Cameron Marc Raecke | violist

The First Law

$L\propto WS$

The First Law concerns the loudness of the sound, represented by $L$. This equation translates to, “The loudness of the sound is proportional to the product of the weight and speed of the bow.” In normal speech: if you add either more weight or more speed to the bow (i.e., use a heavier or faster bow), the sound gets louder. Inversely, playing with either a lighter or a slower bow makes the sound softer. In practice, the weight and speed are often increased or decreased together to change the loudness of the sound.

On the sound chart, we can draw a straight line from the origin to the sound point. The area under that line represents the loudness of the sound (Figures 3, 4, and 5).

Now for some exercises you can experiment with on your own instrument to make these ideas clearer.

Exercise 1

Play long notes with the bow halfway between the bridge and the end of the fingerboard. While playing, keep the bow weight the same, but change the speed of the bow, listening for how the sound gets louder as the bow moves faster, and softer as the bow moves slower (Figure 6). You might find that you need to resist the urge to lighten the bow when slowing it down.

Exercise 2

Play the same long notes, but instead of changing the speed, change the bow weight, listening for how the sound gets louder when the bow gets heavier, and softer when lighter (Figure 7). The change in loudness is reflected in the difference between the areas of the triangles in the sound charts.

Since both the weight and speed of the bow can be controlled to change the sound’s loudness, it’s possible to move the loudness in one direction with a change in weight while moving it in the opposite direction with a change in speed so that these effects negate each other, and the loudness remains the same, which brings us to …

Exercise 3

Still playing long notes halfway between the bridge and the fingerboard, alternately decrease the weight of the bow while increasing its speed, and increase the weight while decreasing the speed, keeping the sound’s loudness unchanged (Figure 8). All of the effects discussed thus far are contained in the First Law.

Though the two triangles in Figure 8 have the same area, you can plainly see that they look different in the sound chart, implying that there’s something different about those sounds, which you can easily verify by ear. For that matter, there’s more about the sound that changed in Exercises 1 and 2 than just its loudness. How can we describe this other aspect of the sound, and how is it displayed by a sound chart?

The Second Law

$D\propto \frac{W}{S}$

The Second Law of Sound Production deals with the density of the sound, which is represented by $D$ and determined by the relative balance between the bow weight and speed, rather than their combined product. To the ear, density is the concentratedness of the sound, how thick, rich, or tightly-packed it seems. The opposite of a dense sound can be described as thin or airy.

According to the Second Law, the sound density increases when we increase the bow weight or decrease the bow speed. To make the sound airier (i.e., less dense), we decrease the weight or increase the speed. We often change both the weight and the speed at the same time to change the density of the sound.

Because the density is proportional to the bow weight divided by the speed, it’s represented in the sound chart by the slope of the line drawn to the sound point. In Figures 9, 10, and 11, that line has been extended and donned an arrow to make this visually clear. The more upwardly the arrow points, the more dense the sound is, and a lower-pointing arrow represents an airier sound. At the same time, we can also still see the area of the triangle, representing the sound’s loudness.

Looking back at Exercises 1 and 2 (and Figures 6 and 7), we find that the aspect of the sound we were changing — other than its loudness — was its density. Figure 12 diagrams Exercise 1 and highlights the density change by including the density arrows. The chart shows that, as we reduce the speed of the bow, the sound’s loudness decreases and its density increases. As the sound becomes softer when you slow down the bow, you should be able to hear that it also seems to become more tight, maybe even crushed (remember not to change the weight when you do this). That’s an increase in sound density.

In Figure 13, you can see that when you perform Exercise 2, both the loudness and the density increase when you increase the weight of the bow. What you should hear — when only increasing the bow weight — is the sound getting louder, but it should also get that same tight, possibly crushed quality it had when you decreased the speed in Exercise 1. That, again, is an increase in sound density, shown by an increase in the slope of the density arrow. Lowering the weight or increasing the speed will, of course, have the opposite effect: to decrease the density. The sound will seem to relax and have a softer edge to it.

Now experiment with Exercise 3 once again to see that you can change the density of the sound without changing its loudness. Using a slower, heavier bow, and then a faster, lighter bow in the right proportion will maintain the loudness at a constant level while changing the density. Finding that proportion is a matter of trial and error. At this point, it might have occurred to you that Exercise 3 has a complement:

Exercise 4

While playing long notes, increase or decrease both the weight and speed at the same time to increase or decrease the loudness of the sound while keeping its density the same (Figure 14). Again, maintaining a constant density requires that you increase or decrease the weight and speed together in the right proportion, which you can find through practice.

When you were experimenting with Exercises 1, 2, and 3, you might have noticed that increasing the weight too much caused the sound to become scratchy and harsh. Likewise when decreasing the speed too much. If you decreased the weight or increased the speed too much, the sound lost its core and became wispy and indistinct. What’s going on here?

The Third Law

$P\propto \frac{1}{D}$

$P$ represents the contact point, which is the exact point on the string the bow is contacting. It’s common to think about this in terms of “bow lanes,” usually five of them: lane 1 is right up next to the bridge, with the lane number increasing as the bow moves farther out until it reaches lane 5, which is at the edge of the fingerboard (all the previous exercises, being halfway between the bridge and fingerboard, were in lane 3). The Third Law says that increasing the density of the sound typically requires a decrease in the lane number of the contact point (i.e., moving closer to the bridge). But it’s a bit more complicated than that, and we need to dig into it.

