Tetrahedra almost tessellate but don't quite do it: https://www.pnas.org/content/pnas/103/28/10612/F1.large.jpg
In music, the frequency ratio of a semitone is ideally 2^(1/12), but without some tiny fudging (called tuning), you can't make harmonies as the frequencies almost but don't quite line up right. I forget exactly how this one works so I may have something off.
Other coincidences that drive me wild: speed of light is almost but not quite 3.0E8m/s And the fine structure constant being almost but not exactly 1/137.
On that subject, if you decided to make your base unit of length the distance light travels in a nanosecond it'd almost be a foot but not quite.
Similarly the earth-sun distance is 8 light-minutes.These feel right, like measuring mass in stone for people, kilos for sugar, carat for diamonds, electron-volts for particles etc.
IIRC correctly, it's not just harmonies; the range of a piano is big enough that if you tune each octave exactly based on that ratio, you'll end up with the first and last octaves sounding off from each other.
Hypothetically, or on an electronic instrument, you could. But if you did all 2^(1/12) ratios, your octaves wouldn't be in tune. Strings on a piano do not behave like an ideal string. Their overtones are not 2X, 3X, 4X, 5X, etc. times the fundamental frequency. Instead, the actual overtones are higher than the ideal frequencies. This is called inharmonicity (https://en.wikipedia.org/wiki/Inharmonicity).
So when tuning a piano, you have to tailor the way you tune it to each different piano if you want that piano's lower strings to be in tune with its higher strings.
I think I've been hearing this for a long time but didn't realize it was real so questioned my perceptual system.
Thank you for the info!
Some of it, of course, sounds like cats screeching but some of it is genuinely astonishing.