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Copyright 2017 by Wayne Stegall
Created December 20, 2017.  See Document History at end for details.

Custom Intonation

A tale of giving my guitar a custom intonation adjustment

Introduction

Although my guitar was comfortable to play, bar chord in higher positions did not sound to my liking.  At some point I thought that I needed to make some kind of alteration.  Ervin Somogyi had said in a article that improperly fitted or less than rigid saddle would diminish the sound.1 Other reading convinced me that saddle and nut could be crafted for a custom intonation.


Theory

Intonation is a short term for frequency accuracy.  On a stringed instrument, the exponential placement of frets is correct.  However in pressing a string to touch a fret the string is lengthened and stretched, both of which alter the frequency.  For this reason the saddle is moved back and the nut forward by a small amount to compensate.























Saddle Intonation

Process

Having bought bone saddle blanks on the internet, when I had a free moment I did my saddle intonation.  I had already decided that the intonation specification published by Greg Byers would be easier than attempting a trial-and-error method.

Figure 1:  Greg Byers's recommended intonation adjustments.2
String
 
Nut Adjustment
 
Saddle Adjustment
1 E

0.3
1.3
2 B
0.7
2.1
3 G
1.0
3.1
4 D

0.5
1.5
5 A

0.5
1.7
6 E

0.5
2.9

Sign of adjustment is indicates direction of string length change.
Nut compensation shortens the open string, saddle compensation lengthens it.


First I wrote a simple C program to print a table of distances from a range of frets to a compensated saddle break point for each string.  The program and its results are shown in figures 2 and 3.  Then I sanded the blank until it fit the groove in the bridge.  Next - because the action of the guitar seemed right to me - I sanded the blank to match the width and height of the original saddle.  Next I used a millimeter ruler to measure from a chosen fret to each string crossing for the correct compensated break point.  The reference fret I chose was meant to use near the full length of the ruler to minimize any error that the non-zero angle of the ruler might introduce.  Then I filed the blank separately for each string to place each break point at the correct compensation point.  The filing was done at an angle so that no part of the blank in the saddle would be filed.  I filed once a little short of the mark, remeasured and repeated the process.  Figure 3 shows the stepped appearance of the finished saddle.

Figure 2:  C program calculating fret distances to compensated saddle for each string.
// sadtable.c

#include <stdio.h>
#include <math.h>

#define SCALELEN 650.0

const char stringname[8] = " EBGDAE";
const double saddlecomp[7] = {0.0, 1.3, 2.1, 3.1, 1.5, 1.7, 2.9};

double stringlen(int fret)
{
    return SCALELEN*pow(2.0,-((double)fret)/12.0);
}

int main(int argc, char* argv[])
{
    int string, fret;
    double dist0, saddledist;

    printf("Distances in mm from frets to compensated saddle\n"
     "for a scale length of %0.2fmm\n\n",SCALELEN);
    printf("Fret");
    for(string = 1;string < 7;string++)
    {
        printf("\t%c%d",stringname[string],string);
    }
    printf("\n");

    for(fret = 7;fret<20;fret++)
    {
        dist0 = stringlen(fret);
        printf("%i",fret);
        for(string = 1;string < 7;string++)
        {
            saddledist = dist0 + saddlecomp[string];
            printf("\t%0.2f",saddledist);
        }
        printf("\n");
    }
    return 0;
}


Figure 3:  Table of fret distances to compensated saddle for each string.
Distances in mm from frets to compensated saddle
for a scale length of 650.00mm

Fret  
E1  
B2  
G3  
D4  
A5  
E6
7
435.12
435.92
436.92
435.32
435.52
436.72
8
410.77
411.57
412.57
410.97
411.17
412.37
9
387.79
388.59
389.59
387.99
388.19
389.39
10
366.10
366.90
367.90
366.30
366.50
367.70
11
345.63
346.43
347.43
345.83
346.03
347.23
12
326.30
327.10
328.10
326.50
326.70
327.90
13
308.06
308.86
309.86
308.26
308.46
309.66
14
290.84
291.64
292.64
291.04
291.24
292.44
15
274.59
275.39
276.39
274.79
274.99
276.19
16
259.25
260.05
261.05
259.45
259.65
260.85
17
244.77
245.57
246.57
244.97
245.17
246.37
18
231.11
231.91
232.91
231.31
231.51
232.71
19
218.21
219.01
220.01
218.41
218.61
219.81

Figure 4:  Finished saddle shows separate compensation for each string.
saddle.jpg

