  In this paper, we define a generalized arithmetic-geometric mean $\mu_g$
among $2^g$ terms motivated by $2\tau$-formulas of theta constants. By using
Thomae's formula, we give two expressions of $\mu_g$ when initial terms satisfy
some conditions. One is given in terms of period integrals of a hyperelliptic
curve $C$ of genus $g$. The other is by a period integral of a certain
Calabi-Yau $g$-fold given as a double cover of the $g$-dimensional projective
space $\mathbf{P}^g$.
