Angle of the saddle or bridge.

[Son of Magni]

Ok, finally the answer is ours!

Using the R450 by plucking the string with varying strength and by plucking the string in different places along its length I can get any outcome I want

Of course the Peterson isn't just picking up the fundamental but all the harmonics as well, plus (as anyone who's used a strobe tuner knows) how hard you pluck the note makes a huge difference. And this according to my R450 who's motor makes a funny sound.

My best guess is that the 2nd harmonic is about 5 cents sharp, which would make the inharmonicity error about equal to the stretch error...

I'm so happy this mystery has finally been put to rest!

 

[Jack Clark]

. . . With my restringing job, though, I play the instrument in the fashion of a double bass, and I was hoping to find more straight-across imaginary frets, but apparently that is not to be.

Well, a lot has happened since I wrote the above. I've since realized that the restringing from loop-ended strings to ball-ended strings resulted in a very large variance in the effective scale length of each string with the original guitarrón bridge. I've since modified the bridge with the new strings in mind and my imaginary frets are pretty much straight across now--except for the first imaginary fret (index finger in half position)--which still requires a slight bit of individual adjustment, but I think that's probably pretty normal.

All of that technical string discussion that followed my opening question was fascinating. I'm glad the thread went that way. Thanks for all the comments.

 

[Jack Clark]

I just happened to re-read this thread, and I, too, was struck by the tone of the responding participants. Even apart from the fascinating content supplied by the responders (certainly not by me), this fairly-short thread might be a good sticky as a model of what civil discussion can be and the results it can yield. Thanks, again, guys.

 

[ctmullins]

Agreed. Civil discourse is all too rare these days.

 

[DoubAlbass]

Hey all!
This is my current bridge setup, I am extremely happy with this, it is exactly what you have all been discussing (albeit long ago) and I had the same query.
It is of great importance that we all note, this is a one-off custom aluminum hollow body ÜBAl, and the face is NOT contoured as is normal with DB construction, it varies in only one plane (head to foot).
Thus the bridge "straddles" the arc and the feet of the bridge are able to resonate on semi-isolated but interacting soundboards.
It does (infact) do precisely what was proffered. However, I can easily see how the bridge height and luthier customization would accomplish the same thing in a well found traditional DB bridge placement.
QED; it does have an effect, and it does work... on this.
However, I doubt a decent DB would appreciate or weather the arrangement.

 

[ixlramp]

Son of Magni is correct to say that guitar string stiffness has a significant effect as well as the stretching due to fretting.

Guitar and bass strings are very flexible for their thickness but the stiffness still has a significant effect in raising the pitch, and the higher up the fretboard you are the string effectively becomes stiffer. You can hear the effect of stiffness in the inharmonicity of the tone, especially a B string played at the 24th fret. If stiffness was insignificant there would be zero inharmonicity and the tone would be as clear and harmonic as the open string.

In this very scientifically-accurate video a thin string (only .017p) is clamped at a halfway point without sideways deflection: zero stretching, and still needs intonation offset:

Here's the video which serves as an introduction to the above and explains equal temperament as opposed to Just Intonation (perfect and natural harmony).