Distilled Water or Tap Water

Nope. I use tap water when I make soap too, and never had any issues. If your water has super high iron I could mayyybe see it giving a bit of trouble but I wouldn't worry about distilled at all.

Well you are going to grind rocks into a powder which is rather dirty anyway , so how could the water make any difference ?

I can say iron oxide will not stain tumble polished rocks since my vibe barrel is plain steel and is often real rusty.
Iron oxide stains remain for a 1000 years on petroglyphs.
However rocks suited for tumbling are not noted for absorbing chemicals.

Surface stains may be another subject.
def:
"Hard water, water that contains salts of calcium and magnesium principally as bicarbonates, chlorides, and sulfates. Ferrous iron may also be present; oxidized to the ferric form, it appears as a reddish brown stain on washed fabrics and enameled surfaces."
Dishwater detergent is designed to remove hard water stains on hard surfaces, perfect for tumbles. Try it sometime.

ph might be a bigger concern. Even bottled water comes in a broad range of ph. Who would dare drink ph 4.5 water ? Or 9.5. Maybe there is a good reason. 4.5 seems like a serious case of the runs.



It would be interesting to have chemist do an analysis on various rock slurries.
Mercury lead arsenic antimony asbestos thallium tellurium copper don't just fly down from heaven.

For pH, consider that most fruit is pretty acidic. Citrus is usually in the 2-3 range.

Bottled water is poorly buffered, so the pH after you drink it will be near 2 (the pH of your stomach).

As for heavy metals not raining down from heaven, I believe 1dave has a different take on that.

And dishwasher detergent... that is pretty harsh stuff, and I’ve had it change the color of rocks (some Gary Green went all brown, but I suspect it doesn’t go very deep).

EVERY ATOM on Earth came down from the heavens. I just learned that all of our uranium came from a 6.6 billion year old supernova - but it didn't all get here at the same time.

While there is truth there, there is also more at play than just pH.  pH only gives one piece of information: the negative exponent of the concentration of hydrogen ion (pH stands for "power of hydrogen"). Well, technically two.  It also allows you to calculate the concentration of hydroxide ion.   At pH7, both are present at 10^-7M, since the product of the two concentrations equals 10^-14. At pH2,, the concentration of hydrogen ion = 10^-2; the concentration of hydroxide = 10^-12.  This means that you have far more hydrogen ion (acid) than hydroxide (base), thus a very acidic solution.  What is far more important than just that concentration, is the presence of buffers, which act as a sponges for both acid and base (usually one more than the other), and thus make the solution resistant to change.  Strong acids (like hydrochloric) release all of their hydrogen ion at once.  If they are diluted 10-fold with water, the pH changes by 1 point; if 100-fold, by 2 points...).  Strong bases, like sodium hydroxide act similarly on the basic side.  Weak acids and bases are so-called because they do not release all of the hydrogen or hydroxide at once.  They have an equilibrium that they "try" to maintain. This means they will start off at a pH that is nearer neutral (see sodas, lemons, vinegar, orange juice on the chart provided by james), but dilution will not affect them as much.  

If equal volumes of equal strength strong acid and strong base are mixed, they will neutralize each other (ph7).  If there is 10 times as much acid, you will have pH6; if there is 10 times as much base, pH8.

Mixing weak acid and strong base will somewhat favor the pH near that weak acid, as the acid will "try" to maintain its favored pH (unless the base is added in overwhelming amounts).  Additionally many buffers have more than one acidic/basic group, and be even more resistant to change.  This is how biological systems protect themselves from damage by acid/base.

Bottled water, which is distilled, with certain minerals added for taste is not buffered, and thus would not be terribly effective at "alkalinizing " the body, unless consumed in enormous quantities.  Well water, which is likely hard water (which is high in calcium carbonate, which is one of the things that often gives it an objectionable taste, along with other minerals which may or may not be good for you) would be more resistant to pH change.  

Dishwashing detergent does help remove the spots by a variety of methods.  One is by changing the pH to one where bicarbonate [HCO₃^-] is more prevalent than carbonate [CO₃^2-], which makes it less likely to bind to calcium ion [Ca²⁺], and precipitate (crystalize).