> 1.0, 1.28, 1.6, 2.0, 2.56, 3.2, 4.0, 5.12, 6.4, 8.0

These numbers are also instantly recognizable as being closely related to the the ISO-standard 1/3 octave bands on a 31 band graphic equalizer:

30 Hz, 40, 50, 63, 80, 100, 125, 160, 200, 215, 315, 400, 500, 630 ...

There is a relation to powers of two because every three steps, the frequency doubles. These numbers are derived differently; basically according to:

                       E.g.:
  2^(1/3) ~= 1.2599    125 Hz   12.5 kHz
  2^(2/3) ~= 1.5874    160 Hz   16   kHz
  2^(3/3)  = 2         200 Hz   20   kHz

The fact that 1/10th powers of 10 approximately double in 3 steps makes those numbers close:

  10^(1/10) ~= 1.259
  10^(2/10) ~= 1.585
  10^(3/10) ~= 1.995

So why is there this three-step geometric sequence (approximately) found among that rearrangement of the powers of two?

Like from 128 to 160 (16 x 10): why is there are 2^(1/3) step?

     160      2^4 x 10            10
     ---  =   ---------    =      --    = 1.25
     128         2^7              2^3

It's simply that 1.25 is close to 10^0.1 is close to 2^(1/3). Three compounded 25% increases are close to doubling.

Note that 2^(1/3) is the same number as 8^(1/9). That's getting close to 10^(1/10).

Which exact power of ten is equal to 2^(1/3).

      10^x = 2^(1/3)

log of both sides:

         x = log10(2^1/3)
  
         x =~ 0.10034333189

Within 0.3% of 1/10th. And which power of 10 is 1.25? log10 1.25 ~= 0.09691001. Some 3% off from 0.1.

There is a relation to music here and musical intervals/scales. If we have a given root note, then 1.25X that frequency (5/4) is an ideal major third interval.

Now music geeks will know that under equal temperament, the major third interval "cuts" the octave into three equal pieces: the equal temperament major third is our 2^(1/3) = 1.2599: slightly sharp.

The closeness between 1.25 and 1.2599 is related to why equal temperament works (captures close-enough values of various musical intervals).