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solipsys.co.uk
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The reason is because log10(2) ≈ 0.3. The shifted powers of two can be written unsorted, as 10^{ n log10(2) } ≈ 10^{ 3n/10 } where {…} denotes the fractional part. { 3n/10 } is periodic with order 10, i.e.

``````{ 3*0 / 10 } = 0.0
{ 3*1 / 10 } = 0.3
{ 3*2 / 10 } = 0.6
{ 3*3 / 10 } = 0.9
{ 3*4 / 10 } = { 1.2 } = 0.2
...
{ 3*9 / 10 } = 0.7
{ 3*10 / 10 } = 0.0
``````

The lexicographic ordering part is a red herring, in fact it’s just the normal sorting order once the powers of two are shifted accordingly.

EDIT: Oh, and for something closer to home for most of the people on here:

``````2^10 = 10^(log10(2) * 10) ≈ 10^(0.3 * 10) = 10^3.
``````

EDIT: Formatted and slightly expanded version of this comment here: https://jo.ie/powers-of-two.html

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Everything you’ve said here is true, and to some extent it does explain what’s going on, but this is what I meant when I said I can do the calculations, and I can show that they’re close, but I still don’t feel that this is “intuitive”. I don’t feel that someone who doesn’t know about logs can see that there’s an inevitability about this, or that with a single, simple observation someone could see how it just flows.

But you’ve said a few things here that will help me synthesise something - thank you.

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Yeah, “Intuitive” is very much a matter of perspective. My own maths education started from a physics perspective, so for me showing that log10(2) is roughly 0.3, feels almost intuitive enough (but it’s still lacking something).

I hope the bit I’ve said on mastodon about Equal Temperament / Just Intonation might be useful toward synthesising a proper explanation, because it really feels close.

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I think it is getting close. Someone else has suggested (using words I’d never use in the explanation) thinking about a logarithmic clock circle. I think that’s where I’m going with this, and your comments have helped clarify things.