This writing concentrates
to the classical guitar intonation. It tries to show the current status of the
developments and spread the needed information to the Lutherists and to the
guitar players around the world. This should be possible through the Internet
and the New Millennium Guitar Magazine.
The information how to
reduce guitar tuning problems has been available for few years. As period in
"nylon string guitar" history it can be considered as a rather recent
development. Before that the builders have produced guitars in a traditional
way and the players have used to the inaccuracies. One could think that the now
when more information is available, the change from old to new is rapid. However
now the time has passed and still few builders have adopted the changes. The
shops are full guitars, which cannot be tuned well enough!
Causes of guitar
intonation
There are many reasons why the guitar is out of tune, so before we go to the
solutions, it is useful to remind the causes for guitar intonation.
Tuning errors, stretching
of the string during playing, we see often an artist tuning the strings even
during a performance!
Bad quality of the strings
itself, thickness variations, aging and wearing of strings ,the normal tension
and high-tension strings have different tuning properties.
Wearing of the frets, a
flat fret is not giving the same vibrating distance as a round one Wrong height
at the nut- or wrong action at the 12th fret cause errors. The tolerances of
fret locations, not correct or more probably completely missing nut
compensation cause errors.
The stiffness of the string
itself causes tuning errors. It should bend close to the ends, but due to the
stiffness of the material it cannot do it. In order to compensate the
non-moving distance, the saddle has to be comensated by moving forwards.
Also the higher harmonics
tend to go sharper due to stiffness, but the problem is smaller compared to
other errors.
Then there are resonances
caused by the guitar itself. The lowest resonance peak is normally from 90 -120
Hz depending of the size and form of the body. The soundboard and the bottom
move to different directions, which causes air to pump in and out through the
soundhole. At the next soundhole resonance the soundboard and the bottom move
to the same direction. These resonances are in the area of 150-250 Hz. The next
a bit higher resonance occurs when the soundboard and the bottom move to
different directions, but the air goes to opposite direction compared to the
1st case. These and other resonances cause audible sound when plugging the
string, it’s like hitting a small drum!
Another problem occur when
the harmonics are close to body resonances. The harmonics are affected by the
body resonances and the strings are not able to vibrate as well as normally. As
a result certain harmonics are attenuated faster and the pitch can be shifted
lower or higher. The changing of fret locations doesn’t help, because even if
one would be able to put one harmonic to correct pitch, the rest harmonics
would be then out of tune! The error has been up to 10 Hz in one
"Hauser" model, but normally with other guitars the shifting has been
about +/- 1-3 Hz at maximum. This effect can be reduced only with better
soundboard construction and internal bracing. Still the biggest problem is to
make a guitar in which all notes sound equally well. Many of the items above
prohibit getting the perfect sound from the guitar. To most of the problems
accurate construction will help, but for the two last there is no clear answer
available. Here we try to concentrate to guitar tuning and especially to
compensations. If they are set correctly, there is more room for other errors.
Guitar compensations
The author of the article in the "Guild of American Lutherie #47 -
1996" concerning "Classic guitar Intonation" was an
exceptionally innovative Greg Byers. He then presented the calculations and
experimental test results for classical guitar compensations. For those who
don't know what compensation is, it is a measure with which the string has to
be prolonged or shortened in order to produce correct pitch. The measures in
the table below are in millimeters, for a typical guitar with 650mm
scale-length using Augustine Regals with Blue Label Basses:
String
Compensation at Nut
Compensationat
Saddle
1
-0,30
1,28
2
-0,73
2,14
3
-0,97
3,14
4
-0,53
1,47
5
-0,45
1,73
6
-0,45
2,91
Table 1. Classical
guitar compensations by Greg Byers
The nut compensation
guarantees that the 1st position gets correct pitch due to stretching of the
string. Otherwise the all those notes will come sharp. Without nut compensation
the frequency errors are always audible.
