21st Century Science & Technology

Violin Building and the World of Harmonic Sound

by Caroline Hartmann

(full pdf, reprinted from 21st Century, Spring 2003)

Intervals in classical musical performance cannot be expressed by a predetermined mathematical ratio, nor can the building of great musical instruments be reduced to a formula.



"A beauty of sound, which knows no bounds," is how a luthier once described the tone of the old Italian (and Tyrolese) violins. And, in fact, these old violins embody a concept of instrument-building, which culminated in the great masterpieces of Antonio Stradivari, which can fill great concert halls without any effort, and can be heard over the entire symphony orchestra. Their richness of tone ranges from majestic fortissimo majestoso, to the sad lament of an andante molto grave, up to the outburst of joy scherzo prestissimo. The only other instrument that can be compared to them, is the trained human singing voice. Yet all these tones come from a little wooden box about 35-37cm long for the violin, 42cm for the viola, and 74-78cm for the cello.

When you look a little more closely at these handmade masterpieces, you can recognize that this kind of instrument is a triumph of both acoustics and physics. There are many myths in circulation about these "fabulous instruments." Were the luthiers of old initiated into some secret of the acoustics of physics, which we no longer know today; is there a secret kind of wood, or art of varnishing, which makes all the difference in the sound; or, is there some other kind of secret, relating to how the wood is cut, which had been carefully protected and was lost after the death of Stradivari?

There are still many unanswered questions today about the construction of the violin family (violin, viola, and cello, which all have the same underlying principle of construction), especially after the death of Stradivari, when the great art of creating such wonderful works sank, further and further, into oblivion–and the wildest speculations and most bitter disputes arose. Most of these discussions, however, quite ignore the key question. The Renaissance, the time that saw the birth of this family of instruments, was a time of many discoveries, not only in music and instrument-building, but also in painting, the plastic arts, architecture, and machine building. The most significant and many-sided artist and scientist of this time was, without question, Leonardo da Vinci.

The situation in the world of music, and especially regarding musical instruments at the time that we call the Golden Renaissance, provides a number of substantial reasons for assuming that the building of the violin was an invention. The idea for building such an instrument must have sprung from the new discoveries of the Renaissance–above all, from Leonardo da Vinci’s investigations into tonality and sound.

However, the Pythagoreans were already acquainted with the paradox, that man recognizes only a few intervals as harmonic, and these intervals, which are heard as harmonic, are created on a stringed instrument by placing the fingers with different spacing between them in each different key. Here, we will pursue this idea somewhat further.

The Unsolved Paradox of Pythagoras

All instruments created by man, use what he has known for thousands of years, that when strings are stretched over a hollow space, more or less beautiful sounds or tones can be created. In India, an instrument of this kind was built around 3000 B.C. Later, Pythagoras (around 500 B.C. ), discovered that it was possible to express the relationship between two tones–called intervals–by rational numbers.

Pythagoras invented a one-stringed instrument, a monochord, which the Pythagoreans used for demonstrations, and as a musical instrument. Today, it is used to demonstrate intervals. For example, if you press down on 1/3 of the length of the string, and then pluck or strike it, the resulting tone will be the interval of a fifth above the tone of that same string when it vibrates freely. The significance of his invention was that man recognizes, or experiences, only a few specific intervals as beautiful. These intervals were called synphon by the Pythagoreans, and are the following:

• Octave (ratio 1:2),

• Fifth (ratio 2:3),

• Fourth (ratio 3:4), and

• Third (ratio 4:5).

In addition, there is also the 5:6 ratio, which is the minor third.

The Pythagoreans possessed an 8-stringed lyre and kitharra. All the stringed instruments taken as a whole, up to the beginning of the 16th Century–that is up until the invention of the violin family–had the following characteristics, which significantly limited the quality of their sound, and did not leave much room for expressing a variety of the scale’s tone colors (for more on this, see Appendix 1):

(1) The fingerboards of these instruments are divided by small ridges, called frets, most familiar to us today from the guitar. The pitch is determined beforehand by these frets, so that for "pure" playing in all the keys one often has to make compromises. Depending on the kind of instrument, there was a certain tempering chosen which allowed for playing in the greatest possible number of keys. One aspect of this, is that the distance from one fret to the next is always different; whence there were naturally many different temperings. When the limits of each instrument’s tempering were reached, it had to be retuned, which was the general practice. The discrepancy between the notes sounded on the frets and the proper pitches, as the musician moved through different keys, is sometimes described as the problem of the Pythagorean comma.

