R=2, p=1.

TMI Department: I classify these as 1D coloring because a single real value is the hand-off from the fractal calculation to the coloring method. The coloring method is not specified, it is something that turns a real number into a color. It is assumed that the coloring method is continuous. It could be a direct calculation of RGB values from the input. If it is a list of colors, there should be interpolation between the colors so that the output color varies continuously with the input. Anyway, let's call the coloring method a palette, and the input an index and hide whatever goes on inside.

Suppose you had the usual iteration count value for color index and compare that to the same image but multiply the index by 0.5 before sending it to the palette. The resulting image would have half the color range, the color bands in the same place, but smaller color difference between bands.

A good color method lets you tweak the color entropy. You can scale the input, apply square root or log functions, etc. Now it becomes fuzzy where exactly the hand-off between fractal calculation and color algorithm. You could undo the compression on the fractal calculation side in the previous paragraph with a re-scaling on the coloring side.

Full disclosure fine print: Notice that that for all z, \( 1-|z|/R_1 < 1-|z|/R_2 \) when \( 0 < R_1 < R_2 \). So reducing R makes the summation, our color index, everywhere smaller. I want to show the same image with different parameters, but every time I change a parameter, the overall color range shifts. I want to remove the effects of the color range shift for comparisons. So, for each of these images, there are also some undocumented parameter tweaks in the palette. I added a simple multiplier to the output of the color index calculation. Each image is computed in two passes, the first pass computes the average value of the escaping pixels, then the reciprocal of that average becomes the multiplier. Thus after the second pass the average color index for escaping pixels, is 1.

OK, that's out of the way. Reducing R to a reasonable number removes most of the color banding present in the iteration count coloring.