Written by Ian Noyce
(First published in the Quarterly Journal of the Guild
American Luthiers in 1977)
Sections in this article
Consonance and the Pythagorean
scale
The Tempered scale
Tuning the Guitar
4th
and 5th fret method
The harmonic
method
CONSONANCE AND THE PYTHAGOREAN SCALE
As long as we play only single note music, it does not matter
greatly how we choose the frequencies of the notes of the
scale we use. However, once we play more than one note at
a time we must consider how the notes we play sound when combined.
The combination of two or more notes which have a pleasing
sound is called a consonance.
The Greek scholar Pythagoras is credited as being the first
person to have observed this phenomenon of consonance. He
used an instrument called a monochord. (see fig 2) to split
a stretched string into two varying segment lengths.
Using this he found that when plucking both segments simultaneously
consonance occurred when the ratios of the string segments
were 1: 1, 1: 2, 2: 3, and 3: 4.
(Although Pythagoras didn't relate these string lengths to
frequency of pitch, we now know that the frequency of a string
is inversely proportional to the length of the string.) (i.e.
L=1/f). An example of this on a guitar is when halving the
open string length, i.e. playing the 12th fret note we get
an octave, or double the frequency. For example, open A is
110 cycles per second and the 12th fret on the A string is
220 c.p.s.
From this we can build up a scale with the following intervals:
The octave has a ratio of 2: 1 or 1: 2 (i.e. if we start with
a note of say 50 c.p.s. the octave below it is 25 c.p.s. and
the one above 100 c.p.s. It is essential to visualize both
the musical and the physical meaning of these intervals. All
musical intervals are based on the ratio of the frequencies
involved and this can be seen on the guitar as the ratio of
string lengths.
The ratio of a fifth interval is 3: 2 of 2: 3 (e.g. to up a
fifth from a given frequency we multiply that frequency by
3/2 or 1.5. To go down a fifth we multiply by 2/3. In terms
of string length, to go up a fifth divide the string length
by 3/2.) (Remember, the inverse relationship between frequency
and string length the higher the frequency, the shorter
the string.)
The ratio of a forth is a 4: 3, of 3: 4 depending on whether
we go up or down a fourth.
Okay, so much for the arithmetic!
We can now build up a scale using this information starting
with the note C of frequency F.
C(f) x 2 gives octave C (2f). Going down a fifth from C(2f)
we get F(4/3f)) 2f x 2/3 = 4/3f.
You my observe this is the same as going up a fourth from
C(f) by multiplying f by 4/3, the rule for going up a fourth.
Going up a fifth from C(f) we get G(3/2f) down a fourth from
G we get (3/2f) x 3/4 =9/8f. This of course is D (9/8f.)
Up a fifth from D we get A (9/8f x 3/2 = 27/16f)
We now have a pentatonic scale:
Note: 
C

D

F

G

A

C

Frequency: 
f

9/8f

4/3f

3/2f

27/16f

2f

Continuing this process yields an eight note scale called
the Pythagorean Diatonic Scale. For simplification lets give
the first C in the scale a frequency of 1. Remember it is
the ratio of frequencies that is important.
Note: 
C 

D 

E 

F 

G 

A 

B 

C 
Frequency: 
1 
: 
9/8 
: 
81/64 
: 
4/3 
: 
3/2 
: 
27/16 
: 
243/128 
: 
2 
Interval: 

