Diagram H
Chambers at the side of the Inner Gates
Scale 5:1
Copyright © 1988

Verse 38 states that the chambers (lishkah) and their entrances are "by" the posts of the (small) gates where the burnt offering is washed. However, the LXX states that these chambers are "behind" the respective gatehouses. "Behind" and "by" may be the same location.
The northern chamber is set aside for the "singers." But although there is dissent between commentaries, the gist of it all appears to indicate that the residents of these chambers, "keepers of the charge of the house," were those priests whom David had ordered and arranged in the service of music. See I Chronicles 25:1-31. There they are called aneshe melakah, men of the work. "Keepers" in Ezekiel 40:45-46 is shamar, overseers, (as in 44:8, 14).
The LXX omits any notice of a chamber beside the Inner east gate.