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I am using GIMP version 2.10.06. I am trying to blur text in the color 36ffaa (RGB: 54, 255, 170) on a transparent background and for this, I am following this tutorial. When I try to make the large background blur in Step 3, I get a ring-like effect:

example

This may not be very visible on some monitors, so here is the above image with every other ring bucket-filled with pink:

example with every other ring made pink

Now, I can make this hard edge go away if I blur the green bit on the black background.

enter image description here

However, I need my design to be on a transparent background, so this isn't an option. Does anyone have a suggestion as to how to prevent and/or get rid of these hard edges?

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What you have is "banding". This happens when you don't get a transition between areas of similar colors. A tell-tale sign is that the histogram look like a hair comb:

enter image description here

Very often this is due to color loss which is itself induced by round-off errors that occur rather quickly when you work with only 8-bit per channel.

Since you are on Gimp 2.10 you can normally mitigate this by using high precision (Image>Precision and choose 32-bit floating-point, linear light). If you remain on 8-bit you can often attenuate banding using Filters>Noise>Spread.

This said you always have to take tutorials with a grain of salt, what worked for the author may not work for you (banding would be less visible with light colors). And since you are on 2.10, you have to divide all the blur values by roughly 6.

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    Increasing to 32-bit floating-point fixed the banding problem! Thank you so much! Diving the blur values by 6 only gave me less of a halo effect, which isn't what I wanted (though it did lead to less banding). I did actually use this same tutorial on previous pictures, but the dpi was MUCH lower on those. Again, thank you for the advice!
    – GCAT01
    Commented Jul 1, 2019 at 20:48

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