Stiffening Cello Bridges with Design
Round 1
Reviewer 1 Report
Overall, very interesting study!
In the introduction it would be good to show a picture of a complete cello in order to get a better picture of where the particular piece fits in. Otherwise, a reader is thrown immediately into the details before getting a broader sense of what you are analyzing.
For Table 1, a suggestion would be to name the indices: xy, yz, zx in order to show the progressing order through all 3 coordinates
For section 2.3, it is hard to visualize which one is the most realistic boundary condition. Perhaps if you include an overall schematic of where this piece fits in the cello as suggested above, this will become more apparent.
For figure 3c, the R^2 is given but the eqn. of the curve is not. Should there be a trendline on Fig. 3d as well or is there no correlation? You mention that it hits a plateau at 20 but maybe the first part of the curve has a trendline.
The mode shapes are hard to follow in A2, especially for those in torsion. Perhaps a front, top and right view would help.
Overall I give you a lot of credit for tackling a 300 year old “we’ve always done it this way” reasoning.
Author Response
Referee 1:
Overall, very interesting study!
In the introduction it would be good to show a picture of a complete cello in order to get a better picture of where the particular piece fits in. Otherwise, a reader is thrown immediately into the details before getting a broader sense of what you are analyzing.
A new picture is added to Figure 2.
For Table 1, a suggestion would be to name the indices: xy, yz, zx in order to show the progressing order through all 3 coordinates
Corrected.
For section 2.3, it is hard to visualize which one is the most realistic boundary condition. Perhaps if you include an overall schematic of where this piece fits in the cello as suggested above, this will become more apparent.
We added the picture of a cello being played in Figure 2.
For figure 3c, the R^2 is given but the eqn. of the curve is not. Should there be a trendline on Fig. 3d as well or is there no correlation? You mention that it hits a plateau at 20 but maybe the first part of the curve has a trendline.
We have added the equation for the trend lines in the two plots.
The mode shapes are hard to follow in A2, especially for those in torsion. Perhaps a front, top and right view would help.
We were aware of the difficulties in the visualization of the mode shapes, so we had made available a video with all the animations. The video was accessible from the Github repository, but it was previously not mentioned in the paper. We have corrected the main text to make this explicit.
Overall I give you a lot of credit for tackling a 300 year old “we’ve always done it this way” reasoning.
Thanks.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
This paper deals with the influence of a systematically varied cello bridge on its mechanical and acoustical properties. A finite element model and systematically varied bridge shapes are being used to analyze how shape affects the properties of the bridge.
This is a fine paper and solid work, I only have minor concerns: The paper does add some new insights to the state of the art in instrument building and cello research. But to me it does not contextualize well, i.e., it does not explain the state of the art well and it does not explicitly reveal what is new. This needs to be done so the reader knows what is new and what is already state-of-the-art.
I suggest publication after "major" revision, which is in fact rather minor than major, but still, I'd like to see that this has been done conscientiously it before I suggest accepting the manuscript for publication.
Last but not least I suggest to make your research data publicly available, i.e., your CAD and COMSOL Multiphysics files, maybe through a repository such as Github.
abstract:
"yet, to the best of our knowledge, there is not a particular explanation as to why the bridge is the way it is" -> I think this statement is exaggerated. You cite quite some literature, and there is plenty more, explaining to some extent, why the bridge is shaped the way it is. Most importantly, the transverse vibration of the strings needs to be transferred efficiently into the top plate, so the polarity (direction of the motion) has to be transformed from left/right to up/down if you will. To achieve this, the bridge must not be too rigid. But at the same time it needs to be rigid enough to deal with the large forces of the strings under tension. This is why we need a skeleton that is not too stiff and rigid but still can handle the force.
"Our results show that varying the shape of the legs has effects that resemble the ones caused by stiffening the material." -> That's pretty obvious and the well-known reason why we change the shape of the bridge. Your finding is less general and more specific: You found a linear relationship between a shape parameter and the displacement along the x-direction.
Introduction:
- In your introduction you list some references that deal with cello (and violin) bridges. But you should review them a little and indicate their findings.
- I think you do not explain well what we already know about the bridge, and where readers new to the field can start researching. I think citing the chapter "Bowed String Instrument" especially the section "The Bridge" in Fletcher & Rossing: The Physics of Musical Instruments, 2nd edn, Springer: 1998, and, of course, Thomas D. Rossing: The Science of String Instruments, Springer 2010 is a must.
- Furthermore, you should (maybe based on the suggested literature) explain in general what is already known about the bridge, which is what I also explained above, concerning the balance between mechanical stability and vibrational flexibility.
Eigenfrequency study:
- Tables 2 and 3 do not really help to make comparisons. For a good comparison it may be better to show 4 plots (free, fixed, spring, spring+) that show the mode frequencies of the original and the model 30 within one plot. These plots would look like Figure 5, which also helps to see relationships between eigenmodes of the bridge and the admitance (or what exactly does Fig 5 show?).
Conclusions:
- You should discuss a bit on the implications of your findings. How will reshaping the bridge affect the bridge hill sound that can still be measured and heard in the built cello? May different shapes affect playability of the instrument in the Helholtz regime and in other regimes?
Author Response
Referee 2:
This paper deals with the influence of a systematically varied cello bridge on its mechanical and acoustical properties. A finite element model and systematically varied bridge shapes are being used to analyze how shape affects the properties of the bridge.
