[BrianHV]

This doesn't match my experience. When I tune by ear, I spend a lot of time compromising to get even a few basic chords to all sound good. But since we're talking about the math, I thought I'd try to put some numbers behind my experience.

  Python 2.5 (r25:51918, Sep 19 2006, 08:49:13)
  [GCC 4.0.1 (Apple Computer, Inc. build 5341)] on darwin
  Type "help", "copyright", "credits" or "license" for more information.
  >>> A = 440
  >>> def alterBySemitones(basenote, semitones):
  ...     return basenote * (2 ** (semitones/12.0))
  ...
  >>> equal_B = alterBySemitones(A, 2)
  >>> equal_G = alterBySemitones(A, -2)
  >>> equal_E = alterBySemitones(A, -5)
  >>> major_third_ratio = 5.0/4
  >>> perfect_fifth_ratio = 3.0/2
  >>> equal_G * major_third_ratio     # B as third of G major chord
  489.9942949771866
  >>> equal_E * perfect_fifth_ratio   # B as fifth of E chord
  494.44133536930485
  >>> equal_B                         # Equal tempered B
  493.88330125612413

There's about a 4.5 Hz difference between just-tuned Bs in those two chords. For reference, the distance between that B and the Bb below it is about 27.72Hz. If your fret spacing is introducing that much error, it's going to be tough to tune the instrument for more than maybe one chord at a time.

[baddox]

It's not intuitive to me how big of a difference 4.5 Hz is, but you really need to use a logarithmic scale (because, for instance, shifting everything up an octave will yield different errors in Hz). In cents, that's about a 16 cent difference, while an equal tempered semitone is 100 cents. Is 16 cents easily noticeable? I don't have a nice tuner handy, so I can't say.

Still, your claim doesn't seem right to me. A perfect tuned guitar with perfect intonation in equal temperament will play the same frequencies as a perfectly tuned piano. Yet, when I play G and E chords on a piano, I don't notice the same tuning issues as I often do on guitar. That's why I assumed the bigger issue on guitars is intonation.

[BrianHV]

Yes, I should have gone logarithmic. But I spent enough time playing with that. Had to get back to work. ;) Thanks for figuring out that it's 16 cents. And yes, 16 cents is very noticeable. I might even say dramatic.

The thing is, tunesmith wasn't talking about a perfectly tuned equal temperament guitar. We're talking about tuning a guitar by ear so that one chord is sounds perfectly in tune (i.e., is in just temperament), then trying to play a different chord. It's going to sound off for the same reasons a just-tempered keyboard would. And as someone who constantly has to resist the urge to tune his B string too high in G major, I can tell you this isn't just a theoretical assertion.

That said, I have played on guitars (especially electric ones) that seem to resist sounding in tune even when the open strings are tuned "perfectly." Maybe that's a fret spacing defect in action. But it doesn't make the tuning-by-ear error negligible.

[baddox]

> The thing is, tunesmith wasn't talking about a perfectly tuned equal temperament guitar. We're talking about tuning a guitar by ear so that one chord is sounds perfectly in tune (i.e., is in just temperament), then trying to play a different chord.

When I reread your comment, I realized that this is what you were talking about. In that case, you're absolutely right. Although it is a pretty bad idea to tune a guitar by ear by playing a single chord. Not only will other chords sound out of tune, but even slightly different voicings of the same chord. A guitar that's designed for equal temperament really needs to be tuned as such. Correct me if I'm wrong, but I think the standard ear tuning technique (where you match the 5th or 4th fret of one string with the open string under it) will give much better equal temperament results.

[ArkyBeagle]

16 cents is a lot. It's 1/6th of a semitone.