  The hyper-Mahler measures $m_k( 1+x_1+x_2),k\in\mathbb Z_{>1}$ and $m_k(
1+x_1+x_2+x_3),k\in\mathbb Z_{>1}$ are evaluated in closed form via
Goncharov-Deligne periods, namely $\mathbb Q$-linear combinations of multiple
polylogarithms at cyclotomic points (complex-valued coordinates that are roots
of unity). Some infinite series related to these hyper-Mahler measures are also
explicitly represented as Goncharov-Deligne periods of levels $1$, $2$, $ 3$,
$4$, $6$, $8$, $10$ and $12$.
