A Set of Vihuelas:
Matching Tone for Polyphonic Music
Daniel Larson, 2005
One
of the most challenging aspects of historical instrument construction is the
concept of the instrument set. The idea
that a musical instrument could be scaled into larger and smaller sizes seems
to coincide with two important developments in Western music culminating in the
15th century.[1] Organum consisting of two lines was expanded into
the more complex part form of polyphony and liturgical and secular forms of
music combined. This union of the
improvised secular songs and dance music where instruments were in common use
with the more formal, composed vocal forms of church music necessitated that
the range of the instruments were expanded to match the range of parts. Just as it is impractical for one human voice
to cover the full musical range, it was equally impossible to design one
instrument that was capable of producing all of those notes. Therefore it was necessary for the instrument
makers like Sebastian da Verona[2] to
develop instruments in different sizes forming a family or set of instruments
that could play the full necessary musical range. By 1511 Sebastian Virdung[3]
was picturing a family of four sizes of crumhorn and in 1528 Martin Agricola[4]
shows a plate of four sizes of geigen, or viols. Evidence indicates that these sets of
instruments were often made by a single maker, or at least were made in one
maker’s workshop, for a patron who could afford the luxury of a matched set of
instruments. Many examples of these sets
are known, including a set of thirty eight violins that Andrea Amati made for
Charles IX of
Although
the lute family was documented by Michael Praetorious in 1615 to have seven
sizes, it is interesting to note that the inventories for the Raymund Fugger
collection made in 1566 and 1580 list instruments sets being either three or
four in number.[8] These descriptions, although meager, indicate
that the instruments were associated together as sets because of matching woods
and decoration, indicating that there was a common theme of design and
proportion that grouped these particular instruments together.
As
the process of rediscovering the traditions of lute making and
reverse-engineering the instruments has developed over the last forty years
there has been considerable attention given to the art of the instrument set
among wind instrument and viol makers, but not as much in the lute world. To be sure, there have been some sets of
lutes made. Lawrence Lundy[9]
exhibited a quartet of lutes at the Boston Early Music in 1989 and Ed Greenhood
made a set of lutes for the Peabody Conservatory in 2000[10],
to site two examples, but the revival of early music has focused more attention
on matching sets of violas da gamba, recorders and reed instruments than plucked
strings. Although much music survives
for groups of lutes, this repertoire is usually played on unmatched instruments
and one can’t help but wonder: what would the music sound like if the instruments
were, in some way, matched?
The project outline
This
is the question which arises with the part music of Enriquez de Valderrábano
published in his 4th book Libro de musica de vihuela, intitulado
silva de sirenas in 1547. Within the pages are sixteen vihuela duets
with the instrumentation arranged thusly:
· Four are for instruments in unison
· Three are for instruments a minor third apart
· Four are for instruments a fourth apart
· Five are for instruments a fifth apart.
In
order to play all of these pieces five instruments are required. For an instrument maker, this is an
intriguing challenge; to make a set of vihuelas that would be capable of playing
this literature as convincing polyphony.
Edward Martin and I discussed this possibility on many occasions and after
the success of his recording of the first 12 fantasias by Luis Milán we both
became enthusiastic about a project of making the necessary instruments and
playing the complete duets of Valderrábano as they were written. Phil Rukavina offered to play the second part
and plans began to develop. The
instruments were completed and delivered to Ed and Phil in January of 2003. The
object of this article is to explain the process of making the instruments and
the decisions that went into trying to maintain the integrity of the project. The object was to create a set of instruments
that would allow the music to sound as if all parts were being played by one
instrument, thereby completing the polyphonic effect.
