A Set of Vihuelas: Matching Tone for Polyphonic Music
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.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 to develop instruments in different sizes forming a family or set of instruments that could play the full necessary musical range. By 1511 Sebastian Virdungwas picturing a family of four sizes of crumhorn and in 1528 Martin Agricola 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 France and a set of viols made by Zenatto of Treviso for a girl’s orphanage in Antonio Stradivari is known to have made sets of instruments for the Duke of Savoy, King Charles III of and the Duke of Tuscany.
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. 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 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, 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:
- The instrument in the Chapel in Quinto
- The instrument in the Jacquemart Andre in Paris, France.
- The instrument in the Royal College of Music in London, England attributed to Belchior Dias
- The instrument in the Citie de la Misique in Paris, France inventory no. E0748
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 Paris would be a good choice but it was discounted because the shallow ribs of this model create an air chamber that is too small to support the fundamental tone of the instrument to my satisfaction.It has become popular to consider the Belchior Dias instrument in the Royal College of Music as a vihuela, so this instrument was considered for the project, but this too was discounted because of the relatively narrow body. One of the features of the set that Ed was insistent about was that the nut and bridge spacing remain consistent on all of the instruments so that the bridge on the smallest instrument matched the width and string spacing on the largest instrument. I felt that as the Dias design was scaled down to the smaller sizes the bridge would be too wide for the bodies and begin to restrict the vibrations of the front to a detriment.So, the only possibility that remained was the anonymous instrument in the Cité de la Musique in Paris, France number E.0748 and this instrument was deemed the best for the project largely because the outline of this instrument is broad enough at the bridge point to support a wide bridge comfortably on the small sizes.
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 
- “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.
|“F” Instrument||“G” Instrument||“Bb” Instrument||“C” Instrument|
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 as a basis with which to judge and adjust the vibration modes between the sizes of instruments with the theory that if I could establish a relationship between the resonances there would follow a similarity in tone and response. This science of vibration modes in instruments is complex with many implications, such as the relationship of modes to air resonance and the influence of struts 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.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.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:
“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 figure2:
“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:
“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.
“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.
“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.
|Mode 1||Mode 2||Mode 3|
|Ring Mode||Bar Mode||“A” Mode|
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|
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.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.
 The Larousse Encyclopedia of Music, “The background to the music” English edition published in the USA 1981, Excalibur Books
 Peter Holman, Four and Twenty Fiddlers: The Violin at the English Court 1540-1690, (Oxford University Press, 1995), p18.In December of 1511 “Maestro Sebastian da Verona” was paid to look for wood with which to make “violette” for the Ferrara court. Presumably he was also to construct the instruments once he had obtained the materials.
 David Munrow, Instruments of the Middle Ages and Renaissance, (Oxford University Press, 1976), p56.
 David Munrow, Instruments of the Middle Ages and Renaissance, (Oxford University Press, 1976), p86.
 David Boyden, The Hill Collection of Musical Instruments in the Ashmolean Museum, Oxford, (Oxford University Press, 1970. p17 – 18.
 Robert Hadaway, Another Look at the Viol, Early Music, vol. 6 no. 4, October 1978, p535.
 W. Henery Hill, Arthur Hill, and Alfred E. Hill, Antonio Stradivari His Life and Work (1644-1737), (Dover Publications,1963), p94
 Douglas Alton Smith, A History of the Lute from Antiquity to the Renaissance, (The Lute Society of America, 2002), p319.
 Boston Early Music Festival Program Book, 1989. p32.
 I am grateful to Ed Greenhood for the information he provided for me.The set consists of four instruments based on the design of the Frei C34 lute in the Kunsthistorisches Museum Vienna: soprano; 44cm, Alto; 57.5cm, tenor; 63.5cm, bass; 72cm, scaled for synthetic strings, i.e. copper wound basses, polycarbonate and nylon, with eight fret necks. All instruments were made with matching materials. Ed used computer using spectrum analyzing software to help match the responses of the fronts and achieve a coherent response from the set.
It must be noted that organology is not an exact science and that there is some discussion as to whether these instruments should be classified as vihuelas or not. The only instrument in this list that is generally considered to be an authentic vihuela is the instrument in the Musee Jacquemart Andre in Paris.
 Donald Gill, A Vihuela in Ecuador, (The Lute Society Journal, 1978), 53-55.
 Maish Weisman, The Paris Vihuela Reconstructed, (The Galpin Society Journal, number XXXV, March 1982), p68 – p77.
 This instrument is classified by the Royal College of Music Museum as a guitar.Since the instrument E.0748 in the Citie de la Musique has come to light is has become possible to consider this instrument as a vihuela, largely due to the similarity in construction of the back of the two instruments. See the published technical drawing number 171 Guitar by Belchior Dias, Lisbon, 1581, Vaulted back, body length 365 mm, belly not original. (2 sheets, 1120 x 770 mm, with additional notes) Drawn by Stephen Barber, 1976
 The Cité de la Musique technical drawing number C32 Vihuela (?), anonymous maker, Spain (?) 17th or 18th century (?), Carlos Gonzalez, 2000, Inv.: E.0784.(+ legend).
 The 4Kg gauge for the 64cm instrument is actually .44mm because, as stated previously, the instrument has a more natural pitch of F#.
 German physicist Ernst E. F. Chladni developed a method of using metal plates to demonstrate the patterns of node and antinode activity in the late 18th c. He would fix a steel plate, square or round, in the center and sprinkle it with metal particles.Then a violin bow was used to vibrate the plate at different points, thereby exciting different patterns of vibration. The metal particles would be bounced off of the areas of the plate that were vibrating, the nodes, and align in the lines between the nodes where there was no vibration, the antinodes. Charles A. Culver, Ph.D., Musicla Accoustics (McGraw-Hill Book Company 1956), p241.
 I am not the first to consider the application of this methodology to the lute family. Such a study was done on lute fronts by Ian Firth long before me. Ian Firth, Acoustical Experiments on the Lute Belly, (The Galpin Society Journal, number XXX, May 1977), p56 – p63.
 A good explanation of vibration patterns in plates can be seen in: Carleen M. Hutchins, Acoustics for the Violin Maker, (The Catgut Acoustical Society Newsletter, Number 28, November 1, 1977), 19.
 The science of analyzing vibration modes in musical instruments and relating that information into useful workshop practice was developed by a number of people beginning in the 1950s. Perhaps the most important of these was Carleen Hutchins. For an introduction to this method please see: Carleen Mary Hutchins, The Physics of Violins, (Scientific American, November 1962).
 Graham Caldersnith, Low range Guitar Function and Design ( The Catgut Acoustical Society Nerwsletter, Number 27, May, 1977), 19.
 George Bissinger and Carleen Hutchins, Tuning the bass bar in a Violin Plate, (Catgut Acoustical Society Newsletter, Number 26, November 1976), 10.
 Most of the science of plate modes has been developed using free plates.That is, the plates were tested before they were glued to the ribs and the instrument assembled and these tests have taken place with hundreds of samples over many years. I did not have the time, or enough instruments to make an in depth study, so rather than study the individual parts I decided to test the instruments fully assembled, assuming that the influences of air, struts, rosette, bridge and varnish would manifest themselves in the final vibration modes.
 The corks were moved around and placed in positions that seemed to optimize the support and vibration of the instrument.
 Gerhard Christian Söhne, On the Geometry of the Lute, Journal of the Lute Society of America, Inc, Volume X111, 1980, p35 – p54.