the-rule-of-31

The Rule of Thirty-One: Realizing 1/4-Comma
Meantone Tuning on Fretted Instruments.
MICHAEL KUDIRKA
The question of where to position the frets on lutes and viols has been a source of debate not only
for musicologists and modern performers, but also for musicians of the sixteenth century. This
period witnessed a change in practice spearheaded by Zarlino to include new consonances based on
higher ratios of the overtone series. The resultant practice of consonant 3rds and 6ths required the
pythagorean tuning system to be updated in order to sweeten its stridently wide major 3rds and
10ths. While the new meantone temperament seems to have been widely adopted by keyboardists,
there is really no agreement as to how fretted string players handled the tuning system. In fact, a
kind of approximation of equal temperament was developed by Vincenzo Galilei to simplify the
issue of the lute’s tuning. This has led many to conclude that all lute music was intended for 12-tone
equal temperament by the middle of the century. This idea has often been confirmed by the
theoretical and technical difficulties in understanding how meantone temperament works on fretted
instruments, an issue that has been further obfuscated by the countless conceptual and mathematical
errors that exist in meantone fretting instructions from the time. To encourage the notion that a
great deal of the fretted string music functions perfectly well in 1/4 comma meantone, I aim to
discuss the tuning system from a theoretical standpoint, review the extant literature on the subject,
and finally to propose a fully functional method of achieving true 1/4 comma meantone tuning on
lutes, viols, vihuelas, and theorbos.
Overview of Meantone Temperament
When speaking of meantone temperament, I am referring to what is often called “1/4 comma
meantone”. This is a tuning system first hinted at by Gafurius in 1496 and Grammateus in 1518,

ut not fully described until the publication of Pietro Aron’s Toscanello in musica (Venice, 1523) 1 . This
tuning system is a modification of the earlier system of pythagorean tuning, in which a chain of
pure fifths is stacked to generate the pitches of the diatonic scale. While this tuning was appropriate
for the consonant 4ths, 5ths, and octaves of 15th century music, the resultant “wide” major third
sounds harsh in the context of 16th century polyphony with its pervasive use of 3rds and 6ths. The
difference between a pythagorean third (the result of four ascending perfect 5ths) and an
acoustically pure major third is called the syntonic comma, and is equivalent to 21.51 cents 2 . To
generate euphonious thirds (5:4), the meantone temperament alters the pythagorean system by
“narrowing” the fifths, each by 1/4 of the syntonic comma. Dy definition, the narrowing of each
of these fifths by 1/4 of the syntonic comma causes the fifth note of the chain (the fourth interval)
to fall exactly on an acoustically pure third.
In Figure 1, this adaptation of the pythagorean chain of fifths can be seen referenced from a pitch
center of “A”, extending outward into sharps on the right, and flats on the left. For compactness,
the chain of fifths is here seen as fifths alternating with fourths. Note that an ascending fifth
narrowed from pure by a 1/4 comma is equivalent to a descending fourth widened from pure by a
1/4 comma.
Figure 1. Chain of 1/4 comma fifths and fourths.
In this figure one may take any succession of four neighboring intervals to find a pure major third
(C-flat to E-flat, A-flat to C, A to C-sharp, F-sharp to A-sharp, etc.). One important property of
this system is that, unlike 12-tone Equal Temperament, the chain does not “close” after 12 pitches.
In fact, because of the flattening of the fifth, the series flattens as it progresses, and “undershoots”
the octave (C-flat to B is less than an octave). This is in contrast to the pythagorean chain of pure
1 Barbour, J. Murray. Tuning and Temperament: A Historical Survey. East Lansing: Michigan State Press,
1951. p. 25-26.
2 Greated, Clive. “Comma.” Grove Music Online. Oxford Music Online.

