I mistakenly bought a 44-32-22 Deore crankset when I was trying to get a 48-36-26 or similar set for touring (with a 11-34 ish cassette in the back). I was considering just keeping the 44-32-22 set though and putting a 12-26 cassette in the back instead; is there anything wrong in doing so? Does the number of chainring teeth up front usually related to the tooth count in the back, or is it just the ratios and gear inches that matter?

thanks!

You should be able to use a compact crankset with a smaller cassette as long as everything is compatible up front. The high gear end of your original plan would not have been that useful anyway. I've never understood the utility in gears close to 120 inches. Pedaling downhill is just not that big a need. A side benefit is that the weight including a shorter chain will be a good bit less, you have more choices as to rear derailleur and the smaller cassette looks less clunky (I know, not exactly a requirement - but nice).

Try what you have now. On a touring bike low gears are the important ones. Going uphill with the xtra weight of a touring load you will be happier with at least a 16 to 20 inch gear. A 22-34 gets you a 17" low gear.
https://www.jbarrm.com/cgi-bin/c42strt.cgi

True; on that note, would a long-cage derailleur still work alright for a sub-30 tooth count?

Yes. If you felt like saving a couple grams, use a short-cage derailleur.

Yes and no. there is a relationship between front and rear sprocket size but it isn't the sum that counts, it's the ratio, or the front divided by the rear that counts. For example a 52/13 combination is identical to a 48/12 or a 44/11 because the ratio for all three is 4:1.

So in your case, either do the math or get rid of roughly half as many teeth on the larger cassette sprocket as you lose from the chainring.

Yes and no. there is a relationship between front and rear sprocket size but it isn't the sum that counts, it's the ratio, or the front divided by the rear that counts. For example a 52/13 combination is identical to a 48/12 or a 44/11 because the ratio for all three is 4:1.

So in your case, an 11-28t cassette will give you a similar low end, and about a 10% lower high, though 44/11 should be plenty good enough for touring, and probably the only time you'll want a higher gear you'll be coasting anyway.

Learn to compute "Gear Inches". It's easy and useful and pretty universal in bicycle gearing evaluation. No. of teeth on the chainring in question divided
by the no. of teeth on the rear cog in question, multiply by the diameter of the rear wheel, in inches. The result is "Gear Inches." If you want to know
how far you would travel per pedal revolution, multiply by pi.

That's correct information, but for comparing gears on the same bike the extra step of multiplying by the wheel diameter is unnecessary since it's a constant.

Depends on what speed range you think you'll be travelling in. With a 44 big you should be able to hit 30+ MPH before spinning out which is plenty of speed for most situations.
I run a similar crank on my commuter, and the practical consequence of that is minimal. There are a few seconds out of each leg of the journey when I spin out, which isn't much to weep about outside a race setting.

In theory you'd get longer life out of your drivetrain if you'd go with a bigger crank and then a bigger smallest sprocket, but that's not exactly a showstopper either.

You are correct, but in order to be "Gear Inches", which I have heard used quite a lot, this thread included,
"Gear Ratio" must be multiplied by the diameter of the driven wheel. In my experience, I don't recall anyone using gear ratio in final drive comparison.
Biking since '92

The same gear range can be achieved in many ways with different combinations of front and rear ring/cog sizes. The disadvantages of setups that involve smaller rings and cogs are that the cassette and chainrings will wear out slightly faster and are not quite as mechanically efficient (maybe up to 0.5% to 1% efficiency difference between using a 15-tooth and an 11-tooth cog on the rear). However, setups with smaller rings and cogs are slightly lighter, and because the spacing between the rear cogs is smaller, shifting will be slightly smoother.