There's one thing I don't understand: Books talk about the inharmonicity of the partials and stretching octaves but I wonder: if I could tune all the strings to their theoretical frequency, i.e. C4=261.626, C#4=277.183, D=293.665 Hz etc. with an electronic tuner. Wouldn't my piano be in tune?

I learned a lot about the topic at this site":

http://www.pianofundamentals.com/book/en/2.5.10

My understanding (limited) is that tuning as you describe will result in the fundamentals being in tune, but that there are other important considerations. When the fundamentals of strings an octave (for example) apart are in tune, the harmonics, which will be out of tune by different degrees in the different octaves, will beat, and at a higher (therefore more audible) frequency.

I also believe (perhaps wrongly) that the ear does not perceive perfectly, and we hear higher pitches as if they are flat, lower ones as if they are sharp. Tuning the higher octaves slightly sharp and the lower ones flat compensates for this physiological propensity of the ear.

I'd love to hear more on this topic from someone with more knowledge.

My piano has a few keys out of tune and a couple of them don't play at all.

> Wouldn't my piano be in tune?
In theory, yes. In practice, it will sound very much out of tune.