fret positioning

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Rough fret positions

12th fret: 1/2 of the string's length

9th fret: 2/5 of the string's length

7th fret: 1/3 of the string's length

5th fret: 1/4 of the string's length

4th fret: 1/5 of the string's length

Fret layout

        D     X     D     X     D     D     X     D     X     D     Xb    D
       :|     |     |  *  |     |  *  |     |  *  |     |  *  |     |     |-  ...  |:
Open C :| C#  |  D  | D#  |  E  |  F  | F#  |  G  | G#  |  A  | A#  |  B  |-  ...  |:
  Nut 0:|    1|    2|    3|    4|    5|    6|    7|    8|    9|   10|   11|-  ...  |: Bridge
   #####|     |#####|     |#####|#####|     |#####|     |#####|     |#####|
   #####|     |#####|     |#####|#####|     |#####|     |#####|     |#####| Piano keys
   #####|     |#####|     |#####|#####|     |#####|     |#####|     |#####|
   #####|     |#####|     |#####|#####|     |#####|     |#####|     |#####|
   #####|     |#####|     |#####|#####|     |#####|     |#####|     |#####|
   ########|###########|########|########|###########|###########|########|
   ########|###########|########|########|###########|###########|########|

Diatonic only (common with dulcimer)

        D           D           D     D           D           D           D
       :|           |           |  *  |        *  |        *  |           |-  ...  |:
Open C :|        D  |        E  |  F  |        G  |        A  |        B  |-  ...  |:
  Nut 0:|          2|          4|    5|          7|          9|         11|-  ...  |: Bridge 

Dulcimer with "blue note":

        D           D           D     D           D           D     Xb    D
       :|           |           |  *  |        *  |        *  |     |     |-  ...  |:
Open C :|        D  |        E  |  F  |        G  |        A  | A#  |  B  |-  ...  |:
  Nut 0:|          2|          4|    5|          7|          9|   10|   11|-  ...  |: Bridge 

Taylor Formula

For precise calculations.

Taylor formula:

string tension
F
string length
l
frequency
f
string diameter
d
string density
Ro
Pi=3.14159
π

F = (2l/f)2*π*d2/4*Ro

Air resonance

A Standard Soprano Ukulele has an air resonance of about 260 Hz, which is Middle C, the third string.

Ukulele Variant

See Tahitian Ukuleles also known as the Tahitian Banjo, These are constructed similar to a banjo with the bridge on a thin disk of wood rather than a drum. Usually strung with 8 strings in 4 courses, tuned like a Ukulele to G4 C4 E4 A4.

Ukulele Tunings

The Ukulele descended from the Rajão, which was tuned in D G C E A. The Ukulele keeps the strings, GCEA.

Common Soprano (reentrant, or high 4th tuning)
G4 C4 E4 A4
Low G Soprano (or low 4th tuning)
G3 C4 E4 A4
D6 or D tuning, (AKA "Mainland"), also reentrant, a step higher than GCEA)
A4 D4 F#4 B4

Common fret board scale lengths

Ukulele Sopranissimo Scale (pocket Ukulele)
11 inches
Ukulele Sopranino or Bambino Ukulele
12 inches (one found as 11.9")
Ukulele Soprano (minimum length mentioned)
13 inches (approximately 330 mm)
Ukulele Soprano (common length mentioned)
13.5 inches
Ukulele Soprano (from "Martin Guitars - A History" by Mike Longworth)
13-5/8 inches (13.625")
Ukulele Soprano Kamaka
13-9/16 inches (13.5625")
Ukulele Soprano Kala/Makala
13-11/16 inches (13.6875", approximately 348 mm rounds to 13.7")
Ukulele Concert
15 inches (also found reference giving 14-3/4)
Ukulele Tenor
17 inches
Ukulele Baritone
21-1/8 inches (also found references giving 19)
Mandolin Gibson F style
13.875 inches
Mandolin Gibson A style
14.125 inches
Guitar Tenor short scale
21 inches
Guitar Tenor long scale
23 inches
Guitar short scale
24.5 inches
Guitar from Gibson, Standard, and Guild
24.75 inches
Guitar from Paul Reed Smith, National, and Dobro
25 inches
Guitar models Fender Telecaster and Stratocaster
25.5 inches
Guitar Baritone
27.5 inches
Guitar Bass Fender short scale
30 inches
Guitar Bass Fender long scale
34 inches

