Stephen Kenyon wrote: ↑
Sat Dec 22, 2018 5:22 pm
The G sharp harmonic will never be perfectly in tune because it is a natural overtone, not an equally tempered (e.g. as per the frets).
Hi Stephen, thanks for your advice, I appreciate it! Yes, I understand the logarithmic scale very well and can calculate the frequency of the beats expected between the natural and tempered G sharp. So yes, it is expected, but in reality thins are more complicated. For example, the distance between the 12th fret and the saddle is longer than a half of the open string, because the 12th fret is played with a higher tension due to the action. However, the action measured at the 4th fret would be 2.5 times lower than at the 12th, if the bridge weren't higher than a fret. So there are more parameters to consider than just tempered vs. natural.
This discovery yesterday took me by surprise, as I didn't expect these notes to match exactly. I simply wanted to test the "432 Hz conspiracy theory" and tuned the guitar to 432, 440, and 448 Hz to compare. The 432 and 448 versions sounded substantially and equally worse than 440 on my guitar, but an unexpected side effect was that the guitar was perfectly in tune at 432, less so at 440, and proportionally worse at 448. Clearly this has to do with the dependence of the open string length on the tension. The string tension can be reduced by (1) tuning down to 432, but it jus sounds worse for some reason (resonances I guess), (2) using different strings, or (3) very slightly lowering the action (not a good option, based on your feedback).
BTW, the effect wasn't limited to G sharp. Other chords also sounded super crisp. The guitar just sounded as if it wasn't tempered, quite shocking, but in a very good way
All I need now is to get the same result at 440 Hz...