fifths which actually “overshoots” the octave (C-flat to B is greater than an octave). Of the three
most common “regular” temperaments (temperaments in which all of the fifths are the same size),
only Equal Temperament “closes” with 12 and only 12 unique pitches. This is significant for
performers because an “open” chain of fifths can (theoretically) extend outward into sharps,
double-sharps, triple sharps, etc. in one direction, and flats, double flats, triple flats, etc. in the other
direction ad infinitum.
While the physical limitations of acoustic instruments make this infinite spiral impossible to realize
in sound, it should not then be assumed that realizing at least some of the so-called “enharmonic
equivalents” is impossible. In fact, our earliest extant fretting document comes from the ninthcentury
theorist Al-Kindī. His instructions for fretting the ‘ud is pythagorean, and generates
sounding pitches beyond a chain of twelve fifths 3 . Even seven centuries later, the vihuelist Juan
Bermudo called for split frets in his detailed instructions for a pythagorean fretting scheme 4 .
This subtle difference in pitch between between the “enharmonic equivalents” generates another
surprising phenomenon: since there are (at least) two unique pitches between a given whole-tone,
the semi-tones must therefor be unequal. In a pythagorean chain of fifths, the “overshooting” of
the octave causes the “sharped” notes to sound higher in pitch than their enharmonics on the flatside
of the chain. Conversely, in the chain of meantone fifths, the “undershooting” of the octave
causes the “sharped” notes to sound lower in pitch compared to their flat-keyed neighbors. In the
meantone system, the smaller semitone is called the minor semitone (or chromatic semitone), while
the larger is called the major semitone (or diatonic semitone).
Primary Sources for Fretting Schemes on the Lute and Viola da Gamba
While tuning instructions abound for keyboard instruments from the 16th century, treatises on
fretting schemes are relatively rare. One of the first comprehensive guides comes from the
aforementioned Juan Bermudo (1555). The musicologist Wolfgang Freis notes that Bermudo’s
Declaración de instrumentos musicales of 1555 represents the work of a theoretician more than a that of
a practical musician 5 . His pythagorean tuning scheme, as well as his invention of a completely new
3 Lindley, Mark. Lutes, Viols and Temperaments. Cambridge: Cambridge University Press, 1984. p. 9.
4 ibid. p. 17-18
5 Freis, Wolfgang. “Perfecting the Perfect Instrument: Fray Juan Bermudo on the Tuning and Temperament
of the ‘vihuela de mano’.” Early Music, Vol. 23, No. 3, Iberian Discoveries III (Aug., 1995). p. 421-435.

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.

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).

discounted if other parts of the fretting are contrary to pythagorean intonation.” 10 . Gerle’s
instructions seem to imply 1/6 meantone, while Ganassi may have intended true 1/4 comma
meantone based on his injunction to tune the 5ths to match those of keyboard instruments, which
were likely to have been in 1/4 comma at that time 11 . Despite the numerous errors and convoluted
instructions, both Gerle and Ganassi clearly distinguish between the major and minor semi-tones in
their fretting scheme, though they fail to give much theoretical information as to why they chose
certain frets to function as “mi” frets and others as “fa”. Because of this, their fretting may function
well for the pieces contained in their respective treatises, but don’t give the player many tools for
altering their fretting based on where they might want to terminate their circle of fifths (i.e. toward
the flat side or the sharp side).
Perhaps the most renowned of all writers on lute temperament in the sixteenth-century was
Vincenzo Galilei. In his lengthy treatise Fronimo (1584), he proposes an early approximation of 12-
tone equal temperament. Rather than the major and minor semitones that occur in both the
pythagorean and meantone temperaments, Galilei gives a method for placing the frets based on the
rule of 18:17. This is a method whereby the lutenist marks off 1/18th of the string length to get
the next higher semitone. This is repeated again from the 1st fret to the 2nd fret, the 3rd to the 4th,
and so on to generate a geometric progression of equally sounding semitones. In fact, 18:17 is
mathematically smaller the a true “twelfth root of 2”, but this difference is neatly compensated for
by the increase in pitch as the string is depressed down to the fingerboard. There is evidence that
Vincenzo’s method was widely adopted and put to use, even to the point where it seems that lutes
and keyboards could not play euphoniously together. Around 1640 Giovanni Battista Doni claimed
that there were two different temperaments in use: one for fretted instruments, and one for
keyboards 12 . The discrepancy was not simply a seventeenth-century phenomenon. As early as 1555
Vincentino remarked that when lutes and viols played with keyboard instruments that divided the
whole-tone unequally, they were never quite in tune together. Approximately twenty years later
Giovanni de’ Bardi wrote to Caccini: “More than once I have felt like laughing when I saw musicians
struggling to put a lute or viol into proper tune with a keyboard instrument ... Until now this highly
important matter has gone unnoticed or, if noticed, unremedied.” 13 By saying “until now” (“fino a
questo giorno”) Bardi seems to imply that up until the time he wrote the letter, there was no solution,
10 Lindley, Mark. Lutes, Viols and Temperaments. Cambridge: Cambridge University Press, 1984. p. 58.
11 ibid. p. 61
12 ibid. p. 47
13 ibid. p. 44