Equal Temperament

To calculate distance from FretA to FretB divide the Scale Length Remaining (FretA to bridge) by 17.817. Three digits of precision is impossible. In fact in the old days they just used 18 instead of 17.817. You might also see 17.835 in old references. This value was used to add in a fudge factor to accommodate the musical tastes of the time (sort of how the standard for the pitch of A above middle C wasn't always 440 Hz (Concert Pitch)).

fret #       factor
 1           0.056126
 2           0.109101
 3           0.159104
 4           0.206291
 5           0.250847
 6           0.292893
 7           0.33258
 8           0.370039
 9           0.405396
10           0.438769
11           0.470268
12           0.5
13           0.528063
14           0.554551
15           0.579552
16           0.60315
17           0.625423
18           0.646447
19           0.66629
20           0.68502
21           0.702698
22           0.719385
23           0.735134
24           0.75

d = s – (s / 2(n / 12))

d
distance from nut to fret number, n
s
scale or string length
n
fret number

Here is a little Python script that will output fret positions given a string length.

#!/usr/bin/env python
# This will print the distance from the nut to each fret
# over a two octave range given an open string length.

import sys
# Set string length here. Units don't matter. Output is in the same units.
string_length = 14.0

fret_length_factors = [ 0.056126, 0.109101, 0.159104, 0.206291, 0.250847,
        0.292893, 0.33258, 0.370039, 0.405396, 0.438769, 0.470268, 0.5,
        0.528063, 0.554551, 0.579552, 0.60315, 0.625423, 0.646447, 0.66629,
        0.68502, 0.702698, 0.719385, 0.735134, 0.75 ]
fret_number = 1
for length_factor in fret_length_factors:
    sys.stdout.write('%3d    %5.2f'%(fret_number, (length_factor * string_length)))
    sys.stdout.write('\n')
    fret_number += 1

sys.stdout.write('\n')

# This should give the same results.
# This calculates the same information in a different way.
for fret_number in range(1,25):
    fret_distance = string_length - (string_length / (2 ** (fret_number/12.0)))
    sys.stdout.write('%3d    %5.2f'%(fret_number, fret_distance))
    sys.stdout.write('\n')

Gives the following table:

fret      distance from
number    nut to fret
  1       0.79
  2       1.53
  3       2.23
  4       2.89
  5       3.51
  6       4.10
  7       4.66
  8       5.18
  9       5.68
 10       6.14
 11       6.58
 12       7.00
 13       7.39
 14       7.76
 15       8.11
 16       8.44
 17       8.76
 18       9.05
 19       9.33
 20       9.59
 21       9.84
 22      10.07
 23      10.29
 24      10.50

Tables for some common fret board lengths (scale length)

Column definitions:
   1: Fret number
   2: Distance from nut to fret in your units, to thousandths.
   3:      your units rounded to 32ths.
   4:      your units rounded to 16ths.
   5:      your units as inches converted to millimeters.
           Ignore this if your units were not in inches.

6 inch scale length

Obviously absurd, but perhaps could play with the aid of a tool. A fingerboard for mice, perhaps.

