Copyright © 2018 by Wayne Stegall
Created March 15, 2018. See Document History at end for
details.
Straight Intonation
Calculate
the
approximate
intonation
error
of
an intonation with a straight nut
and saddle.
Introduction
If the intonation correction at the nut and saddle is different for
every string on a guitar, then the straight nut and saddle of most
guitar
setups are a compromise. I thought it might be interesting to estimate
how much error is present in these setups and whether it is reasonably
audible. The custom compensation is assumed perfect then each
alternative intonation is calculated for error in mm and then in
cents. Cents are an exponential percentage calculation tailored
to representing frequency error. 100 cents = 1 semitone.
The reader is then to infer an approximately linear and/or
inverselylinear relationship between changes in string length and the
associated frequencies.
As a point of reference it may be reasonable to think the error
introduced by equaltemperament tuning relative to perfect harmonic
tuning (just temperament) is tolerated by music listeners.
Therefore I have calculated this error and presented it in
table 1 below.
Table
1:
Equaltemperament
error
for
common
intervals

interval 

just
ratio

semitones

error(cents) 

unison 

1

0 
0.000000 

semitone 

16/15

1 
11.731285 

whole tone


9/8

2 
3.910002 

minor 3rd 

6/5

3 
15.641287 

major 3rd 

5/4

4 
–13.686286 

4th 

4/3

5 
–1.955001 

5th 

3/2

7 
1.955001 

6th 

5/3

9 
–15.641287 

7th 

15/8

11 
–11.731285 

octave 

2

12 
0.000000 


Here octaves are perfect. The smallest error represented by the
intervals of a fourth and a fifth (1.955001 cents) might be a
comparison standard.
Custom and average nut intonation
All normal nut setups are both straight and not slanted.
Therefore the average nut intonation is just that.
(1)

Δ_{nutaverage} =

Δ_{nutstring(1)} + Δ_{nutstring(2)}
+ Δ_{nutstring(3)} + Δ_{nutstring(4)} + Δ_{nutstring(5)}
+ Δ_{nutstring(6)}
6 
In these spreadsheet calculations, the G string appears to be the
primary violator (as everyone who plays already knows) for the
nut. Yet it is still less than the reference error.
String
Name 

String
Number 

Δ_{nutoptimal}
(mm) 

Δ_{nutaverage}
(mm) 

ε_{nut}
(mm) 

ε_{nut}
(cents) 
E 

1 

0.3 

0.583333 

0.283333 

0.754476 
B 

2 

0.7 

0.583333 

–0.116667 

–0.310762 
G 

3 

1 

0.583333 

–0.416667 

–1.110121 
D 

4 

0.5 

0.583333 

0.083333 

0.221939 
A 

5 

0.5 

0.583333 

0.083333 

0.221939 
E 

6 

0.5 

0.583333 

0.083333 

0.221939 
Figure
1:
Fully
compensated
nut


Figure
2:
Average
compensated
nut 

Custom and average saddle intonation
Since most straight saddles are slanted to pass through the closest
intonation of each string, I decided to calculate a slanted average
line through the ideal values. This is done by a calculation
called a linear regression, available on some scientific calculators
and some internet calculators as well. My first calculation
produced
equation 2
below. In the table that follows the G string exceeds and the A
and D strings approach the reference error.
(2)

compensation_{saddle} =
0.148571 × num_{string} + 1.58









String
Name 

String
Number 

Δ_{saddleoptimal}
(mm) 

Δ_{saddleaverage}
(mm) 

ε_{saddle}
(mm) 

ε_{saddle}
(cents)_{s}

E 

1 

1.3 

1.728571 

0.428571 

1.141096 
B 

2 

2.1 

1.877142 

–0.222858 

–0.593670 
G 

3 

3.1 

2.025713 

–1.074287 

–2.863663 
D 

4 

1.5 

2.174284 

0.674284 

1.794982 
A 

5 

1.7 

2.322855 

0.622855 

1.658141 
E 

6 

2.9 

2.471426 

–0.428574 

–1.141856 
Figure
3:
Linear
regression
plots
average
saddle
line
through
ideal
points.


Comparison with special G compensation
With the realization that the G string is a difficulty, many saddles
are straight except for a customized G intonation. A calculation
without G produces
equation 3
below which is more ideal for the other strings. G is now
presumed exactly correct. Now all intonation error is less than
that of a fifth interval.
(3)

compensation_{saddle} =
0.186047 × num_{string} + 1.23023









String
Name 

String
Number 

Δ_{saddleoptimal}
(mm) 

Δ_{saddleaverage}
(mm) 

ε_{saddle}
(mm) 

ε_{saddle}
(cents) 
E 

1 

1.3 

1.416277 

0.116277 

0.309669 
B 

2 

2.1 

1.602324 

–0.497676 

–1.326036 
G 

3 

3.1 

3.100000 

0 

0 
D 

4 

1.5 

1.974418 

0.474418 

1.263122 
A 

5 

1.7 

2.160465 

0.460465 

1.225985 
E 

6 

2.9 

2.346512 

–0.553488 

–1.474808 
Figure
4:
Linear
regression
plots
average
saddle
line
through
all
points
except
G. 

Example error of a misplaced bridge
My need to resetup my guitar's intonation likely resulted from an
error in the factory placement of the saddle. Examination of my
guitar's saddle shows little or no additional intonation for
strings A and D. Therefore shifting the previous
compensation values forward by an amount producing correct intonation
for these strings (≈ –0.5mm) will allow evaluation of intonation error
that I considered objectionable. The result is excessive error
for the B and E(6) strings. G would have been worse if it had not
had a special factory setback.
Figure
5:
Custom
saddle
intonation
shows
little
additional
compensation
for
the
A
and
D strings.


()

compensation_{saddle} =
0.755812 × num_{string} + 1.23023









String
Name 

String
Number 

Δ_{saddleoptimal}
(mm) 

Δ_{saddleaverage}
(mm) 

ε_{saddle}
(mm) 

ε_{saddle}
(cents) 
E 

1 

1.3 

0.941859 

–0.358141 

–0.954149 
B 

2 

2.1 

1.127906 

–0.972094 

–2.591049 
G 

3 

3.1 

2.625582


–0.474418


–1.264044

D 

4 

1.5 

1.500000 

0.000000 

0.000000 
A 

5 

1.7 

1.686047 

–0.013953 

–0.037163 
E 

6 

2.9 

1.872094 

–1.027906 

–2.739930 
Figure
6:
Factory
compensation
in
error
by
≈
–0.5mm.


Final remarks
I have read that an intonation with correctly placed straight nut and
saddle (except for a G setback) is musically pleasing to all but a
few. Only some sensitive individuals.benefit from a custom
intonation. In my case I had the misfortune of a misplaced
bridge and the fortune that it was placed correctly for a custom
intonation to fall in the range of a normal saddle blank.
Disclaimer: Principle is valid for all guitars, however
the exact data will be different for nonclassical ones.
^{1}See related article: Custom Intonation.
Document History
March 15, 2018 Created.