The key to sourdough is the starter—a living blob of active, natural yeasts. Making your own is surprisingly easy, and maintaining it even simpler. King Arthur Flour has an amazing resource for starting your own starter, complete with photographs, and I recommend referring to it.
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Combine equal amounts whole wheat flour and water in a jar. KAF suggests 113 g whole wheat flour and 113 ml water, and that's what I did. Let it sit at room temperature with a lid on but not airtight for 12 hours.
Discard all but 113 g of the starter, and feed it 113 g all-purpose flour and 113 ml water. Let sit at room temperature for 12 hours.
Repeat step 2 for 7–14 days. Observe the activity, but don't fret if it isn't doubling in that timeframe and seems sluggish, despite all the resources indicating that it will take exactly howevermany days (usually places say 1 week). The ambient temperature and humidity play much larger factors than you would think.
Once the starter doubles in about 6 hours and passes the float test (drop a dollop into a bowl of water--if it floats, it's ready to go!), you can make bread with it and move it into maintenance mode. If it's been two full weeks and it's still not doubling, and you're tired of using up bags of flour at such a rapid rate, you can feed it, let it sit for an hour or so, and then put it in the fridge. Get it out in 2–3 days, let it sit for about an hour, then give it a feeding. This sort of halfway slows the process but keeps it moving; it's not truly maintenance mode yet.
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Every time you go to make bread, you will use some portion of the starter. You'll need the starter to be at peak height—doubled—when you use it, so you'll need to feed it in advance (several hours before). Recipes will specify how much to feed.
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Once the starter is healthy and doubles in 5–6 hours, it's ready for maintenance mode. After a feeding, let it sit at room temperature for about 1 hour, then put it in the fridge. You can ignore it until you're ready to make bread next, though often people will not wait more than one week. Truth is, a starter can sit in the fridge for a quite long time and be easily revived.
If you are not making bread or otherwise using the starter, and it's been about a week, it's time to feed. You can probably let it go longer than a week at a time, but a week is generally a good amount of time. Take the starter out of the fridge and let sit at room temperature for about 1 hour. (If you are making bread, see below.)
Discard all but about 100 g of starter, then feed it 100 g all-purpose flour and 100 ml water. You could even maintain just about 50 g (or even less!) but I'm most comfortable having a nice amount in reserve, so I do 100/100/100. (King Arthur Flour recommends 113/113/113.) Let sit for about 1 hour before returning to the fridge.
A note about discarding: If I am feeding my starter just to put back in the fridge, I will discard and then feed. But if I am making bread that day, I rarely discard any, especially if I'm making a large batch and the amount of starter I'll need is well over the amount I've been storing. I just put the entire amount of starter into a bowl, and feed it an equal amount of water and flour to reach the total needed. (But note--and this will read like nonsense until you do it--I do not feed it the same amount of flour and water as the amount of starter. I calculate the difference between the amount of starter i need and the amount I have, then divide the difference by two. So I feed it equal amounts of flour and water, but all 3 values are not equal.)