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   0.337   0+11/32   0+ 5/16    8.554mm
  2   0.655   0+21/32   0+10/16   16.627mm
  3   0.955   0+31/32   0+15/16   24.247mm
  4   1.238   1+ 8/32   1+ 4/16   31.440mm
  5   1.505   1+16/32   1+ 8/16   38.229mm
  6   1.757   1+24/32   1+12/16   44.637mm
  7   1.995   1+32/32   1+16/16   50.685mm
  8   2.220   2+ 7/32   2+ 4/16   56.394mm
  9   2.432   2+14/32   2+ 7/16   61.782mm
 10   2.633   2+20/32   2+10/16   66.868mm
 11   2.822   2+26/32   2+13/16   71.669mm
 12   3.000   3+ 0/32   3+ 0/16   76.200mm
 13   3.168   3+ 5/32   3+ 3/16   80.477mm
 14   3.327   3+10/32   3+ 5/16   84.514mm
 15   3.477   3+15/32   3+ 8/16   88.324mm
 16   3.619   3+20/32   3+10/16   91.920mm
 17   3.753   3+24/32   3+12/16   95.315mm
 18   3.879   3+28/32   3+14/16   98.518mm
 19   3.998   3+32/32   3+16/16  101.543mm
 20   4.110   4+ 4/32   4+ 2/16  104.397mm
 21   4.216   4+ 7/32   4+ 3/16  107.091mm
 22   4.316   4+10/32   4+ 5/16  109.634mm
 23   4.411   4+13/32   4+ 7/16  112.034mm
 24   4.500   4+16/32   4+ 8/16  114.300mm

8 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   0.449   0+14/32   0+ 7/16   11.405mm
  2   0.873   0+28/32   0+14/16   22.169mm
  3   1.273   1+ 9/32   1+ 4/16   32.330mm
  4   1.650   1+21/32   1+10/16   41.920mm
  5   2.007   2+ 0/32   2+ 0/16   50.972mm
  6   2.343   2+11/32   2+ 5/16   59.516mm
  7   2.661   2+21/32   2+11/16   67.580mm
  8   2.960   2+31/32   2+15/16   75.192mm
  9   3.243   3+ 8/32   3+ 4/16   82.377mm
 10   3.510   3+16/32   3+ 8/16   89.158mm
 11   3.762   3+24/32   3+12/16   95.559mm
 12   4.000   4+ 0/32   4+ 0/16  101.600mm
 13   4.225   4+ 7/32   4+ 4/16  107.302mm
 14   4.436   4+14/32   4+ 7/16  112.685mm
 15   4.636   4+20/32   4+10/16  117.765mm
 16   4.825   4+26/32   4+13/16  122.560mm
 17   5.003   5+ 0/32   5+ 0/16  127.086mm
 18   5.172   5+ 5/32   5+ 3/16  131.358mm
 19   5.330   5+11/32   5+ 5/16  135.390mm
 20   5.480   5+15/32   5+ 8/16  139.196mm
 21   5.622   5+20/32   5+10/16  142.788mm
 22   5.755   5+24/32   5+12/16  146.179mm
 23   5.881   5+28/32   5+14/16  149.379mm
 24   6.000   6+ 0/32   6+ 0/16  152.400mm

10 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   0.561   0+18/32   0+ 9/16   14.256mm
  2   1.091   1+ 3/32   1+ 1/16   27.712mm
  3   1.591   1+19/32   1+ 9/16   40.412mm
  4   2.063   2+ 2/32   2+ 1/16   52.400mm
  5   2.508   2+16/32   2+ 8/16   63.715mm
  6   2.929   2+30/32   2+15/16   74.395mm
  7   3.326   3+10/32   3+ 5/16   84.475mm
  8   3.700   3+22/32   3+11/16   93.990mm
  9   4.054   4+ 2/32   4+ 1/16  102.971mm
 10   4.388   4+12/32   4+ 6/16  111.447mm
 11   4.703   4+22/32   4+11/16  119.448mm
 12   5.000   5+ 0/32   5+ 0/16  127.000mm
 13   5.281   5+ 9/32   5+ 4/16  134.128mm
 14   5.546   5+17/32   5+ 9/16  140.856mm
 15   5.796   5+25/32   5+13/16  147.206mm
 16   6.031   6+ 1/32   6+ 1/16  153.200mm
 17   6.254   6+ 8/32   6+ 4/16  158.858mm
 18   6.464   6+15/32   6+ 7/16  164.197mm
 19   6.663   6+21/32   6+11/16  169.238mm
 20   6.850   6+27/32   6+14/16  173.995mm
 21   7.027   7+ 1/32   7+ 0/16  178.485mm
 22   7.194   7+ 6/32   7+ 3/16  182.724mm
 23   7.351   7+11/32   7+ 6/16  186.724mm
 24   7.500   7+16/32   7+ 8/16  190.500mm

