Enter the scale length, number of frets and number of decimal places that the calculation should be rounded to.
Note: If you are fretting an Appalachian dulcimer, then the default mode of "Mixolydian" should normally be used. Read the discussion below for an explanation.
The scale length is the vibrating length of the open string. This is usually the distance from the edge of the nut facing the bridge to the top of the saddle. If there is no saddle, then measure to the point where the string leaves the bridge.
A feature of the modern dulcimer is that its fret pattern doesn't match that of the major scale. This confuses many people, but the reason is obvious when we consider the pattern of tones and semitones formed by the frets. All major scales without accidentals have the same pattern of whole tones (T) and semitones (s) that repeats at the octave:
T-T-s-T-T-T-s
This particular pattern of tones and semitones (the pattern of the major scale) is called the Ionian mode. If we keep the same repeating pattern of tones and semitones, but start at a different place in the pattern (on a different degree of the scale), we get a different mode:
| Mode | Pattern | Tonic relative to major scale | How it sounds |
|---|---|---|---|
| Ionian | T-T-s-T-T-T-s | I | Major (major scale) |
| Dorian | T-s-T-T-T-s-T | II | Minor |
| Phrygian | s-T-T-T-s-T-T | III | Minor (flamenco scale) |
| Lydian | T-T-T-s-T-T-s | IV | Major |
| Mixolydian | T-T-s-T-T-s-T | V | Major |
| Aeolian | T-s-T-T-s-T-T | VI | Minor (natural minor scale) |
| Locrian | s-T-T-s-T-T-T | VII | Minor |
Because each mode starts on a successively higher degree of the scale, II, III, IV etc. the pattern for each consecutive mode is shifted one place to the left.
If you think of a C major scale, then the Ionian mode is the normal major scale you get when you start playing the scale on a C (I). You get the Dorian mode when you start playing it on a D (II) and so on. The mode is independent of absolute pitch, so this works for all major scales, not just C major.
The "standard" fret pattern for the Appalachian dulcimer is T-T-s-T-T-s-T and from the table above, you can see that this is the Mixolydian mode. However, you can play in any of the modes by starting playing at a different fret (see table below). For a Mixolydian dulcimer, you can see that the Ionian mode pattern starts on the 3rd fret, so it is actually set up well to play pieces in Ionian mode (major scales) provided they never go more than 3 frets below the tonic. Oddly enough, Mixolydian mode pieces are not as well accommodated, because if they go below the tonic, you have transpose them up an octave and start them on fret 7. From this, you can see that there is a strong argument for fretting the dulcimer three modes lower than the one in which you wish to play in order to accommodate notes below the tonic. For example, to play well in Aeolian mode, you might actually fret the dulcimer in Phrygian mode.
| Fret | Mode |
|---|---|
| 0 (open string) | Mixolydian |
| 1 | Aeolian |
| 2 | Locrian |
| 3 | Ionian |
| 4 | Dorian |
| 5 | Phrygian |
| 6 | Lydian |
The easiest way to change the mode of a dulcimer is to apply a capo. This also changes the absolute pitch, but you can compensate for this by adjusting the tuning. For example:
By examining the fret of the first and last notes in a piece of music, you can often determine its mode. However, at the beginning a piece there may be leading notes up to the true tonic that are often in a partial bar.
Early Appalachian dulcimer makers often fretted to different modes because the folk tunes they wanted to play were often modal. Our fret position calculator supports all modes. However, unless you have very specific requirements and really know what you are doing, fretting to anything other than the Mixolydian mode will make your life very difficult. In particular, "standard" dulcimer tablature, which assumes Mixolydian mode, will no longer work.
It is also worth pointing out that the modes accessible to us in 12 tone equal temperament are not tuned the same as the modes used by the old dulcimer makers. These makers generally didn't calculate fret positions, but rather found them experimentally by placing a piece of fret wire under a string and moving it to achieve maximum consonance. We can only approximate consonance in equal temperament, so none of our modes will sound as pure.
The dulcimer is not chromatic and traditionally it is missing all accidentals. This means it can only play successfully in a few keys - whatever you tune it to, and a few closely related keys. By adding extra frets to the dulcimer, you can make more keys accessible. Clearly, if you add enough frets, it becomes chromatic, and you can play in any key, but this makes the instrument harder to play (it becomes like a guitar), and I think it detracts a great deal from its fun and charm. Nevertheless, it has become very common to add two extra frets:
I would not add any more frets than this, but the great advantage in making your own instrument is that it is up to you!
In this section we will consider the theory behind caculating the fret positions for instruments in 12 tone equal temperament.
The fundamental frequency of a vibrating string, $f_0$, is:
These proportionalities are expressed in Mersenne's Law:
$$f_0 = \frac{1}{2L}\sqrt{\frac{F}{\mu}}$$
For a given string on a given instrument, $F$ and $\mu$ are fixed, so:
$$f_0 \propto \frac{1}{L}$$
If you double the string length you half the frequency. Conversely, if you half the string length you double the frequency. So we can generate octaves by multiplying or dividing by a factor of 2. This is because the ear responds to sound in a more or less logarithmic fashion.
What factor, $n$, would we have to multiply or divide by to increase or decrease frequency by one semitone? We know that in 12 tone equal temperament the octave is divided into 12 equal semitones therefore:
$$ n^{12}f_0 = 2 f_0$$ $$n = \sqrt[12]{2}=1.059463$$
This is where the magic number 1.059463 comes from when you are calculating fret positions - it is the 12th root of 2.
Now we can calculate the position of the 1st fret as follows:
$$L_{1} =\frac{ L}{\sqrt[12]{2}}$$
Where $L_{1} =$ distance from bridge to first fret.
The distance from the bridge to the second fret is given by:
$$L_{2} =\frac{ L_{1}}{\sqrt[12]{2}} = \frac{ L}{{\sqrt[12]{2}}} \times \frac{1}{{\sqrt[12]{2}}} = \frac{ L}{({\sqrt[12]{2})}^{2}}$$
And generally, for fret $n$:
$$L_{n} = \frac{ L}{({\sqrt[12]{2})}^{n}}$$
When fretting an instrument, we always measure the distance from the nut, and this is given by:
$$D_{n} = L - L_{n}$$
All 12 tone equal temperament fret position caclulators use this formula to calculate the fret positions.