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This is largely not true for most songs. From my experience Bard DOT's will do the same amount of damage whether you have a lute or a gypsy lute equipped. Why is this?
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I think this is due to rounding error. At least right now (at level 26) Chords of Dissonance does 8 damage base, 16 damage with a lute, and 16 with a lute of the Gypsy Princess. I think the formula will automatically round down the damage, so with the LoGP I am really doing 8 * 2.1 = 16.8 damage, but the server just rounds off the fractional part.
I will post again when CoD starts doing 10 damage base (should happen sometime in the mid 30's) and report if it does indeed do 21 damage with the LoGP.