4 equidistant strings/McCarty tulip dulcimer
Instruments- discuss specific features, luthiers, instrument problems & questions
Bridge, I was in a similar fix with my Carstanjen courting dulcimer. An invaluable tool for doping out the intonation problems was this fret position calculator: http://www.liutaiomottola.com/formulae/fret.htm#mozTocId169477. There are a number of calculators on that web page. The one you want is titled "Calculating Fret Spacing for All Frets".
The calculator is set up for a chromatic fret board so the fret numbers are actually semi-tones and you'll have to adjust to accommodate the diatonic scaling of your fret board (e.g. the first fret is 2 semi-tones, the second is 4 semi-tones, the third is 5 semi-tones etc.). The equivalents for the first octave are as follows: 1~2, 2~4, 3~5, 4~7, 5~9, 6~10, and 7~12.
The StewMac web site has a similar fret position calculator ( http://www.stewmac.com/FretCalculator) that additionally provides fret to fret measurements. With this calculator you can specify "dulcimer" as a parameter input and doing so will restrict the output to the diatonic scale of your instrument - well, except it includes the 6+ fret which it labels fret #7 and calls the octave fret #8. An advantage of this calculator is that it is easy to cut and paste your data output into a spreadsheet or print out.
For the "scale length" (i.e. VSL) measure the distance from the zero-fret to the seventh fret and multiply by two, which will give you the uncompensated VSL for the instrument. Adjusting the bridge position by finding a true octave at the seventh fret gives you a compensated VSL that factors in the effect of the string tension increase caused by the act of depressing the string to the fret. A compensated VSL will be slightly larger than an uncompensated VSL and it is the latter that we want to work with - at least for now.
Once you've entered the VSL measure the distance from the nut (or zero-fret) to each of the seven frets and compare them with the calculated values and look for discrepancies (i.e. errors). The seventh fret VSL will of course be spot on automatically because that measure was the parameter used to determine our scale length input. What we want to look for is a pattern in the errors for frets 1 through 6. For example, do all errors in frets 1 to 6 come up short? I suspect they would if frets 1 to 6 play flat as you described.
Where we go from here is dependent upon the error findings and to cover all possible outcomes and remedies would make this reply overly long (if it is not already). So if I've tweaked your interest, provide feedback on what you find and I'll try to dope out a solution - if there is one.
If you use Excel or another spreadsheet program, that would be a good vehicle for keeping your notes and for doing some later data manipulation, hopefully leading to a solution of the intonation issue(s).
updated by @jim-hedman: 05/01/18 06:22:42AM