Exercise 5

Pull a string — as if to pluck — at different contact points, and you’ll find that the string is much easier to pull farther away from the bridge, and it gets more difficult to pull the string the closer you get to the bridge. You should think of this as the string being stronger nearer the bridge (i.e., with lower values of $P$, or lower bow lane numbers), which for us means it’s more capable of handling high-density sounds there. This is the essence of the Third Law.

Increasing the density past a certain point — whichever method you use to do it, whether by increasing the weight or decreasing the speed — causes the sound to become harsh and scratchy, possibly without an identifiable pitch at all (this is usually not a sound we want to be making). When this happens, you’ve crossed over into what I call Scratchtown, where the density is too high for the string to make the clean, clear sound we normally intend; the string simply isn’t strong enough for that density level. There are three ways out of Scratchtown: the first two are to decrease the bow weight or increase the speed; doing either of these things (or both) decreases the density, which takes you out of Scratchtown and back into the Tone Zone, where we find our normal clean, clear sound. This is shown in Figure 15, which depicts Scratchtown as a black wedge in the upper left part of the sound chart. The white area to the lower right is the Tone Zone. I’ve removed the triangle and arrow showing the loudness and density to unclutter the chart.

The third way to return to the Tone Zone is to move Scratchtown itself. The Third Law says that you can play higher-density sounds closer to the bridge (because it’s stronger there); thus, moving your bow closer to the bridge pushes Scratchtown upward, out from under your current sound point (Figure 16).

On the opposite end of the spectrum, if we decrease the density enough, we get to a point where the sound loses its core and becomes unclear. It sounds like the bow is floating across the top of the string instead of gripping it properly (we usually don’t want this sound, either). We have now arrived at Wispyville. Similarly to the case of Scratchtown, there are three roads out of Wispyville: decreasing the bow speed, increasing the weight (both of which increase the sound density), and moving the bow farther away from the bridge, which moves Wispyville (along with Scratchtown and the Tone Zone) downward so that the sound point would then lie within the Tone Zone. Figures 17, 18, and 19 show all three of these sound regions at different contact points.

Nearly all of our playing is done in the Tone Zone, and depending on what contact point we’re using, we have a certain range of loudness levels and densities at our disposal. We can change our contact point to give ourselves access to different regions of the sound chart, depending on what type of sound we feel is appropriate for the music we’re playing at the moment.

Sometimes, our choices are limited by what the music requires us to play, and understanding the Three Laws of Sound Production can help us determine what to do in different situations. For example, imagine you need to play a very long note in one bow. Obviously, this means you would need to move the bow slowly, which restricts you to the left-hand region of the sound chart. In Figures 20 and 21, I’ve added a vertical line indicating a low bow speed to mark the region of the sound chart this note must be played in (the shading of Scratchtown and Wispyville has been lightened for visual clarity). You need to stay to the left of that line, and you can see that there isn’t much room to make use of if you want to stay in the Tone Zone.

Exercise 6

In bow lane 3, play slow notes, eight beats long at 60 bpm. Notice how, as shown in Figure 20, you can’t add much weight without going to Scratchtown. What this does is limit the loudness of the sound; you can see that the loudness triangle in Figure 20 would be quite small.

Exercise 7

The same as Exercise 6, but with one change: play close to the bridge. What this does, as shown in Figure 21, is to bring the three sound regions upward so that there’s an area of the Tone Zone to the left of our maximum speed line in the upper area of the sound chart, which is the high-weight area. The loudness triangle in Figure 21 would be much larger than the one in Figure 20, and you can verify for yourself that you can play much louder this way (it’s also a much denser sound, which you can see in the chart by imagining how much higher the density arrow would be).

If we’re playing at the bridge right now, then is this what’s called sul ponticello (Italian for “on the bridge”)? No. For sul ponticello playing, we do play near the bridge, in lane 1 or maybe 2, but only with a medium density, which puts us firmly in Wispyville. The resulting loss of core in the sound is the essence of the standard sul ponticello sound (Figure 22).

We often need to play music with asymmetrical bowings, that is, bowings where more time is spent moving in one direction than the other. For example, imagine a passage in 6/8 where the rhythm is alternating quarter notes and eighth notes for many measures, and every note is played with a separate bow (Figure 23). In this situation, every quarter note is played with a down-bow, and every eighth note is played with an up-bow, so twice as much time is spent playing down-bow as is spent playing up-bow. In order not to drift toward the tip and run out of bow, we need to use the same amount of bow for each note, which means the bow needs to move twice as fast on the eighth notes as on the quarter notes.

The First Law tells us what will happen when we do this: the eighth notes will be louder due to this increase in speed, producing a hiccup effect, which we usually would want to avoid in this situation. To solve this, the First Law also tells us that we can simply do what we did back in Exercise 3: lighten the bow when moving it faster, preventing the eighth notes from being louder than the quarter notes.

As you can see, the Three Laws of Sound Production — along with the accompanying sound charts — have a great deal of explanatory power, which can help any violist or violinist trying to gain a deeper understanding of how to control their instrument’s sound. Of course, these Laws don’t explain everything there is to know about sound production; for example, they don’t address the fact that the bow will normally need to be farther from the bridge on lower-pitched strings and closer on higher ones, or that the bow will be closer to the bridge when playing in higher positions. A genuinely exhaustive explanation of everything that can be known about sound production could easily be a thousand pages long, so the readers need to be satisfied with what amounts to a somewhat comprehensive first-level understanding of how it all works. My hope is that at least a few readers will find it helpful to apply the ideas contained within these Three Laws to their playing, and I wish everybody happy and productive practicing.