300 mm scale length

This is probably the smallest, comfortable size -- depending on your idea of comfort and assuming you don't need to go much past an octave. This is a metric size that is just a bit shorter than 12 inches (by about 1/4").

 1:      2:   
---------------
  1  16.838 mm
  2  32.730 mm
  3  47.731 mm
  4  61.890 mm
  5  75.254 mm
  6  87.868 mm
  7  99.774 mm
  8  111.012 mm
  9  121.619 mm
 10  131.631 mm
 11  141.081 mm
 12  150.000 mm
 13  158.419 mm
 14  166.365 mm
 15  173.866 mm
 16  180.945 mm
 17  187.627 mm
 18  193.934 mm
 19  199.887 mm
 20  205.506 mm
 21  210.809 mm
 22  215.815 mm
 23  220.540 mm
 24  225.000 mm

12 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   0.674   0+22/32   0+11/16   17.107mm
  2   1.309   1+10/32   1+ 5/16   33.254mm
  3   1.909   1+29/32   1+15/16   48.495mm
  4   2.476   2+15/32   2+ 8/16   62.880mm
  5   3.010   3+ 0/32   3+ 0/16   76.458mm
  6   3.515   3+16/32   3+ 8/16   89.274mm
  7   3.991   3+32/32   3+16/16  101.370mm
  8   4.440   4+14/32   4+ 7/16  112.788mm
  9   4.865   4+28/32   4+14/16  123.565mm
 10   5.265   5+ 8/32   5+ 4/16  133.737mm
 11   5.643   5+21/32   5+10/16  143.338mm
 12   6.000   6+ 0/32   6+ 0/16  152.400mm
 13   6.337   6+11/32   6+ 5/16  160.954mm
 14   6.655   6+21/32   6+10/16  169.027mm
 15   6.955   6+31/32   6+15/16  176.647mm
 16   7.238   7+ 8/32   7+ 4/16  183.840mm
 17   7.505   7+16/32   7+ 8/16  190.629mm
 18   7.757   7+24/32   7+12/16  197.037mm
 19   7.995   7+32/32   7+16/16  203.085mm
 20   8.220   8+ 7/32   8+ 4/16  208.794mm
 21   8.432   8+14/32   8+ 7/16  214.182mm
 22   8.633   8+20/32   8+10/16  219.268mm
 23   8.822   8+26/32   8+13/16  224.069mm
 24   9.000   9+ 0/32   9+ 0/16  228.600mm

