You say you want a revolution?

Lybian guitarist urges his buddies on while they smash the state.

Is there a formula for compensation?

For those of you who are new to this, the compensation is the small increment in length added to the string's scale length to compensate for the slight sharpening that occurs each time you press the string down to fret it. As far as an actual formula that predicts what that increment should be, yes, there is a real formula. The late Daniel Haines, physicist at Bell Labs and later a consultant for the CF Martin Organization, gave me the compensation formula he devised, but I don't use it because it is too cumbersome to use. I find that the same compensation of .15" measured at the centerpoint of the saddle works acceptably well for short and long steel string guitar scales (.10 for nylon string guitars). But if you want to try your hand at following the formula here it is for each string:

Where C, as Dan Haines said, "the amount that the saddle half of the string is longer than the nut half of the string" is the compensation, and it's expressed in inches

H is the string height expressed in inches measured at the 12th fret. Notice that it's squared, suggesting that small changes in the string height result in large changes in the compensation required. That's why your guitar goes out of tune as it settles under tension and the strings rise oh-so-slightly off the fingerboard.

E is the constant that represents Young's modulus, or the modulus of elasticity, of the string core material. Steel is 29,000,000. Nylon is around 290,000: roughly 100 times more elastic (or less stiff) than steel.

A is the cross sectional area of the monofilament string or, on wound strings, its core--measured in square inches. That's why steel string saddles are sloped: the thicker strings want more C. On nylon-string guitars its more complicated: the string thickness doesn't go up as neatly and evenly as they do on steel. And since the variation in the inputs of the strings--given the sum of the other factors--is so slight. So that in practice on nylon the variation in total C is so slight among strings that the classic saddle ordinarily doesn't need to be sloped--although in some cases the classic G-strings in some string sets find their intonation improved by notching them back on the saddle. In many cases that can be avoided by using Tynex Gs--(a denser, thus a thinner G), rather than thick Nylon Gs--which can improve the usually notoriously dull and sour G on many classics. Tynex Gs are found on D'addario Composite sets.

L is the vibrating length of the string expressed in inches. You'd use the scale length in this calculation. It's below the line on the formula, so it impacts the compensation requirement inversely. Hence short scales require greater compensation (i.e. mandolin bridges are usually set back as much as 1/4-inch).

T is the tension of the string--at concert pitch--expressed in pounds. Also, because it's below the line on the equation, the lower the tension, the greater the compensation required. Just watch what happens when you drop your low E string to D for dropped-D tuning. Your existing compensation usually becomes insufficient and the floppy low E string goes sharp as you play up the neck. The precise tension of a string is not so easy to find: In order to calculate T you need another formula, found here:

But in order to satisfy that equation, you have to know the precise mass of the string in grams. Got a gram scale? No? That the most accurate way to plug in this number. But if you don't have a gram scale, the next best thing is to go to here. It's a chart which supplies the tensions of the different strings ....  BUT.... on a 25.5" scale guitar. I guess you can approximate what they are on a 24.9 or a 26-inch scale by setting up a proportion. 

So you see the immense hassle. That's why I just rely on the empirical figure of .15" to the midpoint of a saddle that is slanted 1/8 inch in 3 inches? Works fine for me and my fussiest customers.

What the formula is useful for, is to show how the compensation requirement increases in direct proportion to the string height, the modulus of elasticity of the string core material, and the diameter of the string core--but INVERSELY proportional to the string length and tension. So in the equation everything above the line: string height, elasticity, core thickness increases the C while at the same time, everything below the line, on the divisor: length and tension are acting to reduce the compensation required.

So the resultant compensation amount is the result of that  push-pulling of factors. When they are all accounted for the guitar sounds sweet and heavenly. When they are not all accounted for, the guitar sounds soggy and sour. No big deal, huh?