Intermediate Results

When completed the intonation was greatly improved at higher frets.  However G string seemed to be out at near the nut.  When G was tuned correctly, the A one whole tone up was somewhat off.  Obviously the nut intonation was needed as well.  For the time being, I did a compromise tuning the G string


Nut Intonation

Process

A year later I ordered nut blanks to finish the intonation adjustment.  First I wrote a simple C program to print a table of distances from a range of frets to compensated nut break points for each string.  The program and its results are shown in figures 5 and 6.  Next I measured the unaltered nut position to see where I was.  The nut was already compensated by 0.5mm an amount already correct for the bass strings and a compromise for the others.  I decided to cut curved surfaces on the front of the nut so that nut would have surfaces touching the end of the fingerboard at between each string.  To begin I shortened the fingerboard by 1mm at the nut end so that all of the strings would have compensation that would involve some shortening of the nut.  As I did so, I wondered if I would have to buy another guitar if I messed up.  After shortening the fingerboard, I made a nut to fit with only preliminary cuts for the string grooves.  Then I attached a small strip of sandpaper around a bolt socket with double-stick tape and installed it in a drill motor clamped in a vise.  A mini drum sander attachment for Dremel tools seems more ideal for this use.  Then measuring and marking the nut blank for the correct compensation point, worked the blank against the sandpaper until I formed a curved surface at the measured point.  Then I worked the nut into a more final shape and cut the string grooves closer to their final depth with a rat-tailed triangle file and verified the compensation measurements again.  I then did simplified method of adjusting groove depth.  I cut each groove until the associated string was above the first fret by eye by a minuscule amount when fretted between the second and third frets.  All seemed perfect, then I made a mistake.  I thought the strings should be closer and made them so.  Then everything buzzed.

I then ordered proper nut files and more nut blanks and repeated the process when I received them.

Figure 5:  C program calculating fret distances to compensated nut for each string.

// nuttable.c
//

#include <stdio.h>
#include <math.h>

#define SCALELEN 650.0

const char stringname[8] = " EBGDAE";
const double nutcomp[7] = {0.0, -0.3, -0.7, -1.0, -0.5, -0.5, -0.5};

double stringlen(int fret)
{
    return SCALELEN*pow(2.0,-((double)fret)/12.0);
}

int main(int argc, char* argv[])
{
    int string, fret;
    double dist0, nutdist;

    printf("Distances in mm from frets to compensated nut\n"
     "for a scale length of %0.2fmm\n\n",SCALELEN);
    printf("Fret");
    for(string = 1;string < 7;string++)
    {
        printf("\t%c%d",stringname[string],string);
    }
    printf("\n");

    for(fret = 0;fret<13;fret++)
    {
        dist0 = stringlen(0) - stringlen(fret);
        printf("%i",fret);
        for(string = 1;string < 7;string++)
        {
            nutdist = dist0 + nutcomp[string];
            printf("\t%0.2f",nutdist);
        }
        printf("\n");
    }
    return 0;
}


Figure 6:  Table of fret distances to compensated saddle for each string as generated by program.

Distances in mm from frets to compensated nut
for a scale length of 650.00mm

Fret  
E1  
B2  
G3  
D4  
A5  
E6
0
-0.30
-0.70
-1.00
-0.50
-0.50
-0.50
1
36.18
35.78
35.48
35.98
35.98
35.98
2
70.62
70.22
69.92
70.42
70.42
70.42
3
103.12
102.72
102.42
102.92
102.92
102.92
4
133.79
133.39
133.09
133.59
133.59
133.59
5
162.75
162.35
162.05
162.55
162.55
162.55
6
190.08
189.68
189.38
189.88
189.88
189.88
7
215.88
215.48
215.18
215.68
215.68
215.68
8
240.23
239.83
239.53
240.03
240.03
240.03
9
263.21
262.81
262.51
263.01
263.01
263.01
10
284.90
284.50
284.20
284.70
284.70
284.70
11
305.37
304.97
304.67
305.17
305.17
305.17
12
324.70
324.30
324.00
324.50
324.50
324.50

Figure 7:  Finished saddle shows separate compensation for each string.
nut.jpg

Final Results

Now the intonation seems perfect.  When tuned correctly the guitar has a clearer sound than it had before.




1Ervin Somogyi, "Principles of Guitar Dynamics and Design," esomogyi.com, link.
2Greg Byers, "Intonation," byersguitars.com, link.

Document History
December 20, 2017  Created.