Around the middle of the 6th century B.C. (550's), Pythagoras developed a 7 tone scale (c,d,e,f,g,a,b) that was created by using a series of perfect 5th's, 5 tones of which were each a factor of 3/2 from each preceeding 5th, and one tone was a perfect fifth lower than the initial tone (a factor of negative 3/2, which is the same as a positive 2/3). The following calculations create the Pythagorean scale:
1) First compute the intervals between each successive 5th
F = 2/3 (Lower fifth of "c")
c = 1 (fundamental pitch)
g = 3/2 (1st upper 5th: c,d,e,f,g = 5 notes, g is therefore the 5th)
d = (3/2)**2 (2nd upper 5th: g,a,b,c,d = 5 notes, d is the next upper 5th)
a = (3/2)**3 (etc.)
e = (3/2)**4
b = (3/2)**5
2) Multiply the fractions (** is the exponential function)
F = 2/3
c = 1
g = 3/2
d = 9/4
a = 27/8
e = 81/16
b = 243/32
3) multiply by the factor of 2 necessary to position the note between the fundamental tone and the first octive. This is called "normalizing" to the first octave.
F = 2/3 * 2 = 4/3
c = 1 (already between 1 and 2)
g = 3/2 (already between 1 and 2)
d = 9/4 * 2 = 9/8
a = 27/8 * 2 = 27/16
e = 81/16 * 4 = 81/64
b = 243/32 * 4 = 243/128
4) Order the notes in ascending order by frequency ratio
c = 1
d = 9/8 = 1.125
e = 81/64 = 1.266 -> This ratio will be used in "The Acoustics of Music" section
f = 4/3 = 1.333
g = 3/2 = 1.500
a = 27/16 = 1.687
b = 243/128 = 1.898
5) compute the intervals between each tone in the scale
(dividing fractions is the same as multiplying by the inverse)
c to d = (9/8)*1 = 9/8 -> (pythagorean whole tone)
d to e = (81/64)*(8/9) = 9/8
e to f = (4/3)*(64/81)) = 256/243 -> (pythagorean semi-tone)
f to g = (3/2)*(3/4) = 9/8
g to a = (27/16)*(2/3) = 9/8
a to b = (243/128)*(16/27) = 9/8
b to c' = 2 * 128/243 = 256/243
Why this scale doesn't work in our musical system
This is the limit of the useful extent of the Pythagorean system. If you continue this method to calculate the rest of the chromatic pitches used in our musical scale it fails to yield tones which can be used satisfactorily in our musical system. To illustrate the problem, steps 1-5 from above are applied to create the remaining tones and intervals that are used in our 12 tone system:
f#= (3/2)**6 = 729/64 = 729/64 * 8 = 729/512 = 1.424
c#= (3/2)**7 = 2187/128 = 2187/128 * 16 = 2187/2048 = 1.068
G#= (3/2)**8 = 6561/256 = 6561/256 * 16 = 6561/4096 = 1.602
d#= (3/2)**9 = 19,683/512 = 19,683/512 * 32 = 19,683/16,384 = 1.201
a#= (3/2)**10 = 59,049/1024 = 59,049/1024 * 32 = 59,049/32,768 = 1.802
e#= (3/2)**11 = 177,147/2048 = 177,147/2048 * 64 = 177,147/131,072 = 1.352
b#= (3/2)**12 = 531,441/4096 = 531,441/4096 * 128 = 531,441/524,288 = 1.014
The last tone, "b#", should be equal to 2, the octave. By positioning the note between the 1st and 2nd octave, we get a result that shows that "b#", (enharmonically = "c") , is slightly higher than the fundamental. Let's continue to compute the intervals between the tones after ordering the chromatic pitches and the normal pitches in one 12 tone scale (we actually need to make more than 12 tones because we'll see that enharmonic tones are not equal to one another in this scale):
c = 1
c# = 1.068
d = 1.125
d# = 1.201
e = 1.266
f = 1.333
e# = 1.352 (should be enharmonically equal to "f")
f# = 1.424
g = 1.500
g# = 1.602
a = 1.687
a# = 1.802
b = 1.898
b# = 1.014 (should be enharmonically equal to "c'" = 2)
Finally, we'll compute some of the intervals between the tones (in decimal form to make the work a little easier) and then discuss the problems with this scale.
c to c# = 1.068/1 = 1.068
c# to d = 1.125/1.068 = 1.053
d to d# = 1.201/1.125 = 1.068
d# to e = 1.266/1.201 = 1.053
e to f = 1.333/1.266 = 1.053
e to e# = 1.352/1.266 = 1.068
In our first computation, we arrived at the conclusion that Pythagorean whole tone was 9/8=1.1250, and that the Pythagorean semi-tone was 256/243=1.053. We see that the semi-tone in this scale is not consistent between all intervals. It alternates between the values 1.068 and 1.053. The difference is called the Pythagorean comma. Notice that if you compute a whole tone by multiplying two successive semi-tones you arrive at the 1.125 value. The problem here is that if we want to play a chromatic musical passage, the intervals between the notes in the scale are not equal, and, we don't get back to the octave after 12 tones! It gets really messy when you try to design an instrument that can be used in many tonal keys. How do you position the frets on the guitar so that it sounds correct in all keys?
I have included this discussion on the Pythagorean system of tuning so that you could be convinced that our equal-tempered method of tuning is necessary. If you plan on using the guitar to play classical music, it must be tuned to conform to the tuning used by other classical instruments. That is the equal-tempered method of tuning and that is the method we will use.
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