Fret positions - The theory

For anyone who is interested................

The maths used to calculated fret positions is a Geometric Series and can be calculated for any given scale length of instrument.

The distance between the frets for any given scale length is calculated as follows:

The distance to the 12th fret is always 1/2 scale length.

We can use the equation for the sum of a geometric series to solve for 'r' , the common ratio. This ratio determines the rate at which the fret positions become progressively closer to eachother as you move up the fingerboard.

Thus from the equation S_n = a.( 1 - rⁿ) / (1 - r) (this is the equation used to determine the sum of a geometric series)

We can deduce that S_12[ = a.( 1-r¹²) / (1 - r), where S₁₂ is the sum of distances from the top-nut to the 12th fret position.

'a' is the first term of the series but as we shall see, this is cancelled out from both sides of the equation to solve for 'r'

We can also deduce that the 24th fret position is ALWAYS 1.5 x the distance to the 12th fret position.

Thus: S₂₄ = (1 - r²⁴) / (1 - r), where S₂₄ is the sum of distances to the 24th fret position

And because S₂₄ = 1.5 x S₁₂

a.(1 - r²⁴)/(1 - r) = 1.5 x a.(1 - r¹²)/(1 - r)

This simplifies to:

1 - r²⁴ = 1.5 x (1 - r¹²). (the denominator and the constant 'a' are cancelled out)

This can be multiplied out and rearranged into a simple quadratic equation to solve for 'r'

r²⁴ - 1.5r¹² + 0.5 = 0

factorize to solve:

(r¹² - 1) (r¹² - 0.5) = 0

Ergo, this quadratic equation gives 2 solutions

r¹² = 1

and

r¹² = 0.5

Therefore the solutions are

r = 1 (which is ignored because you cannot use a ratio of 1)

and:

r = (12th root) of 0.5 or 0.5^1/12 = 0.943874312

If you multiply any given scale length by this ratio, it will calculate the distance remaining to the bridge. However, you really need the distance to the fret positions. Not a problem

All you need to do is multiply the scale length by 1 minus the ratio (1-r) to determine the first fret position. Then multiply the remainder by the same factor (1-r), and again, and again, until you have calculated all the fret positions.

This ratio will work for all scale lengths.

All you need to do now is create a simple spreadsheet in Excel and you can calculate the fret positions for any guitar of bass using the scale length of your choice.

The 17.817 rule is derived by forming an inverse equation of the above: 1/(1-r), or 1/(1-0.5^1/12)
and dividing rather than multiplying this by the scale length. It will give you exactly the same results.