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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.

@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@ColinTheMathmo@mathstodon.xyz@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.