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As if there was just material for a single PhD thesis! I would guess that the mathematical theory of bells is pretty much open. The shape of a bell (a solid

As if there was just material for a single PhD thesis! I would guess that the mathematical theory of bells is pretty much open. The shape of a bell (a solid of revolution) can be described by one function u:R->R. Given u, you can compute the vibration spectrum of the sound by standard methos. But if you are given a desired timbre, how to find u? Nobody knows. The most that we can do is optimization starting from a close solution candidate.

sitkack 4y ago
If drums are bells, you should see how Nicolas Bras [1] modifies the sound by damping some of the nodes of the drum. Are you saying all bell are radially symmetric? [1] https://www.youtube.com/watch?v=fXavtyyyj34 https://www.youtube.com/watch?v=fXavtyyyj34
hprotagonist 4y ago
The pitch perception work alone is probably a few degrees, if you get deeper into it.
Someone 4y ago
> The shape of a bell (a solid of revolution) can be described by one function u:R->R If you phrase it that way “surface of revolution” describes bells much better than “solid of revolution”. I think “solid of revolution” is the better term, though. That allows you to model varying wall thickness. You’d need more than a function u:ℝ->ℝ to do that. Also, I don’t a priori see why the shape of a bell would be limited to such functions. If the edge of a bell bends inwards into the bell and goes up a bit, I think I would still call it a bell, especially if it somewhat sounds like one. Finally, rotational symmetry isn’t required for a bell. Your typical cow bell, for example, is elliptical or even rectangular (https://en.wikipedia.org/wiki/Cowbell_(instrument) https://en.wikipedia.org/wiki/Cowbell_(instrument)) ⇒ the design space is a lot larger than you indicate.
- mbustamanter 4y ago
  Surface of revolution is 'okay', 'solid' would mean its interior is completely filled. What you're describing is an active area of research in numerical analysis (adjacent to my own MS thesis topic, buckling of shells). > That allows you to model varying wall thickness That's mentioned in the first chapter of this book [0]. [0] The Finite Element Analysis of Shells – Fundamentals, Chapelle & Bathe EDIT: I'm aware it is not a surface, but the notion is closer than that of a solid of revolution.
  - nomel 4y ago
    > Surface of revolution is okay But surface, by definition, has no concept of thickness, right? Surely there's a better term? > 'solid' would mean its interior is completely filled. I think the easiest way to define a bell would be the difference of two solids of revolution (front surface, back surface). What do you call the difference of two solids of revolution?
    - mbustamanter 4y ago
      I meant it from a numerical analysis/differential geometry perspective (the reason I attached the reference). While not being strictly a surface, it is characterized by a mid-surface, and a thickness t. After having that information, for each point in the mid surface, a normal vector is drawn and scaled by a magnitude of t/2 in both directions of the normal, giving a thickness of t to the object that, I should have made it clearer, is not a surface but characterized by one.
      - enriquto 4y ago
        I don't think that a bell is correctly modelled as a shell of variable thickness. The inner surface is smooth, and the outer surface has ridges. These ridges are apparently important to the timbre, thus it may be more appropriate to consider it as a solid volume of arbitrary shape. Of revolution, if you want to stay with a one-dimensional model.
        4y ago
- a1369209993 4y ago
  > You'd need more than a function u:ℝ->ℝ to do that. Actually, if you normalize the bell height to 1, you can take u from 0 to 1 as the inner surface (tracing down from the top), and u from 1 to 2 as the outer (tracing back up from the bottom). I'm not sure that's mathematically convenient, but it does work. (You could also use a periodic function on 0..pi for inner and pi..tau for outer, which might be nicer math-wise and also allows you to say that the height (pi) is half of whatever the function's period is.)
- mellavora 4y ago
  I've got a fever, and the only prescription is more cow bell!
nonrandomstring 4y ago
This site is quite old and IIRC I had bachelors students building additive synthetic bells from the amazing data he's collected. Also great tales of acoustic archaeology, building bell sounds based on models from the foundry drawings and metallurgy data.
- crustycoder 4y ago
  It's been around for a while but he's still active in the field (he recorded & analysed our bells a couple of years ago) and he's revamped the site fairly recently.
mbustamanter 4y ago
Yes, it's a vibration of an elastic axisymmetric thin shell (surface of revolution, as you meantioned). Going deeper into the theory, to know or characterize the sound it makes one should have a model of free or forced vibration and find the eigenvalues of the resulting linear system. That system is typically (K-omega^2 M) U = F, where K is the stiffness matrix of the material, M the mass matrix, U the displacement matrix, F the force matrix and omega the frequency. Now, to find the displacement given the eigenvalues (spectrum) is possible in a finite-dimensional setting, in infinite dimensions it's trickier. I'm sure there are approximation results of this kind, but not if there is an exact way.

sitkack 4y ago

If drums are bells, you should see how Nicolas Bras [1] modifies the sound by damping some of the nodes of the drum. Are you saying all bell are radially symmetric? [1] https://www.youtube.com/watch?v=fXavtyyyj34 https://www.youtube.com/watch?v=fXavtyyyj34

hprotagonist 4y ago

The pitch perception work alone is probably a few degrees, if you get deeper into it.

