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Precision Strobe Tuners

Guitar Tuning By Harmonics

Jim Campbell

The usual method of tuning a guitar involves tuning each high string by matching the pitch played at the fifth or forth fret of the next lower string. Another method uses harmonics produced between the fifth fret of the lower string and the seventh fret of the next higher. I have seen this method used by tuning the harmonics until the beats disappear, but I found when I tried to do this, the tuning never came out quite right. It turns out, that as a consequence of the equal tempered scale , these harmonics must be mistuned very slightly for this to work out. I have found that I can get a suitable tuning by this method, but that the beat rates required are not always the same.

The need to tune the harmonics produced by each string slightly flat is a result of the difference between the natural harmonic series and the equal tempered scale. Consider the frequency of the first three natural harmonics of each of the guitar strings:

Harmonic: Fund 2nd 3rd 4th
Fret: 12th fret 7th fret 5th fret
E 82.407 164.814 247.221 329.628
A 110.000 220.000 330.000 440.000
D 146.832 293.665 440.497 587.330
G 195.998 391.995 587.993 783.991
B 246.942 493.883 740.825 987.767
E 329.628 659.255 988.883 1318.510

Notice that the difference between the 3rd harmonic of the A string and the 4th harmonic of the low E string is .372 Hz. This means you can tune these strings open, without fretting, by listening to and adjusting the beat rate of .372 Hz, or about one beat every 3 seconds. To tune the whole guitar, start by ringing the 3rd harmonic on the A string (7 th fret) and the 4th harmonic of the low E string (5th fret), and adjusting the A string sharp to a beat rate of .372 beats per second ( 2.7 seconds per beat). Next, ring the 3rd harmonic on the D string (7th fret) and the 4th harmonic on the A string, adjusting the D string sharp to a beat rate of .497 beats per second. Ring the 3rd harmonic on the G string (7th fret) and the 4th harmonic of the D string and adjust the G string sharp to a beat rate of .664 beats per second. The high and low E strings can be tuned directly by ringing the 4th harmonic of the low E string (5th fret) and the open high E string and tuning until the beats stop. Then ring the 3rd harmonic of the high E string (7th fret) and the 4th harmonic of the B string and adjust the B string flat to a beat rate of 1.116 beats per second. The guitar should now sound reasonably well in tune, but on close inspection, it may not actually be so. This is because beat rates used in the above procedure were based on ideal harmonics, the partials actually produced by the string are somewhat different.

A precision strobe tuner can be used to measure how much the partials produced vary from the harmonic series. This is done by ringing each harmonic separately while adjusting the calibration control to stop the pattern rotation. Then the error is read from the calibration setting. These measurements must be setup and made very carefully, since many things can minutely change the fundamental tuning during the measurements. The neck should be supported at a point around the 12th fret and the temperature must be stable. A furnace turning on can vary the string pitch by several cents or more! The following table shows the results of such a set of measurements, along with the resultant frequency of each harmonic, for a new set of Ernie Ball Super Slinky strings on a Gibson Les Paul Special:

Harmonic: 1 2 3 4 5 6
Fret: Open 12th 7th 5th 9th 3 1/2
E Cents 0 -1.4 0.4 -0.2 1 2.6
f 82.407 164.681 247.278 329.589 412.273 495.184
A Cents 0 -0.2 1.4 0.8 2.6 2.2
f 110.000 219.975 330.267 440.203 550.827 660.839
D Cents 0 1.6 1.8 2.4 3 3.4
f 146.832 293.936 440.955 588.144 735.435 882.726
G Cents 0 1 1.6 2.2 3 2.4
f 195.998 392.222 588.537 784.988 981.688 1177.618
B Cents 0 0.2 0.8 1 0.4 0.4
f 246.942 493.940 741.167 988.337 1234.994 1481.992
E Cents 0 0 0.2 0 -0.2 -0.4
f 329.628 659.255 988.997 1318.510 1647.947 1977.308

A careful measurement of the partials produced by a set of guitar strings shows that not only do the partials vary considerably from the harmonic series, but that how much inharmonicity exists depends on the type of string, and how worn and dirty it is. One could use the information from such a set of measurements, and figure the precise beat rate to use for each set of string pair harmonics for a perfect tuning. From the table of actual measurements of guitar strings above, the beat rate required between the 3rd and 4th harmonics of the B and high E string would be .66 beats per second, rather than 1.116 beats per second if the harmonics were ideal. This method would then work until you changed string types, let the strings rust for a while, or stretched the living crap out of them. Worn and dirty strings can produce partials which are off by more than 10 cents from the ideal harmonic series. The best method for finding the correct beat rates to use for each sting is probably to tune the six strings perfectly by some other method and listen for the beat rates actually produced by each string pair harmonic. This information could then be used as a basis of a quick tuning check using string harmonics. But whatever you do, don't try to tune by zero beating the harmonics, it just won't end up sounding right.

Copyright 1997 James A. Campbell
All Rights Reserved