14 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   0.786   0+25/32   0+13/16   19.958mm
  2   1.527   1+17/32   1+ 8/16   38.796mm
  3   2.227   2+ 7/32   2+ 4/16   56.577mm
  4   2.888   2+28/32   2+14/16   73.360mm
  5   3.512   3+16/32   3+ 8/16   89.201mm
  6   4.101   4+ 3/32   4+ 2/16  104.153mm
  7   4.656   4+21/32   4+10/16  118.265mm
  8   5.181   5+ 6/32   5+ 3/16  131.586mm
  9   5.676   5+22/32   5+11/16  144.159mm
 10   6.143   6+ 5/32   6+ 2/16  156.026mm
 11   6.584   6+19/32   6+ 9/16  167.227mm
 12   7.000   7+ 0/32   7+ 0/16  177.800mm
 13   7.393   7+13/32   7+ 6/16  187.779mm
 14   7.764   7+24/32   7+12/16  197.198mm
 15   8.114   8+ 4/32   8+ 2/16  206.089mm
 16   8.444   8+14/32   8+ 7/16  214.480mm
 17   8.756   8+24/32   8+12/16  222.401mm
 18   9.050   9+ 2/32   9+ 1/16  229.876mm
 19   9.328   9+10/32   9+ 5/16  236.933mm
 20   9.590   9+19/32   9+ 9/16  243.593mm
 21   9.838   9+27/32   9+13/16  249.879mm
 22  10.071  10+ 2/32  10+ 1/16  255.813mm
 23  10.292  10+ 9/32  10+ 5/16  261.414mm
 24  10.500  10+16/32  10+ 8/16  266.700mm

16 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   0.898   0+29/32   0+14/16   22.809mm
  2   1.746   1+24/32   1+12/16   44.339mm
  3   2.546   2+17/32   2+ 9/16   64.660mm
  4   3.301   3+10/32   3+ 5/16   83.840mm
  5   4.014   4+ 0/32   4+ 0/16  101.944mm
  6   4.686   4+22/32   4+11/16  119.032mm
  7   5.321   5+10/32   5+ 5/16  135.161mm
  8   5.921   5+29/32   5+15/16  150.384mm
  9   6.486   6+16/32   6+ 8/16  164.753mm
 10   7.020   7+ 1/32   7+ 0/16  178.316mm
 11   7.524   7+17/32   7+ 8/16  191.117mm
 12   8.000   8+ 0/32   8+ 0/16  203.200mm
 13   8.449   8+14/32   8+ 7/16  214.605mm
 14   8.873   8+28/32   8+14/16  225.369mm
 15   9.273   9+ 9/32   9+ 4/16  235.530mm
 16   9.650   9+21/32   9+10/16  245.120mm
 17  10.007  10+ 0/32  10+ 0/16  254.172mm
 18  10.343  10+11/32  10+ 5/16  262.716mm
 19  10.661  10+21/32  10+11/16  270.780mm
 20  10.960  10+31/32  10+15/16  278.392mm
 21  11.243  11+ 8/32  11+ 4/16  285.577mm
 22  11.510  11+16/32  11+ 8/16  292.358mm
 23  11.762  11+24/32  11+12/16  298.759mm
 24  12.000  12+ 0/32  12+ 0/16  304.800mm

18 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   1.010   1+ 0/32   1+ 0/16   25.661mm
  2   1.964   1+31/32   1+15/16   49.881mm
  3   2.864   2+28/32   2+14/16   72.742mm
  4   3.713   3+23/32   3+11/16   94.320mm
  5   4.515   4+16/32   4+ 8/16  114.687mm
  6   5.272   5+ 9/32   5+ 4/16  133.911mm
  7   5.986   5+32/32   5+16/16  152.056mm
  8   6.661   6+21/32   6+11/16  169.182mm
  9   7.297   7+10/32   7+ 5/16  185.347mm
 10   7.898   7+29/32   7+14/16  200.605mm
 11   8.465   8+15/32   8+ 7/16  215.007mm
 12   9.000   9+ 0/32   9+ 0/16  228.600mm
 13   9.505   9+16/32   9+ 8/16  241.430mm
 14   9.982   9+31/32   9+16/16  253.541mm
 15  10.432  10+14/32  10+ 7/16  264.971mm
 16  10.857  10+27/32  10+14/16  275.760mm
 17  11.258  11+ 8/32  11+ 4/16  285.944mm
 18  11.636  11+20/32  11+10/16  295.555mm
 19  11.993  11+32/32  11+16/16  304.628mm
 20  12.330  12+11/32  12+ 5/16  313.191mm
 21  12.649  12+21/32  12+10/16  321.274mm
 22  12.949  12+30/32  12+15/16  328.903mm
 23  13.232  13+ 7/32  13+ 4/16  336.103mm
 24  13.500  13+16/32  13+ 8/16  342.900mm

