Piano keys are familiar and easy enough to draw if you're not trying to be exact, but if you want label the dimensions with their exact measurements (like in a CAD drawing), it turns into a math puzzle. The problem comes from the groups of two and three black keys.

This article explains it like this:

If you've ever looked closely at a piano keyboard you may have
noticed that the widths of the white keys are not all the same
at the back ends (where they pass between the black keys). Of
course, if you think about it for a minute, it's clear they
couldn't possibly all be the same width, assuming the black keys
are all identical (with non-zero width) and the white keys all
have equal widths at the front ends, because the only simultaneous
solution of 3W=3w+2b and 4W=4w+3b is with b=0.

To unpack that a bit: in that equation, 'W' is the width of each white key at the front (which should all be the same), 'w' is the width of a white key at the back, and 'b' is the width of a black key.) The first equation is for the group of two black keys (separating C, D, and E) and the second equation is for the three black keys separating F through B.

Since it's mathematically impossible, a constraint needs to be relaxed. The article describes ways to make the white keys have slightly different widths at the back.

If we set c=e=(W-5B/8) and a=b=d=f=g=(W-3B/4) we have a maximum
discrepancy of only B/8, and quite a few actual pianos use this
pattern as well. However, the absolute optimum arrangement is to
set c=d=e=(W-2B/3) and f=g=a=b=(W-3B/4), which gives a maximum
discrepancy of just B/12. This pattern is used on many keyboards,
e.g. the Roland PC-100.

When actually building a musical instrument (instead of just drawing the keyboard), there is a further constraint, described in this article:

The black keys on a piano keyboard, instead of always being centered on the dividing line between the two white keys they lie between, are spaced so that the twelve keys which make up an octave are spaced equally as they enter the internal mechanism of the instrument.

But this means that the "key caps" for the white keys should be slightly off-center compared to whatever rod or lever they're attached to. The author speculates about how to divide this up using various units.

(They seem quite annoying to 3D print.)

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Long ago, Cree leader Captain Swan visited the Athabasca area. In 1715, he described a scene to Hudson’s Bay Company fur trader James Knight: “... there is a Certain Gum or pitch that runs down the river in such abundance that [Indians] cannot land but at certain places.” This was the first written reference to bitumen in Canada. Bitumen forms when organic matter is buried and subjected to heat and pressure over geological timescales. That organic matter was primarily algae and plants, which had sequestered carbon dioxide (CO₂) from the atmosphere by photosynthesis, thereby locking CO₂ in place, significantly reducing atmospheric CO₂ levels, and helping sustain all aerobic life.

In 1859, John Tyndall explained how atmospheric gases absorb heat from the sun as infrared radiation. His paper details an early understanding of the greenhouse effect. Scientists have long since linked CO₂ emissions—burning refined bitumen and coal—to changing Earth’s climate. A 1912 Popular Mechanics article states, “The furnaces of the world are now burning about 2 billion tons of coal a year. When this is burned, uniting with oxygen, it adds about 7 billion tons of carbon dioxide to the atmosphere yearly. This tends to make the air a more effective blanket for the earth and raise its temperature.” A century on, we’re burning 500% more fossil fuels.

Wishful thoughts will not prevent Earth’s global average temperature from increasing as we combust fossil fuels back into atmospheric CO₂. And while our generation reaps the rewards of inexpensive energy, our grandchildren will face the consequences of repaying this debt. A debt undermining the ancient Haudenosaunee philosophy that today’s decisions should result in a sustainable world seven generations from now.

Building a better world for our children requires energy—yet doing so by burning fossil fuels to the point of climate destabilization twists irony into generational betrayal far removed from sustainability.

