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The French very early on had quad main batteries, when most nations were still working with twins
1x 12" MkI, 373 tons, .4 rounds/min, 2.6% accuracy at 5k
2x 12" MkI, 575 tons, .27 rounds/min (.52 per turret), 2.3% accuracy at 5k
So, in 10 minutes you had:
1x about a 10.0% chance of getting at least one hit at 5k
2x about a 11.0% chance of getting at least one hit at 5k
Now at the other end, in 1940:
1x 12" MkV, 382 tons, 1.92 rounds/min, 2.6% accuracy at 15k
2x 12" MkV, 592 tons, 1.42 rounds/min (2.84 per turret), 2.6% accuracy at 15k
So, in 10 minutes you had:
1x about a 39.4% chance of getting at least one hit at 15k
2x about a 52.2% chance of getting at least one hit at 15k
So, in 1890, probably not a whole lot of difference. In 1940, 2x seems a lot more effective, especially when you consider that often weight isn't the limiting factor but deck space and/or limits on the number of turrets: you can't just choose between 4 2x turrets or 8 singles or even, in a lot of cases, 3 and 6.
The other aspect to consider is gun technology. In 1940 custom battles, I prefer quad 15" MkIV to triple 16" MkIII which are close to the same weight.
4x 15" MkIV with Auto II, Triple Base, TNT IV, Capped II, and Superheavies is 2785 tons, 0.94 rounds/min (3.76/min for the turret), 15k accuracy of 2.5% with 48" of penetration and 39.9 km max range
3x 16" MkIII with that above techs is 2577 tons, 0.74 rounds/min (2.22 rounds/min for the turret), 15k accuracy of 1.4% with 50.5" of penetration and 34.3 km max range.
I think that's a clear win for the more advanced 15" quads. 70% more rounds every 10 minutes, which are more accurate. Sure, the 16" shells do more damage by 36.8% - not enough to make up for the increased number of rounds downrange even if the 16" was equally accurate.
Just my empirical experience. I'm not that good at math.
for me: 10~12" guns are best for pre dreadnought era BB.
11~12" works great for early dreadnought era.
If the enemy ships get too THICK, move on to 13.8~15" guns.
15~16" is the best caliber for modern BBs.
They have excellent pen, destructive power, and reload much faster then Larger guns.
Kind of actually follows the historical calibers.
Ah OK, for my experience it was a bit overkill. I prefer smaller guns in early ages. So in pre-dreadnought era I usually stick to 10, or at most 11 for BB, and 9 for CA, 4 for CL. And fill it with a bunch of smaller 2-3 inch guns, as much as your deck/casement can hold. This goes until Krupp II comes into play and I will have to improve the penetration.
Sadly this game doesn't allow us to design different shell type for secondary guns, like use heavy shell for main gun and light one for secondary.
My brain understands that is efficient. But my heart keeps saying
"Buuuut 10" pew pews on a BB is not sexy...."
I generally just stick to historical preferences and try to make them work. So, e.g. French and alt-history post-WWI Russian ships (before the new variable flag was introduced, at any rate) get quad 16-inchers.