Yes, I know about plucking wheel spokes to check tension by musical pitch:
https://www.bikeforums.net/bicycle-mechanics/356361-rear-gear-upshifting-hops.html

And this is partially related to this thread:
https://www.bikeforums.net/showthread.php?t=356361

I was feeling that my Shimano-equipped road bike's shifting wasn't quite right, so I ignorantly started fiddling with the inline barrel adjusters.

Okay... "fiddling" is too kind of a term. I was spinning those suckers who-knows-how-much to see if I could get the derailleurs to move.

So, rather than taking the bike to the shop and pleading, "I broke my bike, can you make it work again?", I decided to finally figure it out myself. In lieu of a workstand, I hung it by the saddle off of my doorway pullup bar and start spinning the cranks & shifting the levers.

The derailleurs seemed to be aligned, and their travel stops seemed correct. That was easy.

However, I just couldn't get the danged chain to shift reliably from one cog to another. No matter which way I turned the RD barrel adjuster, the chain invented new ways of misshifting.

For some reason, I started looking at the cable, particularly the section that runs along the downtube.

While on the small cog, it was slack. I could tug it at least a whole inch. I figured that, even on the smallest cog, the cable should have at least some tension so that the RD would start shifting right away when the lever is moved.

I think it was partly because of where the bike was hanging -- the door frame & pullup bar probably resonate well enough -- but after finally dialing most of the slack out of the cable with the barrel adjuster (I really, really spun it before!), I could hear the cable make a pitch, about the range of an upper bass guitar string.

*Pluck pluck pluck* I went, raising the cable's pitch by a step or two. BINGO -- the shifter actually began working correctly!

It wasn't yet perfect, but man, it was better than before. Rather than just trying to count turns in the barrel adjuster, I started turning it while plucking the cable, making sure that I actually turned it just enough to change the pitch a bit, thereby changing the tension while shifted onto the smallest cog. It was like I was tuning the cable. Change the pitch another quarter-step or less, flick up & down through the gears, change the pitch again, test again.

Awesome. I un-broke my RD.

The FD's barrel adjuster had also been subjected to my ineptitude, so I started on the other side. I didn't even try test-shifting at first -- I just set it to the small chainring, then started turning the barrel adjuster while plucking its cable along the downtube.

After turning the barrel for a while, I started getting tension again, and the cable started making an audible pitch. Same thing as for the RD -- pluck, check the pitch, shift a few times, turn the barrel, pluck, etc.

Awesome. I un-broke my FD.

Just to be sure, I took it for a few laps around the block. Each click of the right lever once again changed one cog; three clicks, three cogs -- seven clicks, seven cogs. The front shifter was working well again, too. I could even use the mini-click to trim it and avoid rubbing the chain while in the small-small combination.

At last! Success! I can go to bed now... well, after I get the laundry from the dryer...

Good job. Next time though, try working with the HIGH and LOW adjustment screws on each derailleur. Sort of a more direct approach to the same result, then later you could use the barrel adjusters for fine tuning. As long as it works - nothing better than smooth shifting. I always find the Park Tool website helpful when I'm unsure.

Rear DR:
https://www.parktool.com/repair/readhowto.asp?id=64

Front DR:
https://www.parktool.com/repair/readhowto.asp?id=75

Oh, the high & low screw stops were fine -- I couldn't shift off the gears no matter what. I could tug on the derailleur cables and the chain wouldn't fly off. In fact, the rear's High screw had been saving me on the road; since there was so much slack in the cable, if the screw hadn't been spot-on, the chain would've fallen off the small cog.

It was all the shifts in between that I had screwed up. Trust me when I say that I really messed with the barrel adjusters. I didn't know that they only needed small turns, so I practically unscrewed them completely -- I might've had them twenty or so full turns in the wrong direction.

While plucking the cables, I could tell when I was adding just enough tension to "match" the levers to the derailleurs. For example, with a slackened cable, the right shifter had to move a bit just to take up the slack and move the RD, which threw off the indexing for further shifts. But, with the slack already taken out, the shift lever's movement went right to moving the RD, and the index clicks finally matched up with the rear cogs.

Frankly, I'm just relieved that I didn't break a chain before figuring this out...

nice work - you got it exactly right. There should never be a need to touch the H&L limit screws unless you change some other part of the drivetrain out. As long as all you touch is the cables the tension adjustment is all you need.

This is a great example of what a few people here have said -- bike repair is not rocket science and if you look (and listen!) and pay attention to what is going on you CAN figure it out.

As far as breaking the chain goes, unless you put your weight on it thats not very likely to happen, no matter how bad you goof it up

Shift down to the small gear, set the bottom(small gear) limit screw, undo the cable binder bolt, Pull the cable taught, and tighten. Shift to the top of the large gears, and set the limit screw so the chain doesn't fall into the spokes. The musical note the cable produces, really has nothing to do with it? as long as the shifter is all the way forward, and the RD on the small gear when you pull the cable tight, you can't go wrong. on indexed, it's all a matter of finding the right spot for the derailleur to sit while in a given gear. That's what the adjuster is for.,,,,BD

Right; after all, screws don't stretch.

I had my doubts about it, because this is my first road bike, and my first bike with brifters. I had gotten a handle on the Suntour friction shifters on my old Huffy ATB from almost twenty-five years ago (my first multi-speed), and was also able to figure out the shifters on my later bikes.

That's what I was worried about -- I had been riding on it that way. I was careful to not pedal hard if I heard any noises, but sometimes I wondered if wind or traffic noise was blocking the chain noises...