The saddle compensation
takes part of both stretching and stiffness of the string. The stiffness
changes during aging and the new strings are normally easier to tune.
The measures are for the
guitar with a 650mm scale-length, if it is more, then the amount of
compensation is different. The longer scale causes more tension to the strings
and reduces the needed compensation caused by the stiffness of the strings.
The nut compensation
The nut compensation is
possible to construct to new and to old guitars. The luthier can shorten the
fretboard by 1,0 mm using a router and the new bridge-bone is adjusted so that
the compensations are according to table 1 for each string:
The saddle compensation
The saddle compensation is more difficult to implement, because the saddle bone
is straight and rather thin (2,5 mm) compared to the nut. If one sets the
guitar so that the 1st and the 6th string get the exact compensation and the
rest strings are more or less out of tune. The angle of the saddle affects much
to the resulted compensations. The following table shows three typical
variations, which all have been used for a classical guitar:
String
Needed
Compensation
a)
straight saddle
Error
Delta s
b)
1st, 6th correct
Error
Delta s
c)
1st,4th correct
Error
Delta s
1
1,28
1,28
0,00
1,28
0,00
1,28
0,00
2
2,14
1,28
-0,86
1,61
-0,53
1,34
-0,80
3
3,14
1,28
-1,86
1,93
-1,21
1,41
-1,73
4
1,47
1,28
-0,19
2,26
0,79
1,47
0,00
5
1,73
1,28
-0,45
2,58
0,85
1,54
-0,19
6
2,91
1,28
-1,63
2,91
0,00
1,60
-1,31
Table 2. Comparison of saddle constructions
The straight variant has big errors for
strings 3 and 6. The error is possible to compensate with a saddle that is
fine-tuned separately for each string, but more than half of the bone has
to be removed and this weakens the quality and power of the sound. The
use of thicker saddle bone would be better.
This variant has the 1st and
the 6th string set correctly and is also common. The problems
come with the strings 4 and 5 that are much over compensated and cannot be
corrected with fine- tuning.
The 3rd variant has strings 1 and 4 set
correctly. The errors can be corrected with fine-tuning, but the problem
of narrowing the saddle still remains, but is slightly smaller than in the
1st variant.
A new solution/ a two
part saddle
When one saddle is not so good, let’s divide it into two parts and adjust the
angles separately. As a result the strings can get very good compensations. The
proposed solution and its errors are shown in the following table:
String
Needed
Compensation
Two part saddle
Error
Delta s
1
1,28
1,30
0,02
2
2,14
2,15
0,01
3
3,14
3
-0,14
4
1,47
1,2
-0,27
5
1,73
1,75
0,02
6
2,91
2,3
-0,61
Table 3. Two-part saddle
The error with the two-part
saddle is small enough that it is doesn’t especially need fine-tuning. I put an
article about this to the "Helsinki Guitar Society Magazine 2-1999". The
other of my guitars was taken to repair to Kauko Liikanen, who one of the most
respected luthiers in Finland. The change was done by filling the
old saddle-track with rosewood and then cutting two separate tracks with a
router. The implementation is shown in the picture:
Picture 2. Two part
Saddle
The tuning results were
rather accurate when measured with an electronic tuner. Now the first series of
the guitars with a two-part saddle construction are in sale.
Excel calculations
In order to estimate the amount of frequency error at all positions along the
fretboard, an Excel spreadsheet was made. The calculations were based to Greg's
model for fretting the string and othervise to a book called "Die Gitarre
und ihr bau". When comparing the nut compensations to Greg's experimental
values, the values from the tool are very close. The following picture tries to
show the problematic 3rd string, with the following variations:
With correct nut compensation; saddle
compensation set to 0,14 mm short
With correct nut compensation; one-part
saddle compensation set to 1,8 mm short
Without nut compensation; one-part saddle
compensation set to 1,8 mm short
Picture 3. Different
compensations with G-string
The picture shows that the
guitar with the exact nut and two-part saddle compensation is superior to the
others. With the two-part saddle the accuracy remains good along the whole
scale. The one part saddle with or without nut compensation goes more and more
out of tune when the position gets higher. The other strings have naturally
better accuracy.