The question of model
The
first issue was to decide on a model for the set. This is a fundamental problem for the vihuela
maker because there only a few extant vihuelas from which to take examples and
these all represent radically different schools of design concepts. Many paintings of vihuelas exist and it is
possible to design an instrument based on iconography, but I wanted these
instruments to have more basis and credibility than images could offer. So, we began to consider the needs of the
music and musicians and match those needs to the available extant models. There were four instruments to be considered as
historical vihuelas:[11]
· The instrument in the Chapel in
· The instrument in the Jacquemart Andre in
· The instrument in the Royal College of Music in
· The instrument in the Citie de la Misique in
The
Quinto instrument was out of the running as there is so little information
available about it that a reproduction of this instrument would be little
better than an iconographical study. The
instrument in the Musee Jacquemart Andre in
Scaling the instruments
The model having been decided we
had to establish the sizes of the instruments and we decided to use the
original instrument as a basis from which to proportion the others. The E.0748 instrument is currently in a
disassembled state and the bridge is not attached to the front, but there
exists a shadow of glue on the wood that shows the approximate position that
the bridge once occupied would allow a string length of about 64cm. This is a slightly odd mensur for instruments
tuned to a modern pitch of A-440 as it is slightly short for an instrument in
“F”, which would be more comfortable at 65cm or 66cm, and it is too long for an
instrument in “G” which is more comfortable at 60cm or 61cm. String calculations and empirical testing of
a 64cm copy of this instrument indicated that a pitch of “F#” would be most satisfactory,
but the decision was made to consider this an “F” natural instrument for the
convenience of pitches for the other instruments. It is true that pitch is somewhat relative
and it would be possible to base this instrument at “F#” and scale the others
accordingly from the 64cm but practical considerations of tuning encouraged us
to tune this size as “F” natural. We
chose to use this as the lowest-pitched instrument decision based on how we
wanted to hear the music. We thought the set would have a lighter, livelier
sound with higher pitches. It would be
just as valid an aesthetic decision to make an instrument one size larger in
“E” for use as the bass and scale the others accordingly. Gut strings were going to be used throughout
the set so the selection of the lengths was important for the stability and playability. Through my experience with gut strings I have
found that the ideal gut string diameter and tension for the lute first course
is .42mm at 4Kg, so I used a mathematical formula to calculate the string
lengths for the set as:
“F” – 64cm [16]
“G” – 59cm
“Bb” – 50cm
“C” – 45cm
Yielding the following duet arrangements:
Unison duets – “G” instruments
Minor third duets – “G” and “Bb” instruments
Fourth duets – “F” and “Bb” instruments
Fifth duets – “F” and “C” instruments
Ed Martin already owned a 59cm
vihuela in “G” that I made for him previously, but in spite of this instrument
being modeled on a different design the decision was made to use this it for
the unison duets with the E.0748 model.
This is the one compromise in the aesthetic standard of the match set.
After the string lengths were
settled it was an easy job to work out the basic proportions of the E.0748
instrument and apply them to the higher pitches. The actual lengths used for the instruments
are listed in Table 1. The string length
of this instrument was divided by the body length to get a factor of 1.616. Then the string lengths of the other
instruments were divided by this factor to establish the body lengths that are
the same proportion as the original. In
the same way the string length of the original was divided by the neck length
and a factor of 2.41 was found. This
factor was used to divide the string lengths of the other instruments to
establish the proportional neck lengths.
This method can be used to establish the body and neck proportional
lengths for any given string length. Once the body lengths were established the
process of drawing the body outlines began.
Fortunately, shape of the E.0748 instrument is made up of a series of
simple curves based on arches from circles and I spent a pleasant afternoon of
reverse-engineering to work out the proportions and center points.
Table
1
|
|
“F” Instrument
|
“G” Instrument
|
“Bb” Instrument
|
“C” Instrument
|
String Length
|
64cm
|
59cm
|
50cm
|
45cm
|
Body Length
|
39.60cm
|
36.50cm
|
30.94cm
|
27.84cm
|
Neck Length
|
26.55cm
|
24.48cm
|
20.74cm
|
18.67cm
|
Once the proportions were established and the drawings
made, it was a fairly routine workshop project to assemble the neck, rib and
back structures. My concentration was
mostly focused on the question of how to make the instruments sound as if they
belonged together, how to get the tones of the different sizes to blend into a
convincing polyphonic whole. I decided
to draw on my background in violin making for a solution and used the
principals of Chladni patterns[17]
as a basis with which to judge and adjust the vibration modes[18]
between the sizes of instruments[19]
with the theory that if I could establish a relationship between the resonances
there would follow a similarity in tone and response. [20] This science of vibration modes in
instruments is complex with many implications, such as the relationship of
modes to air resonance[21]
and the influence of struts[22]
and rosette, etc. I did not want to get
to preoccupied with developing a new branch of investigation, so I tried to
keep the project simple. Since the
original instrument was the model from which the other designs originated, I
reasoned that if some patterns could be identified on this instrument these
could be used as a comparison to the other sizes.[23] The testing apparatus is simple, consisting
of a table with a six inch audio speaker mounted in a central hole pointing
upward. A sine wave generator feeds a
signal into the speaker through a frequency counter so I can identify the pitch
of the tone being produced. The
instrument rests on corks placed on the table to allow vibration to be as free
as possible.[24] The front of the instrument was then
sprinkled with particles, in this case tea leaves, so that the vibration modes
would be displayed as the range of musical tones was played through the
speaker.
