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.
> 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.
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.
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?
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.
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.
“When struck, a bell produces a number of partials which, if imprecisely tuned, can create an unpleasant sound and which prevents it from harmonizing in accordance with other bells. To address this problem, the Hemony brothers gave their bells a particular profile and thickened it in certain places. The bells were then tuned by hollowing ridges from specific parts of the inner wall until the first few partials were acceptably in tune.”
(Aside: if you want your work to be used for centuries, becoming a professional bell caster seems a good bet)
> 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.)
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.
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.
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.
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?
Yes, Bill did. He's a really nice guy and the subject of bell tuning has been of academic interest for a very long time, because they don't produce a single note and the note you "hear" isn't produced by the bell at all. See for example Nobel Laureate Lord Rayleigh's paper of 1890, linked to on Bill's site.
That is fascinating. I thought that bells had one or a couple dominant / loud frequencies that would mostly determine pitch, and the rest would just mix in to change the timbre but not the main percept. I always found the sound of melodies played with bells kind of uncomfortable because there is too much going on, too many different frequencies interfering with each other.
For literally centuries European bell tuning was all done by rule of thumb, with results varying from to sublime to abysmal. It was all finally figured out by Canon Simpson in 1895/6 who described the relationship between the 5 main components of a bell's strike note - the hum, prime, tierce, quint and nominal.
Bells is also one of the sub-themes in the Polifonia EU-funded research project (disclaimer, I'm involved in a different theme). It's more about documentation of historical bells in open/linked data, rather than acoustics.
In the UK we have 61,626 documented bells (a lower bound) spread across 7,148 towers. https://dove.cccbr.org.uk
In the last year unaffected by COVID, there were 10,701 recorded performances and many more unrecorded ones and practices. https://bellboard.uk/
There are 23,397 Methods and 52,650 Compositions available, where a Method is roughly equivalent to a piece or song in traditional music terms. http://methods.ringing.org/intro.html
I used to enjoy listening to the church bell practice sessions on summer evenings when I lived by a church (not so much being woken on Sunday mornings though). Annoyingly, they eventually shortened and lessened the frequency of practice because newcomers to the village complained that their idyllic second home in the countryside was affected by "noise pollution"— something that has been a regular feature of village life since the bell tower was built in 1545.
The UK government did promise to reduce the grounds for noise complaints about church bells, on the grounds that normally they were there before the complainant arrived, and would still be there long after they were gone, but it appears it was like many government promises...
There's usually two different things going on in UK bell towers - bell ringing which involves ringing the bells full-circle using the internal clappers, and clock chiming which uses external hammers driven my the clock. The clock chiming is much quieter than change ringing.
Ours only chimes on the hour but it's going 24/7 and is very close to the surrounding houses (~40m) some of which are uphill and directly in line with the bells. I had to go in today to fettle it as it wasn't chiming and someone had posted on the village FB group that they were missing it. That's a fairly regular occurrence, most people people like hearing them, even through the night. Any complaints we get are when the clock stops.
It is possible to modify clocks to stop them chiming overnight, although ours is still in its entirely mechanical 1883 form, including me having to wind it every week.
If people want this, then it should be fine. To me, in case described in the article above the bells were to stay because of the church and tradition, not because the citizens wanted them.
It's usually one miserable git who complains, most people who buy houses next to big sticky up things at the end of religious building are smart enough to figure out what's in them.
Very cool. I became fascinated with bells after watching this [0] documentary on how they are made a few years ago (it's in German, but that hardly matters, just watching the process is extremely interesting).
The whole process is a combination of extreme sophistication and extremely simple tools and materials, something that has become quite rare nowadays. And despite an exact calculation of the bell shape and thousands of years of trial and error, tradition, and experience (the foundry in the video has been owned by the same family for 300 years), the cast is still highly dangerous and it still regularly happens that the bell just doesn't come out right and cannot be saved - months of work for nothing.
I think that explains why on the day of the cast (which is just 1 or 2 times per year), everyone involved is extremely nervous. The bell founder and the customer all join in prayer, and the cast traditionally starts with "in God's name" [1].
Little known fact: They made the AC/DC's original Hell's Bell for the Back in Black tour. Touring with 3,000 kgs of bell in a flight case made for some interesting experiences
Even with it being in German, I agree, I still enjoyed it. If you've ever made anything or seen a different thing made, you'll get the concepts of what they are doing. The scale of what they are making is impressive, as well as their jigs for building. Thanks for the link
That mug-tapping demonstration in the "Doublets or warble in bells" page [0] is neat! I'll be looking for an opportunity to show that to someone, ask why they think it happens, explain pedantically, and yell "Science!"
An interesting podcast on bells form Planet Money:
When you've got two companies down the road from each other making the same thing, you can almost guess the history. Originally there was just one company, but something happened. Maybe someone got angry, somebody left, and they started a second business, a competitor.
Today on the show, the rivalry between two companies that make nearly all the world's handbells. And how they eventually made peace.
