4/4 will work well if you want to emphasize strong downbeats, but since you're looking to enhance the sombre mood, try 6/8 time. The triplet beat will give you more notes between downbeats, which can make it sound either sombre or jazzy, depending on the tempo. Here's my suggested bass guitar tab to go with a slower 6/8 time signature:
```
G|--------------------
D|--------------------
A|--3--3--3--2--0--0--
E|--------------------

G|--------------------
D|--------------------
A|--2--2-----3--3-----
E|--------------------

G|--------------------
D|--------------------
A|--2-----------------
E|-----------1--------

G|--------------------
D|--------------------
A|--0--0--0--0--0--0--
E|--------------------
```
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