implying, perhaps, that someone had recently found one. As to what that might be, we’re not sure.
Perhaps it involved a new, proper meantone tuning method for the lute, or perhaps tuning the
keyboards in accordance with the lute (a process derided by Jean Denis in the 1640’s), or possibly
using continuo orchestrations that kept the lutes and viols always separated from the keyboards and
harps.
While Vincenzo Galilei’s 18:17 equal semitone is highly functional and reliable it does create rather
wide Major 3rds and can make simultaneous performance with a keyboard a painful experience. It
seems possible that Galilei, Bardi, and the many other advocates for equal-tempered lutes in the 16th
century were either unaware of the lute’s potential to achieve true meantone tuning, or were unable
to execute the more difficult mathematical calculations necessary to find the fret locations (though
this seems unlikely in the case of Galilei). His complaint seems to be that he finds the tastini (little
frets), to be a cumbersome annoyance. The (fictional) student Eumatio asks his teacher Fronimo:
“why don’t you use, on your lute, frets spaced to give unequal intervals, and some additional little
frets to take from the major 3rds and 10ths some of their acuteness, as I have seen used by some
[players] ...” 14 These “little frets” can effectively split a given fret at one location and raise or lower
the intonation to generate a “mi” fret on one course where the other courses require a “fa” fret.
Even as late as the mid-seventeenth century Jean Denis recommended that lutes use little ivory frets
(“touches d’yuoire”) to allow for the staggered “mi” “fa” frets in a given position across different
strings 15 .
1/4-Comma Meantone Fretting for the Modern Performer
Perhaps the most helpful model for achieving 1/4 coma meantone on the lute is that of the 31-
division octave. This system divides the octave into 31 equal steps and corresponds to the pitch
ratios of the 1/4 meantone temperament 16 . The correspondence is not exact however, though it
does approximate it to within fractions of a cent (well within the bounds of human perception
based on biologically determined psychoacoustic constants). Using the 31-division method, the
minor semitone is two steps wide while the major semitone is three steps wide. A whole tone is thus
5 steps wide, and the difference between enharmonic “equivalents” is 1 step wide, 1/31st of an
14 ibid. p. 46
15 ibid. p. 47
16 Duffin, Ross. How Equal Temperament Ruined Harmony (and Why You Should Care). New York: W.W.
Norton & Company, 2007). p. 55.

octave, or 38.71 cents. Similarly, the Major 3rd is 387.1 cents wide, compared to the just Major 3rd
which is 386.3 cents 17 . The octave in 31-division can be visualized as a wheel with 31 spokes. Note
that all pure Major 3rds are 10 steps apart.
Figure 4. Pitch Wheel of 31-division octave with meantone congruities.
This fine-grained equally-divided octave closes the infinite loop of ascending fifths, albeit rather late
in the cycle (F-double-sharp is at step 25, and F-triple-sharp at step 27 which finally lines up with its
true enharmonic equivalent A-double-flat). These obviously ridiculous extensions into chromatic
deep-space combined with the psychoacoustic indistinguishability between the 31-division Major 3rd
17 ibid. p. 163.

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.

