Friday, September 14, 2012

Guitar Tuning - harmonics plus tweaking

kw: music, musical instruments, instruction

I played guitar for forty years before I obtained an electronic tuner. At first, I kept my guitar in tune by ear using the piano; my mother was a fine pianist and our piano was kept in tune. Before going off to college, I got a tuning fork, the A-440 one. I learned to tune the A string by sounding its 4th harmonic, striking and setting the fork, and tuning for zero beats. Then I tuned the rest of the strings using harmonics, which I'll describe shortly.

As my ear got keener, I noticed that tuning with pure harmonics led to certain chords, such as D, not sounding quite right, and I tinkered with the tuning a lot before I learned more about what I was doing. Finally I learned enough about equal temperament and how the harmonics relate to it to get my guitar tuned right, no tinkering needed.

This fingerboard diagram, the first octave cropped from a larger illustration in Wikipedia Commons, shows how the 3d and 4th harmonics relate everywhere but between the G and B strings:
The Nut is on the left, and the bridge would be off to the right a distance equal to the width of this image. The harmonics at the 7th fret (the rightmost set of orange dots) are the 3d harmonics, and those at the 5th fret are the 4th harmonics. You produce a harmonic by very lightly touching a string above the fret, being sure not to depress the string, and plucking the string near the bridge, preferably pretty close. The closer you pluck to the bridge, the greater the amplitude of higher harmonics. The 4th harmonic is strongest when you pluck 1/8th of the way up the string from the bridge, or about 3.2 inches (80 mm or so). This is also a good place to pluck for a strong 3d harmonic, which avoids having to adjust when you are trying to get two strings to sound on different harmonics so you can hear if they are in unison, or nearly so.

"Nearly so" is very meaningful in this context. This table shows why. Note first that the 4th harmonic of the A string is 440 Hz when it has been tuned to the tuning fork. Now, when the D string is in proper tune according to equal temperament, its 3d harmonic is almost half a Hz higher. By the same token, the 4th harmonic of the low E string is almost 0.4 Hz lower than the 3d harmonic of the A string, when the E is properly tuned.

Early on, I would tune to exact harmonics, using the A string to tune the E and D, then the D to tune G, the low E to tune the high E', then the high E' to tune the B. Working out the math, we find that with A at 110 Hz, the two E strings are then 82.5 and 330 Hz, the D is 146.67 Hz, G is 195.55, and B is 247.5 Hz. We'll see in a moment that tuning a pair of strings to harmonics this way results in the lower string being 2 cents sharp with respect to the higher string. A "cent" is 1/100th of the halftone interval. Thus if E and E' are 2c sharp, B becomes 4c sharp, while D is 2c flat and G is 4c flat. Thus the G-B interval is off by 8c.

One would think that these little errors, which amount to fractions of a Hz, would be too small to hear. Not so. An error of one musical cent or two is not easy for everyone to hear, but almost anyone can hear an 8c mistuning. This table shows the frequency and cent differences, string pair by pair, and how I compensate for them:

As you can see, for the lower strings, the frequency differences are less than 1Hz, so it is hard to adjust by counting beats. In my experience, I cannot hear the beat when it is less than 1Hz, and I can barely hear the B4 to E'3 beat. What I do, then, is adjust until there is a difference I can barely detect. For example, I first tune the low E to the A using E4 and A3, and get it exact by nulling the beats. Then, knowing that the E is a tad sharp, I flat it until I can just detect the difference. I do so for each pair. This procedure leads to a final tuning so close to equal temperament that the G and B sound good together, and the chords D, A, C and E all sound equally good.

Of course, now that I have an electronic tuner, I can get a guitar in tune pretty quickly, but I have found that the green light stays on over an interval of plus-or-minus a cent or two. Thus, after I tune electronically, all the strings are probably within a cent or so of being right, but if I tweak a little by ear, I can make my guitar sound just a little bit better.

One caveat. This does not take inharmonicities into account. Because of small nonlinear effects the harmonics are slightly sharp compared to the open note, but for a guitar using steel strings the difference is half a cent or less at the 4th harmonic, and negligible at the 3d harmonic. Nonetheless, the result is a slight stretch of the tuning. You may know that pianos are usually stretch tuned, and over the 88-key range, the highest notes are 10-15 cents sharp as compared with the lowest notes. Piano strings are stretched tighter than guitar strings, and the harmonics get sharper as a result. Stretch tuning makes notes a few octaves apart sound better together. This is also true for the guitar, in principle, but it is very rare to play notes more than 3 octaves apart on the guitar. I haven't found a noticeable difference.

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