(2) As for the sound, the resonance chambers of these instruments were for the most part quite flat, or as is the case with fiddles, lutes, or many viols, arched according to certain specific geometrical forms (a cylinder), or with a shape taken from forms in nature. This, from the start, put a limit on the capacity of providing for a "real" or peer-quality accompaniment to the trained bel canto voice. Moreover, the bridge of the instrument is not curved, so that the bow cannot avoid touching all the strings at once, which means that only chords can be played. This kind of limitation can be easily recognized in the accompanying painting of the angel by Fra Angelico (p.19).

The new instrument family of the violin, viola, and cello were revolutionary relative to both these points. The characteristic vaulting curves of these instruments have remained unchanged until today, the instruments showing the same proportions down to the smallest detail. Unlike almost all of man’s other inventions, this form has stayed unchanged for 550 years.

Moreover, the paradox of the colors of the tonal scale is solved with genius: They simply eliminated the frets, so that the player himself can determine the pitch and how he will play it. Other than the human singing voice, there is no other instrument which allows this. What a revolutionary breakthrough in music! The instrumentalist could finally "sing" with his instrument, as we know today, from hearing the great violin, viola, or cello virtuosi. These two points also prove that there is no way that the violin family could have developed stepwise from some other instrument.

The luthier Max Möckel, who worked around the turn of the 19th Century in St. Petersburg and Berlin, did not rest until he had investigated the true origin of the sonorous and architectonic beauty of the violin. His idea was to investigate whether, in the light of the knowledge of the Renaissance, it might not be possible to discover what part had been played by Leonardo da Vinci, Luca Pacioli, and Albrecht Dürer in the revolution in instrument building. Thus, he began to look for clues to support his hypothesis in the works of these great artists, and he came to the following conclusion:

Is there really an Italian secret? Yes and no. If we think of it as some kind of recipe, hidden somewhere in some old chest, then no. . . . We must put ourselves into the time in which the violin was invented, and the ideas out of which each of the old masters created their works . . . The most significant minds, to name but two of them, Leonardo da Vinci and his friend Luca Pacioli, had shortly before concerned themselves, in their work of so many facets, with mathematical problems, and when they saw the triangle and the pentagon, they did not see them as merely simple geometrical figures, but they saw in the pentagon, for example, the secret eye of God, a living sensuous image, with its infinite number of unfoldings, for everything that is becoming.

With this hypothesis as a starting point, Möckel developed a procedure for building the violin, viola, and cello, whose standard was what Luca Pacioli called the Divine Proportion. (In the Divine Proportion, the division of a line or a geometrical figure is such that the smaller dimension is to the greater as the greater is to the whole.) From that time on, he built many excellent instruments according to this method. . . .

We present this piece as a contribution to the pedagogical effort of the LaRouche Youth Movement, which is presently struggling to master the paradox of the Pythagorean comma. Their crucial, related purpose is to attempt to revive some aging intellects of the Baby Boomer generation, who have denied these youth a future by their immoral abandonment of the principle of truth. Readers may find an audio archive of a panel presentation on the Pythagorean Comma by members of the LaRouche Youth Movement, at the website www.theacademy2004.com (click on "2003 Schiller Institute/ICLC President’s Day Conference," on the home page).

This article originally appeared in the German-language Fusion, (July-Aug.-Sept. 2002). It was translated into English by Richard Sanders. The author, Caroline Hartmann, is an organizer with the LaRouche political movement in Germany. She plays the violin in the Schiller Institute orchestra.

Anna Shavin (left) and Jennifer Kreingold at the LaRouche Youth Movement panel presentation during the President’s Day Conference Feb. 16, 2003 in Reston, Va. Kreingold sang a sequence of natural thirds, to demonstrate that they do not close at the octave. [EIRNS/Stuart Lewis]

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