Using the adjusted measurement, multiply the “sounding length” by each “string length factor” in
Figure 5. The factors are derived by a similar formula as that which finds 12-tone equal tempered
intervals, except instead of finding the 12th root of 2, we find the 31st root of 2.
F = (2 1/31 ) n ÷ 220hz
In this formula, “F” is the frequency of the note, “n” is the degree of the scale you are looking for
(0-31), and 220hz is the “pitch center” of your quarter-comma system (see “Figure 1). This can be
any frequency, but in the example, 220hz is used. The resultant frequencies for all the degrees of
the 31-note division can then be translated into string length factors by dividing the fundamental
frequency by the frequency of the pitch above it:
“string length factor” = 220hz ÷ F
Then simply multiply each “string length factor” by the total “sounding length” to get the distance
from the bridge to the nut.
“distance from bridge to fret” = “string length factor” × “sounding length”
Here it is important, though, not to place frets based on the measurement from the bridge, because
we are working from an imaginary bridge (to compensate for pitch sharpening). You must subtract
the new sounding length from the total sounding length to get the distance from the nut.
“distance from nut to fret” = “sounding length” − “distance from bridge to fret”
The data from these calculations is summarized in the following chart. A lute with a “sounding
length” of 650mm was used, though the true distance from nut to bridge was in fact 655mm. Also,
the open 3rd course (pitch “A” 220hz) was used as a reference, but due to the equal-division system,
any pitch may be used with position “0” as the root.
Note Name
Step Number
(n)
Frequency (F)
(in Hertz)
String length
factor
Distance from
bridge to nut
Distance from
nut (D)
A 0 220.0 1 650 0
1 225.0 0.9778 635.6 14.4
A# 2 230.1 0.9561 621.5 28.5
Bb 3 235.3 0.9349 607.7 42.3

Note Name
Step Number
(n)
Frequency (F)
(in Hertz)
String length
factor
Distance from
bridge to nut
Distance from
nut (D)
4 240.6 0.9141 594.2 55.8
B 5 246.0 0.8943 581.3 68.7
6 251.6 0.8744 568.4 81.6
7 257.3 0.8550 555.8 94.2
C 8 263.1 0.8362 543.5 106.5
9 269.0 0.8178 531.6 118.4
C# 10 275.1 0.7997 519.8 130.2
Db 11 281.3 0.7821 508.4 141.6
12 287.7 0.7647 497.1 152.9
D 13 294.2 0.7478 486.1 163.9
14 300.9 0.7311 475.2 174.8
D# 15 307.7 0.7150 464.8 185.2
Eb 16 314.6 0.6993 454.5 195.5
17 321.7 0.6839 444.5 205.5
E 18 329.0 0.6687 434.7 215.3
19 336.5 0.6538 425.0 225.0
20 344.1 0.6393 415.5 234.5
F 21 351.8 0.6254 406.5 243.5
22 359.8 0.6115 397.5 252.5
F# 23 367.9 0.5980 388.7 261.3
Gb 24 376.3 0.5846 380.0 270.0
25 384.8 0.5717 371.6 278.4
G 26 393.5 0.5591 363.4 286.6
27 402.4 0.5467 355.4 294.6
G# 28 411.5 0.5346 347.5 302.5
Ab 29 420.8 0.5228 339.8 310.2
30 430.3 0.5113 332.3 317.7
A 31 440.0 0.5000 325.0 325.0
Figure 5. Chart of 31-division data, based on a 650mm sounding length.

Note the relationship between the “A” and the “C#”. Here we can mathematically observe the
imperceptible discrepancy with a pure major third. The pure third would have a frequency of 275
Hz, but the 31-division only misses the mark by .1 Hz, that is, it would only beat once every ten
seconds (far longer than any lute string would resonate). Similarly, the string length for the C#
expressed as a simple ratio would be 4/5 (.8), whereas in the 31-division system it is .7997, a
difference that is vastly within the natural margin of error both acoustically and physically
considering the minute imperfections of the string and fret material themselves.
The next step in finding fret locations on the lute is to mark off distances from the nut for the
desired pitches, but to do that, one must first decide which pitches to use. Just as one could
theoretically have a plethora of split keys on the keyboard to account for all 31-divisions, one could
place enough frets on the lute to account for every possible spelling of a note. There is no evidence
that sixteenth-century players did this however. Rather, for practicality, they chose a particular range
of the spiral of 1/4 comma fifths to situate their instrument in. This subset of pitches is sometimes
called the “disposition” and can be defined by where the wolf 5th lies 19 . For practical reasons, the
disposition of a lute is largely dependent on the tuning of its open strings. By this, I am not
referring to any objective pitch reference but rather to what Wolfgang Freis calls “imagined
tunings” 20 . This is a process familiar to any performer on a transposing instrument where fingerings
are played as though in a particular key or mode. The functionalities of various pitches then are
preserved, though the objective pitch standard is not. Considering this, lutes, can be “imagined” to
have any pitch as their basis for the open strings, though, in practice, we find that sixteenth-century
composers used only a limited subset of the possibilities. For simplicity’s sake, I have included
charts for the pitches of the two most common “imagined tunings”, that of a lute in “A” and a lute
in “G”. In practice, the “A” tuning would be appropriate for the vihuela, the baroque guitar (minus
the 6th course), and the theorbo. The “G” tuning would be appropriate for the Renaissance lute, the
archlute, and the so-called “English” theorbo.
19 Lindley, Mark. "Mean-tone." Grove Music Online. Oxford Music Online.
20 Freis, Wolfgang. “Perfecting the Perfect Instrument: Fray Juan Bermudo on the Tuning and Temperament
of the ‘vihuela de mano’.” Early Music, Vol. 23, No. 3, Iberian Discoveries III (Aug., 1995). p. 425.

