I made a universe, ages ago, with its own rules of algebra. Then I forgot about it. Now I look at its container and dust off the top, where I’d excitedly scrawled:
(¹²√2)³ ≅ ⁶⁄₅ (¹²√2)⁴ ≅ ⁵⁄₄ (¹²√2)⁵ ≅ ⁴⁄₃ (¹²√2)⁷ ≅ ³⁄₂ !!!
I shrug. It must have meant something to me when I was younger, but I am more even-tempered now. I open the lid, and hear … music !!!
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(Let me be the first to admit the Western-music-chauvinism in this toot. Not all human music uses 12-tone equal temperament.)
The diagram at https://en.wikipedia.org/wiki/Equal_temperament is a fascinating demonstration of how close different kinds of equal-temperament tuning systems come to hitting the “just” intervals with their simple ratios of frequencies.
