“The Kertsopoulos mathematical model and geometrical progression of the classical guitar, presented for the first time in such a detail.” Vol. 1 of 7.
The classical guitar’s construction based on tradition and influenced by “aesthetics” criteria and historical trends, the mathematical model and the geometrical progression.
The mathematical proportions and the geometrical progression involved in the designing of the classical guitar’s outline-perimeter, bridge and sound-hole in relation to the string length.
A seven-article series ranging from the “aesthetics” criteria and the historical trends with the mathematical model and the geometrical progression presented in this first article, to a full explanatory constructional presentation and acoustical analysis based on the approximately 180-year life of the modern instrument’s unrecorded secrets.
Is this a revelation of the inner soul of the guitar or a fascinating journey in discovering the real genius of Antonio de Torres (13 June 1817 – 19 Nov. 1892)?
For many centuries and even as far back as the ancient periods, the criteria in “aesthetics” have been mainly connected with the mathematical proportions-ratios and/or the geometrical progression existing in defining the form-shape and context of a concrete object. A perceptual representation of harmony is conceived by such a concrete object possessing these qualities, accompanied by a simultaneous optimum performance of the object in its functional and liturgical properties.
A repetition in the presence of the above triad: a) aesthetics b) liturgical functions and
c) mathematics has also been common in the construction of musical instruments. The necessity of applying mathematics for providing the optimum in the aesthetic beauty of a musical instrument, with the joint scope to obtain the desired perfection in its sound production, remains a subject of historical and acoustical importance.
It involves the continuing struggle of many researching minds, working in a chained manner with successful past experiences, striving to capture the musical essence of the immaterial medium that excites the senses: sound. Of course, it has been known for thousands of years that this medium exists in space and time, respecting the physiological behavior of our perception but also behaving in an independent physical way, regardless of the perception’s understanding, need or happening.
Ernst Gottlieb Baron writes in 1727:
“The essence depends entirely on the luthier. He knows the appropriate mathematical proportions, so that the cavities, height, depth, length and width fit together uniformly. This uniformity (egalite) is the reason that an instrument, whether it be of Italian, German or French wood, sounds good.”1
Characteristic of the spherical connections interrelated between the sciences and the arts, as applied especially to the early renaissance is the following, as expressed by the great architect of that period, Leon Battista Alberti:
“Outline, we will call a certain correspondence of those lines that measure the dimensions numerically: length, width and height. The rules of outline can be best extracted out of that in which Nature clearly and impressively reveals herself to be significant and admirable. And actually, I always find Pythagoras confirmed that she remains similar in all her creations…The self – same numbers, which affect the harmony of voices to sound pleasant in the ears, also fill the eyes and the soul with extra-ordinary joy. And thus, I will borrow the laws of the outline from the musicians, who explored these numbers”2,3,4
Moreover, the spherical knowledge of the many variables and coefficients that exist in a specific field of art was a necessity and common rule for the luthier, artist or musician, who did not live in our modern times. In Isaac Newton’s time, a scientist was also a philosopher, for science was only a natural outcome of philosophy, not being able as yet to stand independently on its own feet. It was well after Newton’s time that the distinction between science and philosophy became a reality.
Today, J.S.Bach is regarded to have been a great composer, however few people know that he constructed, invented and played many instruments as well as improvised successfully on all of them, all in addition to his other obligations. His interest in acoustics and mathematics was significant, or he would never have been able to succeed in creating the “Well – Tempered Clavier,” which led to the well-established “equal temperament.”
Inevitably, the above rule became an exception in our modern times of “specialization” giving way to mass production needs and various competitive marketing demands. It naturally skips our minds, that the above mentioned distinct peculiarity of poly-liturgical necessity, involving so many different and various parameters of daily functional happenings, could have been the very basic reason for those centuries offering us the great masters in any field of art or science.
The social and economic changes that impose different and/or even limited demands on the individual remain a controversial subject; for the author is aware of the greats of the 20th century. The point is made comparatively to hint at the importance the author gives in analyzing the work of the great artists or luthiers that decisively marked the arts, and lutherie in general, in past centuries.
One plunges into historical and sociological habits and trends with the hope that through a knowledge and analysis of the facts, a unique synthesis may evolve that will be compatible to one’s own daily life and do better justice to his or her work, including general society’s increase in valid knowledge.
The ideas, thoughts and traditions that circulated and influenced the musical world, the arts, lutherie, interpretation and so on, were under a moving and changeable process, with constant, simultaneous experimentation taking place. Tradition remained strong and unchanged whenever it reached perfection in the arts.
