Physics and the Bow: Forces of Fingers and Thumb

May 18, 2021, 1:49 PM · In the early 1980's while a performance major in violin at the University of Iowa, I used my background in mathematical physics to analyze the demands on the fingers and thumb of the right hand in bowing. To that end, the simplest problem to study was bowing in a straight line at constant speed and contact point with the goal of a uniform sound throughout the bow stroke. Applying Newton's laws to this simple problem, the physicist in me was not surprised by the results.

But as a music student, I was shocked by the confusion in the pedagogical literature as well as some violin teaching on this subject. Having struggled with bow technique and especially with sound production, I became a better player almost overnight when I understood how this works, eventually presenting the results at a regional ASTA conference in the 1990's and at the Violin Society of America conference in 2000. That paper can also be accessed at the following link: The Science of the Art of Bowing.

This work only focuses on the physics of bowing, identical for everyone and not on the physiology. In 2014, Fan Tao, the head of research and development at D’Addario strings gave a talk at the National ASTA Conference about this work and the work of John Schelleng on the bowed string.

While everyone who plays knows that something is changing during the bow stroke, much of the confusion is based on the following simple experiment described in "The Art of Violin Playing" by Carl Flesch:

"The weight of the bow is unevenly distributed between the frog, middle and point (or tip)... If, as an experiment, we limit ourselves to holding the bow totally loosely in our fingers, without letting any pressure influence the result, we find that, at the frog, the bow is too heavy, and thus inhibits the vibrations of the string; at the point it is too light to set the strings into vibrating motions; and that it is only in the middle where the bow, by its own weight, is able to produce a tone. It is therefore necessary for the bow to be 'pressed down' at the point and to be 'lifted' a little at the frog. The pressure (or additional weight) is provided by the index finger, the lifting at the frog by the little finger. The first is a corollary of pronation (inward turning), the latter of supination (outward turning)..."

Unfortunately, as demonstrated by Newton's laws, the suggestion that since the bow is lighter at the tip and heavier at the frog means that you have to add weight at the tip and lift the weight off at the frog, is completely false. If you don't believe this, repeat this experiment while physically holding the bow at the tip! This misunderstanding has been reinforced by many more contemporary pedagogues as for example in "Principles of Violin Playing" by Ivan Galamian:

"...An unhappy consequence of this lever characteristic is the violinistically-very-awkward fact that an equal pressure or weight applied throughout the bow results in an unequal pressure on the strings. Consequently, wherever an even dynamic is needed, the pressure-weight combination applied has to be uneven. The pressure must be stronger toward the point to counteract the loss of weight in the bow and correspondingly decreased toward the frog where the bow’s weight is heaviest."

It easier to discuss this with the bow on the G string and horizontal, that is, parallel to the floor but the essence of the results are the same on any string.

vertical forces on bow

The only forces in a vertical plane acting on the bow are the force of the string on the hair, the downward weight (W) of the bow acting at its balance point, and the forces of the fingers and thumb. Any force in this vertical plane can be split into horizontal and vertical components, that is, in the direction of the bow and perpendicular to the bow. The force of the string on the hair is made up of its vertical component (N), equal and opposite in direction to the "bow pressure," and a horizontal component that is the friction between the hair and the string. Since it is the vertical forces that determine bow pressure, in order to achieve a uniform sound through the entire bow stroke at constant speed and contact point, Newton's laws imply that the sum of all vertical forces must be zero, otherwise the bow would move vertically. Therefore, since N and W are constant, the sum of the downward forces of the fingers minus the upward force of the thumb must be constant to get a uniform sound throughout the bow stroke. The forces in the fingers and thumb of course have horizontal components, but they balance the friction from the hair and do not determine bow pressure. When adding to the weight of the bow (loud playing), the sum of the downward forces of the fingers is greater than the upward force of the thumb. Conversely, when subtracting from the weight of the bow (soft playing), the force of the thumb is greater than the sum of the forces of the fingers.

