Here's a purely 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 intensity = (cos(a)+1)/2 where a is an angular offset in the spectrum.
 - The rainbow is an octave of light, from 400THz-800Thz, or 750nm-375nm.
 - We can see 12 rainbow colours, like the 12 musical notes.
 - The colours red, green and blue correspond to tonic, major third, dominant.
 - The notes have relative frequencies of approx pow(2, i/12), for i in 0..11
 - The 12 notes are almost equally spaced, and correspond to 12 resonances.
 - The 12 resonances are simple fractions involving the primes 2, 3 and 5.

In order of complexity:
 
 2  3  5  name                freq    order  colour

 0  0  0  tonic               1        0     red
-2  1  0  dominant            3/4     -5     blue
 2 -1  0  sub-dominant        4/3      5    
-2  0  1  mediant             5/4      4     green
 2  0 -1  mnr.submediant      4/5     -4
-4  1  1  leading             15/16   -1
 4 -1 -1  upper leading       16/15    1
 1  1 -1  mnr.mediant         6/5      3     yellow
-1 -1  1  submediant          5/6     -3
-3  2  0  supertonic          9/8      2
 2 -3  0  subtonic            8/9     -2
-5  2  1  dominant lead       45/32    6
 5 -2 -1  subdom. upper lead  32/45   -6

In pitch order: 

 2  3  5  name                freq    order  colour

 5 -2 -1  subdom. upper lead  32/45   -6     light blue
-2  1  0  dominant            3/4     -5     blue
 2  0 -1  mnr.submediant      4/5     -4     indigo
-1 -1  1  submediant          5/6     -3     violet
 2 -3  0  subtonic            8/9     -2     magenta
-4  1  1  leading             15/16   -1     hot pink
 0  0  0  tonic               1        0     red
 4 -1 -1  upper leading       16/15    1     orange
-3  2  0  supertonic          9/8      2     yellow
 1  1 -1  mnr.mediant         6/5      3     grass
-2  0  1  mediant             5/4      4     aqua
 2 -1  0  sub-dominant        4/3      5     cyan
-5  2  1  dominant lead       45/32    6     light blue

Note that the "sumdom. upper lead" and "dominant lead" notes are most distant
from the tonic, and very similar; they differ by less than 1/5 of a semitone,
They lead strongly to modulations: the dominant major and sub-tonic minor.
They are commonly regarded as a single note.  If using a single frequency, its
value should be the intermediate, sqrt(2), from which the harmonies differ by
under 1/10 of a semitone.  This is the only harmonic "compromise" in the
chromatic 12-tone scale.

 - 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