Any experimentation that was outside of the tradition had to be one step clearly ahead of it, before it could be adapted and either change or revise tradition. One step short, clearly meant failure of the experiment.
Video Caption: Smaro Gregoriadou is interpreting on a Kertsopoulos constructed classical guitar respecting the “Kertsopoulos mathematical model and geometrical progression”.
Somehow, it is somewhat discouraging to realize that today’s technology cannot hint at the true details of success regarding a Guarneri, Amati or Stradivarius violin. A logical consequence if one realizes that the real details of their success lied inside the brains of the builders, and in the actual process of each unique construction.
The actual process of the construction is of great importance in lutherie. It progresses lively and depends fully on the conscious and even sub-conscious intuitive applications of the luthier, who feels the intimate reactions of his wood during the process. It is a happening. Of course, the happening will include measurements, proportions, mathematics, and data that can be analyzed and synthesized and even fed into computers.
The happening itself though, cannot be grasped, measured or analyzed, lacking the possibility to the third party that comes after the construction process to be able to conceive and comprehend the real cause-reason of acoustical success. There seems to be a natural boundary by space and time to guard the unique process of the construction, similar in a sense to the eventual birth processes of all natural organisms.
Additionally, every day is a new and different world with various unique conditions of temperature; relative humidity and many other factors involved, influencing the gluing, stress distributions or even the varnishing processes. Such factors are not only considered seriously by each maker, but also make it impossible for a third party to exactly trace the real and detailed causes that are responsible for the construction of a beautifully sounding instrument.
Although we cannot exactly trace the happening itself, an effort to deeply comprehend the constructional applications of the proportions or geometric dependences of a musical instrument of a great luthier can lead one into the different essential channels of thought in the maker’s mind.
This has been the case in the violin family of instruments. Once the mathematical model of the violin was established, it provided answers to many other constructional applications. Which, although they seemed independent, in the long run were interconnected with the mathematical model itself. Throughout the history of the luthier’s profession, sadly enough, the communicational climate has been chilly, or even frozen in many cases, because the field was characterized by extremely high competition.
The result being the guarding of the successful applications in construction by luthiers, or within the society of their fellow associates that existed in many cases, to defend these findings from competitive intruders. This caused successful methods and applications to became “secrets of the trade,” to be transmitted only to the trusted ones or to their heirs.
An historical example of this is the lute society of the “Tieffenbrucker”5 (Italy) in the renaissance, which had as a goal, along with the guarding of the important secrets of the trade, the economical blockade of every new luthier who was not an approved member of the society. This was expressed with the sale and circulation of wood strictly to the members of the society, since the society held strong influence with the wholesale traders of wood and material.
Even more, a lutenist would teach the student how to play the lute but would never say a word to the student about the stringing of the lute. The cat-gut strings had a special technique to clean them, hanging them by the ceiling with specific weights for each gauge of string to maintain their molecular structure in straight lines for uniformity. As well, they were strung on the lute with a winding process at specific angles.
The student had to search and find everything in his or her own self-taught way, without being shown anything by their teacher concerning the stringing process. After all, the stringing process was related more to lute construction and less to lute playing, so it enjoyed the secrecy imposed by the builders of the lute themselves.
Video Caption: Yorgos Kertsopoulos is interpreting on a Kertsopoulos constructed flamenco guitar respecting the “Kertsopoulos mathematical model and geometrical progression.”
To summarize the above, all of these issues created a climate that led to the well-guarded transmission of accumulated knowledge in historical lutherie. Moreover, one can see how this climate is related to the “aesthetic” importance held in the application of the mathematical design and proportions involved, which simultaneously was at all times expected to lead to the successful acoustical construction of an instrument.
So then, what was the situation back in 1976? This was the year that the author started to search for the mathematical model of the traditional classical guitar. Could the lack of recorded, documented reference (at that time) in regard to the mathematical proportions and geometrical design of the guitar become capable enough to disprove their existence?
The international bibliography covering the instrument’s history and construction did not mention their existence, and the interviews with important luthiers up to that time negatively answered the often-asked question by interested parties and guitar construction experts and reporters.
The five-year extensive research and design work that was undertaken by the author until 1981 included the investigation and study of the work of all great makers, keeping however the work of Antonio de Torres as the reference point. The concrete aims were to determine the specifically valid shape, which marked the instrument’s character and not to purposely invent a mathematical model unrelated to the “aesthetic” tradition of the instrument.
Finally, the existence of intricate and synthesized mathematical proportions and ratios evolved, being supported by a geometrical progression, constituting the design of the perimeter-outline, bridge and sound hole, all determined by the chosen specific string length of the classical guitar.