The real life is not as
bright, even if we know how to set the compensations, there are places to make
small errors during construction. The next picture tries to show that effect
when frets and saddle are set at the accuracy of +/- 0,2 mm.
Picture 4. The effect of
tolerances
The tuning accuracy is
immediately worse, when the tolerances are brought in. The fret locations and
the nut compensations should be done at least with the accuracy of +/- 0,2 mm. The
amount of nut compensation can be measured easily with a micrometer in order to
guarantee exact result. In order to make perfect tuning, the saddle could be
even fine-tuned after completion of the guitar.
Practical measurement
results
In order to prove the advantage of the better compensations different guitars
were measured. Two guitars were from Liikanen, two others were my own
prototypes. All guitars used nut compensation; three had a two-part saddle and
one guitar was with one part saddle. The measurement accuracy in the
"SpectraPro" spectrum analyzer was 0,336 Hz with the used settings
(sample rate 44,1 KHz, FFT size 32 768 K, Decimation ratio 4). The measurement
results when the open string is tuned correctly are shown in the following
table for the 1st and the 12th position:
Model Hauser, 1-Saddle
Model Lens,
2-Saddles
Alhambra,
2-Saddles
Landola,
2-Saddles
String
1st
12th
1st
12th
1st
12th
1st
12th
1
0,8
0,5
0,2
1,6
0,5
-2,6
0,3
-2,2
2
1,1
-1,0
-0,7
-1,1
-0,2
-1,8
0,1
-3,4
3
1,5
3,0
0,8
0,4
-0,5
1,2
0,1
-1,5
4
-0,2
-0,5
-0,4
0,7
-0,1
-0,9
-0,1
-0,5
5
-0,4
-0,6
0,3
1,3
0,5
-0,7
0,2
1,8
6
1,0
-0,5
-0,7
0,8
-0,1
-0,5
0.0
0,0
Table 4. Measurement
results of tuning errors (Hz)
As expected the Liikanen
model Lens with the new 2-part saddle was the best. The Model
"Hauser" with a 1-part saddle was good, but the 3rd string is 3 Hz
out of tune. In this guitar the saddle bone had been fine-tuned by cutting part
of it allowing slightly better compensation.
The 2-part saddle looks
better than the 1-part saddle, the listening tests show that the voice quality
is better than the measurement results. The accords seem to stay and the
typical modulation results are not present.
Later analysis showed that
the Alhambra had some rounding at the nut at the
1st position. This can have happened after adjusting of the height at the nut
or just wearing of the nut. After small filing the trouble was moved and the
result better. Here I must say that the edges of the saddle are sharp and
sometimes (not often) the string will brake.
More practical
measurements!
In overall the error is more than expected! In order to get better view of the
strings behaviour a new measurement was made for all strings. Each string was
measured up to the 12th position and the spectrum was analysed up to the 5th
harmonic. The results showed that many harmoniscs were often shifted from the
correct pitch. For intance the string 6 ( E ) had shifts between A - A#, where
the lowest 113 Hz body resonance is located. The string 4 (D) or the same notes
on string 3 (G) had bigger -1,5--+1,5 Hz shifts between A#- H, where the
flatter 231 Hz and the 253 Hz body resonances appear.
Picture 5. Frequency
errors around the body resonances
Both pictures show that the
harmonic frequencies are shifted on both sides of the resonances. The shifting
is depending of the construction of the guitar, in this example the Alhambra has been measured. .The higher
harmonics are affected as well, so the guitar sound is a compose of harmonics,
which are slightly out of tune! Another example of the effect to the higher
harmonics is shown below:
Picture 6. Harmonic
errors for string E
Allmost all harmonics have
shifts and probably the amount of those is even quite typical. But as it was
mentioned, up to10 Hz deviation has been measured, when the pitch of the 1st
harmonic was changed from G to G#! There are of course some notes between which
are not troubled and all problems are not audible.
