the-rule-of-31

7-course vihuela da mano lacking the interior Major 3rd, seem utterly unsuited to the euphonious
performance of contemporary music by his countrymen Luis Milan and Miguel de Fuenllana, and
apparently had no real impact on performers of the time 6 .
While Bermudo seems like an isolated theoretician, Luis Milan was his exact opposite, a practically
minded player and composer who gave meantone tuning instructions in perfect accord with his own
compositions. A thorough analysis of Milan’s El Maestro by Antonio Corona-Alcade in 1991 shows
that Milan’s fret placement was quite specifically connected to the mode of a given Fantasia 7 .
Specifically, Milan prefaces each Fantasia with information about which mode the piece uses, and
also instructions as to how to adjust the frets so that the major and minor semi-tones are arrayed in
the correct sequence on the fingerboard. While this is invaluable for determining Milan’s use of
meantone temperament, unfortunately, the instructions use language which is non-mathematical. In
other words, he specifies when frets should be moved to either “mi” or “fa” positions, but fails to
provide a detailed mathematical system by which these placements can be found. The only fret
whose physical placement along the plane of the fingerboard is unambiguous is the IVth. This fret
generates the Major 3rd above the open string and must make a unison between the fretted fourth
course and the open 3rd course. This interval, when tuned pure, forces all the other open strings to
be tuned as wide fourths (1/4 comma) to cycle back properly to the outer strings which are exactly 2
octaves apart (an un-negotiable interval in any temperament).
Figure 2. Open strings of the vihuela da mano.
Figure 2 shows that there are exactly four fourths that when stacked can generate a pure major third
if each is widened by a 1/4 of the syntonic comma. In this way, the open strings of the vihuela, and
for that matter the lute (which has the same tuning scheme), inherently requires tempering of the
6 ibid. p. 432.
7 Corona-Alcade, Antonio. “‘You Will Raise a Little Your 4th Fret’: An Equivocal Instruction by Luis Milan?.”
The Galpin Society Journal. Vol. 44 (Mar., 1991). p. 2-45.