Though the middle A may be at 440 Hz, a grand piano will be increasingly flat in the lower octaves and increasingly sharp in the higher octaves.

All the strings have overtones sharp to their fundamentals. A piano tuner takes this into account when tuning.

A piano tuner told me this is referred to as temper.

Though well tempered scales are another matter.

Is the word temper correctly used here?

Why would the strings have overtones sharp to their fundamentals?

I have actually seen this effect displayed on a strobe tuner as outer rings of overtones rotating clockwise to the stationary fundamental.

Photo -  Bruce T

Whilst cleaning my trombone, I put water through one of my tuning slides, which varies from a small diameter tube to a large diameter tube (I wonder how they did that in the factory? Never mind!). When I put water through the small end of the tube, the water came straight out again the same end, but when I put the water through the larger end, it came out of both ends of the tube (probably due to how fast I was running the tap that stopped it from just coming out through the small end). What stopped the water coming out of the large end when I put water through the small end? I imagine it's something to do with pressures.

And as requested, attached are some pictures of the tuning slide in and out of its place.