Intonation Puzzle

Have you noticed that guitars play out of tune? When I began building guitars 20 years ago this frustrated and puzzled me. I investigated all the published solutions and found that none provided an acceptable answer. I spent the next ten years grappling off and on with a half-solution that left me just as unsatisfied, though I knew pragmatically that I was close. Finally, with a serious effort and the generous help of my friend Cem Duruöz, the conceptual framework of a real solution gradually became clear as we hammered out the necessary mathematics and experimental work. This work was published in the GAL quarterly American Lutherie, Vol. 47, Fall 1996. An abreviated and simplified version appears here, but if you wish, you can take a look at the original article in PDF format (2 MB).

Experimental Exploration of Soundboard Design

Let me set the stage for a research proposal: As lovers of the classical guitar, we have an immediate emotional response to the sound of our instrument, yet no one fully understands the factors responsible for a guitar’s unique sound. Each instrument conveys diverse musical qualities of color, dynamic range, and sustain, resulting from the interaction of the acoustical properties of the wood and the design and construction techniques used. Ask a roomful of luthiers what accounts for the sound of a guitar, and you will get a roomful of answers. Only recently, with the advent of new technologies, has it become possible to sift through these myriad answers in an objective way.

The traditional methods of building the modern classical guitar can be traced directly back to the vihuela and the four-course guitar of the Spanish Renaissance. These instruments were small and fragile, and their gut strings were doubled to increase sound output. They subsequently underwent a series of transformations until, by the second half of the 19th century, the guitar took on essentially modern form at the hands of Antonio de Torres. A carpenter from Andalucia, Torres set a new standard for the instrument in Spain, not only for classical music but for the world of Flamenco as well. It is well to remember that for most of the 20th century, the principal luthiers and their followers all based their designs on the Torres guitar.

As we begin a new century, the number of serious performers on the classical guitar and the level of virtuosity and musicianship is at an all-time high. The repertoire is exploding with new offerings. Unprecedented demands are being placed on the instrument, both by the music being composed and by the large concert halls in which guitarists often perform. This has necessitated expanding the dynamic and timbral range of the instrument beyond what was previously considered acceptable, or even possible.

Consequently, much experimentation in materials and design has been carried out in recent years, though we are far from having established a new paradigm for the instrument. The traditional spruce soundboard is often replaced with Western red cedar or other species with similar acoustical properties. Laminates incorporating synthetic materials have been explored. Luthiers have also experimented with the traditional placement of the soundhole and shape of the bridge and bouts.

Ultimately, however, most makers focus their experimentation on the design of the bracing under the soundboard, and to the thicknessing of the soundboard itself. The soundboard must be flexible yet strong, in order to respond effectively to the pull of the vibrating strings. Torres’ design had transverse bars above and below the soundhole, and seven struts splayed across the lower bout like the ribs of a fan. In its many variants, “fan bracing” is still the standard and most widely used system of soundboard construction. However, there is also active exploration and development of alternate bracing systems that may feature lattice, radial, or various hybrid patterns. Soundboards are often thicknessed in particular ways to compliment different bracing systems. The lower bout may be thicknessed uniformly, or contoured in some fashion. (Soundboard thicknesses customarily range between 1 mm and 3 mm.) There is lively controvery among luthiers over which system is best. Nevertheless, there is general agreement that the bracing and thicknessing of the soundboard are at the heart of guitar acoustics.

The subject of guitar acoustics is notoriously complex, starting with the physical properties of the woods which are combined to form the instrument. No two pieces of wood are identical in these properties, and the various woods are combined in ways that are geometrically and structurally complex. Finally, strings are added, differing in dimensions, tensions and composition. In recent years, physicists have applied systems analysis to the guitar and simple physical models of how sound is amplified in the guitar have been presented. Because it is such a complex, non-linear system, none of them have come close to explaining the subtlety and nuance in sound quality that distinguishes one guitar from another.

The process of learning to build high quality instruments is largely a matter of combining woodworking skills with the traditions of lutherie, rudimentary acoustical principles, trial-and-error, and intuition. Handbuilders commonly work on only a few guitars at a time. My own methods favor assembling one or two guitars concurrently, and the potential for making direct comparisons of modifications in their construction is limited. Professional builders typically do not have the time, resources, or expertise to conduct carefully controlled experiments in soundboard construction. It is my proposal to do just that, combining my familiarity with the methodology of science, and the art of intuitive lutherie.

