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Why are the lower-pitched chimes both longer and wider? Do the width and length both affect the pitch?

As usual, my sister-in-law, Taylor, was asking some excellent questions. We had just finished a thrown pottery class together, along with my wife and her other sister, and were standing on the front lawn admiring a very large, very beautifully tuned set of wind chimes. The chimes themselves were long and wide, about 2 inches in diameter and ranging from something like 18 to 36 inches in length. As much a visual statement as an aural one.

Why were these lower-pitched chimes both longer and wider than the higher-pitched set we’d seen the previous day? We’d been on a wind chime kick during this trip — Taylor was on the lookout for a set of wind chimes after taking a class on conducting sound baths, and my wife and I had our own set at home that I was enamored with.

My best guess was that diameter had more do to with the amplitude than the pitch. Each chime in an individual set seemed to be the same diameter, so it was certainly possible to adjust the pitch by only adjusting the length. Maybe only the length affected the pitch, but since people are better at hearing higher pitches than lower pitches, you also need to make the chimes wider, or you wouldn’t be able to hear the lower tones? It was kind of bothering me that I wasn’t sure. But something else occurred to me as we turned away to rejoin the rest of the group. These were just metal tubes, the kind that hardware stores sold in ten-foot lengths.

Taylor… I think we could make these. There must be a formula to calculate the pitch based on the length of the tube. Someone must have figured this out.

Quite a lot of my most interesting projects have started with a similar sentiment. Often it turns out that I’m right — someone has figured out the fundamental challenge, and packaged the solution to make it easier for others to utilize. Finding those solutions is one of my favorite experiences. The world is full of people who’ve become obsessed with a single problem, learning its ins and outs, and who’ve decided to share their discoveries. I’m regularly astonished by how often those people are willing to share their discoveries for nothing in return.

And I had a hunch that this was another one of those times. Wind chimes are fascinating, both aesthetically and mechanically, and humans have been creating them for thousands of years. So I started searching around the Internet for information about the formulas for chime pitch, and I stumbled into what has since become one of my favorite websites.

This website is a masterwork of documentation. It has information at every level of detail, from pre-built measurement tables to straight physics formulas. It has such a tremendous wealth of information that a community has formed around it, drawn by its author’s passion and commitment, providing even more information and tools.

Lee’s website taught me several crucial things about wind chimes. First and foremost, it taught me the answer to Taylor’s question: The length and diameter do both matter, but not in the way I would have guessed!

Increasing the length of the chime decreases the pitch. This makes intuitive sense to me, because the chime is essentially making sound by bending in half in alternating directions over and over again. This means that a longer chime will be producing a longer wavelength, which translates to a lower pitch. I was surprised to learn, however, that not only does increasing the diameter also effect the pitch, but it also lowers it. In retrospect, this also makes sense, I just didn’t have the correct mental model at the time. A larger diameter means more mass, more mass means more inertia, and more inertia means that the chime takes longer to slow down and speed up when it’s changing directions at the end of a bend. And more time to change directions means a longer wavelength and a lower pitch!

So why were the lower-pitched chimes larger in diameter than the higher-pitched ones? It turns out, the answer is most likely that larger diameter (and thicker-walled) chimes just… sound better! Because they have more mass, they resonate for longer and have clearer harmonics. If you want to make a higher-pitched set of chimes, then you need less mass, as explained above, but apparently it’s otherwise almost always preferable to use wider diameter tubes.

Armed with this knowledge — and a plethora more — it was time to start making some decisions. First, we needed to pick some pitches. To do this, I started out by playing some scales on a piano and had Taylor pick ones that she liked the sound of. We’d learned (through the website, of course!) that below C4, the chimes almost certainly wouldn’t have enough energy for a listener to hear the fundamental frequency, which results in the perception that the chime is producing a higher pitch. But we also wanted some low, resonant tones, so in the end we landed on a scale that started on G3 (only a bit below C4) and moved up the Mixolydian scale by thirds, forming a G dominant 9th chord. We spaced the notes out by thirds to try to avoid having pitches that were too close together, which might result in harmonics that clashed, creating a beating effect.

Then it was time to a make a design and choose materials. We were a bit low on tools (though we ended up receiving a last minute toolkit loan from a friend who was excited about the project), and a lot of the designs seemed to be somewhat reliant on having a drill press at hand. So we tried to put together a design that would require, in essence, the least amount of precise drilling and fastening. We ended up choosing ¾ inch copper pipe for the chimes, tied to an 8-inch nickel-plated ring from and art supply store, with a slate coaster (for mass) fastened to a light wooden disk (also from the art supply store). Oh, and a copper-colored aluminum sheet for the wind sail!

