right
arm was able to noticeably absorb high-frequency sounds thus negatively
affecting sound quality. My hypothesis is that in the human body mechanical
energy can be absorbed (damping) by tightened muscles. The expectation is
that tightened muscles show the same type of damping properties as the silicon-lead
mass used in the experiment. The use of dispensable muscles always comes
in pairs. Every superfluous tension is accompanied by counter-tensions executed
by the so called antagonist muscles. If the hypothesis is true, superfluous
muscle tension in the right arm should be avoided for reasons of sound quality.
(Further experiments to test the hypothesis could be conceived and conducted).
Thus, it seems to me that, in view of energy transmission for sound generation,
the right arm is the body part which is closest to the cello. Right arm
relaxation is an important feature of high-quality sound production and
an important goal in the development of one's bowing technique.
3 RELAXED BOWING TECHNIQUE
3.1 A PROPOSED METHOD
How can relaxed bowing be achieved? Bowing is a very complicated movement.
Already the bowing of one single string is difficult due to the continuous
changes in forces executed by the arm. Bowing is even more difficult, since
four different strings have to be bowed, and sometimes two strings together.
Ideally, if bowing a single string, the bow moves in one bowing plain. At
least seven different bowing plains can be distinguished for simple bowing:
A, A-D, D, D-G, G, G-C and C. But in fact the number of plains is even larger,
as some bowing techniques require very rapid string alternation over 3 or
4 strings.
Although we can describe some exterior characteristics (visual analysis:
movement directions, attitude, etc.) of bowing, relaxed bowing is not a
matter of how it looks like from the outside. It is about how it feels internally
inside our bodies. A cellist may play in a way which from the outside looks
as if he makes the correct bowing movements. He might however execute the
downward bowing forces by pressing the bow down rather than by relaxation
(see Section 2.2). Yet, the shape of bowing movement is important. It may
be an indicator to a teacher detecting unrelaxed bowing. It may also assist
in developing one's own relaxed bowing technique. This is argued below.
Since bowing is such a difficult thing, we should look into bowing methods
by means of which relaxation is the most easily achievable. In my experience,
the easiest and most effective attitude is with a high elbow, kept high
whether playing at the tip or the frog (Figure 7). To easily understand
the proposed way of bowing, rather than to look at the provided sketches,
it might be better to listen to my teacher, Max Werner. Max Werner used
to say: "Always remember that a tennis ball should be capable of smoothly
running down your arm, whether playing the A string or C string, whether
playing at the tip or the frog." The advantage of this attitude is
that in this manner the arm movements are as simple as possible. The arm
shapes are similar during up-bow and down-bow, but also in all bowing plains.
Also the relative movements of the arm segments are integrated in a very
fluent manner.
fig below
Figure 7, High-elbow bowing attitude of the right arm (Dashed line: low
elbow).
It is difficult to motivate this statement without a cello at hand to demonstrate
things. One should take his own instrument and just give it a try. The best
is to also try a most contrasting movement as well: Down-bow on the D string
with the lowest possible elbow. It is then observed that closer to the tip,
the upper arm comes to a stand-still, while it is being twisted around its
axis (which is difficult) until the bow-tip is reached. Contrary, with a
high elbow, the arm just gradually stretches.
The most important feature of the described bowing technique, however, is
that it enables to experiment with arm relaxation and thus to develop this
capability. As a mechanical engineer I would explain this as follows. The
bowing arm can be described as a number of segments which are connected
by a number of ball joints and one line joint (the elbow) (Figure 8). Segments
and joints are listed as follows:
Segment
Connected by (joint type)
Body
Muscles (ball joint)
Shoulder blade
Shoulder socket (ball joint)
Upper arm
Elbow (Line joint)
Forearm
Wrist (ball joint)
Hand
Finger joints (ball joint)
Fingers
A ball joint allows motion in any direction. The human hip is another example
of such a joint. A line joint allows movement in one direction only (example:
a door in its hinges). If the elbow is held high while the arm is in a bowing
position, the elbow line joint is oriented vertically. This arm position
is being maintained by lifting the upper arm with the shoulder muscles.
