I used to have an old upright, with mildew on the felts, and rust on the strings. I did basic repairs and tuned it myself (I had an ‘A’ tuning fork, and the pegs were pretty stiff so I usually used a small adjustable spanner even though I had the proper tool).
There is nothing magic. You start at the bottom and work upwards. All the strings interact, so tuning one string can cause the others to sound funny. In general deep strings seem to affect the higher ones, rather than the other way around, so you do a quick first pass from bottom to top to get everything in the ballpark, and no obvious ‘wolf’ harmonics. Then you do a second pass to get it closer. And if you are a proper piano tuner (I was not) you are probably close enough.
Tuning double and triple strings is a lark. These are not supposed to be the same pitch. If they are detuned a bit, then you get the clear attack, sustain and decay phases; the decay happening when the strings are no longer locked to a common frequency (Huygens entrainment). If you could tune one perfectly, then you would get no decay, and the thing would hum on. Fortunately, that is pretty rare - you tune the worst of the multiple strings until they are close enough to sound as a single note, and that is usually it.
I got an electric piano because it sounds like my piano might sound for a week or so after being tuned; the bass sounds like an enormous piano I could never afford; and I can play it with headphones on so no-one else has to hear the accelerando in ‘Clair de Lune’ going all to hell. I would still like a ‘real’ piano, but I find it difficult to say exactly why.
