The equal tempered scale divides the octave into twelve equal semitones. The effect of increasing a given pitch one octave is to double the frequency of the note. For example, the frequency of A4 is 440 Hertz. The frequency of A5, an octave higher, is 880 Hertz or two times the frequency of A4. To divide the octave into twelve equal parts, we must determine a constant that, when multiplied successively to the frequency twelve times, doubles the initial frequency. The constant that accomplishes this is the twelfth root of two or 1.059463 (rounded to six decimal points). If the frequency of A4, 440 Hertz, is multiplied by 1.059463 you obtain the frequency of A#4 (that is A Sharp 4) which is 466.16. Multiplying 466.16 Hertz by 1.059463 yields the frequency of B4 which is 493.88. Repeating this process twelve times in all culminates in the calculation of the frequency of A5 which is 880. Note that the ratio between any two successive notes is equivalent to the twelfth root of two. For example, the ratio of the frequencies of C5 and B4 is 523.25 divided by 493.88 which is 1.0595.
In calculating fret positions, the objective is to divide the vibrating string length in the same way the octave has been divided above. For example, the open fifth string of the guitar normally tuned to A2, when fretted at the 12th fret will produce A3 an octave higher and the frequency will double from 110 Hertz to 220 Hertz. The constant 1.059463 can be used to directly subdivide the string length and determine the fret positions. The scale length (excluding any adjustment for compensation in the actual positioning of the saddle) is divided by 1.059463 to calculate the distance from the saddle to the first fret. This calculated distance is then divided by 1.059643 to determine the distance to the second fret and so on. The result is that the position of the twelfth fret will be exactly half the distance of the scale length and that the frequency of the note played at the twelfth fret will be double the frequency of the open string.
More typically, we calculate the position of the frets measuring from the nut end instead of the saddle. For this, a little math is in order. As noted above, the distance from the saddle to the first fret, d1, is calculated by dividing the scale length, s, by the twelfth root of two. The twelfth root of two is the same as two raised to the one twelfth power hence:
d1 = s/(2^(1/12))
To measure from the nut end, we determine a distance, r, which is the distance from the nut to the first fret. Note that r is equal to the scale length, s, less the distance d1,
r = s – d1
Combining these two equations,
r = s – s/(2^(1/12))
For this method, we want to determine the ratio of scale length, s, to r. Solving for this ratio yields the following result:
s/r = 1/(1-(1/(2^(1/12)))) = 17.817
To determine the position of the first fret, the scale length is divided by 17.817. This distance is then subtracted from the scale length to determine the remaining string length. This remaining length is divided by 17.817 to calculate the distance from the first fret to the second fret. This distance is added to the distance from the nut to the first fret to determine the overall length from the nut to the second fret. This process is repeated for each fret. The constant 17.817 rounds to 18, hence the “rule of 18” that is cited as the method used historically to calculate fret positions. The length is divided by 18 as an approximation.