[ the craft of the guitar maker ]By TREVOR SEMPLE
6. The FingerboardTHERE was a time when mankind believed that the universe, perfectly created by god or gods unknown, could be understood through simple mathematical relationships. Music then was seen as some kind of metaphysical mathematics, a system of simple and perfect proportions that mirrored the beauty and symmetry of that universe.
Tradition assures us that Pythagoras was the man who set out to establish some of these mathematical proportions. He is credited as the first man to systematically study vibrating strings using a monochord. He developed, we are told, the tuning system that bears his name. In the course of his life, he may well have set stylus to paper, but since no writings of any kind have survived, he has assumed some of the legendary qualities of King Arthur, and may well have been given credit for things he never even dreamt of.
Whatever the truth of these legends, it is clear that for several centuries BC various Greeks were developing theories of music. Their experiments took the theories of the early philosophers and shook them to the roots, like huts in an earthquake, in the way that some of the great scientific discoveries have done from time to time since. What they discovered was this: simple musical intervals just did not fit together. It was like working with some of the pieces from several different jigsaw puzzles however you tried, there were always pieces left over.
The impact of scientific discovery that started so long ago has continued in the same direction ever since. We have come to see a universe of incomprehensible size, complexity and chaos. We no longer believe in perfection. Some of the final nails for the coffin were accidentally provided by Heisenberg and Einstein, who seemed to rob us of the few beliefs we were still clinging to. It remains to be seen if mankind can find a new understanding that allows us once again to value ourselves and the natural world around us.
I sometimes look at all the irregularities of a guitar fingerboard and wonder whether it just might prove not only that God exists, but that She has a wicked sense of humour! Leaving the metaphysics aside, let us now look at the gremlins that cause all the trouble. If you start at any given note of the musical scale, and climb (or descend) by intervals of a pure fifth, no other note within the range of our hearing will ever again fall on any pure octave of the starting point. If you start again, using the interval of a pure third, no note will now ever overlap either with the series of fifths or octaves. This is easier to visualise using a diagram.
The process of resolving these awkward intervals into music, we call intonation. It is a relatively easy affair with instruments of flexible intonation like the wind instruments, the violin family or the human voice. The problems are much more difficult with keyboards and fretted instruments. Whatever intervals you choose are then fixed until you stop and retune. In the case of keyboards or microfrets, this is not something to take on in the middle of a concert. You therefore have two options. Either you spread the errors fairly evenly across the whole range of possible keys, or you pile them all up in what you hope is an obscure corner, like sweeping dust under the carpet. Over the years, many ingenious solutions were worked out, and tuning systems for harpsichords and organs number several hundred.
Although the same methods might have been used by lute or guitar players, the practical problems were insurmountable for centuries. Tied gut frets do not lend themselves to micro-fretting. The calculations to enable stepped metal frets were not possible until the latter part of the 19th century, when it was possible to measure frequencies accurately.
So it was that lutenists and guitarists became the trendsetters, admittedly largely by default. They adopted equally tempered intonation some 300 years before it was widely accepted by musicians in general. Broadwoods, the piano makers, did not start tuning pianos this way until about 1860, over 30 years after the death of Beethoven. At about the same time, one or two Victorian eccentrics were developing ways of doing the reverse on the guitar. One such attempt was the 'enharmonic guitar, a creature of such horrendous complexity that most musicians would share about equal enthusiasm for lion taming.
Although equal temperament has become the single accepted tuning system for western music, it does entail serious compromises and the resultant murky harmonies. The purity of true intervals is quite staggering, and sometimes this beauty can be glimpsed in the open tunings of some folk guitarists. It is quite possible to use a classical guitar like a monochord and to explore micro-intervals, and experiments in this direction can give rich rewards. Equal Temperament: intervals are 'tempered' or pulled out of true so that: 12 fifths fit into 7 octaves. 3 major thirds fit into 1 octave. 7 major thirds fit into 4 fifths. Pure intervals give rise to Just Intonation. Thirds and fifths are tuned by ear to give the least beating when two notes sound together. This arises from the least interference between the combined harmonics. Such "pure" intervals are perceived as particularly restful and pleasing, but unfortunately they no longer fit together with each other. The compromise: tempered fifths are flattened by around 1/10th of one per cent (2 1/2 cents). Tempered thirds are sharpened by a massive 1 percent (12 cents). A succession of 12 pure fifths is now larger than 7 octaves by one "Pythagorean comma".
©1990 Trevor Semple
"Classical Guitar Magazine", January 1991