Calculating Fret Spacing For A Mountain Dulcimer

The Table

D-Fret    ET *  | Deg JI-05   cent |    Mix     Aeo     Loc     Ion     Dor     Phr     Lyd
------  ------  |----------------- | -------------------------------------------------------
   0      0.000 |   I   01    0.00 |   0.000   0.000   0.000   0.000   0.000   0.000   0.000
   0+     0.056 |  ii   03   11.43 |   0.062   0.040   0.040   0.062   0.052   0.040   0.062
   1      0.109 |  II   03    3.80 |   0.111   0.100   0.111   0.100   0.111   0.100   0.100
   1+     0.159 | iii   05   15.26 |   0.167   0.156   0.147   0.167   0.167   0.147   0.156
   2      0.206 | III   05  -14.03 |   0.200   0.200   0.200   0.200   0.200   0.200   0.200
   3+     0.251 |  IV   03   -2.01 |   0.250   0.250   0.250   0.250   0.259   0.250   0.250
   3      0.293 |   v   03  -10.03 |   0.289   0.280   0.289   0.297   0.289   0.280   0.297
   4      0.333 |   V   03    1.90 |   0.333   0.325   0.333   0.333   0.333   0.333   0.325
   4+     0.370 |  vi   05   13.35 |   0.375   0.360   0.360   0.375   0.375   0.360   0.375
   5      0.405 |  VI   05  -16.02 |   0.400   0.400   0.400   0.400   0.407   0.400   0.400
   6      0.439 | vii   05   17.18 |   0.444   0.438   0.431   0.438   0.444   0.438   0.438
   6+     0.470 | VII   03  -12.03 |   0.467   0.460   0.467   0.467   0.467   0.467   0.473
   7      0.500 |   I   01    0.00 |   0.500   0.500   0.500   0.500   0.500   0.500   0.500
   7+     0.528 |  ii   03   11.43 |   0.531   0.520   0.520   0.531   0.526   0.520   0.531
   8      0.555 |  II   03    3.80 |   0.556   0.550   0.556   0.550   0.556   0.550   0.550
   8+     0.580 | iii   05   15.26 |   0.583   0.578   0.573   0.583   0.583   0.573   0.578
   9      0.603 | III   05  -14.03 |   0.600   0.600   0.600   0.600   0.600   0.600   0.600
  10      0.625 |  IV   03   -2.01 |   0.625   0.625   0.625   0.625   0.630   0.625   0.625
  10+     0.646 |   v   03  -10.03 |   0.644   0.640   0.644   0.648   0.644   0.640   0.648
  11      0.666 |   V   03    1.90 |   0.667   0.662   0.667   0.667   0.667   0.667   0.663
  11+     0.685 |  vi   05   13.35 |   0.688   0.680   0.680   0.688   0.688   0.680   0.688
  12      0.703 |  VI   05  -16.02 |   0.700   0.700   0.700   0.700   0.704   0.700   0.700
  13      0.719 | vii   05   17.18 |   0.722   0.719   0.716   0.719   0.722   0.719   0.719
  13+     0.735 | VII   03  -12.03 |   0.733   0.730   0.733   0.733   0.733   0.733   0.736
  14      0.750 |   I   01    0.00 |   0.750   0.750   0.750   0.750   0.750   0.750   0.750
  14+     0.764 |  ii   03   11.43 |   0.766   0.760   0.760   0.766   0.763   0.760   0.766
  15      0.777 |  II   03    3.80 |   0.778   0.775   0.778   0.775   0.778   0.775   0.775
  15+     0.790 | iii   05   15.26 |   0.792   0.789   0.787   0.792   0.792   0.787   0.789
  16      0.802 | III   05  -14.03 |   0.800   0.800   0.800   0.800   0.800   0.800   0.800
  17      0.813 |  IV   03   -2.01 |   0.812   0.812   0.812   0.812   0.815   0.812   0.812
  17+     0.823 |   v   03  -10.03 |   0.822   0.820   0.822   0.824   0.822   0.820   0.824
  18      0.833 |   V   03    1.90 |   0.833   0.831   0.833   0.833   0.833   0.833   0.831

Using the Table

The table can be used for calculating fret spacing Both both equal temperament (ET) and 5 limit just intonation (JI).

ET is the most commonly used system for modern Western music, and probably is what most builders are interested in. To obtain the ET fret spacing for a fret, multiply the VSL by the multiplier in the ET column of the table.

For 5 limit JI, multiply the VSL by the multiplier in the appropriate mode column of the table. For a mountain dulcimer that is to be played in a 1-5-5 tuning like DAA, you would use the "Mix" column for the bass string and the "Ion" column for the middle and treble strings.

If you don't care how the table was computed, and just want to use it, you can stop reading here. The information that follows is a rationale and information for the curious.

How ET multipliers were calculated

ET is based upon the 12^th root of 2. There are twelve steps in a chromatic octave. If you choose a frequency for the tonic; I arbitrarily chose A4 (440 Hz) and multiply it by 1.0594630943593, you get 466.16376151809197 Hz. Multiply that by 1.0594630943593 and you get 493.8833012561285. This step is repeated until all twelve steps of an octave are calculated. At that point you have 880.00000000004695 which rounds to 880 Hz. The octave of a note is that note multiplied by 2 so 440 * 2 = 880.

