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.)

Find yourself watching tons of great videos on [insert chosen video sharing platform], but also find yourself reluctant to flood the Tildes front page with them? Then this thread is for you.

It could be one quirky video that you feel deserves some eyeballs on it, or perhaps you've got a curated list of videos that you'd love to talk us through...

Share some of the best video content you've watched this past week/fortnight with us!

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I'm in the planning stages of a custom setting for a new campaign I'm aiming to start next year with my current table. We're doing PF2's Kingmaker and AD&D's Temple of Elemental Evil in the meantime.

The game is to be Viking themed, in that the starting locale and civilization will be structured in similar ways to the coastal Scandinavian settlements and there will be an on/off season. During the on season, they will board boats and sail many hundreds of miles across water to distant lands to find dungeons and ruins to loot, with a clock they have to keep an eye on; the expedition can only afford to be out for so long, and they need to ultimately make a profit. During the off season, they will be home and can spend time locally engaging in low-tier politics, explore the untamed parts of the continent, or both.

I'm intending for gods to play a more concrete and available part in this game and have been chewing on how best to represent that mechanically. I discovered that one of D&D 5e's supplements for a Magic: The Gathering setting, Mythic Odysseys of Theros, does something similar and has mechanics for tracking Piety with a given deity, which comes with boons at specific breakpoints. I liked the idea, though I'd be making my own boons for my pantheon rather than use these as-is, especially since I wouldn't be running this game in 5e, but rather in AD&D 1e.

I have a group chat with a few of my players that I can trust for this kind of thing to bounce ideas off of for various things, so I put this forth to them and got their thoughts. They universally thought the example boons from 5e were too personal and individual for the kind of stuff Norse gods would get up to, and there wasn't really a way to track a given deity's disdain of you in a similar manner. They also didn't like that you could track the Piety with a discrete score and could reliably measure when your next boon would be.

What we settled on doing is utilizing my custom tarot effects we're already doing in my regular AD&D campaign, but having it apply in certain regions or during certain stretches of adventure. This would allow for randomly coming across an avatar of a god and earning a minor boon or bane for assisting or denying them.

This is a recurring post to discuss programming or other technical projects that we've been working on. Tell us about one of your recent projects, either at work or personal projects. What's interesting about it? Are you having trouble with anything?

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.

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!

What have you been watching and reading this week? You don't need to give us a whole essay if you don't want to, but please write something! Feel free to talk about something you saw that was cool, something that was bad, ask for recommendations, or anything else you can think of.

If you want to, feel free to find the thing you're talking about and link to its pages on Anilist, MAL, or any other database you use!