A member of the wonderful community that has formed around Lee’s website created a simple calculator that we used for determining the length of each chime.

It even allowed us to enter the length that the pipe was sold in (in our case, 10 feet) so that we could see how many pipes to purchase and how much would be left over. This helped us decide on our pipe diameter — we specifically chose a ¾ inch pipe because it allowed these pitches to fit in a single 10-foot length!

At first, construction was a breeze. Lee has created detailed videos walking through various parts of the construction process, and we set up a little factory line with myself, Taylor, and my wife as we cut pipes to length, filed away their sharp edges, and drilled holes to hang them on. In what felt like no time, we had our chimes, and were ready to hang them!

In software engineering, there’s a rule called “the ninety-ninety rule”:

The first 90 percent of the code accounts for the first 90 percent of the development time. The remaining 10 percent of the code accounts for the other 90 percent of the development time.

— Tom Cargill, Bell Labs

This is a clearly tongue-in-cheek “rule”, but the humor speaks to something real about building things. Humans are full to the brim with cognitive biases, and the way that we see the world tends to nudge us toward the assumption that if something visually appears to be close to completion, it probably is. In software, this might mean that when we’ve written 90% of the code, we perceive the problem as 90% solved. In chime-building, apparently, it can mean that when we have all the individual pieces of the chimes prepped and laid out, we perceive the chimes as 90% constructed. After all, all that was left was to tie some knots!

In both scenarios, the “last 10%” is about putting the pieces together. This is so often where we find another 90% of work because this is where our assumptions are tested! When three engineers have been working on three separate modules, and it comes time to integrate them, that’s when we learn whether all three of those engineers have been working under the same set of assumptions. And when building wind chimes, and it’s time to fasten the pieces together, that’s when we learn what assumptions we’ve made about how these pieces will play with each other.

Sure enough, I had made some assumptions, and not all of them were well-founded. Firstly, I had assumed that a simple overhand knot would be sufficient to fasten the chimes to the nickel-plated ring that we used as the support ring. Lee’s website mentioned rings as support systems only in passing, and had almost no detail about them, focusing instead on wooden support disks (which are, presumably, a better choice). We had gone with a ring because I was worried that — with our limited tools — we would struggle to drill perpendicular, evenly spaced holes in the support plate. I was now coming to regret this choice tremendously.

My experience with knots is almost exclusively confined to rock climbing, where the rope materials, construction, and thickness are specifically engineered to hold knots when the rope is holding about two human beings worth of mass. I was woefully unprepared to attempt to transfer that knowledge to this new scenario, where a much thinner, simpler, and smoother cord was attempting to hold a knot with a heavy metal pipe attached to one end. The results were, if I’m being honest, devastating. No matter which knots I tied, the lightest jostling of the ring would send them all unraveling in moments. And more secure knots, like the figure eight, were very challenging to tie tightly at specific lengths in such a small cord.

Finally, after hours of researching and trialing new knots, I landed on a double Davy knot. It held up very well under significant jostling and bouncing, and it was also relatively easy to tie very tightly against the ring itself, providing a decent amount of friction against sliding. This turned out to be very important, because I had made a second assumption: the knots would largely stay in place on the ring.

This second assumption proved even more dangerous than the first. While enough wind would eventually jostle the chimes enough to loosen the knots I had tried before the double Davy, they tended to encounter a similarly catastrophic failure far before that happened. After only the slightest adjustment in the ring’s balance, which would happen essentially every time a moderate gust blew, all the knots would rapidly slide around the ring to a single point. This would sometimes be enough acceleration to then trigger the knots to come undone, but even when it didn’t, the result was a completely useless set of chimes.

I was so surprised by this initially because I had failed to imagine the chimes set as a dynamic system. A slight tilt in the support ring led to one or two knots sliding a few millimeters — on its own, this was no problem. But that slide redistributed the weight on the ring, causing it to tilt further in the same direction, which led to more sliding, then more tilting, until the ring was essentially vertical, and the knots were falling at speed toward the bottom point.

As mentioned, the first step toward resolving this issue was to find knots that provided enough friction to stay in place during a slight tilt. That was accomplished with the double Davy knot. The second step was to limit the amount that the ring could tilt, so that we could avoid the cascade. Eventually, after quite a bit more trial and error, this was accomplished by attaching the support hook (the single point that the entire set would hang from) to the ring with five more cords, their knots interspersed within the knots holding the chimes.

Then, suddenly, the chimes worked. I held up the set, gave the wind sail a nudge, and then… music.