Help with computing fractional octave center frequencies

Question

Given frequency X, I want to compute frequency Y, given interval is Z-octaves. Z will normally be fractional.

Y = X + X * 10^log10(Z) returns 33.333 given X=25 and Z=1/3.

I'm expecting ~31.5.

Answer 1

One octave-up is multiplying frequency by $2$.

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Correct formula should be: $$ Y = X\cdot 2^{Z}; $$ for $X=25 $ (Hz), $Z=1/3$: $$ Y = 25\cdot 2^{1/3} \approx 25\cdot 1.259921 \approx 31.498. $$

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For example:
$1$ octave from $25$ Hz is $50$ Hz;
$3$ octaves from $25$ Hz is $200$ Hz
etc.

Answer 2

Hint: Write $$y=25+25\cdot 10^{\lg(1/3)}$$ and use that $$\lg(1/3)=-\lg(3)$$ and $$\lg(x)=\log_{10}{x}$$