Post by freemankeller on Mar 4, 2017 22:09:53 GMT -5
In another thread, CTGull mentioned that he didn’t really understand intonation. That is not a big surprising – an explanation can either be super simple (but not very good) or pages of complex physics and math (which probably isn’t very helpful either). I fool myself into thinking that I have a modest understanding of how it all works so let me try to explain, somewhere between the two.
This is the equation for the frequency of a vibrating string.
(that slide is from a lecture by D’Addario engineers on string theory). We are used to seeing it rearranged to solve for Tension, that is the basis for your little tension calculator apps and all the D’Addario tension information.
If you stretch a string you incrementally increase the Length (which lowers the Frequency), but you dramatically increase Tension. There is some high level math related to the physical and engineering properties of the string, but you can prove it to yourself – bend your 2nd string at the 8th fret from G to A (one step). BB King does it all the time, but its darn hard – it takes a lot of strength. And as the equation predicts, if the Tension goes up by a lot, so does the Frequency.
When we fret a note we slightly stretch the string just as when bending, only not nearly as much. But the frequency does go up – the note sharpens.
The equation above is not totally accurate – it’s a good approximation that works for ideal strings but it leaves out the effect of the strings stiffness. The stiffer the string, the more sharp it goes when it is bent or fretted. The strings stiffness is proportional to it CORE diameter and the way strings are built, the B string has a larger core than the G string (and probably the A). The B string sharpens more than its neighbors when fretted.
You can test this on any guitar with a movable saddle – put the saddle exactly at the scale length (where the theory says it ought to be) and fret a note – if you check the pitch with an accurate tuner you’ll see that it gets increasingly sharp as you up the neck. A quick way to check is to pluck the harmonic at the 12th fret – that should be exactly the octave note and then fret the 12th fret (which should be the octave note) – they aren’t the same. The harmonic is true, the fretted note sharp.
What can we do about this?
Again, the math says that if we make the string from the fret to the saddle longer (and keep the fret in the same position overall) we can lower the amount of sharpening (I won’t go thru the math, but the final equation is
Comp = a^2(ES/TL) +2(EI/T)^1/2)
Where a is the action at the fret, L is the length of the vibrating string, T is tension, and E is the modulus of elasticity. Point is, you can calculate how much to lengthen the string.
Guitar makers have figured out either by doing the math or experimenting that for an average scale steel string guitar if you lengthen the high E about 1/16 and the low E about 3/16 its pretty darn good. So almost every maker “compensates” their saddle by that amount, more or less. For many players who only play cowboy chords that’s good enough (audiologists tell us that most people can’t hear less than 5 cents difference between two notes).
You probably can hear that pesky B string however, particularly if you play much up the neck. Because of its bigger core it needs more compensation. With a 3/32 wide saddle blank if you put the high E and the G near the front of the saddle and push the B to the very back it probably is good enough – that’s what you see on most factory compensated saddles.
Obviously you can calculate the amount of compensation for each string using that equation – if I was designing an odd scale guitar from scratch that is certainly what I would do.
However, there is a method that works pretty well and is easy to do. Do your setup as usual, make your saddle and curve it to your f/b radius but leave it flat on top. Do all the normal setup stuff lowering the action until it is almost to your target – just a hair high. String it up and tune to concert. Now put a small wire, a piece of B string is just right, on the top of the saddle under a string. Move it back and forth comparing the harmonic to the octave (you’ll have to retune each time you move it). When the two notes are the same, mark both sides of the wire with a really sharp pencil. Now do the next string and the next and the next. Finally file the top of your saddle so the break point of each string is between the pencil marks, round it off and polish. Most of the time you can blend the break points together and it will look smooth and pretty.
A couple of minor points here – when you take the wire out and/or finalize your action you will lower it a little – that actually requires slightly less compensation. You can try to fudge that into making the saddle – I’ve never worried about it., Obviously different string gauges and/or different tunings will affect the compensation – heavier strings will require more, tuning down will require less.
