Optimisation techniques for nut and saddle compensation

[Dominic]

Hi Trevor, what package do you use to calculate the optimisation problem to get nut and saddle compensation values. I could be getting out of my depth here but I want to work out how to do this set of calculations. It kind of like doing the cross word for me. I did a bit of economic optimisation at uni but that was theoretical and we never got to apply it. I'm in no rush so I have time to do more reading first.
Cheers
Dominic

[Trevor Gore]

Optimisation is arguably more about the technique than the package and there are lots of variants in the technique. I didn't go into detail in the book (and neither am I going to anywhere else) because the subject is quite large and there's plenty written on it already (like, for example, there is on tuning hand planes or blade sharpening). I would recommend using the package you're experienced in already, because it will likely handle the mathematics, which is mostly to do with many iterations rather than great complexity. Matlab is one package which has the capability, or you could even try Excel; basically any program that will support mathematical analysis, from Fortran onwards!

[Dominic]

Ok thanks, i have access to mathlab and a few other packages so I'll do some research on whats best. I thought you would run it through a program which would run the iterations and solve for you. Or write some VBA code perhaps?

So you thank you once again. Most fun I've had with maths for ages.
Cheers
Dom

[kiwigeo]

Boffins!!

[Dominic]

Thanks Martin, boffin (or pointy head) works for me.
Dom

[Dominic]

Hi Trevor, with your string test rig you mention in a foot note that you should allow for guitar compression in the rig. Do you just measure the deflection of the tail bolt once the string is brought to pitch? And then is it just a trial and error process to get around 0.5mm deflection?
Thanks
Dom

[Trevor Gore]

The idea I ran with was to make the test rig approximately the same longitudinal stiffness as a guitar. Hence the ~0.1mm overall compression/deflection of the test rig under a single steel string load, which is pretty much what I get on a guitar (~0.5 mm for 6 strings). Most of the rig deflection is in the flexibility of the tail bolt and its mounting. Mine came out about right (pure luck) so I didn't have to do anything to adjust its stiffness. I measured the rig shortening from the "nut" to the tail bolt. Trimming the stiffness would be just trial and error, as you say.

You could do a calc. and if the longitudinal stiffness of a guitar is very much greater (say greater than a factor of 20) than the longitudinal stiffness of the string, (which it likely is for steels and almost certainly is for nylons, but I haven't done the calc.) you need not worry about it. Just make your rig stiff, too. It's important to make sure that the resonant frequency of your test rig (and the bench it's mounted on) is well away from the frequencies you're interested in. If not, you'll likely get coupling and frequency shifting, which screws things around.

[Dominic]

Thanks once again Trevor, let me know when you get fed up answering these questions. But I am really enjoying myself and have nearly finished although I would love to get your model working one day.
Cheers
Dom

[Dominic]

Hi Trevor, I have been thinking about this string rig as I make my own today. If a guitar compresses longitudinally by 0.5mm or so, that would occur between the nut and the saddle so the scale length is shortened slightly. And you account for this in compensation. But if all the flex in your rig occurs behind the saddle, the scale length is unchanged and I don’t see the reason for building in flex. Unless it comes into the calculation for the fretted part of the longitudinal stiffness test. Am I missing something?
Cheers
Dom

[Trevor Gore]

On a guitar, the vast majority of the compression is actually the bridge/saddle rocking forward and a bit of top compression. There is hardly anything in the neck/fretboard, so the scale length (as in fret spacing) doesn't really change. In the Build book you'll note that I say to add back in an allowance for this sort of "compression" when you position the bridge/saddle.

In theory, you build flex into the rig for this reason: Suppose the guitar soundboard was really flexible compared to the string stiffness. When you fret the string you'd get no tension increase (because the guitar top would flex). You'd also get this effect if the string were tensioned with a pneumatic cylinder (effectively a constant load device) rather than a mechanical tuner. If this were the case, because you get no tension increase on fretting you'd get no intonation problem and therefore no need to add compensation. On the other hand, if the guitar were really stiff, when the string was fretted you'd get all the tension change due to the fretting deflection. The stiffness of the guitar is somewhere between these two extremes of one giving you no intonation problem, the other giving you a considerable intonation problem. Hence the need to approximate the stiffness of the guitar in the stiffness of the rig.

However, in practice, if the guitar is very much stiffer longitudinally than than the stiffness of the string, which it probably is, even for steel strings, the vast majority of the theoretical tension increase is going to be manifest in the string regardless. So you'll need to do a 100% compensation.

So the bottom line, unless you're planning on building really floppy guitars, don't worry about it! (I hope you can make sense of all of that!)

[Dominic]

Thanks Trevor, that all makes sense.

I have a question about the height of the string above the fret point on the rig (d). I think I read that bending stiffness is linear so that would imply it does not matter how high the string is set in the rig (within reason).

Is this correct or is it better to stick to heights approximating the finished action?

Cheers
Dom

P.S. I hope these questions and answers will be a useful resource for others down the track if they decide to apply these more technical approaches to their building.

[Trevor Gore]

In Section 1.5.9 I do a lot of analysis on string bending stiffness and come to the conclusion that in most cases it can be ignored. So, again, one less thing to worry about!

On my rig I have the action at ~5mm, just to be more-or-less in the high end of the standard range. In theory, you should be able to go a lot higher than that and still stay linear.

[Dominic]

Cool, thanks. I figured the higher you go the smaller the percentage measurement error and its hard to measure under a string with great accuracy because it is floppy.

Just about finished setting up the rig, and then I am going to test its resonant frequency and make adjustments if necessary. Then I can start doing some testing.

Cheers
Dom

[Trevor Gore]

I measure the gap with feeler gauges. You can usually hear it when you're "there".

[Dominic]

Hi Trevor, I have just about finished putting all the equations for the nut and saddle model into excel. Its been great fun but I have a few questions as I tidy things up and create my full tables and check my results.

1. The first is in equation 4.7-14 which shows the radius of the arc of the fretboard under string s. I would assume that it is the same for each string. The same would be the case with the elliptical component I would imagine. Only the Ys change due to different actions across the 1st to 6th strings. Is this correct?

2. For my above radius calculation I am getting around 220m for a low relief. This sounds reasonable to me but I have not thought about these numbers before. Does this sound in the ball park?

Thanks Trevor
Dom

[Trevor Gore]

Hi Dom,

1) Yes, that's right. I use the same radius under each string, but you don't have to.

2) Yes, that's in the ball park for low relief. But it's quite sensitive (as you'd expect).

Spreadsheeting on a Friday night, huh?

Did you know that people have been rusticated for less?

[Dominic]

So it also assumes the fretboard is either flat across its width, has a compund radius such that the strings remain equal distance from the fret board for it length, or that its curvature is >16" which you mention give the same impact as a compound radius.

Anyway, thanks again Trevor
Cheers
Dom

[Trevor Gore]

Yes, that's right.

[kiwigeo]

Thanks Martin, boffin (or pointy head) works for me.
Dom

Got one pointy head in my family already....one of my brothers is a maths teacher (in Bhutan). I loved maths and excelled at it through to my last year at high school when I got stuck with a really bad teacher who killed my interest in the subject totally.