A nice thing about ratios, is that a ratio relationship remains the same with any number. So 440 / 440 = 1, 440.0 / 466.16376151809197 = 0.94387431268168931, 440.0 / 493.8833012561285 = 0.89089871814033139, ... 440.0 / 880.0 = 0.5.

A formula for calculating fret spacing is (1.0 - 1.0/((ratio numerator / ratio denominator)) * VSL so we can obtain the multipliers in the ET column in the table by using a VSL of 1. This multiplier can then be used with any VSL (multiplier * VSL).

How JI multipliers were calculated

Just intonation is based upon harmonic relationships and their ratios. A perfect 5^th is the ratio 3/2, a perfect 4^th the ratio 4/3, a 3^rd 5/4, etc. The lower the numbers in the ratio, the better our ears like it.

The ratios used for the 5 limit JI scale in the table are: 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1.

The numbers in the JI-05 column are the prime limit for that ratio. The perfect 5^th is a ratio of 3/2 and 3 is the highest prime number in that ratio. The flat 7^th (vii) has a ratio of 9/5 and the highest prime number is 5. The ii has a ratio of 16/15. 16 is a multiple of 2 and 15 is a multiple of 3 so the highest prime number in the ratio is 3. Etc. Since the highest prime limit for any of the ratios is 5, the scale is considered to be a 5 limit JI scale.

This gives us the ratios and fret spacing from the nut, which is the fret spacing for the Mixolydian mode on a mountain dulcimer. If you have this fret spacing and wish to play Ionian mode (from fret 3) the ratios are wrong. To calculate the fret spacings for each mode, calculate the frequencies for each ratio and shift them for the mode and multiply the small numbers by 2 to put them in the appropriate octave. Then calculate new ratios with those frequencies and then use those ratios to calculate the multipliers for the mode. This corrects for the offset of the note at the nut, relative to the note at the tonic.

The luthier's dilemma

Building a JI fretted mountain dulcimer has a serious complication that jumps out at you from the fret spacings. If you build it with the intention of playing with a 1-5-1 (DAd, Cgc, etc.) tuning you would need to have a different fret spacing for each string. Yes it is the same for the bass and treble string but not for the middle string so you will need 1^3rd width frets on each string which is a lot of work.

The easiest solution would be a 1-5-5 tuning (DAA, CGG, etc.) where the bass frets would be for Mixolydian and the middle and treble string would be for Ionian. If you are building it to be played drone style, where finger picking, flat picking and chords will never be a consideration, all you need is the Ionian fret spacing, but it might be wise to only put frets for the treble string since any time someone tries to do those things, they will think that you are a poor excuse of a luthier.

Tuning a JI mountain dulcimer

Tune a string that will be Mixolydian with a tuner, or to an instrument that you will be playing with, then tune the non-Mixolydian strings relative to the notes on the Mixolydian string. Remember that you can not use a capo because your drones will not be a perfect 5^th apart with the capo.

Harmonics and difference frequencies

The 2^nd harmonic of 440 is 880 (440 * 2), or the octave. Every note has harmonics, and they are part of the richness of the notes that come from musical instruments. When you combine two frequencies in a mixer (your ear is one) you get the two fundamental frequencies plus the sum and the difference frequencies. With a JI tuning in the key of A, the tonic is 440 Hz and the perfect 5^th is 660 Hz. 2 * 660 = 1320 and 3 * 440 = 1320. That is a JI ratio and sounds good to your ear.

The ET frequencies for the tonic and 5^th are 440 and 659.26 Hz. 2 * 659.26 = 1318.52 and 3 * 440 = 1320. The difference for 1330 and 1318.52 is 11.48 Hz. That difference note is what you hear when you tune up by comparing the sound of two strings to each other and you tune to reduce the "beat" note to zero. It is also a distracting, or even annoying, sound to some people when they listen to ET instruments. It is the reason that just intonation sounds so good. The ET note frequencies are a compromise that are not perfect ratios and are full of beat notes, rather than notes based upon ratios that have pleasing harmonic relationships. To some people, ET is a buzzing bee hive. Most of us are accustomed to the sound and have learned to ignore it.

A JI instrument will sound out of tune when played with ET instruments, but for a solo performer, or for playing with AJI instruments, including voice, it could be worth someone's effort to build a JI mountain dulcimer.

DulciFrets.py

The calculations were performed with the python script DulciFrets.py. There are provisions for you to even add your own JI limits if you wish, change the output to include data that I chose to not print out, or you might just want you check my math/methods to satisfy your curiosity.

It can be downloaded from the following links:

DulciFrets.tar.gz MD5: 1b9fff77c7b926e43548cde43ca829e7

DulciFrets.zip MD5: 3da9acf5f0f522ddc2a6d706f22ba9cb

Python is platform independent so anyone will be able to use it (Linux, Mac, Win). You can download Python (free) at www.python.org and run DulciFrets.py with it. Python is a great, free, script language that can be used to develop platform independent programs. Who knows, you might even decide to become a geek when you start using it :0)

Last update 11-Jul-2009