20 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   1.123   1+ 4/32   1+ 2/16   28.512mm
  2   2.182   2+ 6/32   2+ 3/16   55.423mm
  3   3.182   3+ 6/32   3+ 3/16   80.825mm
  4   4.126   4+ 4/32   4+ 2/16  104.800mm
  5   5.017   5+ 1/32   5+ 0/16  127.430mm
  6   5.858   5+27/32   5+14/16  148.790mm
  7   6.652   6+21/32   6+10/16  168.951mm
  8   7.401   7+13/32   7+ 6/16  187.980mm
  9   8.108   8+ 3/32   8+ 2/16  205.941mm
 10   8.775   8+25/32   8+12/16  222.895mm
 11   9.405   9+13/32   9+ 6/16  238.896mm
 12  10.000  10+ 0/32  10+ 0/16  254.000mm
 13  10.561  10+18/32  10+ 9/16  268.256mm
 14  11.091  11+ 3/32  11+ 1/16  281.712mm
 15  11.591  11+19/32  11+ 9/16  294.412mm
 16  12.063  12+ 2/32  12+ 1/16  306.400mm
 17  12.508  12+16/32  12+ 8/16  317.715mm
 18  12.929  12+30/32  12+15/16  328.395mm
 19  13.326  13+10/32  13+ 5/16  338.475mm
 20  13.700  13+22/32  13+11/16  347.990mm
 21  14.054  14+ 2/32  14+ 1/16  356.971mm
 22  14.388  14+12/32  14+ 6/16  365.447mm
 23  14.703  14+22/32  14+11/16  373.448mm
 24  15.000  15+ 0/32  15+ 0/16  381.000mm

22 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   1.235   1+ 8/32   1+ 4/16   31.363mm
  2   2.400   2+13/32   2+ 6/16   60.966mm
  3   3.500   3+16/32   3+ 8/16   88.907mm
  4   4.539   4+17/32   4+ 9/16  115.280mm
  5   5.519   5+17/32   5+ 8/16  140.173mm
  6   6.444   6+14/32   6+ 7/16  163.669mm
  7   7.317   7+10/32   7+ 5/16  185.846mm
  8   8.141   8+ 5/32   8+ 2/16  206.778mm
  9   8.919   8+29/32   8+15/16  226.536mm
 10   9.653   9+21/32   9+10/16  245.184mm
 11  10.346  10+11/32  10+ 6/16  262.786mm
 12  11.000  11+ 0/32  11+ 0/16  279.400mm
 13  11.617  11+20/32  11+10/16  295.082mm
 14  12.200  12+ 6/32  12+ 3/16  309.883mm
 15  12.750  12+24/32  12+12/16  323.854mm
 16  13.269  13+ 9/32  13+ 4/16  337.040mm
 17  13.759  13+24/32  13+12/16  349.487mm
 18  14.222  14+ 7/32  14+ 4/16  361.234mm
 19  14.658  14+21/32  14+11/16  372.323mm
 20  15.070  15+ 2/32  15+ 1/16  382.789mm
 21  15.459  15+15/32  15+ 7/16  392.668mm
 22  15.826  15+26/32  15+13/16  401.992mm
 23  16.173  16+ 6/32  16+ 3/16  410.793mm
 24  16.500  16+16/32  16+ 8/16  419.100mm

