Help needed with intonation calculation

[BenH]

Hi all,

I've been crunching the equations from 4.7.3.2 in Maplesoft Maple, and am getting somewhere. Problem is I'm not getting plausible results and have stared myself blind looking for errors. I wonder if any of you can see where I went wrong?

Attaching an image of the plot of the Intonation error length vs the fret# with no nut or saddle compensation, and also a pdf of the calculations. The calculations should have a similar looking format to those in the book.

Thanks for looking

P.S. AAaaaaaaaaaAAAAaaaaaahhhhh!!!!!! D.S.

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intonation.pdf

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[Trevor Gore]

Hmmm. I've never plotted it out like that before. But, if I'm reading this right, if you shorten the nut to first fret distance by ~0.6mm, you should have a straight-ish line, intercept at 0.0008 (i.e 0.8mm if you're working in SI. Perhaps you should dimension your axes ). Then saddle compensation should reduce the slope of the line, but I'd be checking your sign convention (+ vs -) for movements of the nut and saddle closer/further from the sound hole . So, if I'm reading this right, you seem to be heading in the right direction. Too late to look at maths tonight...

[BenH]

Thanks for looking Trevor,

Unfortunately it's plenty of day left on this side of the pond, so time to get cross-eyed again.

I think the last plot might have been misleading since it didn't really start at 0. So here it comes again along with two plots shifting the nut 0.6mm in either direction. Definitely something funky going on here. I'll get some rest and go through it all again.

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[Trevor Gore]

Amongst other things that may be wrong ( ), you appear to have lost some zeros somewhere. If you take 0.6mm off the nut to first fret length, the graph should come down 0.0006m to start at ~0.0004m not from ~0.04m on your "intonation" scale (which should actually be compensation; intonation error is measured in tonal units, usually cents).

Your delta N reads as 0.6. Best be sure what units that is (mm, cm, m etc.) to check for any lost zeros.

The intonation part of the calc is relatively easy in this thing. The difficult bit is modelling the curvature of the neck in a way that gives the relief and action accurately and that also corresponds to the way builders/guitarists actually measure and refer to relief and action. So, to get the intonation part sorted, first model zero relief in the neck (straight line geometry) to make the action-per-fret geometry simple, and work on the pitch rise/intonation error part until you've ironed out the issues there. Then address the curved neck geometry.

I still haven't looked at your .pdf...

[BenH]

The dN value of 0.6 was actually calculated as 0.6*10^-3 so that part should have been correct.

I think I got the straight neck geometry correct since it's resulting in very plausible values. With an 18th fret action of 2*10^-3 and a nut action of 0.1*10^-3, I get a 12th fret action of 0.001569572252 and a 1st fret action of 0.0002649615061. So far so good I think.

The change in length (Equ. 4.7-24) when the fret is stopped at fret 12 (scale length 650*10^-3) is: 0.0000116645. A mere 0.011mm... Is that reasonable? (g0 set to 0.5*10^-3 and Fp to 0.75). Change in length for fret 1 for same values is 0.0001081388.

k (for high E string, 0.012" diameter) has been given a value of 210*10^9*(25.4*(0.12e-1*(1/2))*10^(-3))^2*Pi, which comes out to 4877.409600*Pi.

T for the same string is 10570*10^(-3)*G, where G is 9.80665, for a total of 103.6562905

μ (equ 4.7-34) comes out to 0.0005644890182 for above value of T, with a frequency of f=329.63 and L=650*10^-3

If all of the above seems reasonable I can at least scratch that from my error list. If no, maybe we've found a winner.

[BenH]

Above is for no nut and saddle compensation

[Trevor Gore]

I've not done full check on your numbers here, so this is just a "sniff" test:

I think I got the straight neck geometry correct since it's resulting in very plausible values. With an 18th fret action of 2*10^-3 and a nut action of 0.1*10^-3, I get a 12th fret action of 0.001569572252 and a 1st fret action of 0.0002649615061. So far so good I think.

..smells about right...

The change in length (Equ. 4.7-24) when the fret is stopped at fret 12 (scale length 650*10^-3) is: 0.0000116645. A mere 0.011mm... Is that reasonable? (g0 set to 0.5*10^-3 and Fp to 0.75). Change in length for fret 1 for same values is 0.0001081388.

Yes, that's reasonable (I get 0.0113mm for an action of 1.5mm at 12)

k (for high E string, 0.012" diameter) has been given a value of 210*10^9*(25.4*(0.12e-1*(1/2))*10^(-3))^2*Pi, which comes out to 4877.409600*Pi.

That one doesn't smell quite right...

T for the same string is 10570*10^(-3)*G, where G is 9.80665, for a total of 103.6562905

That's fine.

μ (equ 4.7-34) comes out to 0.0005644890182 for above value of T, with a frequency of f=329.63 and L=650*10^-3

That's OK, too.

So, have another look at that stiffness number.

[BenH]

Thanks Trevor,

I had a couple of looks at k, but I arrive at the same number for a 0.012" string. Not sure where I'm going wrong.

From page 4-109 k = E*A

I assume E for steel to be 210*10^9 N/m^2

I calculate A for a 0.012" string to 7.29659e-8 m^2

So E becomes 15322.839 N, or the 4877.409600*Pi N given above.

I might be getting brain fatigue here

[Trevor Gore]

I assume E for steel to be 210*10^9 N/m^2

That's the text book number for bulk steel. In practice it seems it can be a fair bit higher due to the nature of the drawing process for guitar strings giving a lot of grain alignment. Using the text book number should get you in the ball park, though.