Here's a mathematical rainbow spectrum plot.

Reasoning:

 - All colours in the rainbow spectrum should be shown with equal intensity.
 - In RGB, yellow(1,1,0) has greater intensity than red(1,0,0) or green(0,1,0).
 - So, we need a less intense yellow for our rainbow.  I used cosines.
 - The rainbow is an octave of light, from 400THz-800Thz, or 750nm-375nm.
 - The refractive index varies with the wavelength of light.
 - The rainbow is a uniform plot over wavelength (not freq. or "note").
 - Several "rainbow echoes" are seen, due to resonance with higher octaves.
 - Each "rainbow echo" is half the width of the previous, as expected.
 - And so, the colours in the visible rainbow are compressed at the blue end.

Here is a picture of rainbow echoes, these are due I suppose to harmonic resonance:

![rainbow with echoes][1]

Combining these ideas, here is a calculated rainbow.

![enter image description here][2]


  [1]: http://i.stack.imgur.com/P4Cwa.jpg
  [2]: http://i.stack.imgur.com/7Mbk6.png