the-rule-of-31

fourths to avoid a discordantly wide third between the open 4th and 3rd courses 8 . Once this pure
third is tuned (by ear), then the fourth fret can be placed to give a true unison between the fretted
4th course and the open 3rd course. The net result is a fourth fret that is exactly 4/5ths of the
distance from the bridge to the nut.
The tuning scheme of these open strings exactly reflects the one encountered when tuning the open
strings of the violin family. As Ross Duffin has noted, to create a pure major 3rd between the low
“C” of the viola and the high “E” of the violin, each of the fifths must be narrowed a a quarter of
the syntonic comma.
Figure 3. Open Strings of viola and violin (adapted from Duffin, 2007) 9 .
Clearly, the lute and vihuela tuning is related to the violin and viola’s tuning by a process of inversion
and octave displacement. Fundamentally, the outer thirds of the viola and violin are transferred to
the inner pair of strings of the lute while the stack of narrow fifths is inverted to a stack of wide
4ths.
Two major figures who gave more specific instructions for meantone fretting than Luis Milan were
Hans Gerle (1532) and Silvestro Ganassi (1543). Both make the mistake of placing their fifth and
seventh frets at pythagorean ratios (3/4 from the bridge and 2/3 from the bridge respectively), but
Lindley dismisses this as a mistake common to all but the equal semi-tone fretting instructions of
the sixteenth century. Lindley states that “these rudimentary steps were so hoary with authority -
and so easy to execute and then modify with a slight adjustment by ear - that they should be
8 Lindley, Mark. “Luis Milan and Meantone Temperament.” Journal of the Lute Society of America. xi
(1978). p. 45.
9 Duffin, Ross. How Equal Temperament Ruined Harmony (and Why You Should Care). New York: W.W.
Norton & Company, 2007).