Someone 4y ago

> The shape of a bell (a solid of revolution) can be described by one function u:R->R If you phrase it that way “surface of revolution” describes bells much better than “solid of revolution”. I think “solid of revolution” is the better term, though. That allows you to model varying wall thickness. You’d need more than a function u:ℝ->ℝ to do that. Also, I don’t a priori see why the shape of a bell would be limited to such functions. If the edge of a bell bends inwards into the bell and goes up a bit, I think I would still call it a bell, especially if it somewhat sounds like one. Finally, rotational symmetry isn’t required for a bell. Your typical cow bell, for example, is elliptical or even rectangular (https://en.wikipedia.org/wiki/Cowbell_(instrument) https://en.wikipedia.org/wiki/Cowbell_(instrument)) ⇒ the design space is a lot larger than you indicate.

mbustamanter 4y ago
Surface of revolution is 'okay', 'solid' would mean its interior is completely filled. What you're describing is an active area of research in numerical analysis (adjacent to my own MS thesis topic, buckling of shells). > That allows you to model varying wall thickness That's mentioned in the first chapter of this book [0]. [0] The Finite Element Analysis of Shells – Fundamentals, Chapelle & Bathe EDIT: I'm aware it is not a surface, but the notion is closer than that of a solid of revolution.
- nomel 4y ago
  > Surface of revolution is okay But surface, by definition, has no concept of thickness, right? Surely there's a better term? > 'solid' would mean its interior is completely filled. I think the easiest way to define a bell would be the difference of two solids of revolution (front surface, back surface). What do you call the difference of two solids of revolution?
  - mbustamanter 4y ago
    I meant it from a numerical analysis/differential geometry perspective (the reason I attached the reference). While not being strictly a surface, it is characterized by a mid-surface, and a thickness t. After having that information, for each point in the mid surface, a normal vector is drawn and scaled by a magnitude of t/2 in both directions of the normal, giving a thickness of t to the object that, I should have made it clearer, is not a surface but characterized by one.
    - enriquto 4y ago
      I don't think that a bell is correctly modelled as a shell of variable thickness. The inner surface is smooth, and the outer surface has ridges. These ridges are apparently important to the timbre, thus it may be more appropriate to consider it as a solid volume of arbitrary shape. Of revolution, if you want to stay with a one-dimensional model.
      - 4y ago
a1369209993 4y ago
> You'd need more than a function u:ℝ->ℝ to do that. Actually, if you normalize the bell height to 1, you can take u from 0 to 1 as the inner surface (tracing down from the top), and u from 1 to 2 as the outer (tracing back up from the bottom). I'm not sure that's mathematically convenient, but it does work. (You could also use a periodic function on 0..pi for inner and pi..tau for outer, which might be nicer math-wise and also allows you to say that the height (pi) is half of whatever the function's period is.)
mellavora 4y ago
I've got a fever, and the only prescription is more cow bell!

nonrandomstring 4y ago

This site is quite old and IIRC I had bachelors students building additive synthetic bells from the amazing data he's collected. Also great tales of acoustic archaeology, building bell sounds based on models from the foundry drawings and metallurgy data.

crustycoder 4y ago
It's been around for a while but he's still active in the field (he recorded & analysed our bells a couple of years ago) and he's revamped the site fairly recently.

mbustamanter 4y ago

Yes, it's a vibration of an elastic axisymmetric thin shell (surface of revolution, as you meantioned). Going deeper into the theory, to know or characterize the sound it makes one should have a model of free or forced vibration and find the eigenvalues of the resulting linear system. That system is typically (K-omega^2 M) U = F, where K is the stiffness matrix of the material, M the mass matrix, U the displacement matrix, F the force matrix and omega the frequency. Now, to find the displacement given the eigenvalues (spectrum) is possible in a finite-dimensional setting, in infinite dimensions it's trickier. I'm sure there are approximation results of this kind, but not if there is an exact way.