BRIDGES
Bridges of baroque string instruments, especially those known as Stradivarian models, seem to me the best place to begin understanding why and how musical instruments produce sound the way they do.
This is because the Stradivarian models are exceptionally well-developed bridges, and their shape, as well as the points where mechanical waves originate and travel to, are very clear.By looking at the Stradivarian models while bearing in mind how mechanical waves propagate, one can appreciate how intelligently designed those bridges were, and how each type of instrument had a differently designed bridge, tailored to the characteristics of that type. They truly were pieces of technology.
So, thanks to this physical principle we can get a pretty good idea of what happens in each type of bridge. I should add, though, that to fully understand the shape of a bridge, you need to play the instrument well enough to feel, under your bow, the differences between one bridge shape and another.
Let's start!
These are the Stradivarian models that I could find. Four for cello, three for viola and two for violin.
Here are the two real baroque bridges I have seen, compared to their model. Both are made of flamed maple. These bridges have suffered quite a lot over their lifetimes, but you do not need to look very closely to see that they are pretty much what their models call for.
The first one can be seen in the book The "Secrets" of Stradivari by S. F. Sacconi (Libreria del Convegno, Cremona, 1979).
The second one, which is decorated, is the bridge of the Tuscan Tenore by Stradivari. I traveled to the Musical Instruments Museum at the Accademia Gallery in Florence to examine that instrument and its bridge firsthand. The profile of that bridge was indeed in wedge form, opening at an angle of about 5.5 degrees. Even though the bridge is now completely bent after all these years, I could verify with a ruler that the front was made uncompromisingly flat up to the bridge's current height, resulting in an upper curve that is thicker at the sides than in the middle.
This is a radiograph of the Tuscan tenore. Clicking on the picture will open it in a new window, where you can see how the outer edge of the bridge meets the outer edge of the so-called bass bar.
The next bridge is the one I have on my baroque viola:
And now let's take a look at what happens in the bridge when a string starts vibrating.
For a general overview, I have added markings to the images below of my Stradivarian bridge model:
The red point represents one of the inner strings vibrating—say, the D string—and the rings represent the combined trajectories of all the waves produced by that vibration, propagating in all directions until they encounter a boundary. I have then drawn the paths of a few of those waves in different colors. You can see how they rebound when they hit the wood-air boundaries all around the bridge.
The base of the bridge feet is a wood-wood boundary, connecting one solid medium to another. Here, most of the incoming waves are refracted due to the similar densities of the materials. This is why it is so important that the feet of the bridge fit the front blade perfectly. If the fit isn't perfect, the wood-wood contact area shrinks, and fewer waves make it into the instrument.
Let's take a closer look at the waves reaching the feet of the bridge via the most direct path:
In pink, we see the waves that reach the soles of the feet directly.
In blue, we see the waves that reach the soles, not directly, but after rebounding just once off the bridge surfaces.
And when you look at the waves whose path contains two reflections, or just a few more, the amount of waves reaching the feet of the bridge is massive. We can say without exaggerating that, in Stradivarian models, most of the waves reach the feet via the shortest possible path.
You can get a very clear idea of how waves reflect off a convex surface by thinking of billiards. And you can see it by pointing a laser pointer at a bottle and observing where the reflection goes. And if you have a glass ball—as I do—you will have even more fun.
The Waves Absorption System (W.A.S.)
It is a system formed by the combination of the curved prominence at the hip of the bridge, the inward curve right below it, and the groin. Its function is to ensure that virtually all the waves reaching it are, as it were, 'absorbed' into the feet of the bridge. It reflects incoming waves in such a way that most of them reach the soles of the feet, especially the outer area, where the bass bar is located, and the area closest to the sound post.
Unlike the baroque bridge, where waves reach the feet by the most direct path, in modern bridges you will not find a single wave that directly reaches the feet—which, given the nature of the modern instrument, is precisely how it needs to be.
The interior surfaces, especially the heart, get in the way of the waves' path to the feet. In baroque bridges the heart helps waves reach the feet, while in modern bridges it hinders them. That's why I think of modern bridges as closed and baroque bridges as open.
See how different the feet of modern and baroque bridges are: baroque feet are thick and wide (open), whereas modern feet are thin with narrow ankles (closed).
While it is true that cello and double bass bridges have reasonably wide legs, the direct path to the feet remains closed.
In modern bridges we can still see a remnant of the W.A.S: the prominence is somehow still there, thought of only as weight or mass according to the theories of modern musical acoustics, or as mere decoration out of tradition. The inward curve is turned into the ankles of the violin and viola bridges, and in the bridges for cello and double bass it is just stretched down to create the typical feet of the modern bridge.
About the Stradivarian bridge models:
The models for the viola are the most open ones, because they have flat-ending arms and carry the waves by the most direct path, with no disturbance, especially the lower one with the pointed hole, which greatly facilitates the transit of the waves towards the feet, under the hole. I find this appropriate, since the middle voices need considerable clarity, and that is especially true for the viola, given its small size relative to its range.
Out of the four models for the cello that I know, three of them have a heart and fingered arms and the fourth has a hole and flat-ending arms. That cellos have bridges with fingered arms does not surprise me, to balance a perhaps too-quick response, for their size would match, as the violin's does, their range. Fingered arms call for heavier bowing than flat-ending arms, which produce a clearer articulation with lighter bowing. I just don't know which kind of cello would require that fourth model with a hole and flat-ending arms, or what function it would fulfill. Yet, regarding the waves' path towards the feet, all four are completely open. Clarity for the basses as well.