Tarkovsky's Andrei Rublev has an extended sequence where a bellmaker's young son agrees to pour a bell for a local lord. You see the entire process of making the mould and pouring the metal. It has a town-fair atmosphere. But at the end, there's tremendous pressure on the young bellmaker to have the bell ring properly and have no cracks. And it does. The protagonist, the lapsed monk Andrei Rublev, regains his faith, seeing the result of the young bellmaker's hope.
One of my favorite sequences in any movie. I will never forget the visuals of workmen sweltering in front of flaming portholes in the bell's cast. Enduring a little inferno, to create the sound of heaven! Thank you for reminding me of it. :)
I can't recall as it has been a number of years, but my impression was that the young bell maker was lying about his father having given him the "secret" to making the bell, hence the anxiety about the success of the project. Am I misremembering that, or was I just not reading it correctly?
“Someone wrote a PhD thesis on bells” is a clickbait title. It implies the value of the content is merely the surprising or shocking concept that the content even exists. But looking at the page, that’s not what the content is about. It’s an earnest study into bells. Are we meant to point and laugh?
I agree, title was a bit clickbait but I was surprised that one would dedicate their life to something that specific. Hyper-specialization at its finest, and the passion is something I can appreciate
Thanks! We've changed the title above to the page title now.
The submitted title was "Someone wrote a PhD thesis on bells". Sometimes a title like that has the virtue of getting an obscure, interesting submission onto the front page. The sin is venial when the submission is particularly good (and mortal when it's dreck, which alas is more often the case.)
The doctoral candidate didn't "dedicate their life" to the topic, merely wrote a thesis on it. So, just an intense part of their life. Perhaps they'll remain focused on bells, perhaps not. Probably all PhD theses are 'hyper-specialized', unless they're surveys by an author who won't be continuing as an academic researcher. It's not surprising that people would want to understand these complex inharmonic sounds we've been making for centuries.
'Bells' are pretty un-specific as far as specialization goes. The average person actually knows what a bell is. The PhD theses in mathematics are likely far less applicable and far more specific. Check out the titles on arxiv and let me know if you think any of these are less specific than 'bells'.
Don't be so sure. At least for mathematicians, the "bell" may be just an excuse to make the subject cool. Many of the underlying problems are very general. For example, the inverse spectral problem (vibration modes of a complex shape) is the basis of the interaction between proteins, enzymes, and the huge molecules of cell walls. At the cell scale, Brownian motion shakes everything in all possible frequencies, but molecules can only vibrate at a specific set of frequencies (their spectrum). The act of "matching" different molecules happens when they have some common frequencies, often with localized modes of vibration.
You may think that you are just modeling church bells, but everything that you do has immediate application to the understanding of some organic chemical interactions, which are a deep mystery yet.
Here is my bell related science-fiction recommendation: Doomsday Book by Connie Willis.
"The drop had been scheduled for noon. If she had come through on time and Probability was right about the slippage, it would be six o'clock in the evening, which was too late for vespers. And if it were vespers, why did the bell go on tolling?
It could be tolling for mass, or for a funeral or a wedding. Bells had rung almost constantly in the Middle Ages—to warn of invasions or fires, to help a lost child find its way back to the village, even to ward off thunderstorms. It could be ringing for any reason at all."
Somewhat tangential; there's a company, clock-o-matic, that makes hardware and software to play carillons (or single bells) programatically using MIDI files.
The bells studied in the PhD thesis are primarily used for English Change Ringing, which started in the early 1600s and the compositions played on them (methods) are based on mathematics, specifically Group Theory, rather than being tunes in the normal sense - the aim is to ring all the possible permutations of the bells.
Performances (Peals) consist of 5040 changes (sequences), have a strict set of rules and take around 3 hours 15 minutes to complete, and must be rung entirely from memory, with no written aids.
In the early 1960s a CS algorithm for efficiently enumerating permutations was published, the Steinhaus–Johnson–Trotter algorithm. It's in Knuth, for those who have access to a copy. Unfortunately for them, it had been known by English Change Ringers as Plain Changes, since around 1621.
Ooh no, don't ever call bellringers that, unless you want to cause grave offence. Campanologist is a name for wussy academics, bellringers ring bells. And drink beer ;-)
74 comments
[ 338 ms ] story [ 2599 ms ] threadIf 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))
⇒ the design space is a lot larger than you indicate.
> 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.
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?
https://funwithbells.com/nigel-taylor/
https://www.youtube.com/watch?v=JEJkMalMVv4
I disagree. Mathematically, I wouldn’t know how to define a solid other than “anything with a volume”.
Certainly, a bell, when defined as a surface of revolution, doesn’t have an interior in a mathematical sense (https://mathworld.wolfram.com/Interior.html), does it?
There’s the convex closure, but that would, for the prototypical bell, add volume ‘outside’ the bell, too.
Also, I think varying wall thickness in bells for centuries was the only known way to tune them. https://en.wikipedia.org/wiki/Pieter_and_François_Hemony#Lif...:
“When struck, a bell produces a number of partials which, if imprecisely tuned, can create an unpleasant sound and which prevents it from harmonizing in accordance with other bells. To address this problem, the Hemony brothers gave their bells a particular profile and thickened it in certain places. The bells were then tuned by hollowing ridges from specific parts of the inner wall until the first few partials were acceptably in tune.”
