I have softened water

Water softening removes magnesium and calcium from your water. All microorganisms, including sourdough cultures, need some magnesium and calcium to grow, however, they may get enough from your softened water and flour, especially if you use whole wheat flour.  If you have trouble, you can substitute mineral water for your tap water, but I doubt it will make a difference.

There is a lot of hocus-pocus in the literature about sourdough.  It was once thought that urban air pollution killed off the wild yeast in the air and that one had to be in a rural environment to make sourdough.   Turns out the wild yeasts are in the flour itself, so the air isn't so much of a factor.

 I have found that the critical factor in making sourdough is to have some acidity in the dough and a lot of patience. I imagine the sourdough experts would agree, but they may comment as well.  Two ways to get some acidity are to use orange juice instead of water when starting the starter and adding a small amount (a few grains) of commercial yeast.  Commercial yeast grows in the starter, generating some acid, which then increases the growth rate of wild yeasts.  

 Hope this helps,

I had some trouble getting my first sourdough culture started, and it turned out that the water I was using, which is from a well and then charcoal filtered, has a fairly high pH, around 8.5 to 9. I had a hard time believing it, but it definitely was the case. I did some testing with a pH meter and found my water cooler machine water (Poland Spring) was totally neutral. It may also have some of the minerals the other post mentioned. Anyway, the trouble I had was that there was some other micro-organism (Peter Reinhart says maybe leuconostoc bacteria associated with spoilage biomechanisms?) that was taking over the culture in the first few hours, creating a very bitter and bad smelling highly bubbly mess, after which it would totally die. Then it would take almost a week of 12 hour refreshments for it to start up for real after which the BBA schedule would work as stated. I found some sites that talked about putting some kind of acid, whether orange juice, pineapple juice, wine, ascorbic acid, lemon juice, etc., into the initial culture. After some investigating with a pH meter, I discovered that I could get a culture to start very reliably according to the BBA (Reinhart) method if I just put any type of acid in the initial water to bring that water to a pH of about 4 or a little lower before adding the flour to the culture. This would make the environment pH after adding flour about 5.5 or 6. Then it would drop down fairly quickly over the next few days. I think the problem was that the offending organism could survive and dominate for a few days in the higher pH environment, but it was knocked out or never established itself in the lower pH environment. In the end, if I want to start a culture, I use Poland Springs, not my alkaline filtered well water. I also drop in a little bit of ascorbic acid - maybe 250mg in 6 oz of initial culture water to start the pH off a little bit on the acidic side to give the whole culture a head start in the acid direction. I think there is an argument for doing this same thing when you refresh a culture, particularly if you are using a very large ratio of new water and flour to old culture. For example, some instructions for cultures you can buy on the internet suggest that you toss in a teaspoon or two of vinegar during a refreshment if the culture is not doing well. However, I haven't had any problem with 5x1 refreshments using my Poland Spring water. I haven't tried using my well water since the ill-fated initial attempts. During the period when the organisms are multiplying but before they've really taken hold, the pH will be back up toward neutral for a while, and other organisms may be able to take hold. Once the culture grows, the pH drops way down to around 3, and other things won't grow, which is why my great grandmother was able to keep a culture going in the kitchen and feed the ranch hands sourdough pancakes every morning. No refrigeration needed, which is a neat thing. However, I keep mine in the refrigerator for as much as a couple of months and still can refresh it easily with a couple or three rounds of feeding.

 I agree with the poster who says that there is a lot of hokus pokus and all kinds of misinformation. The above is not based on a very careful scientific study, but I can tell you that I consistently had failures with my well water, while I had not one failure in getting a culture started using Poland Spring water and a little acid - didn't really matter what kind of acid as far as I could tell as long as the pH of the water going into the initial culture was down between 3 and 4.