We could say that the bridge models for the violin are the most closed of all Stradivarian designs, but Stradivarian models are not closed at all. I would rather say that the models for the violin are the most nuanced. Of the two, the shorter features a little hole above the heart. This hole is on the most direct path of the waves coming from the inner strings, and is sending down to the feet a portion of the waves coming from the outer strings which, without that hole, would not yet travel towards the feet. The taller has very high armpits and a very high heart cutting deeply into the direct path of the outer strings.
I could say that disturbing the path of the waves towards the feet makes sense, for the violin already has a pretty easy response and quite a bright sound.
However, I believe that it goes further than merely having bridges for each kind of instrument. All Stradivarian models favor the inner strings: the waves coming from those strings have the most direct path to the soles of the feet, with even a portion of those waves reaching them completely undisturbed. Regarding the taller model for the violin, the high armpits and high heart cut off the direct path to the soles of the feet. Essentially, however, the bridge still functions similarly to those for the viola or cello, though its wave paths are lengthened by additional reflections. For the inner strings, the path involves a single reflection from the torso straight down to the corners of the foot heels. For the outer strings, the path relies exclusively on the reflections off the armpit directly below them and off the upper curve, before they finally find a wide-open path.
But the shorter violin model, while maintaining roughly the proportions of the viola and cello models, introduces a small hole right above the heart. This removes the undisturbed path for the inner strings while simultaneously routing waves from the outer strings downward—acting as an equalizer, suppressing middle tones while enhancing the bass and treble registers.
So it seems to me that, through these distinct models for each instrument type, Stradivarian bridges balance both the sound across the instrument family and their role within the ensemble.
This is a design for a five-string double bass bridge. It has flat-ending arms and is as open as can be, because double basses need all the help they can get to sound clear.
Once again, the following illustration uses colors to highlight the waves that reach the bridge's feet via the shortest, most direct paths. Shown in red and pinkish are the waves that directly reach the soles of the feet, all originating from the inner strings (strings 2, 3, and 4).
Shown in blue are the waves originating from the outer strings—in this case, the first string. These waves do not reach the soles directly; instead, they arrive via reflections off the outer ends of the feet, which correspond directly to the projection of the bass bar's width.
The term 'bass bar' now seems inadequate to me. Given that the frequencies reaching it most directly are actually trebles, I would call it the treble bar. Interestingly, the Dutch call the bass bar zangbalk, which translates roughly to 'singing bar.' In reality, it is the bass frequencies that reach the sound post most directly.
We have already seen that the heart of the modern bridge is in the way of the waves from the top to the feet, while the hole/heart of the baroque one makes way for them.
If you take a baroque bridge with a hole and turn the hole into a heart, the instrument will sound louder, a bit darker, and for sure more robust. Within reason, the bigger the heart becomes, the louder the instrument sounds. If you go too far, you will find not much difference between a really big heart and a far too big one.
Here you can see how more or less the same waves will rebound differently on a hole than on a heart. This explains the differences in the sound they produce.
You can also see the path of a wave, colored in green, which has entered one arm. See how the wave in the flat-ending arm is reflected directly back while the one in the fingered arm stays there longer.
The profile of the bridge—meaning its shape when viewed from the side—is also critically important. I prefer bridges with a wedge-shaped profile featuring straight back and front, because they offer better articulation and a more direct response. Straight, divergent lines are indeed very good for sending waves down to the feet of the bridge:
Often, bridges are made with a belly: the front is curved in its upper section and eventually becomes parallel to the back. To our eyes, this creates a convex surface, but to the waves, it acts as a concave boundary—forcing them to take a much longer path on their way to the feet. As a result, the articulation suffers in clarity, and the instrument demands significantly more weight from the right hand, making it harder to play. It appears that this profile was chosen primarily because it helps the bridge last longer.
The more I take the time to try to see where every possible wave coming from the strings will go, the more I see how good the Stradivarian models are at bringing those waves to the feet of the bridge. That direct path of the waves to the feet means that what the player does to the strings is transmitted more effectively to the instrument, making the instrument much more responsive.
And indeed, the bridge on my viola works really very well. Even too well: it responds so well that my arm and bow are far too heavy for it. I once played a modern cello, mounted with a bridge in this manner, using a very light, early baroque violin bow. Despite the fact that a cello is not supposed to be played with a very light violin bow, it did feel very well in balance.
That is because modern and "baroque" instruments from the 19th century onward do not have blades with a thick middle part anymore. Baroque instruments did have such thicker blades before being transformed, which balances the response of the instrument with the weight of the bow and the player's arm. The intensity of the waves that these bridges bring to the instrument is somewhat too much for the thin blades of the modern instrument. In the section about Jakob Stainer, I write about those thick blades of the ancient instruments.
This model of bridge stands between the modern and the baroque, leaning closer to the modern bridge. It has the same feet, the same remnants of the W.A.S. and the armpits in the same place. The main differences are that it is made of flamed maple and has no heart. Further, it has a little hole under the absent heart, which, as it were, breaks the flat surface beneath it, where most of the waves would rebound back. It is also placed low enough to not hinder waves reflecting off the armpits from reaching the feet. Without a heart, this bridge is much more open than modern bridges. The inner strings do not have any direct path to the feet, but the outer strings do. Flamed maple—being, unlike stripped maple, of irregular structure—produces a darker sound, allowing the bridge to be of a regular thickness without sounding too bright. Open bridges made of stripped maple tend to sound quite bright.
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