In a 2013 experiment, University of Berkeley researchers found that breathing in a CO₂ concentration of 1,000 parts per million (ppm) indoors causes a measurable decline in intellectual capacity; at 2,500 ppm, initiative and strategic thinking declined to a dysfunctional level, which has since been corroborated by other researchers, including a 2023 meta-analysis on the short-term exposure to indoor CO₂ levels versus cognitive task performance. These cognitive effects become particularly concerning when viewed against atmospheric trends. On June 2, 2025, atmospheric CO₂ surpassed 429 ppm, a significant increase from the 318 ppm measured at Mauna Loa on June 15, 1959.

https://i.ibb.co/yFcXJqCy/graph.png

The graph illustrates a troubling acceleration in CO₂ emissions. At the current growth rate of 3.8 ppm per year, atmospheric CO₂ could reach 1,000 ppm in six generations (150 years). A 2021 study published in Nature emphasized the urgent need for action, stating that global oil and gas production must decline by 3% annually until 2050. Moreover, to limit warming to 1.5 °C above pre-industrial levels (1850–1900), an additional 25% of oil reserves must remain untouched.

Against this backdrop, political leaders advocate for increased fossil fuel extraction. Danielle Smith wants to unlock Alberta’s “$14 trillion in oil wealth” to “benefit millions of Canadians for generations.”

Short-term economic benefits derived from resource exploitation have repeatedly led to gradual, often unheeded, environmental degradation. This pattern repeatedly culminated in ecological and economic crashes, devastating the very communities who initially profited. Notable cases include Mesopotamian salinization, the Classic Maya collapse, the Ancestral Puebloan collapse, Norse Greenland settlements, Easter Island’s deforestation, the Dust Bowl, the Aral Sea’s desiccation, and the Grand Banks cod collapse. While some nations have sustainably managed resource wealth, the immediate economic pressures and political incentives that drive extraction often overshadow long-term planning.

The question is not: “How many Canadian generations will benefit?”

The question is: “How many generations will suffer, globally?”

Will we learn from history? Will we set an example for the next seven generations?

Or will we build more oil and gas pipelines, condemning our descendants to an unsustainable future?

Hereby released into the public domain. Feel free to adapt, correct, and send to representatives.

Yes, I realize the answer is “everything.”

I intend the question to be less about simply naming answers and more about surfacing their associated complexity.

Walk us through something you know well so that we can appreciate how not simple it actually is!

What have you been listening to this week? You don't need to do a 6000 word review if you don't want to, but please write something! If you've just picked up some music, please update on that as well, we'd love to see your hauls :)

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Tildes is a very serious site, where we discuss very serious matters like forums, stardew valley and whatever. Tags culled from the highest voted topics from the last seven days, if anyone was boggled of the mind.

But one of my favourite tags happens to be offbeat! Taking its original inspiration from Sir Nils Olav III, this thread is looking for any far-fetched offbeat stories lurking in the newspapers. It may not deserve its own post, but it deserves a wider audience!

Any clever ways to connect to the Internet safely to update drivers, security, etc? I'd only want to connect to Intel, AMD, Microsoft, etc, and then would physically disconnect the lan card. I know, dangerous, but I'm trying a piecemeal approach with a flash drive and getting mixed results. I tried to update to Service Pack 2, and it bricked the computer on restart, back to flashing Vista.

I'm starting to use AI increasingly, and am getting some value out of it. I'm curious if paying for paid tiers of the big players (in particular, ChatGPT and Claude) provides significantly better responses.

I'm aware that the paid tiers offer more features and benefits than just higher response quality. For me, those are just nice-to-haves, and not my primary concern.

My main uses of AI are software development and foreign language learning. So far, I've used the free versions of ChatGPT and Claude, as well as "proxies," including Github Copilot and Duck.ai. For both my use cases, I've found the responses usually good and helpful. I just maintain a healthy skepticism about the correctness of the answers, and challenge, test, and double check where needed (especially testing suggested code when developing software).

Have you found response quality to be noticeably and significantly better with paid tiers? I was just randomly thinking, and it occurred to me that the cost of an AI subscription is in the same ballpark as a subscription to a language learning service like Duolingo. So, if I can get value from AI that approaches what I'd get from a dedicated language learning service (even if it doesn't quite match or exceed it), then also getting the value of general AI in the same subscription should make things quite valuable and worth it. Not to mention possibly getting better software development assistance in the same package.

I don't have a lot to say myself, yet. I signed up for the beta mostly because people talk so positively of what Digg was like in the past. I never experienced that, as Digg was already on it's way out when I discovered it.

I'd love to hear from those who remember the golden age of Digg, as well as those just curious about the reboot themselves... What are you thoughts, expectations, hopes and concerns?