When the bow is moving in a straight line at constant speed, Newton's laws also reveal that the sum of the moments of all forces is zero with respect to any point, otherwise the bow would rotate. The moment or torque of a force is the product of that force and the perpendicular distance or moment arm, from the line of action of the force to the point of reference. For example, for a bicycle wheel off the ground and supported only by its axle, if you push perpendicular to the spokes in the plane of the wheel, it rotates in the direction implied by the moment of the force with respect to the axle as the reference. The farther the line of action of the force is from the axle, the greater its moment arm and therefore the more leverage that same force has in causing rotation. If the line of action of the force goes through the reference point, the moment is zero. This would be analogous to trying to turn the wheel by pushing only on the axle.

Now, if we take the point of contact of the thumb as the reference, the force of the thumb has zero moment since its moment arm is zero, and the moment of the bow weight is a constant since its line of action is at a fixed distance from the thumb. For constant bow pressure, the moment of the vertical string force N is increasing even though N is constant as the bow moves from frog to tip since the distance from the string to the thumb is increasing. The only way to keep the bow moving in a straight line is for the fingers to provide a net moment or torque that balances the moments of the bow weight W and the string force N.
From the player perspective, the necessary torque from the hand is clockwise at the frog and counterclockwise at the tip and it is the sum of the moments of the fingers that is changing while the sum of the forces of the fingers minus the thumb must remain constant to get a uniform sound.

Although this is useful information, there are an infinite number of ways to achieve the desired force of the hair on the string while being consistent with the physics that are the same for everyone. I then decided to search for an optimal solution, namely, how to produce the desired force of the hair on the string with the smallest forces of the fingers and thumb.

Clearly, since the sum of these forces must be constant, increasing the force in the thumb requires an equal increase in the sum of the fingers, causing unnecessary squeezing of the bow and adding damping to the subtle vibrations of the bow. Therefore, the least-effort solution corresponds to the smallest force in the thumb. Remarkably, it turns out that this solution is essentially unique. That is, except for a very small section of the bow stroke guaranteed to be shorter in length than the spread of the fingers on the bow, the roles of the fingers and thumb are uniquely determined. Outside of this small region, only the index finger and thumb or little finger and thumb are the only vertical components involved, providing the counterclockwise moment approaching the tip or clockwise moment toward the frog, with their difference constituting the constant force perpendicular to the bow producing the constant vertical force of the hair on the string. The reason the middle fingers are not involved is because they have smaller moment arms relative to the thumb and therefore have less leverage than the index and little fingers. With more leverage, their forces can be smaller than those of the middle fingers, thereby reducing the force in the thumb. Since the transition region between only index finger and thumb and only little finger and thumb is so small in the case when adding to the weight to the bow, and non-existent when subtracting weight from the bow, it is just simpler never to involve their vertical components in sound production.

The maximum required clockwise torque at the frog is not very large and is easily developed between the little finger and thumb, as the down bow proceeds, the clockwise moment decreases linearly reaching a point where the torque is zero. In general, this point depends on the angle of the bow (other strings), bow pressure, the bow weight and its balance point. As bow pressure decreases, this point moves closer to the tip.

Beyond this point, the required moment is counterclockwise and continues to increase linearly as the tip is approached, requiring an increasing force of the index finger with an equal increase in the thumb force in order to produce a uniform sound. This required torque can become quite large when using large bow pressure since the hand has to balance the ever increasing large clockwise moment of the force of the string on the hair, equal and opposite to the force of the hair on the string.

A pedagogue who seems to have some understanding of this is Simon Fischer:

"...The amount of [thumb] counterpressure should always be as much as necessary but as little as possible. At the heel (frog), the thumb supports and helps balance the bow in the hand, but even when playing loudly the thumb hardly needs to exert counterpressure. At the point (tip), the thumb must exert counterpressure, especially when playing loudly. Playing with considerable counterpressure at the point, but failing to release it during the journey back to the heel so that the hand ends up locked when playing in the lower half of the bow, is a common error..."

This application of Newton's laws to the problem of bowing in a straight line at constant speed and contact point has demonstrated that for a uniform sound, it is only the torque on the bow that is changing during the stroke and that this necessary change has nothing whatsoever to do with the mass distribution in the bow in producing a uniform sound. Furthermore, the essentially unique "least effort" solution defined by the smallest force in the thumb, has revealed which fingers should be involved in sound production if the goal is using the smallest possible forces in the fingers and thumb.