The method really only adds a small amount of time to the fabrication of a new saddle – I’ll routinely do it if I know the player spends much time above the fifth fret.
This is the equation for the frequency of a vibrating string.
(that slide is from a lecture by D’Addario engineers on string theory). We are used to seeing it rearranged to solve for Tension, that is the basis for your little tension calculator apps and all the D’Addario tension information.
If you stretch a string you incrementally increase the Length (which lowers the Frequency), but you dramatically increase Tension. There is some high level math related to the physical and engineering properties of the string, but you can prove it to yourself – bend your 2nd string at the 8th fret from G to A (one step). BB King does it all the time, but its darn hard – it takes a lot of strength. And as the equation predicts, if the Tension goes up by a lot, so does the Frequency.
When we fret a note we slightly stretch the string just as when bending, only not nearly as much. But the frequency does go up – the note sharpens.
The equation above is not totally accurate – it’s a good approximation that works for ideal strings but it leaves out the effect of the strings stiffness. The stiffer the string, the more sharp it goes when it is bent or fretted. The strings stiffness is proportional to it CORE diameter and the way strings are built, the B string has a larger core than the G string (and probably the A). The B string sharpens more than its neighbors when fretted.
You can test this on any guitar with a movable saddle – put the saddle exactly at the scale length (where the theory says it ought to be) and fret a note – if you check the pitch with an accurate tuner you’ll see that it gets increasingly sharp as you up the neck. A quick way to check is to pluck the harmonic at the 12th fret – that should be exactly the octave note and then fret the 12th fret (which should be the octave note) – they aren’t the same. The harmonic is true, the fretted note sharp.
What can we do about this?
Again, the math says that if we make the string from the fret to the saddle longer (and keep the fret in the same position overall) we can lower the amount of sharpening (I won’t go thru the math, but the final equation is
Comp = a^2(ES/TL) +2(EI/T)^1/2)
Where a is the action at the fret, L is the length of the vibrating string, T is tension, and E is the modulus of elasticity. Point is, you can calculate how much to lengthen the string.
Guitar makers have figured out either by doing the math or experimenting that for an average scale steel string guitar if you lengthen the high E about 1/16 and the low E about 3/16 its pretty darn good. So almost every maker “compensates” their saddle by that amount, more or less. For many players who only play cowboy chords that’s good enough (audiologists tell us that most people can’t hear less than 5 cents difference between two notes).
You probably can hear that pesky B string however, particularly if you play much up the neck. Because of its bigger core it needs more compensation. With a 3/32 wide saddle blank if you put the high E and the G near the front of the saddle and push the B to the very back it probably is good enough – that’s what you see on most factory compensated saddles.
Obviously you can calculate the amount of compensation for each string using that equation – if I was designing an odd scale guitar from scratch that is certainly what I would do.
However, there is a method that works pretty well and is easy to do. Do your setup as usual, make your saddle and curve it to your f/b radius but leave it flat on top. Do all the normal setup stuff lowering the action until it is almost to your target – just a hair high. String it up and tune to concert. Now put a small wire, a piece of B string is just right, on the top of the saddle under a string. Move it back and forth comparing the harmonic to the octave (you’ll have to retune each time you move it). When the two notes are the same, mark both sides of the wire with a really sharp pencil. Now do the next string and the next and the next. Finally file the top of your saddle so the break point of each string is between the pencil marks, round it off and polish. Most of the time you can blend the break points together and it will look smooth and pretty.
A couple of minor points here – when you take the wire out and/or finalize your action you will lower it a little – that actually requires slightly less compensation. You can try to fudge that into making the saddle – I’ve never worried about it., Obviously different string gauges and/or different tunings will affect the compensation – heavier strings will require more, tuning down will require less.
The method really only adds a small amount of time to the fabrication of a new saddle – I’ll routinely do it if I know the player spends much time above the fifth fret.
