Tuning a guitar is a contradictory subject. One would think that a note is a note, tune it up on a meter and let that be all there is to it. If you never played two notes in harmony, together, it would. But we play chords, and harmony melody lines, giving us two or three and even four notes that need to blend well together in our ears to sound good. Notes that are perfectly in tune, by the meter or tuner, won't sound good in harmony with each other because the laws of physics break down somewhere in the upper and lower registers of the overtones of each note. If we lived in a perfect environment where there was no air, or gravity, or noise, and all materials were space age perfect combinations of atoms, it might work.
When we pick any string we make it vibrate from one end to the other. The bigger the string, the slower it will vibrate. The smaller the string, the faster it will vibrate. The vibration settles in at a frequency of cycles per second. A tuner with a meter will read that frequency and let you tune your guitar without having to hear it. It's better not to hear it, we have found that our ears change from day to day and outside interference constantly plays tricks on our tuning ability. Tuning by the meter saves time, ears, and it gets it right on the money everytime.
We cannot tune all the strings to "0", or "440" on the tuner because each string has more than one frequency at which it will vibrate. You do hear one basic frequency, but the string divides itself in half, and then in half again as it vibrates. The tones that are produced by this dividing principle are what we term "harmonics". There are harmonics above the original frequency, and below it. They are octaves of the original. For each string you pick, you will get several octaves of harmonics above, and several below. These harmonics are not nearly as loud as the true note, so you think you don't hear them, and they should not have any role in this process. But you can detect them and feel them, they wait in the dark to attack your ears when you play harmony notes.
If these harmonic overtones were perfect reproductions of the original frequency we would not have a problem here. But for some reason, nature, or the guitar, or cosmic influence, keeps these harmonics from reproducing themselves perfectly. A true octave, or harmonic of a "440" would, and should be "880" cycles.On your guitar it lies, and becomes 879 for some unkown reason. When the harmonic cuts itself in half, the lower harmonic, it reproduces itself as 221, not 220.
Now you have three frequencies going into your ear, all trying to get heard, but they don't all divide themselves mathematically even. Your ear drum doesn't know what to do about all this. It can handle one note with some harmonics like this being out of whack, but when you play two notes in harmony, like a chord, you give your ears two frequencies, and eight or ten harmonics that are out of tune with each other to deal with. It is just too much. Your eardrums vibrate on one frequency, then the other, trying to get one sound that is common to all. It can't find one, so it does the only thing left, vibrate at several all at once. This sounds like distortion too you. You can hear a buzz, or an uncomfortable whizzing sound in your head.
There is only one way to solve this problem. Accept the fact that your guitar will make this happen and retune the original notes so that they will make harmonics that are mathemically acceptable to each other. This is "compensating" for the faults of the harmonics. People with great ears, on a good day, can do this without a tuner, by ear. Most people can't and never will.
The numbers on the tuning chart are a result of many attempts of trial and error at this compensating theory. This seems to be the working model to date. You will notice that the notes are above and below "0", or "440" on the tuner scale. Each note has a place, interval, in the scale of the chord we are tuned to, like E9th, or C6th.
Each note has a different relationship to the root note, thus giving it a need to be "compensated" this way or that. Everything is a compromise. This is merely the best compromise we can make working with what we have.
In the past we have had these charts with everything being tuned from "0" and below. We have since found that the relationships, compensated numbers, are still correct, but the overall tuning is better if everything is raised slightly so that all the numbers are on average above and below "0", or "440". Basically we have moved all the numbers from the old charts, up 2 1/2 Hertz. This change is not all that noticable when using the bar, but it is much better when playing open notes. It is especially better when playing an "F" chord on the first fret which used to be considerably flat if you tried to play right over the first fret.
Yet, even with all these numbers, you will still have places and chords that sound a little out of tune. The future of steel guitar may find every string with compensated pullers on them for all the different combinations of intervals we have to play. The price will get appreciably higher.
We still play a steel guitar by ear. Being in tune is important, but it won't solve all the problems that most players have with intonation, better yet, outonation. The tuning chart may improve your playing 5%, the rest is still in the hands and the technique. You can't buy that, you can only earn it.