Lets temporarily adopt the (non-standard) definition that for all cardinals $\kappa$ and $\nu$ we have $[\kappa,\nu) = \{\mbox{cardinals } \eta \mid \kappa \leq \eta < \nu\}.$ Note the word cardinals. Furthermore, write $2^\kappa$ for cardinal exponentiation (rather than ordinal). Under these definitions, it is trivial to see that $$\mathrm{ord}[\aleph_\alpha, \aleph_{\alpha+\beta}) = \beta.$$
Now define a function $f$ such that
$$f(\alpha,\beta)=\mathrm{ord}[2^{\aleph_\alpha}, 2^{\aleph_{\alpha+\beta}}).$$
Then GCH proves that $f(\alpha,\beta)=\beta.$ My question is, in the absence of GCH, what can ZFC tell us about $f(\alpha,\beta)$? Is it necessarily independent of $\alpha$? And, does our ability to pin down $f$ fall prey to Easton's theorem, or something like it? (I presume so.)