Muscles in the upper and forearm are not needed at all. In this manner the
lower and upper arm are relaxed "automatically" just like a door
which - being hinged to vertical line joints - does not need an arm to remain
closed or opened. Bringing down the upper arm to a slightly lower position
(Figure 9) will now result in the execution of the necessary downward force
by the gravity of the forearm alone, while no use is made of any downward
aimed muscle tension in the arm at all. (It should be understood however
that the production of the torque and the pulling force mentioned in Section
2.1, to be developed by the appropriate muscles in the right arm, cannot
be avoided. This is needed to create equilibrium.)
fig below
Figure 8, Mechanical structure of the bowing arm.
fig below
Figure 9, Bringing down the right upper arm to execute the downward force.
Cellists should be aware that they dispose of five body segments by means
of which the bowing movement may be built-up:
! Fingers
! Wrist
! Elbow
! Shoulder socket
! Shoulder blade
Physically, it is possible to carry out the bowing movement without making
use of movement of the latter. I believe that a lot of cellists indeed do
not use their right shoulder blade for this purpose. However, the involvement
of this shoulder blade may favour continuity and similarity of movement
in all potential bowing plains, and hence improve sound.
3.2 SOME EXERCISES
Perhaps the following exercises are useful. To start bowing in a relaxed
manner, the right arm may be hung down besides the shoulder, completely
at rest, though loosely holding the bow. While describing a large circle
starting to move to the right, the right hand moves towards the D string
- the bow sort of landing from the sky in a down-bow movement, coming down
near the frog. The aimed at final arm shape is the "high elbow position".
This exercise can be repeated while landing in the middle of the bow or
at the tip.
A useful experiment to get a feeling for the movement potential of the right
shoulder blade is to deliberately bring it forward in up-bow on the C string,
when close to the frog. (The left arm may also involve its shoulder blade,
especially when playing in high positions. The two moving shoulders together
embrace your cello.)
Start a down-bow movement at the frog, in the high-elbow position. Experiment
with right arm relaxation by slightly lowering the shoulder and upper arm
in the mf area. Make whole up- and down-bowings from frog to tip and vice
versa. Increase playing strength by lowering the upper arm. Listen to and
improve sound quality aiming at a rich and focused sound. Do not apply vibrato.
4 CONCLUSIONS
Based on the analysis in this paper, the following general conclusions can
be drawn:
! A cellist must execute horizontal (pulling) and vertical (downward) forces
as well as torques (in wrist and forearm) on his bow to keep it balanced.
! The physical analysis shows that both light and heavy persons (man and
woman) dispose of sufficient arm weight to apply the downward force required
by gravity alone, without ever pressing the bow down. In fact the right
arm must be lifted continually by both light and heavy cellist.
! To manage the downward force, the right arm may just be relaxed in a controlled
manner. A cellist may as well press the bow down, but this probably negatively
affects sound quality.
! The bow is an energy transmitter but may also act as a brake.
! The hypothesis is made that the bow's function as a brake can be largely
avoided by a relaxed bowing technique. This hypothesis is supported by preliminary
experiments which illustrate the effect of damping in the right arm.
! Maintaining a high elbow results in a simple and fluent bowing movement
on all strings. This helps to develop a relaxed bowing technique.
! Due to the specific structure of the arm, the high elbow position enables
to experiment with, and to practice, right arm relaxation in relation to
sound quality.
ACKNOWLEDGEMENT
Max Werner, former solo-cellist in the Netherlands Radio Chamber Orchestra,
member of the Gaudeamus Quartet, and teacher at the Enschede Conservatory
is gratefully acknowledged for his teachings.
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Roland V. Siemons
Mechanical engineer and amateur cellist
Haaksbergerstraat 205
7500 AE ENSCHEDE
The Netherlands
Phone: 31 53 4307651
E-mail: siemons.btg@ct.utwente.nl
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