bitflipper
That pdf was a good read, Drew. Thanks. 45 years later I now know why my first piano-tuning experiment was doomed to fail. 
 
One thing I still don't understand, though: "This exponentially organized structure of octave-repeating notes is in immediate conflict with the linear spectrum of the harmonics."
 
Inharmonic overtones, I can understand. But how can the harmonics of one note be in conflict with the harmonics of its octave? 

See figure 3. The black notes on a piano are significantly off using a just intoned C tuning. The red (a) is linear multiples of a C tuned to 11Hz - i.e.the harmonics.

michael diemer
Bitflipper, my understanding is that the human ear is the problem. It hears really high notes as flat, and really low notes as sharp. I don't do any stretching until I get up to the highest and lowest octaves.

 
Stretch tuning is used because since a string is not perfectly flexible, it's higher harmonics become increasingly sharp. This is why on a guitar wound strings are used for the low strings - it allows one to add mass to the string while still having good flexibility. More mass allows for the string to be tuned to a low pitch with tension similar to the higher strings. For a vibrating string, lower tension/longer length/more mass = lower pitch and vice versa for higher pitch. A piano uses rather long strings under high tension, so the strings can't be super skinny to be tuned to pitch.
 
So on an instrument like piano - with a very wide range and lots of high harmonics - the higher octave harmonics of a lower string go out of tune with the fundamentals of the same notes a few octaves higher.

Makes sense. Maybe it's one of the reasons synthesizers sound bland compared to acoustical sources: the harmonics are too neatly aligned. But the human-hearing hypothesis rings true, too. There's no doubt we have increasing difficulty judging pitch the further we get from the middle of the keyboard, which is probably why I prefer a midrangey headphone cue mix for vocal tracking.

All I know is C3 PO comes before R2 D2

As you were