Musical  Arches

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 Bartok.jpg (41414 bytes) Kodaly.jpg (40978 bytes)


On the left is Bartók Béla.  He used a symmetrical plan for several pieces of music, which employed of five movements.  This arrangement is often referred to as an arch.  The most symmetrical of these pieces is the fourth string quartet.  The main themes of the first and last movements are identical; in fact the last page of each movement is almost the same.  The theme is like a little arch of notes.

The themes of the second and fourth movements are also very similar, differing only in tempo.  Each looks rather like a perspective view of a bridge with three arches receding into the distance.  Supported in the centre of all this is a an eloquent and passionate piece of music.  It contains elements from the sounds of nature and from folk music.

It also seems to contain an ever-decreasing series of golden sections, like a sunflower.  The total number of crotchet beats in the whole work is 2584 - a Fibonacci number.  This isn't pointless numerology - the composer was well aware that these numbers occur in nature.

On the right is Kodály Zoltán, a composer, and a collaborator and friend of Bartók.   

These busts are close together on Margit-sziget in Budapest.  After Bartók heard Dósa Lidi singing, in 1904, he became very interested in folk music, as did Kodály, and in time their music became a bridge between "art music" and "folk music".  

In the case of Bartók, the music is perhaps also a bridge connecting nature, art and science, three areas of great interest to him.


Composing a piece of music presents technical problems, like building a bridge.  You have some idea of the span of time that needs to be filled. You have to decide on the number of  sections or movements.  You also have to decide on a means of relating these movements to create a coherent whole . This process goes on right down to the smallest details, until the construction is as satisfactory as you can make it.

The artistic person can create whatever she or he likes, as long as a living can be made from it, or by some other means.

Building a bridge is different. Economy is far more important than in art.  The bridge designer is forced to work with what nature offers, rather than with a blank sheet of paper.  The width of the crossing, the type of ground, the depth of water, the range of the tides, and the navigational requirements, for example, are not under the designer's control.  But he or she can use art and science to produce the best solution to the problems.  And solving the problems may lead to great ideas, just as the search for a line with the right rhyme and rhythm may lead to an idea that a poet might not otherwise have had.


At the beginning of Beethoven's quartet Opus 131, or Bartók's Music for Strings, Percussion and Celeste, we seem to be hearing not so much tuneful, foot-tapping music as the building of a great structure.  The Bartók piece is literally built up, bit by bit, towards  a  tremendous climax, from which it descends quite rapidly on the other side.  Erno Lendvai has shown that this climax occurs at about the golden section of the movement.

The Beethoven piece gives a great sense of density, in terms of music per minute.  A musician could explain this, but as with bridges or nature, we can enjoy things without understanding most of the details.  But nothing is lost if we learn more - science does not spoil pleasure - it provides an extra dimension.

Both pieces were created for specific instrumental groups, whereas Bach, in some cases, did not specify any instruments at all, believing that the structure was all.  So when you buy a recording of the  Musical Offering, you will hear a particular choice of ensemble.

In all these examples, the music exists only for itself: no story is being told.

Wagner's Ring cycle, on the other hand, seems to be engineered quite differently, as if it were a huge machine, designed to create the frame of mind in which the quite implausible stories become not only acceptable, but inevitable, so much so that they can seem to dwarf the real world.  Few people would say that this music is of equal interest at all points, and a non-Wagnerian might think that much of it is only there as a framework for the points of high emotion.

Yet another structural type is found in those minimalist pieces that repeat many times without change, rather like a trestle bridge with many spans.


Many kinds of art and science involve structure and symmetry.  In another part of this web-site there are pages about symmetry and broken symmetry in physics.

Perfect symmetry in art would be rather boring.  On the other hand, an art in which nothing is related to anything else would be very odd.  Even in a painting by Jackson Pollock, there is relationship between the components, and the paintings are recognisably different from each other.  They have not much structure, and not much symmetry, except in the sense that a "truly random" object is about the same at any point.


On the other hand, poetry that rhymes and has metre is both well structured and highly symmetrical.  The most symmetrical poem would consist of the same word repeated many times.  

The symmetry is reduced by breaking it into lines, and grouping these into stanzas.  The lines are generally all different.  But the poem is held together by the rhyming scheme and the metrical pattern.  So we have an example of broken symmetry.

There may be occasional imperfect rhymes or feet that are not a part of the rhyming scheme.   But these may even be used to emphasise a point.


Prof Ian Stewart, in "Life's Other Secret (Penguin), shows how broken symmetry and simple rules can generate complicated structures and behaviour.  So symmetry and patterns arise in nature as well as in art and engineering. What are the differences between "popular" music, "folk" music, and "classical" music?  In fact there are so many forms of music that this probably cannot be answered.  But at least we can ask how they differ in complexity of structure and the degree of ornamentation of those structures.


In an article about Bartók, Kodaly wrote that art and science are not unrelated, and that someone can achieve greatness in both. This is surely true of the best bridge builders, whether the bridge is a humble footbridge or a great span. The article is quoted in "Bartók Remembered", by Malcolm Gillies, published by Faber and Faber. In fact, no activity exists in a vacuum: as as CLR James wrote - "He knows not cricket, who only cricket knows". C P Snow made a cause out of "The Two Cultures". As there are thousands of cultures and sub-cultures, from angling to zebra finch breeding, it is not obvious that this simple classification has quite the value that he thought.


In Budapest, you can see a house which was once occupied by Bartók, and another which was once occupied by Kodaly.  You can see manuscripts of the song and the intermezzo from the Háry János suite.  If these were found in a thousand years time, they might be incomprehensible.  If the finders did not know whether to read them from right to left or from left to right, or whether high notes were at the top or bottom, and what system of pitch notation was used, they would get no further.  And without a knowledge of the instruments of a 20th century orchestra, they would not be able to hear these pieces as they were intended.

The score is rather like the genetic code.  It has all the information, but only if you have the correct environment, consisting of a suitable group of players and their instruments.  And the players need to know the genre.  Listening to classical players trying to play jazz is not always an enjoyable experience.  Trying to recreate extinct animals may fail for the similar reasons.  The code is not enough, as Ian Stewart says in his book - "Nature's Other Secret".

Perhaps in our chromosomes there are parts which were once used in some other organisms. Just so, in the music of Bartók and Kodaly, there are parts from other countries than their own, and through those parts, traces from countries and times far away, for no musician is an island. Engineering, mathematics and science, too, as we know them, have been developed through thousands of years in many places. It can be exciting to travel physically to other places, but sometimes travel in the mind is possible when contemplating the arts and sciences. Most of us can understand things only at a simple level, but even so, we are privileged to do so. We should never say "I am brilliant because I can understand Shakespeare.": we should rather say, "Shakespeare must have been brilliant, in order to communicate to a dullard like me."

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