Origin of the Wolf Tones
The reasons for these so called wolf tones has been studied in many
publications. One good source of information is the "ACUSTICA"
magazine Volume 49 (1981), which explains how the effective lenght of string is
changed close to the body resonances. When the string resonance is at lower
frequency than that of the coupled structural resonance, the bridge moves in
the same phase as the forces acting on it; the effective lenght of the string
is increased and the resonant frequency is therefore decreased. For a string
resonance above the resonant frequency of the coupled resonance, the bridge
moves in opposite phase to that of the forces acting on it, so that the
effective lenght of the string is decreased; its resonant frequency is
therefore increased. For relatively weakly coupled string resonances the
maximum damping occurs when the frequency of the string resonance coincides
with the structural resonance.
According to the theory the
effect can be controlled by reducing the Q-value of the body resonances or by
increasing the effective mass at the bridge support. Probably the new
soundboard designs have been improved in this sense. There is information in
the New Millennium Guitar Magazine articles like the interview of Jim Redgate
of his Lattice Braced Guitar and the one of the Dutch Jeroen Hillhorst. The
Hillhorst soundboard design is not revealed, but it certainly contains
interesting solutions, not ot mention the lifted fretboard and a bouble back. The
Finnish Kauko Liikanen has the "Lens resonance system" where the area
around the bridge has been build to very stiff. There is no traditional bracing
the saddle- and two-part bridge are well compensated. Could it be that once again
we are facing a revolution in guitar construction?
Conclusions
The original purpose of
this paper was to show how easy it is to build a guitar which goes well in
tune. Many practical measurements were made during writing, some at home, once
at the lutherists workshop. The spectrum analyzer results were compared with
each other and to the 3D views, which showed the attenuation of harmonics along
the time. The original purpose was not completely ashieved, but about the
accurate nut and saddle compensations we can state the following:
The accurate nut and
two-part saddle compensations are definitely needed, it gives more space for
other errors!
All strings should go
rather well in tune, if the tolerances are minimized!
The guitar sound comes
softer; you will hear the difference and you'll play better!
There are still problematic
notes that are slightly out of tune due to guitars own resonances. Development
work is still needed to correft these problems.
The PC based spectrum
analyzer "SpectraPro" was enough accurate tool, when analyzing the
problem areas.
The studies continue by
measuring advanced guitars and the wolf notes if any, probably more
measurements of different string sets
References
[1] Guild of American
Lutheries issue 47-1996, Classic Guitar Intonation - Greg Byers "Classic
Guitar Intonation" by Greg Byers Finding perfect intonation through deep
math and jiggling the string length at both ends. For some luthiers the quest
for perfection knows no bounds. The rest of us are just jealous.
[2] Helsinki Guitar Society, Kitaristi 1-1999, Kitaran sävelpuhtauden
ongelma - Kauko Liikanen
[3] Helsinki Guitar Society, Kitaristi 2-1999, Vanhan kitaran sävelpuhtaus -
Rauli Parkkali
[4] Die Gitarre und ihr bau,
Verlag Edwin Bochinsky
[5] AUDIBILITY OF INHARMONITY
IN STRING INSTRUMENT SOUNDS, AND IMPLICATIONS TO DIGITAL SOUND SYNTHESIS Hanna
Järveläinen, Vesa Välimäki and Matti Karjalainen Helsinki University of
Technology
[6] Modelling of Tension
Modulation Nonlinearity in Plucked strings, Tero Tolonen, Vesa Välimäki, Matti
Karjalainen IEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 8, NO3, 2000
[7] Kitaran akustiikasta ja sen mittaamisesta, Matti Stenroos Teknillinen
korkeakoulu 1999
[8] Theory of String
Resonances on Musical Instruments, C.E. Gough, ACUSTICA" magazine Vol 49
(1981)
Guitar making 
