And thanks to you, too, Michael. Two new things learned today!
 
Long ago I got a tuning wrench that came with an autoharp. Noticing that it also fit my piano's tuning pegs, I decided to try tuning it myself. How hard could it be, right?
 
This was long before digital tuners were available, so I tried to do it entirely by ear. What I ended up with was a horrible mess. In my defense, I was around 19 at the time and still believed I could do anything I set my mind to. What I ultimately set my mind to was paying a piano tuner to save my butt. I was lucky he didn't charge me extra. And to add further insult, he did it all by ear.

I also tried tuning on my own when my mother gave me her little spinet. I bought a cheap tuning kit, and, like you, Bitflipper, I just sank deeper and deeper. but then I went to piano tuning school. It took 4 months of daily work to finally get it. I finished the course one month early. Most people don't realize that most of the notes in a piano are comprised of three strings. The bass have two and the lowest notes just one. so even one note can be out of tune with itself! and some pianos are essentially untunable. They have false beats, so you think you have it when actually you don't. They can really be beasts. Then there's action regulations. Over a thousand separate adjustments. Now you you know why they charge so much.

dmbaer
On the other hand, the Scientific Notation system denotes middle C as C4, which is probably the *least* used choice in virtual instrument/FX documentation.  Most of the time it's C3, but sometimes it's C5, which can be argued is correct since MIDI note 0 is also C0 in that system.

Yes, there is a bunch of other systems. For a brief description refer to this article
http://www.flutopedia.com/octave_notation.htm

Some might find this discussion of the difficulty of properly tuning pianos interesting (warning - technical paper):
 
http://arxiv.org/pdf/1203.5101v2.pdf
 

Interesting paper. I bookmaked it for further study, but an initial comment: I used stretching when tuning the high and low notes, to compensate for the tendency of the ear to perceive high notes as flat and low notes as sharp. So you sharpen the high notes ands flatten the low ones to compensate. You do it until it feels right. And yes, when tuning some pianos you deviate from the "correct error" of a given interval so that it sounds right. This is purely intuitive and varies from piano to piano. I'll check out the article in depth later. Thanks for posting it.

I read the whole article. Really interesting. Wish I could grasp the math! Anyway, it seems that ear-tuners, which is what I am, take into account the musicality of Western keyboard music, which is based on equal temperament, so that they instinctively know how to compensate for the errors caused by the underlying waveform conflicts, so that the result sounds "musical." My recollection is that these errors are caused by imperfections in the materials, in this case piano wire, an extremely hard and durable metal (which nevertheless does break. I've broken strings during tuning, and it's not a pleasant experience. I would cringe when stretching those upper notes). Our instructor, Robert Perkins (don't know if he's still in business, I went to his school in the 1980's) definitely had us trust our ears when in doubt. But some ears are better than others. These are the tuners who end up tuning the concert grands for the virtuosi of the performance world. Mastering this process involves internalizing the rates of beat patterns in the various intervals of the temperament octave. for example, one interval might involve beats going at 3 per 5 seconds, another one per five seconds, and so on. How do you memorize this? Well, one trick not involving electronic devises (to which he was vehemently opposed) is to use a metronome. you set it at a certain speed, and memorize that speed. sometimes you have to count every other tick. Then you match the rate the beats are going to that speed you have memorized. Sound difficult? It is. This is why it took me 4 months to finally get it. It's a Zen thing. It has to become part of you. but it's a great feeling when you know you can tune just about any piano with only your ear. (The other thing is tuning it so that it stays in tune a reasonable amount of time. This is called "setting the pins. Another thing that separates the best from the rest).

michael diemer
 Anyway, it seems that ear-tuners, which is what I am, take into account the musicality of Western keyboard music, which is based on equal temperament, so that they instinctively know how to compensate for the errors caused by the underlying waveform conflicts, so that the result sounds "musical." 

From it I sort of took that you were sort of somehow coming up with an ideal compromise factoring a whole host of tuning issues across the entire range of the instrument. (!!!) And it's much more complex than it might seem.
 
And, needless to say, that skilled piano tuners are due much respect. 

They are indeed. When I went to a piano concert, and the tuner went up at the break to make some adjustments, I could only look on with a sense of awe. I was good enough for most situations, but not that level. 
Another fine art of piano technology, and one I did not learn, is voicing. This involves stabbing the felt of the hammers (the things which actually strike the notes) with a tool that has these needles sticking out of it. This softens up the felt so a nice tone can be produced, in terms of loud/soft and dull/bright. This skill is by no means universal among piano technologists. 

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? 

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. Although there is some fudging even in lower octaves. You're supposed to check the fourths and fifths by playing thirds. As you go up the keyboard, those thirds start to sound nasty, so you're supposed to alter them slightly. I never got into that too much. I was just your basic household tuner. Never made any money at it, but I do have a nice 100 year old Hazleton Baby Grand to show for it, which I reconditioned myself. And am thinking of selling now, as I never play it. Heck, I don't even tune it anymore.