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Noyce Guitars
Mount Clear
Ballarat, Victoria
Australia, 3350

Equal Temperament

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 G-B 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.