The reason we avoid E# and B# is to get nice-sounding chords by only using the white keys. This way, the C-E chord has a ratio of 2^(4/12) which is approximately 5/4; the C-F chord has a ratio of 2^(5/12) which is approximately 4/3; and the C-G chord has a ratio of 2^(7/12) which is approximately 3/2.

In fact, before we understood twelfth roots, people used to tune pianos so that the ratios above were exactly 5/4, 4/3, and 3/2. This made different scales sound different. For instance, the C major triad might have notes in the ratios 4:5:6, while a D major triad might have different ratios, close to the above but slightly off.

There's also the question of whether the difference between these makes a difference in the sound. There's two answers to that. On the one hand, it's a standard textbook exercise that the difference between pitches of a note in two different tuning systems is never large enough for the human ear to hear it. So, most of the time, the tuning systems are impossible to distinguish.

On the other hand, there are certain cases in which the human ear can detect very very small differences when a chord is played. To give a simple (though unmusical) example, suppose we played a chord of a 200 Hz note and a 201 Hz note. The human ear, to a first approximation, will hear a single note of approximately 200 Hz. However, the difference between the two notes has a period of 1 second, so what the human ear actually hears is a 200 Hz note whose (EDIT) amplitude wobbles every second. This is very very obvious, it's a first sign of your piano being out of tune, and in different tuning systems it happens to different chords.

Notes sound good if they're approximately simple rational multiples of each other. Hence you want your scale to contain multiples.

Since the simplest multiple is x2 we use that for the octave. As for why we break it up into 12 semitones, the reason is that 2^(7/12) is approximately 3/2 and as a bonus 2^(4/2) is a passable approximation to 5/4.

Here's a tweak I made that I think keeps to the spirit.

import random, time, winsound
timebias = 0.2
pitchbias = 0.7
changebias = 0.75

current = [(4.,1.),(5.,1.),(7.,3.),(None,1.), (4.,1.),(5.,1.),(7.,3.),(None,1.),(4.,1.),(7.,1.),(12.,2.),(11.,2.),(9.,2.),(9.,2.),(7.,1.),(None,1.),(2.,1.),(4.,1.),(5.,3.),(None,1.),(2.,1.),(4.,1.),(5.,3.),(None,1.),(2.,1.),(5.,1.),(11.,1.),(9.,1.),(7.,2.),(11.,2.),(12.,4.)]

timeshift = 1;
while 1:
timeshift = timeshift + timeshift * random.uniform(1 - timebias, 1 + timebias)
if timeshift > 1.0 + 2.0 * timebias or timeshift < 1.0 - 2.0 * timebias:
timeshift = random.uniform(1.0 - timebias / 2.0, 1.0 + timebias / 2.0)
key = random.randrange(0, len(current) - 1)
if random.random() > changebias:
if current[key][0] is not None:
current[key] = (current[key][0] + current[key][0] * random.uniform(-1.0 * pitchbias, pitchbias), current[key][1])
else:
current[key] = (current[key][0], current[key][1] + current[key][1] * random.uniform( -1.0 * timebias, timebias))
time.sleep(random.random())
for (p,d) in current:
if p is None: time.sleep(0.2*d * timeshift)
else: winsound.Beep(int(440*2**(p/12.)), int(200*d*timeshift))

Basically, each loop it tweaks the song slightly from the one before it, randomly. The three different bias settings on the top dictate how the song evolves. But besides just changing the song, the rate of any play varies randomly (according to the timebias as well).

The timebias applies to changes of timing. So the tempo of the play, the rate of change of the length of a note and the length of pauses are all shifted by the timebias randomly. increasing this number will create more dramatic swings in time changes from run to run (as well as the overall bounds of the tempo).

The pitchbias applies to pitch changes. Increasing it will let the algorithm drift from the normal song much faster. Too high will cause obvious swings in notes. Too low, and it'll take forever to get a decently maddening change (but perhaps that's part of the master plan).

The changebias indicates the chance that on a particular loop, the pitch of a random note will change, or if the duration will change. This change is carried on to all future plays (and will have a ripple effect)

The result is quite maddening, as parts of the song will randomly trend back towards the correct notes. And notes you could have sworn were wrong will appear normal later. And back and forth it goes. Just repeating, and changing until you get driven mad (or bored) enough to ^C...

Basically, it's a genetic algorithm without a binding fitness function. Its random changes will just propegate infinitely towards chaos. But for a very long time it will have the "feel" of the original song...

I couldn't help myself. I had to have a go at making it, too.
http://jsfiddle.net/GVTk2/

Didn't check it on anything other than chromium, and I can't guarantee it won't eventually use all your memory and crash.
It's horrible in many ways: switches key, misses the frequency of notes, changes from 2^(1/12) ratio between semitones, pauses at random and changes note length.

Take a listen, there's always a chance it'll stop :D

/edit ambiguity. Come to think of it, skipping notes is the one thing I didn't do. Note that it starts reasonably close to being in tune and slowly degrades.

I can't get this to work in Wine. Could you please put up a recording? Thank you :-)

Just to add: (1) The pointless "1*" is because I experimented with other sizes of error too. (2) A slight modification of this lets you, e.g., have the pitch drift downward by 1/10 of a semitone per note, which for me at least is very noticeable and unpleasant even though each individual interval is OK.