December 11, 2008
This is just a place for some internet research with some fairly good explanations of practical fret placement for the standard 12tet guitar. I doubt these are the only proceedures used and in fact I know of a few more, some of which may be merely academic, I will add them as time permits.
(So far as I know, Vincenzo Galilei, a lutenist and the father of Galileo advocated and first published the method in the late 16th century, acknowledging its prior use by lutenists earlier in the century. And also, so far as I know, no one has ever actually used a mesolabium to construct a fretboard (apparently the first geometrically correct derivation) . These are much better methods.)
http://www.cybozone.com/luthier/instruments/fretscale.html
Anthoy J. Huvard
For any given vibrating string length they would simply divide the length of the string by 18... yielding the distance from the nut to the first fret. By subtracting that figure from the original string length they arrived at a new shorter scale measurement which was then divided once again by 18 and resulted in the distance between the first and second frets. They continued in this manner until the entire scale was determined. Over the years several variations on this theme have been developed... The divisor has been refined, (based on a complex mathematical formula that utilizes the 12th root of 2) resulting in more accurate scales; other approaches have been tried as well.
I use the factor 17.817 as the constant in this formula, but others may use a different quantity.
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http://homepage.ntlworld.com/coleman005/Fretting%20Calculations.htm
Acoustic Guitar Notes
Howard Coleman
This method started out as the "Rule of 18" and was improved in accuracy until it became what I call "The Rule of 17.817". It is mathematically equal to the vibrating length minus the 12th root of 2. It gives perfect results for my guitars with low action and 0.012-0.053" strings. If fret positions need to be adjusted to get the 12th fret in the middle of the string length then the calculation has not not been carried out with sufficient decimal places. This has allowed a small error to creep in and accumulate. You need three decimal places in the calculation (not the measuring) to prevent this.
Experience and common sense are the luthier's greatest assets in this area, and a good luthier will get to know what allowances can be made in his or her own particular methods of construction to accommodate this method of fretting.
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http://www.talkbass.com/forum/archive/index.php/t-56195.html
pilotjones
It turns out that the successive division method is "easier method" that approximates the actual exponential formula. (It is easier because it does not use exponents, just division, and so is easier to figure.) The actual formula, which produces the exact position (accurate within the digit limits of your spreadsheet, calculator or program) from the bridge to any fret, is:
dist(x) = s / (2 ^ (x/d) )
where
x = number of the fret being evaluated
dist(x) = distance from bridge to
fret x
s = scale length (distance nut or zero fret to bridge)
d = number
of divisions per octave ( = 12 in western music)
This produces the positions for the standard equal-tempered scale. The nut-to-fret positions and fret-to-fret positions are then produced by simple subtractions.
http://www.xs4all.nl/~huygensf/doc/stevinsp.html
Galilei
wanted the semitones to be equal, and he chose the value 17:18. Now the octave,
when taken to consist of twelve such semitones, turns out to be, 18^12 : 17^12
approximately 11.57 : 5.83 instead of 11.66 : 5.83 = 2 : 1. The defect is 9 in
nearly
1170...
From a review of Mark Lindley, Lutes, Viols and Temperaments, 1984 -
" ...equal temperament on lutes and viols is never anachronistic." -( meaning generally it is never chronologically out of place, meaning I suppose, it could have been in use at any time - at least during the period of fretted lutes and viols in Europe.)
Also,
"...some players always adapted flexible intonation and were able to play with keyboard instruments in meantone temperament." (i.e. lutenists were able to adjust the tied gut frets closer to meantone)
Further.
"It seems that many good players adjusted their frets by ear rather than conforming to any exact regular spacing, "as they often do today". (21st century lutenists, using adjustable tied gut frets)
The best rule for equal-tempered fretting, supported by his evidence, is to shorten the distance to the bridge by 1/18th for each successive fret..." (p93)
(For the purposes of this article, 1/18 or 1/17.817 or some close approximation may all be considered as "The Rule of 18" for an equal temperament fretboard. Likely 1/18 was used first and slight adjustments were made in the tied frets depending on how it sounded at the time. Even today, when fixed metal frets are embedded and someting closer to the 17.817 is more likely to be used, there are practical aspects of the luthier's art which generally dictate slight differences."