This is a fine paper and solid work, I only have minor concerns: The paper does add some new insights to the state of the art in instrument building and cello research. But to me it does not contextualize well, i.e., it does not explain the state of the art well and it does not explicitly reveal what is new. This needs to be done so the reader knows what is new and what is already state-of-the-art.
I suggest publication after "major" revision, which is in fact rather minor than major, but still, I'd like to see that this has been done conscientiously it before I suggest accepting the manuscript for publication.
Last but not least I suggest to make your research data publicly available, i.e., your CAD and COMSOL Multiphysics files, maybe through a repository such as Github.
The data is already available and can be seen in the section data availability. We have edited the main text to make this clearer.
abstract:
"yet, to the best of our knowledge, there is not a particular explanation as to why the bridge is the way it is" -> I think this statement is exaggerated. You cite quite some literature, and there is plenty more, explaining to some extent, why the bridge is shaped the way it is. Most importantly, the transverse vibration of the strings needs to be transferred efficiently into the top plate, so the polarity (direction of the motion) has to be transformed from left/right to up/down if you will. To achieve this, the bridge must not be too rigid. But at the same time it needs to be rigid enough to deal with the large forces of the strings under tension. This is why we need a skeleton that is not too stiff and rigid but still can handle the force.
We have edited the abstract and included in the introduction the referees explanation. We nevertheless think that our main point is still valid. Indeed we understand roughly why the bridge has to be the way it is, a skeleton rather than a solid piece of material, but there is no actual scientific explanation as to why one shape should be preferable to another beyond simply tradition. To the best of our knowledge there is no scientific explanation why we have two models for the cello bridge, the french and the belgian, and why those two models are “better” than any of the historical examples previously used.
"Our results show that varying the shape of the legs has effects that resemble the ones caused by stiffening the material." -> That's pretty obvious and the well-known reason why we change the shape of the bridge. Your finding is less general and more specific: You found a linear relationship between a shape parameter and the displacement along the x-direction.
Corrected.
Introduction:
- In your introduction you list some references that deal with cello (and violin) bridges. But you should review them a little and indicate their findings.
Corrected.
- I think you do not explain well what we already know about the bridge, and where readers new to the field can start researching. I think citing the chapter "Bowed String Instrument" especially the section "The Bridge" in Fletcher & Rossing: The Physics of Musical Instruments, 2nd edn, Springer: 1998, and, of course, Thomas D. Rossing: The Science of String Instruments, Springer 2010 is a must.
We thank the reviewer for the comments. Leaving out Fletcher and Rossing was rather a conscious decision. There are a lot of assumptions going on in the old literature that we don’t feel are fully justified, not least because most of the articles cited are impossible to find online and are more than dated. One example of this is the quote of Savart where he states that “if we take a piece of wood cut like a bridge and glue it on to a violin…” (our emphasis). As anyone who has ever worked with violins knows the bridges are NOT glued to the violin. Errors like this show a fundamental lack of understanding of the instrument that we don’t want to continue spreading. The same quote goes even further to say that “It is astonishing that by feeling our way we have arrived at the shape currently used, which seems to be the best of all that could be adopted.” This is not a scientific statement. If anything, it is fully loaded with teleological implications, without giving any mechanistic explanation why the bridge is the way it is.
- Furthermore, you should (maybe based on the suggested literature) explain in general what is already known about the bridge, which is what I also explained above, concerning the balance between mechanical stability and vibrational flexibility.
We have rewritten the introduction to include this. However, the fact that the works cited in Fletcher & Rossing are not available online make it very difficult to have a proper discussion of the results without seeing them firsthand. We honestly think it's time to either re-do those experiments and publish them in proper peer reviewed journals or just move on from this acquired “knowledge” (see Fig. 10.35 of F&R second edition for an example of this).
Eigenfrequency study:
- Tables 2 and 3 do not really help to make comparisons. For a good comparison it may be better to show 4 plots (free, fixed, spring, spring+) that show the mode frequencies of the original and the model 30 within one plot. These plots would look like Figure 5, which also helps to see relationships between eigenmodes of the bridge and the admitance (or what exactly does Fig 5 show?).
Figure 5 shows indeed the admittance of the bridge for clamped feet conditions. We have kept the tables to have a different and complementary way of studying the data.
Conclusions:
- You should discuss a bit on the implications of your findings. How will reshaping the bridge affect the bridge hill sound that can still be measured and heard in the built cello? May different shapes affect playability of the instrument in the Helholtz regime and in other regimes?
This is an excellent yet very difficult to answer question. The use of a different cello bridge model is indeed audible, and that is why the Amorim family came up with the design in the first place, because they found that instruments sounded better with a modified cello model. We have done experiments where the bridge of violins is changed and this can be clearly seen in the bridge admittance of the instrument, in particular around the 2000Hz range. We do not have yet the experimental results for the cello though, but we are working on it.
In terms of playability, scientific studies are even more difficult to perform. For computing the force in the cello needed to play the instrument in a realistic setup a complex series of sensors needs to be connected to the cello, and to the best of our knowledge the problem is still not solved. Claudia Fritz has a setup but is still not operational. We will work with her to answer this in the future.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Thank you for your detailed response letter and the updated manuscript. The paper looks good now and the video of the cello bridges reminds me of cute little dancers.
line 185: "the influence in the sound" -> "on"
line 187: "shit the position" -> "shift"
line 213: Instead of "Moreover, a short video containing animations can be viewed here."
you should write -> "viewed on YouTube" so that people know what they're clicking on
"https://youtu.be/JTxqf4hO2vA"
Author Response
We thanks the reviewer for their kind comments. We have fixed the typos.
