  For a positive integer $d$, a non-negative integer $n$ and a non-negative
integer $h\leq n$, we study the number $C_{n}^{(d)}$ of principal ideals; and
the number $C_{n,h}^{(d)}$ of principal ideals generated by an element of rank
$h$, in the $d$-tonal partition monoid on $n$ elements. We compute closed forms
for the first family, as partial cumulative sums of known sequences. The second
gives an infinite family of new integral sequences. We discuss their
connections to certain integral lattices as well as to combinatorics of
partitions.
