the-rule-of-31

and the Just Major 3rd, make this system a perfect candidate to be the basis of a 1/4 comma fretting
scheme. In a way, it’s not unlike Vincenzo Galilei’s equal tempered system, only that the gradations
are much finer, and many of the intervening “steps” are skipped to generate the major and minor
semitones.
Step-By-Step Guide for Meantone Fretting
First the player must find the string length from which to calculate the divisions. Treatises from the
sixteenth and seventeenth centuries instruct one to use the complete distance from the nut to the
bridge. There is a serious problem with this method however. As one frets notes onto the
fingerboard, the string is stretched slightly, causing an increase in tension, and thus, an increase in
pitch. The increase is a non-trivial amount and can throw the series of geometrically calculated
major and minor semi-tones into complete disarray. To compensate for this, I recommend applying
a technique proposed by Eugen Dombois 18 . Dombois’ technique is to find what he calls the
“playing scale”. This is a length slightly shorter than the complete length of the string. Using this
slightly shorter length for the calculations will cause the fret placement to be shifted minutely in the
direction of the nut (away from the bridge). Thus, when fretting a note, the increase in pitch from
depressing the string will be compensated for by the subtly longer sounding length. Unfortunately,
Dombois’ recommendation entails a trial-and-error method which requires one to calculate all the
fret positions, place them, and then judge the quality of the temperament by ear. This laborious
process can be simplified by the following method: play the 3rd partial on the 6th course. This
harmonic can be found at 1/3 of the string length from the nut and will produce a pitch one octave
and a fifth above the open string. Slide your VIIth fret to a position directly under where the
harmonic is found, but then adjust it so that the pitch of the fretted note matches the pitch of the
harmonic (though an octave lower). You will find that the position of this fret is slightly closer to
the nut than the position of the harmonic. Multiply the distance from the nut to this repositioned
VIIth fret by 3 to get the “sounding length”. This distance will be slightly shorter than the true
distance from the nut to the bridge, though how much shorter will vary from one instrument to the
next.
18 Dombois, Eugen. “Correct and easy fret placement.” Journal of the Lute Society of America. vi (1973). p.
31.