@TruthSandwich got me interested in the vibrational modes of bells. They're not harmonics with frequencies 1, 2, 3, 4, ... times the lowest frequency: they're much more complicated! That's why bells sound clangy. This chart shows how they sometimes work.
The lowest frequency vibrations are called:
• the 'hum' (the lowest frequency)
• the 'prime' (with frequency roughly 2 times that of the hum)
• the 'tierce' (roughly 2.4 times the hum, so a minor third above the prime)
• the 'quint' (roughly 3 times the hum, so a major fifth above the prime)
• the 'nominal' (roughly 4 times the hum, so an octave above the prime)
and so on. If you think these names are illogical, join the club! One reason it's tricky is that the loudest vibration is not the lowest one: it's the 'prime'.
The numbers I just gave you should be taken with a big grain of salt. They really depend on the shape of the bell, and you'd have to be great at designing bells to make them come out as shown here. It's not like a violin string or flute, where the math is on your side.
This quote helps explain the chart:
"Modern theory separates the modes of vibration into those produced by the "soundbow" and those produced by the remaining bell "shell". The bell vibrates both radially and axially and the principal vibrational modes are shown in the diagram together with their classification using the scheme proposed by Perrin et al. This scheme consists of the mode of vibration (RIR - Ring Inextensional Radial, RA - Ring Axial, R=n - Shell driven), the number of meridians (where “m” is half the number of meridians) and the number of nodal circles (n)."
Starting to sound like orbitals in quantum mechanics!
@johncarlosbaez@TruthSandwich when I was a kid I noticed that I could get a bell-like sound by stacking perfect 4ths on the piano, any idea why that seems to work?
@tobinbaker - I don't know why stacked fourths remind you of bells. But the pianist McCoy Tyner popularized the use of stacked fourths in jazz - it's called 'quartal harmony' and it still sounds cool.
@johncarlosbaez@TruthSandwich Also keep in mind that at least in christian church there are two types of bells musically: orthodox, or Eastern bells, and Western bells. Google for a catholic or an evangelist church vs. orthodox church, and you'll get what I mean. To be super simple, with Western bells you can play a melody, with Eastern ones you cannot because well, there are too many notes and there is no separate audible "main tone" (sorry, I'm not a mathematician, I just have perfect pitch).
Another cool thing: in a bell, most of the vibrational modes are 'doublets'. So they're actually 2 modes with frequencies that are close, but not equal unless you're really careful when designing your bell!
The reason is that with an even number of nodal meridians - see the charts for what that means - there are actually 2 different ways for the bell to vibrate. Annoyingly, these 2 ways are not really shown in the charts, though you might think that's what they're showing.
What's cool is that in quantum mechanics we use the term 'doublet' to mean 2 vibrational modes of an atom with frequencies that are close but not equal. But instead of sound, the atom emits light!
I've just been learning about this right now so I could be making mistakes. Here's where I got my info:
The second one is a labor of love by an organization that "helps churches acquire surplus and/or redundant bells to be hung for English-style full-circle bell-ringing." I love it!
But I don't like how the English say you're "redundant" when they fire you from your job - as if they suddenly noticed they'd hired two copies of you. Now I'm imagining sad redundant bells hanging around on the dole.
@djc - yes, at least in the linear approximation. It complicated not only because the bell has a wacky shape, but also because the surface of the bell is free to vibrate except where it's hung. This is different from a violin string or drum which has a fixed boundary, giving "Dirichlet boundary" condition for your Laplace operator.
So there's a ton of math here, for those willing to explore it.
@johncarlosbaez As someone who likes to dabble in spectral geometry, I now wonder whether it's possible to hear the shape of a bell.
Assuming the linear approximation is not far from the truth, what are the correct boundary conditions? Can't be just Dirichlet at the center, right?
Thanks to your post I just realized bells are nice examples of why one would want to study the spectrum of a Riemannian manifold of cohomogeneity one - that might come in handy for the one or other research proposal ;)
@pschwahn - hmm, whether you approximate it as 2-dimensional or treat it as 3-dimentional, I think the boundary that's free to move obeys Neumann boundary conditions, but the point at which its hung is held fixed. But actual experts would know more!
It seems like a really hard problem designing a bell-like shape that has reasonably nice-sounding eigenvalues.
@johncarlosbaez@djc they are not. You are finding the eigenvalues/eigenvectors of shells. Those couple membrane and "plate" effects and are commonly modeled as two fourth order equations coupled
@johncarlosbaez@djc. In the case of bells, they are commonly axisymmetric and the equations simplify enormly. You still have some coupling between the membrane (tensile) and bending effects.