24 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   1.347   1+11/32   1+ 6/16   34.214mm
  2   2.618   2+20/32   2+10/16   66.508mm
  3   3.818   3+26/32   3+13/16   96.990mm
  4   4.951   4+30/32   4+15/16  125.760mm
  5   6.020   6+ 1/32   6+ 0/16  152.916mm
  6   7.029   7+ 1/32   7+ 0/16  178.548mm
  7   7.982   7+31/32   7+16/16  202.741mm
  8   8.881   8+28/32   8+14/16  225.576mm
  9   9.730   9+23/32   9+12/16  247.130mm
 10  10.530  10+17/32  10+ 8/16  267.474mm
 11  11.286  11+ 9/32  11+ 5/16  286.676mm
 12  12.000  12+ 0/32  12+ 0/16  304.800mm
 13  12.674  12+22/32  12+11/16  321.907mm
 14  13.309  13+10/32  13+ 5/16  338.054mm
 15  13.909  13+29/32  13+15/16  353.295mm
 16  14.476  14+15/32  14+ 8/16  367.680mm
 17  15.010  15+ 0/32  15+ 0/16  381.258mm
 18  15.515  15+16/32  15+ 8/16  394.074mm
 19  15.991  15+32/32  15+16/16  406.170mm
 20  16.440  16+14/32  16+ 7/16  417.588mm
 21  16.865  16+28/32  16+14/16  428.365mm
 22  17.265  17+ 8/32  17+ 4/16  438.537mm
 23  17.643  17+21/32  17+10/16  448.138mm
 24  18.000  18+ 0/32  18+ 0/16  457.200mm

32 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   1.796   1+25/32   1+13/16   45.619mm
  2   3.491   3+16/32   3+ 8/16   88.678mm
  3   5.091   5+ 3/32   5+ 1/16  129.319mm
  4   6.602   6+19/32   6+10/16  167.680mm
  5   8.027   8+ 1/32   8+ 0/16  203.888mm
  6   9.373   9+12/32   9+ 6/16  238.064mm
  7  10.643  10+21/32  10+10/16  270.321mm
  8  11.841  11+27/32  11+13/16  300.768mm
  9  12.973  12+31/32  12+16/16  329.506mm
 10  14.041  14+ 1/32  14+ 1/16  356.631mm
 11  15.049  15+ 2/32  15+ 1/16  382.234mm
 12  16.000  16+ 0/32  16+ 0/16  406.400mm
 13  16.898  16+29/32  16+14/16  429.209mm
 14  17.746  17+24/32  17+12/16  450.739mm
 15  18.546  18+17/32  18+ 9/16  471.060mm
 16  19.301  19+10/32  19+ 5/16  490.240mm
 17  20.014  20+ 0/32  20+ 0/16  508.344mm
 18  20.686  20+22/32  20+11/16  525.432mm
 19  21.321  21+10/32  21+ 5/16  541.561mm
 20  21.921  21+29/32  21+15/16  556.784mm
 21  22.486  22+16/32  22+ 8/16  571.153mm
 22  23.020  23+ 1/32  23+ 0/16  584.716mm
 23  23.524  23+17/32  23+ 8/16  597.517mm
 24  24.000  24+ 0/32  24+ 0/16  609.600mm

36 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   2.021   2+ 1/32   2+ 0/16   51.321mm
  2   3.928   3+30/32   3+15/16   99.762mm
  3   5.728   5+23/32   5+12/16  145.484mm
  4   7.427   7+14/32   7+ 7/16  188.640mm
  5   9.030   9+ 1/32   9+ 0/16  229.374mm
  6  10.544  10+17/32  10+ 9/16  267.822mm
  7  11.973  11+31/32  11+16/16  304.111mm
  8  13.321  13+10/32  13+ 5/16  338.364mm
  9  14.594  14+19/32  14+10/16  370.695mm
 10  15.796  15+25/32  15+13/16  401.210mm
 11  16.930  16+30/32  16+15/16  430.013mm
 12  18.000  18+ 0/32  18+ 0/16  457.200mm
 13  19.010  19+ 0/32  19+ 0/16  482.861mm
 14  19.964  19+31/32  19+15/16  507.081mm
 15  20.864  20+28/32  20+14/16  529.942mm
 16  21.713  21+23/32  21+11/16  551.520mm
 17  22.515  22+16/32  22+ 8/16  571.887mm
 18  23.272  23+ 9/32  23+ 4/16  591.111mm
 19  23.986  23+32/32  23+16/16  609.256mm
 20  24.661  24+21/32  24+11/16  626.382mm
 21  25.297  25+10/32  25+ 5/16  642.547mm
 22  25.898  25+29/32  25+14/16  657.805mm
 23  26.465  26+15/32  26+ 7/16  672.207mm
 24  27.000  27+ 0/32  27+ 0/16  685.800mm

