Problem With Texture Sheet Alpha for Particle

I'm using a texture sheet animation for particles and for some reason the alpha doesn't work when the blending is set to additive.

It works fine with alpha blended, but with additive the particles are square and show the edge.

I know alpha is probably ignored on the additive shader, but how do you make a shader that's additive with a texture sheet not look like a bunch of hard-edged squares?

Here's my node tree and a screen shot of the resulting effect. The same node tree with alpha blended works, but I need the particles to be additive and not alpha blended.