(Aside: if you want your work to be used for centuries, becoming a professional bell caster seems a good bet)
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.)
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.
[1] https://www.youtube.com/watch?v=fXavtyyyj34
https://www.hibberts.co.uk/building-a-bell-sound/
https://polifonia-project.eu/
In the last year unaffected by COVID, there were 10,701 recorded performances and many more unrecorded ones and practices. https://bellboard.uk/
There are 23,397 Methods and 52,650 Compositions available, where a Method is roughly equivalent to a piece or song in traditional music terms. http://methods.ringing.org/intro.html
https://www.churchtimes.co.uk/articles/2018/2-february/news/...
Over _all the night_. Seriously, this appears to be quite a rational complaint.
Ours only chimes on the hour but it's going 24/7 and is very close to the surrounding houses (~40m) some of which are uphill and directly in line with the bells. I had to go in today to fettle it as it wasn't chiming and someone had posted on the village FB group that they were missing it. That's a fairly regular occurrence, most people people like hearing them, even through the night. Any complaints we get are when the clock stops.
It is possible to modify clocks to stop them chiming overnight, although ours is still in its entirely mechanical 1883 form, including me having to wind it every week.
The whole process is a combination of extreme sophistication and extremely simple tools and materials, something that has become quite rare nowadays. And despite an exact calculation of the bell shape and thousands of years of trial and error, tradition, and experience (the foundry in the video has been owned by the same family for 300 years), the cast is still highly dangerous and it still regularly happens that the bell just doesn't come out right and cannot be saved - months of work for nothing.
I think that explains why on the day of the cast (which is just 1 or 2 times per year), everyone involved is extremely nervous. The bell founder and the customer all join in prayer, and the cast traditionally starts with "in God's name" [1].
[0] https://www.youtube.com/watch?v=OQ8ZZV5UjcI
[1] https://youtu.be/OQ8ZZV5UjcI?t=1918
[0] https://www.hibberts.co.uk/doublets-or-warble-in-bells/
When you've got two companies down the road from each other making the same thing, you can almost guess the history. Originally there was just one company, but something happened. Maybe someone got angry, somebody left, and they started a second business, a competitor.
Today on the show, the rivalry between two companies that make nearly all the world's handbells. And how they eventually made peace.
https://www.npr.org/2021/12/23/1067333584/bell-wars-classic
https://www.youtube.com/watch?v=eMgmJEaYs70
"The boy, named Boriska [...] claims that before his father died, he revealed to him the secret of casting a bronze bell"
"Amid his tears, Boriska reveals that his father never imparted to him the secret of bell making"
The submitted title was "Someone wrote a PhD thesis on bells". Sometimes a title like that has the virtue of getting an obscure, interesting submission onto the front page. The sin is venial when the submission is particularly good (and mortal when it's dreck, which alas is more often the case.)
https://arxiv.org/list/math/new
Don't be so sure. At least for mathematicians, the "bell" may be just an excuse to make the subject cool. Many of the underlying problems are very general. For example, the inverse spectral problem (vibration modes of a complex shape) is the basis of the interaction between proteins, enzymes, and the huge molecules of cell walls. At the cell scale, Brownian motion shakes everything in all possible frequencies, but molecules can only vibrate at a specific set of frequencies (their spectrum). The act of "matching" different molecules happens when they have some common frequencies, often with localized modes of vibration.
You may think that you are just modeling church bells, but everything that you do has immediate application to the understanding of some organic chemical interactions, which are a deep mystery yet.
https://www.youtube.com/watch?v=vsZ5RIZrNKw
"The drop had been scheduled for noon. If she had come through on time and Probability was right about the slippage, it would be six o'clock in the evening, which was too late for vespers. And if it were vespers, why did the bell go on tolling?
It could be tolling for mass, or for a funeral or a wedding. Bells had rung almost constantly in the Middle Ages—to warn of invasions or fires, to help a lost child find its way back to the village, even to ward off thunderstorms. It could be ringing for any reason at all."
https://www.youtube.com/watch?v=qbn_Fzcxw3o&ab_channel=petya...
The Mathematics of Bell Ringing https://www.youtube.com/watch?v=44jXUo6KaVs
Performances (Peals) consist of 5040 changes (sequences), have a strict set of rules and take around 3 hours 15 minutes to complete, and must be rung entirely from memory, with no written aids.
In the early 1960s a CS algorithm for efficiently enumerating permutations was published, the Steinhaus–Johnson–Trotter algorithm. It's in Knuth, for those who have access to a copy. Unfortunately for them, it had been known by English Change Ringers as Plain Changes, since around 1621.
https://www.bbc.co.uk/programmes/b006sgsh
On the other hand, the American church predilection for electronic bells is an abomination.
My desk telephone has real bells in it for the ringer. It's from the 1950s, naturally.