The Mudcat Café TM
Thread #25849   Message #2482555
Posted By: JohnInKansas
02-Nov-08 - 01:19 PM
Thread Name: Making Catspaw's 2-String Stick Dulcimer
Subject: RE: Making Catspaw's 2-String Stick Dulcimer
Fret locations the easy(?) way.

THE THEORY

For a "tempered scale" instrument, the ratio of the frequencies for two notes a semitone apart is the "twelfth root of two."

Frequency 1/Frequency 2 = 2(1/12) = 1.059463

The string length (nut to fret) is inversely proportional to the frequency for notes played on the same string, so the fret for the note a semitone above the adjacent note must be 1/1.059463 times as far from the nut as the fret for the next higher note.

1/1.059463 = 0.943874

The number 0.943874 is the cosine of an angle of 13.125751 degrees.

The tangent of an angle of 13.125751 degrees is 0.233181.

THE METHOD

1. On a large piece of paper, draw a line the length of your fretboard from nut to bridge (or in the case of a dulcimerish instrument from upper nut to lower nut).

2. Label one end of the line "A" and the other end "B0".

3. Multiply the string length – distance A to B0 by 0.233181. Call this number "T."

4. Draw (construct) a line going through point B0 perpendicular to the line An – B0 and measure carefully distance "T" from B0 on the line and mark point "C."

5. Draw the line from point A to point C. (Call this the "construction line" if you want a name for it.) This line will be 13.125751 degrees from the fretboard line – or as close to it as you'll get with the care you've used to draw it.

6. With center at point A, and radius to go through point B0, draw an arc up to the construction line A – C.

7. Where the arc hits the line A – C, mark "construction point C1." Then draw the perpendicular to line A – B0, from point C1, back down to the line A – Bo. Where the perpendicular hits the fretboard line is the location of the fret one semitone above the open string, i.e. the location for the first fret. Call that point B1.

8. Continue with an arc centered at A through point B1, to intersect the line A-C (at point C2), and drop the perpendicular back down to the fretboard to get point B2, for the second fret.

9. Center A through B2, to the line A-C, (point C3) and drop back down to the fretboard for B3, the location of the third fret.

10. Continue in this manner until you have as many fret locations as you want. Note that it's usually hard to get a good "tone" more than about an octave and a half up from the open note on a string, so more than about 30 frets is probably "overkill." Common dulcimer construction would require you to go to B29, but after omitting "the black keys" you'd "install" frets at only 17 of the places found (or 19 if you want the 6.5 frets).

You will need a compass long enough to make an arc with the radius equal to the length of your open string. Hobby shops and/or hardware stores should have "trammel points" that you can attach to a "yard stick" that should suffice if you draw carefully. Note that you don't really need a "pencil point" to actually draw the arcs. You can use "pointy ends" on your "beam compass" just to prick the paper on construction points on the "construction line. Dropping the "perpendiculars to the fretboard" is easily done with a reasonably large "drafting triangle" – or better, with a pair of triangles – if you don't have a "machine" for making precision drawings.

END NOTE:

Once you lay it out on paper, it's a lot easier than it looks from just the description.

This method, of course, gives fret locations for a complete chromatic scale. For a dulcimer, or other "strum stick" instrument, you'll want to omit the appropriate fret locations for the sharps/flats; but you do have to lay them all out on the paper.

John