@johncarlosbaez just a side note on redundancy: being made redundant doesn't equate to being fired from your job, it means that the job itself no longer exists (so you genuinely are redundant).
For an imperfect analogy that maybe a bit closer to home, the only way to get rid of a tenured professor (unless they do something actually illegal) is to get rid of their department. I once heard of someone getting so fed up with their actual department that they were tempted to create a new one-person department to avoid their colleagues, but didn't because it left them open to this form of dismissal.
@johncarlosbaez this bit caught my interest "It is believed that some medieval founders cast eccentrically shaped bells so that the frequency pairs were so far apart they would not beat." but the cited paper (Group Theory and The Bell, Journal of Sound and Vibration (1973) 31:411-418. R. Perrin, T Charnley.) seems to be paywalled everywhere.
eccentrically shaped medieval bells searching i go.
@tsrono - cool! I'll see if my university has access to the 𝐽𝑜𝑢𝑟𝑛𝑎𝑙 𝑜𝑓 𝑆𝑜𝑢𝑛𝑑 𝑎𝑛𝑑 𝑉𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 and let you know if I succeed in getting this article. But if anyone else can get it, let us know!
Elsevier says this journal "supports open access", but it costs $4070 to publish an open-access article there. 🤢
I like the idea of medieval bell makers deliberately breaking the doublets.
@tsrono - ugh, I could get the journal through my university if I'd remembered to install the VPN software on my new laptop when on campus, but I didn't know you had to be on campus to install it, and I'll be far away until September. 😿
@abdullahkhalids@tsrono - thanks! For some reason I never use Sci-Hub, though I use LibGen regularly to get math books that would take forever to get any other way.
@tsrono@johncarlosbaez Did some digging and had to show to someone: In one of the experimental papers cited here (Grützmacher et al.: Akustische Untersuchungen an Kirchenglocken/Acoustic study of church bells, Acustica 16, 1965) one finds this cool image of the sound field of a model bell, displaying both phase and amplitude of the sound waves simultaneously.
The picture is itself taken from a work of Schroeder (Ein optisches Verfahren zur amplituden- und phasengetreuen Darstellung stationärer Schallfelder, Acustica 13, 1963).
@pschwahn@tsrono - Cool! I need to finish up my series of posts on tuning systems and turn them into a paper, but I'm not careful I'll get seduced by the math and physics of bells. I need to hold off on that.
It's not just the resonance of the bell itself that has some interesting maths behind it, but the whole practice of English-style change ringing is applied group theory! It's essentially finding particular Hamlitonian cycles in the Cayley graphs of symmetric groups. One major open problem is finding a so-called "bobs-only extent of Erin Triples" - here's a paper on the topic co-authored by the late Andrew Johnson, who solved the closely-related problem of a bobs-only extent of Stedman reasonably recently: https://ejgta.org/index.php/ejgta/article/download/516/pdf_95
Brian Eno got interested in bells for his album "January 07003: Bell Studies for the Clock of the Long Now". He made it to help the Long Now Foundation, back before a bunch of twits gave "longtermism" a bad name by acting like being an asshole now is good in the long run.
Here's the story:
"One of their projects involves the construction of a clock, designed by W. Daniel Hillis, intended to keep time for 10,000 years. The first prototype of the clock is working and on permanent display at the London Science Museum. Eno, a member of the foundation, created this album both as a tribute to the project, and as an artistic expression of the vast swathes of time represented by the sheer "long-termness" of the philosophy behind the clock. About the workings of the record, Eno stated that he "naturally wondered what kind of sound it could make to announce the passage of time" and that he "had nurtured an interest in bells for many years, and this seemed like a good alibi for taking it a bit deeper"."
"The sounds on the album are entirely synthesizer-based, although Eno studied the actual physics of bell tones in order to simulate the kinds of bells familiar to modern ears. He also tried to imagine what bells might sound like in the future, which took him "out of the bounds of current physical and material possibilities .... imagine bells with quite different physical properties from those we now know". To that end, mathematical algorithms were also used to generate some of the sounds; Eno also made use of his generative software."
Check it out! The first pieces sound like realistic bells, but some of the later ones have titles like "Virtual dream bells, thick glass".
@johncarlosbaez Eno's bell drone reminds me of the technique used in very opening "Mutations" by Jean-Claude Risset [0]. The opening consists of a few tones, followed by a bell sound. The bell sound is constructed using additive synthesis, and the tones heard before it are the partials that make up the bell. This tone set makes up the musical "scale" used throughout composition, which creates an interesting relationship between timbre and tonality.
From the sound of it, Eno may be doing something very similar with his bell drone track, extracting the partials from a bell, stretching them out, and then stacking them on top of each other.
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