9/8 

9/8 

256/243 

9/8 

9/8 

9/8 

256/243 

N.B. the intervals are found by obtaining the ratio of the
frequency of one note to that of the note below it. e.g. the
interval F to G is found by dividing 3/2 by 4/3 i.e. 3/2 x
3/4= 9/8.
The ratio 9/8 = 1.125 whereas the ratio 256/243 = 1.053 representing
a smaller interval.
As we know, the intervals between the third and the fourth,
and between the seventh and the eight steps of the diatonic
scale are the two semitone of half step intervals.
In the above scale the 9/8 interval is called the Pythagorean
scale whole tone and the 256/243 diatonic semitone.
Now, when the use of other intervals such as the major and
minor thirds, and their inversions, the minor sixth and the
major sixth (all having simple arithmetical ratios of 5: 4,
6: 5, 8: 5,5: 3 respectively), came into common usage, it was
found that these intervals sounded either unpleasantly sharp
or flat by a small amount. e.g. The major third was sharp,
and the minor third interval sounded flat.
Several scale systems were experimented with over many years,
all of which had various problems with some of the intervals,
particularly as music developed in the direction of more free
modulations into remote keys.
THE TEMPERED SCALE
As all the scales tried previous to the tempered scale involved
an octave of 12 half steps, with slightly different intervals
between these half steps, it was eventually decided to try
a 12 note scale with exactly even intervals, and this is done
as follows.
Starting with a note (let's call it C) of nominal frequency
of 1, a scale can be built up with exactly the same interval
between each note. (Let's call that interval .)
The following chart sets out the results of this.
In explanation: Starting with C as a frequency of 1, as done
above, then C# has the relative frequency ,D
is
times this or
xor
.
D# is
times the result, or
and so on.
You could check this using the Pythagorean relationships
stated previously by multiplying all the intervals together
and they also come out at 2 as follows:
Note: 
C 

D 

E 

F 

G 

A 

B 

C 
Frequency: 
1 
: 
9/8 
: 
81/64 
: 
4/3 
: 
3/2 
: 
27/16 
: 
243/128 
: 
2 
Interval: 

9/8 

9/8 

256/243 

9/8 

9/8 

9/8 

256/243 


1.125

x 1.125 
x 1.0535

x 1.125

x 1.125

x 1.125

x 1.0535

=2.00

To find the value of the interval
in the figure, and remembering that the octave ratio is 2: 1,
we can then say:
= 2 so
must be the twelfth root of 2.
i.e.
=
= 1.0594631
(At this stage I must apologise for those who have become
bewildered by the above juggling, but anyone who studied maths
at school up to 10th year should understand it, and I therefore
assume that enough of you will appreciate it to warrant its
inclusion.)
Now we can look at how frets are placed on a guitar!
As stated earlier, string length (L) is inversely proportional
to frequency or pitch, (f). i.e. L 1/f
so we simply choose the string length (i.e. theoretical distance
from the nut to the bridge saddle) and start dividing by or
1.0594631.
Fret No. 
Distance of fret from bridge
in mm. 
0 (nut) 
650.000 