Figure 6. Fret positions for “A” tuning.
In these figures one can see the functional difference between the “mi” frets and the “fa” frets.
“Mi” frets function as leading tones and major thirds of triads in general and can be found two steps
(minor semi-tone) in the 31-division scale above the natural pitch of the same same. “Fa” frets are
one step of the 31-division above their “enharmonic equivalents” and can be found three steps
(major semi-tone) above their lower diatonic neighbors.
Figure 7. Fret positions for “G” tuning.
The mode or key that a piece is written in will not only define what “imagined tuning” to use, but
also what “disposition” to select. Clearly, figures 6 and 7 show that “mi” positions will favor
dispositions toward the sharp end of the chain of fifths, whereas the fa positions will favor
dispositions toward the flat end of the chain of fifths (see figure 1). The positions of frets II, III,
IV, V, and VII are invariant since they create octaves and unisons with the open strings. Though
Antonio Corona-Alcade has done an exhaustive analysis of the modes, tunings, and implied
dispositions of Luis Milan’s El Maestro, choice of disposition for other sixteenth-century repertoire
must currently by done on an ad hoc basis by the performer. Fortunately, a common practice for
modal-polyphonic music on the lute and vihuela was to use a mode whose final is identical to one of
the open strings. Thus for the “A” lute tuning one sees tonic triads of:
A Major D Major G Major B Major E Major
A minor D minor G minor B minor E Minor

And for the “G” lute tuning:
G Major C Major F Major A Major D Major
G minor C minor F minor A minor D minor
The result of these different “imagined” tunings is that the “A” lute will tend to favor sharp key
dispositions, while the “G” lute will tend to favor flat key dispositions. With the appropriate
selection of “mi” and “fa” fret positions, one need only add tastini in a few positions to provide
leading tones in certain modes. The most likely candidate would be a little fret glued in for the 4th
course in the “mi” position when the rest of the first fret is needed for “fa”. This would be
necessary for the F# leading tone in “G Aeolian” on the G lute, or similarly, the G# leading tone
for “A Aeolian” on the A lute. While Vincenzo Galilei seemed to deride this practice of gluing in
tastini, his comments in Fronimo strongly suggest that it was a common practice at the time 21 .
I’ve found that an effective fretting scheme for much sixteenth-century repertoire consists of a “fa”
position for the first fret (with a little fret glued or taped in for “mi” on the fourth course), and a
“fa” position for the sixth fret. Another helpful addition would be a little fret glued in one 31-
division step below the third fret to provide a leading tone to the open 3rd course. This can be
essential in pieces like Francesco da Milano’s “Fantasia 10” from his third book of solo lute pieces.
Figure 8. Fantasia 10 (libro terzo) by Francesco da Milano (opening) 22 .
21 Lindley, Mark. Lutes, Viols and Temperaments. Cambridge: Cambridge University Press, 1984. p. 46.
22 da Milano, Francesco. Libro Terzo. Venice: 1547

