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@ColinTheMathmo

Sent an "intuitive" explanation via the comment form. Only realised it was you after I clicked the follow on mastodon link. In brief, it's because log10(2) ≈ 0.3, and the fractional part of {0.3 n} is periodic with order 10.

solipsys.co.uk/new/PowersOfTwo

@j That's all true, but I'm not certain that it's "intuitive". It does enhance what I already have, though, so it's useful in trying to synthesise something closer to my target.

Thank you.

@ColinTheMathmo Hence the "quotes". Nevertheless, it's the most intuitive explanation I've managed to come up with (it can also be summarized as "because 8 is approximately 10", but that's the physics side of me talking)

Maybe a musical analogy might help? It reminds me a little of how equal temperament is used to approximate just intonation.

@ColinTheMathmo Actually, it's almost exactly the same concept. The lexicographically sorted and shifted powers of two differ only by small ratios 5/4 and 32/25, which is approximated as by powers of the irrational 10^(1/10).

Similarly, in just intonation, the small ratios between notes (9/8, 10/9, etc.) is approximated by powers of the irrational 2^(1/12)

@ColinTheMathmo The only major difference is that 2¹⁰ is not exactly equal to 10³, whereas 2^(12/12) = 12, so just intonation and equal temperament match up at octave boundaries.

@j These are all useful comments, and are definitely part of the story, but helping to explain how and why the lex ordering is getting involved is more tricky. Lots of people are waving their hands and saying "its just this", and they're right, but the audience I have in mind won't be able to see it. They can't (easily) follow calculations, so it's tricky.

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