The Sound of Bells – The acoustics, sound, tuning and history of church bells
hibberts.co.uk
hibberts.co.uk
"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 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].
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.
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))
⇒ 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?
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.
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.
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.)
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.
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 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."
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.
https://www.youtube.com/watch?v=qbn_Fzcxw3o&ab_channel=petya...