Proposal

I have in mind constructing a set of perhaps six guitars with removable backs. Aside from the soundboards, they will be indentical in all respects. Both spruce and cedar soundboards will be used, with different thicknessing configurations. In addition to traditional solid wood tops, I would also explore a laminated soundboard of my own design. It will be necessary to build a special mold and workboard to support these instruments, which will facilitate removal of the backs to gain access to the interior bracing.

With this set of instruments in hand, it will be relatively straightforward to make systematic modifications to the soundboard bracing. Variables can include the number and placement of braces, their shapes, sizes, and material composition. Series of modifications can be made, with the goal of illuminating some of the complex interactions between different components of the traditional Torres fan bracing system. Modern variants of this system can also be scrutinized and compared. In addition, I would like to explore lattice bracing, the chief rival to fan bracing, and focus on the resulting musical distinctions between the two systems. Modifications will be replicated in the six experimental guitars in order to observe interactions between bracing and soundboard species and thicknessing. Careful planning of the experiments will allow standard scientific usage of statistical inference to evaluate results.

In order to assess results I would proceed in two directions. First, I would regularly ask for subjective evaluations from guitarists. Though you don’t always agree in your judgments, you guitarists are obviously essential critics in this process. Second, I want to record the sound of the instruments. This is important. With hardware and software available today, it is possible to take a high quality digital recording and transform it into a very detailed frequency spectrum analysis of the sound. The components of such a system include a pair of microphones and preamps (one to record in the direction of the listener and one in the direction of the performer), a high quality analog to digital (AD) converter, computer interface hardware, a computer, recording and editing software, spectrum analysis software, storage and playback capability, and headphone or speaker monitors.

This analysis would comprise the heart of the testing procedure. The recorded waveform represents amplitude through time and gives an overall picture of the onset and decay of a plucked note. Transfomed into a frequency spectrum, it becomes a graphical representation of the overtone series -- the building blocks -- of a note at a brief moment in time. A series of such “snapshots” pieced together sequentially presents a very complete visual representation of the recorded sound, and includes the essential elements of timbre, sustain and dynamics. With this information, the auditory details of the sound of the guitar can be visually analyzed and compared (mapped) to what we hear.

This may seem far-fetched to some of you, but consider why this is important. I'm sure you will agree the adjectives we use to talk about our subjective experiences of sound are very imprecise. What is a warm sound, a dark sound, a clear sound? Rough or smooth, raw or refined (are these even opposites)? Even a concept like sustain means different things to different observers. Graphical analysis of the same auditory experience provides great precision, is easily shared information, and can be related directly to the decisions made at the workbench. Furthermore, the recordings are readily archived, thereby providing a means for comparing experiments made at different times. A recording protocol can be developed to maximize the consistency of the procedure and usefulness of the results.

In addition to the spectrum analysis of played notes, it is also very useful and quite straightforward to record the resonant frequency series of each soundboard configuration. The potential sound of an instrument is embodied in this series of vibratory peaks, or resonances, unique to each guitar. Hitting the guitar on the bridge with an appropriate rubber mallet (ouch!) will elicit this information, which, when recorded, will make it possible to relate changes in bracing patterns to changes in resonant frequencies. Some luthiers (including myself) use the first two or three frequencies in the series (roughly obtained by tapping or singing to the guitar and listening for a resonant response) to guide their work. Experimental results should enhance the effectiveness of this technique.

Naturally this work should help me optimize my own designs, but it will also be possible to obtain results of more general application. Specific principles of guitar design, which are now largely intuitive and controversial, can be confirmed or refuted within the parameters of these experiments. In addition, the experimental work proposed here provides an avenue for determining which systems of soundboard design show most promise, and can more quickly reveal the inherent limitations of others. Understandably, this work cannot explain all the complex interactions that result in the sublety and nuance that distinguish the best classical guitars. It can, however, go a long way toward establishing reliable guidelines for how to imbue a guitar with the qualities one wishes to impart.

A Plea for Funding!

I think you can see this proposal is not something I can undertake in my spare time. I have in mind taking a year-long sabbatical from building to take on this project. Unfortunately, funding for this falls through the cracks of most granting organizations. It is in a nether world between the arts and sciences, not easily catagorized and pigeon-holed. Perhaps the classical guitar community can take an interest. With your help I would like to make this happen. I think it is something from which we can all benefit. Obviously, I would broadcast my results widely, so that all luthiers who take an interest can make use of the information generated. I would also encourage input from players and luthiers alike on what direction the experiments might take. I welcome your feedback on this. Can we get a Consortium together to defray the costs? Can you suggest sources of funding I may not have considered? Please share your thoughts with me. Thanks!