42 inch scale length

 1:      2:        3:        4:       5:
-------------------------------------------------------------------------------
  1   2.357   2+11/32   2+ 6/16   59.875mm
  2   4.582   4+19/32   4+ 9/16  116.389mm
  3   6.682   6+22/32   6+11/16  169.732mm
  4   8.665   8+21/32   8+11/16  220.080mm
  5  10.536  10+17/32  10+ 9/16  267.603mm
  6  12.302  12+10/32  12+ 5/16  312.458mm
  7  13.968  13+31/32  13+15/16  354.796mm
  8  15.542  15+17/32  15+ 9/16  394.758mm
  9  17.027  17+ 1/32  17+ 0/16  432.477mm
 10  18.428  18+14/32  18+ 7/16  468.079mm
 11  19.751  19+24/32  19+12/16  501.682mm
 12  21.000  21+ 0/32  21+ 0/16  533.400mm
 13  22.179  22+ 6/32  22+ 3/16  563.337mm
 14  23.291  23+ 9/32  23+ 5/16  591.595mm
 15  24.341  24+11/32  24+ 5/16  618.266mm
 16  25.332  25+11/32  25+ 5/16  643.440mm
 17  26.268  26+ 9/32  26+ 4/16  667.202mm
 18  27.151  27+ 5/32  27+ 2/16  689.629mm
 19  27.984  27+31/32  27+16/16  710.798mm
 20  28.771  28+25/32  28+12/16  730.779mm
 21  29.513  29+16/32  29+ 8/16  749.638mm
 22  30.214  30+ 7/32  30+ 3/16  767.439mm
 23  30.876  30+28/32  30+14/16  784.241mm
 24  31.500  31+16/32  31+ 8/16  800.100mm

For a 16" fret board (with fractions of an inch rounded to 16ths) use:

fret   |   distance from nut to fret
number |   in three different precisions
-------+----------------------------------
  #    |   hundredths   32nds     16ths
------------------------------------------
  1    |   0.90         29/32      14/16
  2    |   1.75       1+24/32    1+12/16
  3    |   2.55       2+17/32    2+ 9/16
  4    |   3.30       3+10/32    3+ 5/16
  5    |   4.01       4+ 0/32    4+ 0/16
  6    |   4.69       4+22/32    4+11/16
  7    |   5.32       5+10/32    5+ 5/16
  8    |   5.92       5+29/32    5+15/16
  9    |   6.49       6+16/32    6+ 8/16
 10    |   7.02       7+ 1/32    7+ 0/16
 11    |   7.52       7+17/32    7+ 8/16
 12    |   8.00       8+ 0/32    8+ 0/16 ___ 1 octave
 13    |   8.45       8+14/32    8+ 7/16
 14    |   8.87       8+28/32    8+14/16
 15    |   9.27       9+ 9/32    9+ 4/16
 16    |   9.65       9+21/32    9+10/16
 17    |  10.01      10+ 0/32    10+0/16
 18    |  10.34      10+11/32    10+5/16 _ _ _ one and a half octaves
 19    |  10.66      10+21/32   10+11/16
 20    |  10.96      10+31/32   10+15/16
 21    |  11.24      11+ 8/32   11+ 4/16
 22    |  11.51      11+16/32   11+ 8/16
 23    |  11.76      11+24/32   11+12/16
 24    |  12.00      12+ 0/32   12+ 0/16 ___ 2 octaves