On the “F” instrument front I was able to distinguish
four symmetrical and fundamental modes operating at four frequencies as
pictured below in figure 1:
Figure 1
“F” instrument
before adjustment
The first, I called the “ring mode” because it
manifests in a circle, was evident at a fundamental frequency of A-55 and at
the octave of A-110. The second, I named
the “bar mode” because the antinode, (line in between vibrating areas), lined
up with the central strut, was apparent at a frequency of C-130 and the third, dubbed
the “A mode” because the central vertical antinodes form an “A” with the
bridge, was generated at a pitch of B-493.
The patterns were pretty well established, but I tried to make them more
definite by manipulating the front. My
method was to scrape the open areas of vibration in an attempt to make them
more active and in that way create more defined antinode patterns. The results are shown in fibure2:
Figure 2
“F” Instrument
after adjustment
The process was moderately successful, with the caveat
that the ring mode was moved from A-55Hz to A#-58Hz, and the distribution of
particles in the modes seemed a little more even.
Having established some patterns on the “F” instrument
I ran the musical spectrum through the “G” instrument to see what modes were
evident. To my satisfaction, the same
mode shapes were apparent and, with a little scraping they more or less matched
those on the “F” instrument. The results
are seen in Figure 3:
Figure 3
“G” Instrument
after adjustment
I found that the ring and bar modes were exactly one
step apart between the two instruments and the “A” modes were separated by a
half step. I began to see a pattern
develop and, encouraged, went on to investigate the Bb instrument with the
results illustrated in figure 4 after scraping adjustment.
Figure 4
“Bb” instrument
after adjustment
On the “Bb” instrument the ring mode was moved up two
whole steps from the “G” instrument to E-82hz and the second, bar mode, was
also moved two whole steps to F#-185Hz.
The pattern of the “A” mode began to break down on this instrument. The two central vertical lines seemed to want
to meet each other at the center point of the center bar as on the larger two
instruments, but I think the width of the bridge began to restrict the
vibrations more than scraping could correct, resulting in an incomplete
pattern. Still, at E-659Hz it was
exactly two steps above the “A” mode of the “G” instrument which maintains the
overall trend of the interval separation.
I think that, if the bridge width were more in proportion to the width
of the front this “A” mode would be the same form as those on the “F” and “G”
instruments.
On the small, “C” instrument the modes were much more
difficult to identify. I think the
stiffness caused by the bars, rose and bridge began to restrict the development
of clear, even patterns. However, I was
able to identify three major patterns that more-or-less corresponded to where
the forms should be, based on the patterns of intervals of the other three
instruments, as shown in figure 5.
Figure 5
“C” instrument
after adjustment
The first mode, which is sort of a ring mode, was
formed at G-98Hz or three half steps up from the ring mode on the “Bb”
instrument. The second mode did not
exactly form at the central bar as on the other instruments, but remained a
sort of bar / ring pattern an octave above the first mode and only a half step
away from the bar mode of the “Bb” instrument.
The “A” mode formed at G#-830, or two whole steps away from the same
mode on the “Bb” instrument, thereby restoring the interval relationships
established on the other sizes.
All of this discussion of modes and intervals is a
confusing, even to me, so I will try to simplify the data in the following two
tables.
Table 2
Mode Frequencies
|
|
Mode 1 |
Mode 2 |
Mode 3 |
|
|
Ring
Mode |
Bar Mode |
“A” Mode |
|
|
|
|
|
|
F
Instrument |
A#-58 |
C-130 |
B-493 |
|
G
Instrument |
C-65 |
D-146 |
C-523 |
|
Bb Instrument |
E-82 |
F#-185 |
E-659 |
|
C
Instrument |
G-98 |
G-196 |
G#-830 |
Table 3
Separation of
Instrument Modes in Half Steps
|
|
Mode 1, Ring mode |
Mode 2, Bar Mode |
Mode 3, “A” Mode |
|
F Instrument to G Instrument |
2 |
2 |
1 |
|
G Instrument to Bb Instrument |
4 |
4 |
4 |
|
Bb Instrument to C Instrument |
4 |
1 |
4 |
Conclusion
In an effort to make instruments that have a similar
tone and response I was looking for simple relationships between the primary
resonances and these do seem to be present in the set. Although not perfect, I think there is enough
of a pattern in the modes of the group to say that the scaling of the set displays
important tonal relationships. It is
reassuring to think that relationships of lengths and breadths of instruments
correspond to the universal truths of musical intervals[25]. I will leave it to others to judge the
musical qualities of the instrument.
From a structural point of view the project is a great success.