by Roland V. Siemons

Complete version with all illustrations in pdf!!!

Complete version in Word 6 Document (.doc) format

(Please note that none of the drawings or figures are included in this text version.)


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.







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).



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.


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.



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.


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.



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:
Connected by (joint type)
Muscles (ball joint)
Shoulder blade
Shoulder socket (ball joint)
Upper arm
Elbow (Line joint)
Wrist (ball joint)
Finger joints (ball joint)

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.


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.


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.


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.

Roland V. Siemons
Mechanical engineer and amateur cellist
Haaksbergerstraat 205
The Netherlands
Phone: 31 53 4307651

Please read copyright information before downloading ICS materials!!!

Direct correspondence to the appropriate ICS Staff
Webmaster: Marshall St. John
Director: John Michel
Copyright © 1995-97 Internet Cello Society