A friend of mine came over and picked up my guitar to play. He immediately started tuning in a way I have never seen before. He was using harmonics to tune the guitar within itself. Being that he was more experienced, I wanted to learn this exotic tuning technique. He was using the 5th and 7th fret harmonics to tune adjacent strings. After some practice, I was able to master the tuning method and felt good about using an advanced technique. Later, I found the technique written in several publications, and felt more confident on the validity of the tuning method.
Well, you may find this hard to believe, but using the 5th and 7th fret harmonics to tune adjacent strings is incorrect. The guitar is an equal tempered tuned instrument. Equal tempered tuning was developed at about the time of Bach to allow the relative interval spacing between notes to be identical in any key. With equal tempered tuning, a musician can play in various keys and not have to retune the instrument. Equal tempered tuning is a compromise though. Interval ratios are no longer perfect ratios, but shifted slightly so to give uniform interval spacing. The frets of a guitar are spaced using the mathematical relationships of equal tempered tuning. Tuning the open strings in a non-equal tempered fashion results in conflicts with the fretted notes.
The 5th fret harmonic sets the string vibrating in four equal lengths, which will quadruple the frequency of the note. The 7th fret harmonic sets the string vibrating in three equal lengths, which will triple the frequency of the note. The 6th E string when tuned to concert pitch should be at 82.41 Hertz (Hz). Playing the 5th fret 6th string harmonic will produce a note at 329.64 Hz. The 5th A string should be tuned to 110 Hz. Playing the 7th fret 5th string harmonic will produce a note at 330 Hz. There is a difference of 0.36 Hz. This may not seem like much, but the 5-7 fret harmonic tuning method is generally done one adjacent string after the other, resulting in accumulation of error. The 1st string is tuned to some reference. The 6th string is tuned to the 1st. The 5th string is tuned to the 6th. The 4th string is tuned to the 5th. The 3rd string is tuned to the 4th. And finally, the 2nd string is tuned to the 1st. Assuming the 6th and 1st strings are tuned properly, the 3rd string will end up being tuned to 195.34 Hz, off 0.66 Hz from the proper 196 Hz. The 2nd string will be tuned to 247.22 Hz, off 0.28 Hz from the proper 246.94 Hz. If you then did a check comparing the 3rd string 4th fret with the 2nd open string, there will be a difference of 1.11 Hz, or about 8 cents. This is definitely a noticeable difference. No matter how careful you are, things will not match up, and you may begin to think your guitar just will not tune right.
The lesson to be learned is not to use the 7th fret harmonic for fine tuning. Its tripling of pitch causes problems with equal tempered tuning. The 5th and 12th fret harmonics are fine to use for fine tuning since these produce octaves. Octaves can always be compared by ear since the ratio of frequencies will be perfect.
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