So that is why there is often a angle lower on the left and higher on the right on a guitar saddle?
(re compensating is because we press between frets)
Not exactly. Because open strings don't use frets. Frets are the same height whether its the treble or bass side.
A note is determined by the tension, and length of the string segment.
When you pluck/strum, your string goes from | to () in the primary node, and also the harmonic nodes.
The math for lengths is based on that, but the math assumes the string can pivot freely to form sine waves.
But an actual string doesn't work that way. If you hold a string straight out, it doesn't just flop down because it has a stiffness.
That stiffness of the string makes the effective node length shorter because part of the string isn't swinging back and forth freely.
The stiffer the string, the shorter the effective node length, because the stiffness resists going into the sine wave shape especially on the the ends, but also on the harmonics.
So, if you think about a guitar, esp a steel string. The fatter the string, the stiffer it is.
So, on the bass end, the slanted saddle makes the effective length longer because it's the stiffest strings, while the treble needs less because it's not as stiff. You'll usually see a second slant on the 2 high strings, because it changes from wound to unwound on the 2 high strings.
Nylon/flouro carbon is not nearly as stiff as steel, but it still has a stiffness.
There's a thickness issue too, because the math works from the center of the diameter of the string, but the string moves from the side of the string that's touching the nut and the saddle, so there's another kind of compensation for that, where the nut and saddle or the slots cut are not parallel with the body slanting down toward the treble side because those strings are thinner.
That's kinda why you compensate for the strings you're using, and not a "universal" compensation, but then that's microtuning. And in the real world of playing.. you introduce other issues, that I believe exceed the tolerance you compensate for. Not to say, starting more accurate isn't better, but it's never perfect in every place because it can't really be.
Like.. frets. We press the string between frets to lock the node length. But, the fret has a height. When we push the string into the valley between the frets, we change the effective tension, which throws the math off. If you hold a chord, and then reach for an additional note with your pinky, the notes held in the chord flat because you're reaching toward the saddle reducing tension on those notes, but the note you hit with your pinky sharps because your pinky pulls the string toward the other fingers holding the chord. We consciously know this happens because we make vibrato exactly in this way by wiggling the fingers holding down the string. But it happens when we play because our fingers don't work strictly up and down, because our knuckles are fixed position, and our fingers spread, so tendons naturally pull fingers spread back together.
