Dave White wrote:
Micheal,
Good point . . . BUT if someone's first instinct is to measure/plot scale lengths for a mult-scale fretboard perpendicular to the FB centreline at the nut extremes of the fingerboard then the other aspects of adapting to making such an instrument are going to be a real struggle for them I would suggest. Conventionally a perpendicular fret is used but you don't need to have one and if nothing is perpendicular then it gets even more interesting !!
There are two ways round this - either set your scale lengths at the fretboard edges when you have worked out the required taper and nut edge angle (which means that the outer string scale lengths will be marginally shorter than these) or do the simple geometry/algebra based on the string insets and recalculate the outer FB edge scale lengths. Well three ways actually - draw the outside string paths and plot the scale along these.
Let:
LS be the long scale length(measured along the string)
SS be the short scale length (measured along the string)
w be the length of the nut edge of the fingerboard from the outer edges of the FB
b be the distance along the nut edge that the bass string centre is from the outer edge of the FB
t be the distance along the nut edge that the treble string centre is from the outer edge of the FB
Then (if the outer strings run at a constant distance from the fingerboard edge) the scale length along the bass edge is:
FBB = LS + b * (w-b-t)/5
and the scale length along the treble edge is:
FBT = SS - b * (w-b-t)/5
In your example FBS would be 28.938" and FBT would be 27.868".
You can do the algebra to allow for the outer strings not being parallel to the FB edges or do it the third way above.
Technically having all the frets on a FB perpendicular to the centre lines and the strings splayed away from the FB centre line means the scale along the string paths are "inaccurate" but the "errors" are minute in the extreme.
You and I are saying the same thing, you have to know the distancce from the centerline of the fretboard to the nut intersect of the first and sixth string and the the same at the saddle so that you can calulate the angle the stings are relitive to the centerline of the fretboard and plot the fret incraments along the centerlines of the strings. First string=short scale, sixth string=long scale.
My biggest point here is; to do this right you have to know where the string centerline is at the nut and saddle. A small change in the angle between the string centers can cause major intonation issues.
I agree that for the common still string builder (my self included) the first thought is to plot both scales starting a the nut/string intersection and lay the divisions out perpendicular to the centerline of the fretboard. I have an inprocess drafting job that is directly related to this conversation. As soon as I started laying the two scale outs I noticed this.