Video Caption: Dionysia Blazaki and Yorgos Kertsopoulos interpreting on two guitars constructed by Y.Kertsopoulos, respecting both the “Kertsopoulos mathematical model and geometrical progression.”
The mathematical model, which is defined by these proportions, did not easily evolve because of the fine constructional adjustments that were found in the different analyzed instruments. Furthermore, these adjustments were different for every guitar, whether Torres or another maker made it.
The geometric progression supports the mathematical model containing both simple and intricate mathematical proportions and ratios, as well as constructional approximations and dependencies of an interesting historical and acoustical context.
It will be seen that the whole essence of the guitar’s shape and context is directly related to the ratios and relations existing in the physical behavior of sound as expressed by the harmonic spectrum, the specific tuning of the instrument, the ideal behavior of strings and the diatonic system.
This multi-interconnection in these relations, combined with the acoustical function of the perimeter-outline, create “standing wave zones,” specific enhancements of ideal resonant frequencies in the chamber and control of “wolf tone” production, giving satisfying answers to so many questions put forward by many in the past.
1) What determines the location and size of the sound hole and the bridge?
2) Why this particular shape?
3) What is the real magic behind the 65-cm scale length?
4) What is the “aesthetic” and acoustically ideal curvature of the outline?
5) Why does every mysteriously experiment lead back to the center of the tradition, that center being Antonio de Torres?
6) Why did Hauser and so many other luthiers copy instinctively, and through experience, the construction work of Antonio de Torres?
One can always argue that all of the proportions and ratios evolving from the mathematical model and geometrical progression of the classical guitar is a game of luck, a plain coincidence of factors and somehow the author related them to the guitar itself, when in fact they do not belong to the tradition of the instrument.
As it will be seen throughout this seven-article series, this model is so intricately and accurately constructed mathematically, and it will be proven so acoustically modulated and refined, that it excludes its coincidental relationship to the traditional classical guitar.
What remains is the understanding of its mathematical construction, followed by an acoustical analysis, which will try to enlighten and put focus on its acoustical value and validity. This will include the behavior of the model along with the results that will be produced when adjustments are made to it, and what the results will be if it is not applied at all.
Before proceeding, it is necessary to describe and define a few matters concerning the behavior of sound in general, the ratios that exist in the harmonic series, the diatonic system, the equal temperament and the unique function of the guitar as a performance instrument. All this, will be coming up in the next seven volumes of the series.
Dedication of Jose Ramirez III
“In Madrid the 8th of April of 1983
The work realized byY.Kertsopoulos on the mathematical solutions of the dimensions of the guitar deserves all my respect and admiration, not only from its scientific scope but also for being a serious study that enlightens some dark matters concerned with the forms of the instrument not defined sufficiently by tradition.”
With my congratulations,
1. E.G.Baron, Historisch-Theoretisch und Praktische Untersuchung des Instruments der Lauten…(Nuremberg, 1727) p. 90,p>
2. L.B.Alberti, De re aedificatoria, IX, 5
3. In the early renaissance and even later, music was regarded as the “theory of numbers” and the Pythagorean conception the main point of reference. B.Munxelhaus, Pythagoras musicus (Bonn, 1976) p. 15.
4. Gerhard Christian Sohne who also refers to these historic points has done an important work of reference concerning the geometric construction of the lute. Gitarre + Laute (April 1980).
5. Tieffenbrucker is a large multigenerational family of luthiers originally from Bavaria active in Venice and Padua, Italy.
A classical-flamenco guitar soloist, composer, professor, luthier and string constructor, the Greek – Canadian Yorgos Kertsopoulos is also a researcher, inventor, author and founder of “the Guitar’s mathematic model and Geometric progression” (DAS MUSIKINSTRUMENT, Heft 9/ Sep. 1984, in German and English “The Physics of the Guitar”- Presentation of the same at the International Music Messe-Frankfurt, College of Furniture-London and seminars to respected luthiers and guitarists of England invited by GUITAR magazine)…. his work is so unique in its scientific approach but also so full of traditional truths, it includes everything…Jose Ramirez (at the press conference given by Y.Kertsopoulos at the Music Messe in Frankfurt-1983). His work as “Kertsopoulos aesthetics” involves the constructional revival of the different forms, tunings and sound timbres the guitar possessed in its history…These guitars possess antique sounds, sounds with an ancient charm. Amalia Ramirez 7 Feb. 2009.
Yorgos is the author of: “Space-Time Theory Vol. A’ The Philosophy of Space-Time”