Why Our Stringed Instruments Sound Out of Tune and A Way to Help Correct the Problem

by Bill Palmer

Some months ago, I promised Phil Mann that I would write this little article because this is such a misunderstood topic and most explanations are based on folklore, not fact.

Several years ago, I was given the rather undelightful task of having to tune accordion reeds. This was made somewhat easier than one might think by the fact that my late father had been extremely aware of the problems involved, and he had provided me with a Conn Strobotuner. This was long before the advent of the inexpensive electronic tuner so easily obtainable today. I was quite confused about why certain instruments sounded in tune with the accordion and others did not. Later, I undertook a study of tuning and learned several things from the experience.

To understand tuning, you must understand temperment, and to understand temperment, you must understand "Just" intonation. (This is pronounced like the first syllable in "justice," not as a rhyme with "fist.") Just intonation is a system of tuning that is based on the normal series of harmonics that are generated by a vibrating body. In the case of a guitar or a banjo, this vibrating body is the string. These harmonics are expressed in terms of ratios of small numbers. They are as follows:
	Fundamental (original frequency) 1/1
	Octave				 2/1
	Fifth				 3/2
	Fourth				 4/3
	Major 3rd			 5/4
	Minor 3rd			 6/5
	Small Minor 3rd		         7/6
These harmonincs continue out unto infinity.

Whenever you pluck a string, it will vibrate in all of these modes at once. Normally, the fundamental will be the loudest of the tones, and each of the higher harmonics will be increasingly less strong. It is the amount of each of these harmonics that gives an instrument its unique timbre or tonal quality.

When one of these "just" intervals is played against the fundamental, there is no "beat" between the fundamental and that interval. Just intervals sound "clean." Many choral experts theorize that a capella singers use just intonation.

We tend to hear in terms of just intonation.

In instruments that have only 7 notes per octave, i.e. diatonic instruments, such as the harmonica, bagpipe, Celtic harp, lyre, etc. there is no problem tuning the instrument in such a way that all the intervals are just. However, when instruments are chromatic, such as the piano, the guitar or the banjo, things soon go awry.

The traditional way to tune a keyboard instrument starts with just intervals. A particular note, such as Middle C is played on a reference instrument, such as a tuning fork. Then the fifth below the C is tuned so that it does not beat against the C, giving us a just F. Next the octave above the F is tuned, and then the Bb below that. Next, the G above the C is tuned, then the G an octave below that. This process continues with the D, the A and the E. The tuning goes "down a fifth, up an octave" and repeats.

When we start getting to the B, itself and then the remaining accidentals, F#, Eb, C# and Ab, We start getting into trouble. For some unknown reason, the old keyboard tuners of long ago decided that B was going to be the note that would get the "leftovers."

"Wait a minute," you ask. "What leftovers?" Remember. I said that things were pretty simple until you started going chromatic? Here's why. If you try to set up a chromatic scale based on A=440, but you do all of your divisions by a 3/2 ratio, B ends up with a frequency of 247.5 Hz. If you are figuring your ratios from C=261.6255 (normal middle C from an electronic tuner) B becomes 248.3398 Hz. However, if you are tuning from an electronic tuner, B is 246.941 Hz. This is somewhat confusing. How can a note have 3 different pitches?

This is due to a practice called "temperment." In order to create a chromatic scale that is the same on all instruments (basically), since 1885 or so, piano tuners have used a scale that is based on a mathematically determined octave. In this scale, an octave equals 1200 cents. A semitone, that is, the distance between two adjacent notes on the piano, or one fret on a fretted instrument, is equal to 100 cents. The interval of a fifth is 700 cents. However, a just fifth is 702 cents. When you add up all of the 2 cent leftovers that gives you an error of 24 cents. The idea of temperment is to spread these 24 cents out equally through the octave.

An electronic tuner, contrary to what most musicians believe, does not put you in tune. It puts you in temperment. Temperment is the amount of "out of tune-ness" that is necessary so that we can play in all keys, and sound equally in tune (or equally out of tune, if you will!) The octave and the unison are the only "just" intervals left in our equal tempered tuning system.

Over the centuries, there have been at least 64 different schemes to accomplish this, each with differing results. The piano tuner resolves this by tuning just, then checking a set of key intervals to make sure that all of the major thirds are out by an equal amount, etc.

In other words, the problem is that, to a great degree, we hear in terms of just intonation, but we play instruments that use equal temperment. We hear in tune and play out of tune.

The Guitar and Banjo Tuning Method.

To tune your stringed instrument correctly, there are some things you can do to make these adjustments.

First, place your bridge as accurately as possible. If the bridge is placed so the first string is very slightly sharp at the 12th fret, the upper octaves will sound more in tune than if it is placed perfectly or flat. Piano tuners routinely tune octaves starting at the F an octave and a fourth abouve middle C very slightly sharp--"one long roll sharp." Also, make sure that your strings are reasonably recent, but properly set, so that they do not stretch too readily.

For the guitar. Tune your first string E, either with an electronic tuner or by using an A=440 tuning fork. Fret the first string at the 3rd fret, and use this G to tune the open G string. Make sure there is no beat at all between these two notes. Next, tune the fourth string by fretting it at the second fret, and tuning this note E to the first string E. Again, there must be no beat. Tune the open low E on the 6th string to the E on the second fret of the 4th string. Tune the fifth string open A to the second fret of the third string G. Then tune the open second string to the B on the second fret of the fifth string. Now make all of your checks. Test the D at the third fret of the B string against the open fourth string. There should be no beat. If there is, go back and check your D string against the E strings, by fretting it at the second fret. It may take several cycles of tuning to get the instrument in tune, possibly just to equalize the string tension on the neck and body. However, on most instruments, this method will work out perfectly, even on those with incorrectly placed bridges.

You may also start this method by tuning the A string against an A=440 tuning fork. From there, tune the B, then the G, the D and the E strings.

This method spreads the errors out equally over the fretboard.

For the banjo. Tune the open G string (third string) to an electroninc tuner or to your guitar player. You may also tune your G string at the second fret to an A=440 tuning fork. Tune the G at the 5th fret of the first string against this open 3rd string, Tune the open 5th string to the open 3rd string. Tune the open 4th string to the open first string. Tune the 8th fret of the second string to the open 5th string. Then check the 3rd fret of the second string against the open D strings. Repeat until the tension on the strings and the neck is equalized. This is far more critical than on the guitar because of the length of the neck of the banjo and its flexibility. Note: Because the banjo is so "lively," you might find it necessary to mute the strings you are not tuning, just to keep the natural overtones out of the way.

For more information on temperment, there is a massive book by Owen Jorgensen called "Tuning." It is concerned with temperments of keyboard instruments and the various schemes used for them. It is a wealth of information on a poorly understood subject. You can get this from college book stores or you can find it at some of the better remainder houses. My copy came from Half Price books and cost me $20.00. It may be available in your local or college library. There is also lots of information about temperment in the Grove Dictionary of Music, However, Jorgensen's research is far more exhaustive and has pointed out some common errors in the Grove work.

Visit Bill's Home Page for more interesting information.

Bill Palmer, A.I.M.C. Merlin the Magician from the Texas Renaissance Festival Cactus Willy Translator of the Punx Trilogy Author of "How to Play Folk and Bluegrass Banjo"