Notice how the marked “fa” and “mi” pitches occur on the inner pair of strings on the 3rd fret.
Without a little fret for the G#, the sounding result would be a pure major third (A-flat to C). The
lower G# makes all of the E Major triads sound euphonious and is effective throughout the work.
To continue with the tutorial, we will use a common disposition for the “G” lute, that is, from D-flat
to F-sharp. To begin the marking process, first move all frets back to the nut:
Figure 9. Lute fingerboard with frets removed.
To find the proper locations for the frets on the fingerboard I recommend taking a sheet of paper
and marking off the series of 31 divisions based on calculations from your lute’s “sounding length”.
“Mi” and “fa” placements can then be marked based on the appropriate number of steps up from
the open string (see Figure 4):

Figure 10. Lute fretboard template (reduced size).
The left side of the template corresponds to the endpoint of the string at the nut as the frets ascend
to the right. Small pencil marks can then be made directly on the fingerboard at the appropriate
locations.
Figure 12. Template aligned to 1st course.

Figure 13. Template aligned to 6th course.
Using the pitches for a D-flat to F-sharp disposition we mark the 1st fret as “fa”, and the 6th fret as
“fa”. We may also include a small dash under the fourth course for a “mi” position 1st fret tastini
yielding “F-sharp”, a minor semitone above the open string F.
(This space has been left blank intentionally.)

Figure 14. Pencil markings for fret placement.
Figure 15. Frets in place with small piece of fret-gut taped under 4th course (tastini).

Once the frets are placed, one can tune using the familiar method of tuning unisons between the 5th
fret of one particular course to the next higher open course (except between courses 3 and 4 which
use the 4th fret for the pure major 3rd). Consonances can be checked with octaves between an open
course and the 7th fret of the next higher course. Also, octaves can be checked between the 2nd
fret of courses 6, 5, and 4, and the open courses 3, 2, and 1 respectively. In fact, this tuning method
is exactly what Ganassi recommends, though his errors regarding the placement of frets V and VII
(pythagorean rather than tempered 4ths and 5ths) would have completely fouled up the process 23 .
Conclusions
The debate over whether to tune by geometry or by ear goes back at least as far as Zarlino. For
better or worse, tuning instructions based solely on “hearing” are subject to all kinds of inaccuracies
and misunderstanding, as can be seen in the deeply flawed, but oft cited, method of Robert
Dowland (1610) 24 . On the other hand, the ear is the final judge and the geometrical models must
lend themselves to the sounding result rather than the other way around. With careful calculation
and appropriate choice of disposition, there is no reason why a true 1/4 meantone fretting scheme
cannot be effectively implemented on lutes, viols, and theorbos. Often, performers opt for the 1/6
comma temperament to avoid readjustment of frets for different modes, but as Milan shows us in
El Maestro, adjustment of frets for different pieces was indeed done at the time. It seems that the
choice of 1/6 comma is often made to give one the sense that the finer distinction between
“enharmonic equivalents” may allow one to freely fret an A-flat when a G-sharp is called for, but I
would assert that whether the difference is 1/5th of a semitone, or 1/9th of a semitone is irrelevant:
they’re both out of tune. If one is playing in F minor for example, one must fret the instrument to that
disposition, and not just use a “softer” version of the meantone temperament in order to rationalize
the use of a 1/6th comma G-sharp as an A-flat.
Unfortunately, most fretting guides from the sixteenth-century (as well as modern ones) neglect to
consider the importance of the disposition of the meantone cycle. Even experts such as Mark
Lindley and Eugen Dombois give little or no information on how to adjust the fretting scheme to
handle tonalities other than the so-called “standard” disposition of E-flat to G-sharp. While Lindley
23 Lindley, Mark. Lutes, Viols and Temperaments. Cambridge: Cambridge University Press, 1984. p. 95.
24 ibid. p. 83+89.

does give some alternate measurements for “mi” and “fa” frets, he does not give readers a
mechanism for constructing their own complete fretting scheme based on the 31-division. This is
essential since no writer on the subject currently gives an adequate explanation for finding the
location of tastini. Though Vincenzo Galilei may have railed against them, the little frets are quite
simple to use and can allow the lutenist to play accurately in a wide variety of meantone dispositions
with confidence that the pitch relationships will be as accurate as those that are so readily achieved in
keyboard instruments.
References
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