1.0594631 
= 
613.518 
1 
613.518 

" 
= 
579.084 
2 
579.084 

" 
= 
546.583 
3 
546.583 

" 
= 
515.905 
4 
515.905 

" 
= 
486.950 
5 
486.950 

" 
= 
459.619 
6 
459.619 

" 
= 
433.823 
7 
433.823 

" 
= 
409.474 
8 
409.474 

" 
= 
386.492 
9 
386.492 

" 
= 
364.800 
10 
364.800 

" 
= 
344.325 
11 
344.325 

" 
= 
325.000 
12 
325.000 

" 
= 
and so on 
As you can see the 12th fret distance is exactly half the
scale length.
If we continued through the 24th fret, or second octave fro
a given string, the distance would be of
650 or 162.500 mm.
TUNING THE GUITAR
Let's assume the guitar being tuned is properly set up so
that the intonation is correct. (More on this later.)
The two most common methods of tuning are
(1) the 4th and 5th fret method and
(2) the harmonic method.
Both of these methods are often misunderstood through confusion
regarding perfect (or Pythagorean) intervals and even tempered
intervals.
1. The 4th and 5th fret method.
Theoretically this is the simplest method as it simply involves
tuning unison intervals. The A string can be tuned to an A
tuning fork, then the bass E is fretted at the fifth fret
and tuned in unison with the A. The D string is tuned in unison
to the fifth fret on the A, the G to the fifth fret on the
D, the B to the fourth fret on the G and the top E to the
fifth fret on the B string. In practice this can be difficult
for a number of reasons, the most common ones being:
a. 
Any errors are accumulative. 
b. 
Any falseness in strings will probably introduce errors,
depending on the skill of the person tuning. 
c. 
If the bridge is not properly adjusted or compensated,
errors will definitely occur. 
2. The Harmonic Method.
This is probably the most misunderstood method, and in fact
it is inherently inaccurate! The reason for looking at perfect
intervals and even tempered intervals was partly to throw
light on this method of tuning.
Harmonics are produced when a vibrating string is made to
vibrate in multiples of its fundamental pitch. For this reason,
harmonic intervals are always "perfect" or pure,
and this method, when done exactly, does not work on a guitar,
which is made to tune to the equal tempered scale.
The common tuning method is as follows:
Tune the A string to a tuning fork, the tune the bass E to
the A by playing the fifth fret harmonic on the E string (produces
a note E, two octaves higher than the open string) with the
7th fret harmonic on the A string (also produces a high E).
This is represented on the A and D strings, then the D and
G strings. Then the fourth and fret harmonic of the G string
(a B note) is played with the fifth fret harmonic of the B
string (also a B, two octaves higher than the open B). The
fifth fret harmonic on the B string is then used to tune the
7th fret harmonic of the top E.
If you follow this method accurately, the guitar will just
not play in tune, and the GB interval will be particularly
bad. In fact, the G will be a pure major third away from B,
and the tempered major third is a much wider interval than
the perfect third.
In order to see how much difference there is between tuning
the guitar to perfect intervals (such as when using the harmonic
method) and tempered intervals, let's look at the open strings
of the guitar, when tuned correctly to tempered intervals.
Whenever two notes tuned properly are played together, "beats"
will be heard, (i.e. an alternate increase and decrease in
volume will be heard. A perfect interval has no beats). The
beats occur when adjacent open strings on a guitar are played
together, as follows:
String: 
6 
5 
4 
3 
2 
1 
Note: 
E 
A 
D 
G 
B 
E 
Interval: 
Fourth 
Fourth 
Fourth 
Major Third 
Fourth 

Beats: 
0.3 / sec 
0.5 / sec 
0.6 / sec 
0.8 / sec 
1 / sec 

It should now be apparent that the harmonic method of tuning
is inaccurate, but if you understand why, you can compensate
for its inadequacies and still use it.
I find a composite method of tuning works for me as follows:
Tune the E, A, D, and G strings using the harmonic method
described earlier but widen each interval by the finest margin
possible, e.g. While tuning the D by playing the seventh fret
harmonic on the D with the fifth fret harmonic on the A, pull
the D up until it is perfect i.e. no beats, then raise it
ever so slightly to widen the interval just a touch (The difference
here is two hundredths of a semitone, or two cents  pronounced
"sonts"). Then, having tuned the lower four strings
this way, check them by playing the 12th fret harmonic on
the E to check the E note on the 2nd fret of the D string,
and likewise for the G string using the 12th fret harmonic
on the A. A further check using the fifth fret unison method
is also helpful. With a little practice you'll find that you've
tuned correctly with the slightest adjustments made initially
to the harmonic method. To tune the top E, use the 12th fret
harmonic on the G string and the third fret of the E string,
and tune the B string similarly using the 12th fret harmonic
on the D string. As these last 2 strings involve unison notes
in tuning, then no compensation for the harmonic method is
necessary. Then, if the guitar is in tune, the fifth fret
harmonic on the bass E string should be in unison with the
open top E string.
If you've done everything correctly, and the guitar is still
out of tune, then it's likely that the strings are faulty
or the bridge compensation is not correct.
If the above information is confusing then it's probably
due to the fact that I've tried to cram in too much information.
For those who'd like to read up on this subject more fully,
below is a list of recommended material:

The Acoustic Foundations of Music,
by John Backus, published by W.W. Norton and Co. Inc.,
NY;

Complete Guitar Repair
by Hideo Kasimoto (Oak Publications); and

John Carruther's column in Guitar Player.
