"Thereere are men and gods, and beings like Pythagoras." - Pythagoras I'll give Pythagoras a pass on the self love... I think he earned it. He's the father of tuning as we know it, flaws and all.
Today I'll give a quick primer on a topic that is a completely baffling mystery to many of my customers of all skill levels - guitar intonation. Really, It's quite simple. The string, fret, and height of saddle make a right triangle. The strings are of different thicknesses to allow them to be tuned to different pitches while maintaining similar tension. When you press a string down to a fret you are shortening it's length and therefore increasing the frequency that it will vibrate. For example, when you press down a string at the twelfth fret you are shortening the string by half and the string will vibrate twice as quickly creating a pitch that is an octave above the open string's tone. What we are doing when we adjust the intonation of a guitar is compensating for the different masses of each string and the height in which the action is set by moving the exact location of the saddle at the bridge. More mass (thicker string) needs a slightly longer scale length to produce the same note played on a thinner string at the same tension. OK. If that made your eyes completely glaze over... take out your guitar and tuner. Play the open low E string and tune it perfectly. Again. Perfectly. Now with the guitar on a strap or in your lap (with no pressure on the neck) play the low fretted at the twelfth fret. Is your tuner saying it is perfectly in tune? If yes, that string is properly adjusted. If not, we have some work to do. If you are playing a traditional electric guitar this is a really simple fix. There should be a little screw at the back of the low E saddle. Tightening it will make the scale longer, and loosening it will make it shorter. If the fretted note read sharper than the open string, you need to make the scale longer. If flatter, make it shorter. That's it. Easy as that. You Must Re-Tune the open string Every Time You Make An Adjustment. If you don't, you will be chasing your tail for hours. Now do the other five strings and your guitar should sound significantly sweeter when you play chords higher up on the neck. If chords in the open position sound out of tune, the string nut needs some attention from a qualified tech. I can't talk you through this process as it is a destructive adjustment and there is no going back if you go too far. If the intonation on your acoustic guitar is out, it is usually fixable by adjusting the neck and saddle height, and sometimes changing string gauges will help. One more quick note- If while adjusting the intonation on an electric guitar you notice the strings oscillating between sharp and flat or going out of tune after the note is sustaining, try lowering the height of your pickups. The magnets of your pickups may be strong enough to physically pull the string out of tune quite drastically. I find that the neck position pickup is more often the culprit in this situation but the bridge can do it too. I hope this was helpful. This is a really brief run through of something that could go on for many pages. If you have any further questions drop me a line.