Fretting Accuracy
What accuracy is needed in placing the frets for precise tonal production?
This question can be a vexing one, when we are trying to establish a method for cutting the fret slots. Here is a quantitative way to answer the question:
There is a tool on the Bear Meadow website to calculate fret spacing, and will give a reading of cents error, e, for any deviation, d, you supply.
But for those who like the math, here is a quantitative way to answer the question. The formula expressing fret placement error in terms of tonal errors is:
e=1200*ln((L+d)2/L2)/ln(2)
where:
e= error (in cents)
L= correct distance from fret to saddle
d= deviation from correct L (in inches, cm, furlongs, etc)
ln= natural logarithm
So set L to various fret-saddle values on your scale, then play around with various values of "d" to see which ones approximate but don't exceed an error of 5 cents (generally accepted as the smallest tonal error detectable by the ordinary human ear).
Paul Buerk, a well-known guitar maker, has produced this table as an example of the sorts of errors one will see within 5 cents of the correct fret placement for all the chromatic guitar frets in that scale. Note, for instance, that at the 24th fret a deviation of .020" gives a tonal error of 5.5 cents.
fret errors in a 25" scale
|
25" Scale |
||||||||||||||
| Error in Cents for a given error in length of: | ||||||||||||||
| Fret | Dist to Nut | (0.05) | (0.04) | (0.03) | (0.02) | (0.01) | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | ||
| 1 | 1.40 | (3.67) | (2.94) | (2.20) | (1.47) | (0.73) | 0.00 | 0.73 | 1.47 | 2.20 | 2.93 | 3.66 | ||
| 2 | 2.73 | (3.89) | (3.11) | (2.33) | (1.56) | (0.78) | 0.00 | 0.78 | 1.55 | 2.33 | 3.11 | 3.88 | ||
| 3 | 3.98 | (4.12) | (3.30) | (2.47) | (1.65) | (0.82) | 0.00 | 0.82 | 1.65 | 2.47 | 3.29 | 4.11 | ||
| 4 | 5.16 | (4.37) | (3.49) | (2.62) | (1.75) | (0.87) | 0.00 | 0.87 | 1.74 | 2.62 | 3.49 | 4.36 | ||
| 5 | 6.27 | (4.63) | (3.70) | (2.78) | (1.85) | (0.92) | 0.00 | 0.92 | 1.85 | 2.77 | 3.69 | 4.62 | ||
| 6 | 7.32 | (4.90) | (3.92) | (2.94) | (1.96) | (0.98) | 0.00 | 0.98 | 1.96 | 2.94 | 3.91 | 4.89 | ||
| 7 | 8.31 | (5.19) | (4.15) | (3.11) | (2.08) | (1.04) | 0.00 | 1.04 | 2.07 | 3.11 | 4.14 | 5.18 | ||
| 8 | 9.25 | (5.50) | (4.40) | (3.30) | (2.20) | (1.10) | 0.00 | 1.10 | 2.20 | 3.29 | 4.39 | 5.49 | ||
| 9 | 10.13 | (5.83) | (4.66) | (3.50) | (2.33) | (1.16) | 0.00 | 1.16 | 2.33 | 3.49 | 4.65 | 5.81 | ||
| 10 | 10.97 | (6.18) | (4.94) | (3.71) | (2.47) | (1.23) | 0.00 | 1.23 | 2.47 | 3.70 | 4.93 | 6.16 | ||
| 11 | 11.76 | (6.55) | (5.24) | (3.93) | (2.62) | (1.31) | 0.00 | 1.31 | 2.61 | 3.92 | 5.22 | 6.53 | ||
| 12 | 12.50 | (6.94) | (5.55) | (4.16) | (2.77) | (1.39) | 0.00 | 1.38 | 2.77 | 4.15 | 5.53 | 6.91 | ||
| 13 | 13.20 | (7.35) | (5.88) | (4.41) | (2.94) | (1.47) | 0.00 | 1.47 | 2.93 | 4.40 | 5.86 | 7.32 | ||
| 14 | 13.86 | (7.79) | (6.23) | (4.67) | (3.11) | (1.55) | 0.00 | 1.55 | 3.11 | 4.66 | 6.21 | 7.75 | ||
| 15 | 14.49 | (8.26) | (6.60) | (4.95) | (3.30) | (1.65) | 0.00 | 1.65 | 3.29 | 4.93 | 6.58 | 8.22 | ||
| 16 | 15.08 | (8.75) | (6.99) | (5.24) | (3.49) | (1.75) | 0.00 | 1.74 | 3.49 | 5.23 | 6.97 | 8.70 | ||
| 17 | 15.64 | (9.27) | (7.41) | (5.56) | (3.70) | (1.85) | 0.00 | 1.85 | 3.70 | 5.54 | 7.38 | 9.22 | ||
| 18 | 16.16 | (9.82) | (7.85) | (5.89) | (3.92) | (1.96) | 0.00 | 1.96 | 3.91 | 5.87 | 7.82 | 9.76 | ||
| 19 | 16.66 | (10.41) | (8.32) | (6.24) | (4.16) | (2.08) | 0.00 | 2.07 | 4.15 | 6.22 | 8.28 | 10.35 | ||
| 20 | 17.13 | (11.03) | (8.82) | (6.61) | (4.41) | (2.20) | 0.00 | 2.20 | 4.39 | 6.59 | 8.78 | 10.96 | ||
| 21 | 17.57 | (11.69) | (9.35) | (7.00) | (4.67) | (2.33) | 0.00 | 2.33 | 4.65 | 6.98 | 9.30 | 11.61 | ||
| 22 | 17.98 | (12.37) | (9.89) | (7.41) | (4.94) | (2.47) | 0.00 | 2.46 | 4.93 | 7.38 | 9.84 | 12.29 | ||
| 23 | 18.38 | (13.13) | (10.49) | (7.86) | (5.24) | (2.62) | 0.00 | 2.61 | 5.22 | 7.83 | 10.43 | 13.03 | ||
| 24 | 18.75 | (13.91) | (11.12) | (8.33) | (5.55) | (2.77) | 0.00 | 2.77 | 5.53 | 8.29 | 11.04 | 13.79 | ||
Last Updated on 2/6/03
By Paul Buerk