Having seen some discussion on the roles of the fingers and thumb in bowing, I thought it might be helpful to provide the results of this work.

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May 19, 2021 at 10:20 AM · Very interesting and very challenging. My physics is over fifty years dormant which limits my comprehension. I think you should try to make a video presentation of this with demonstrations on the violin.

One question: Is the elasticity of the string a factor in your model? If not, should it be?

May 19, 2021 at 12:58 PM · The very small deformations of the bow and string have been neglected in this constant velocity, zero acceleration assumption. Though real, their effects are small relative to the large scale mechanical results.

May 20, 2021 at 04:31 AM · Thank you, Francis T.

My Physics and Math courses ended after the freshman year; I switched to music.

I did not understand all of that, and did not attempt your attached technical paper. If one designed a robot to play violin, one would need to program those equations into the computer control, but expert violinists don't need anything beyond arithmetic and an intuitive analog feedback mechanism. Humans may be the only creatures that can hit a moving target at a distance with a rock, spear, or football. But if one moving battleship tries to hit another moving battleship with conventional artillery, it takes a rangefinder,compass, and computer to handle all the trigonometry.

"moment" = torque = leverage ?

There is more than one way to hold or use the bow.

You had the thumb between the second and third fingers. I have not seen anyone do that besides Spivakovsky and a colleague with a damaged hand.

For me, I am not conscious of any force on the thumb, nor do I try to do anything with it. For me, what I teach and do is that each finger has one simple job. The thumb prevents the bow from falling out of the hand. the first finger supplies the leverage. The second finger forms the pivot point-opposite the thumb. The all important fourth finger supports the weight of the bow for all off-the-string technique and prevents it from falling off the other side of the violin. The third finger causes problems if you try to use it for anything. And another thing not often mentioned holds the bow--the string!

Getting the bow to move in a straight line, parallel to the bridge, is a different problem. We try to do that because if it tracks at an angle, Vector forces (Yay!-I used some jargon) will move the hair away from the optimum point of contact. At an advanced level, we can learn figure-8 bowing, slightly angled keeps the hair from drifting over the fingerboard.

If our joints are locked up- rigid fingers, wrist, elbow, shoulder, straight bowing is impossible, the bow moves in an arc. the mechanical analogy that I learned was to think of the drive mechanism on a steam locomotive. The linear force at the piston is converted to a circular force on the driving wheel. The bearings on the connecting rods in between are all loose, they slip, move. What we do with the bow is sort of the reverse of that, converting circular motions of the joints into a linear motion of the bow. We let the elbow move, the wrist bend, and for some players, the bow can pivot inside the hand.

May 20, 2021 at 03:49 PM · The picture of the hand was superimposed by the editor.

Pronation usually causes the vertical contact point of the middle finger to be slightly ahead of the thumb.

May 22, 2021 at 01:14 AM · Professor Testa, I think we could use a demonstration. A video would be a great way to translate the physics to our realm of experience.

May 23, 2021 at 10:29 PM · Hi Francis. I applaud your work and sympathise with your perception of the "misunderstandings" promulgated by many string teachers.

As a mechanical design engineer I often work with inventors and have an interest in the history of the inventive process. My experience is that invention is often, if not always led, not by correct science, but rather by a "story" that the inventor finds useful as a focus for thoughts and investigations.

The world will be forever indebted to the misunderstandings of the alchemists and metaphyscians who, in their search for their own particular philosophers stone, invented much of what is now known as science.

As a viola player I regard "practice" as an inventive process where I explore and experiment with the outcome (the music) that results from different physical actions.

Like yourself, I have a background in "physics" and at times I find it useful to allow this background to guide my musical explorations. But I dont think my musical soul relates much to physics and often I find other storys more useful.

All that being said I think a good case could be made for incorporating the study of some fields of analytical physics in a string pedagogy degree. But only as one of the possible stories that could be useful for a particular student.

May 25, 2021 at 12:53 AM · Hi Jim. I completely agree with your concept of practice as an "inventive process." While it's probably driven by our similar analytical backgrounds, we are not alone. I recall an interview with Nathan Milstein where he described playing the violin as a "lifelong effort of inventing ways to become better."

Thanks for the kind words.

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