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Where brain and heart meet: Johann Sebastian Bach

This version: 18 August 2003

I have a theory.

What makes truly great art? I believe it is the perfect fusion of the creative spirit and the conscious mind. Much so-called art of today is, I believe, the product solely of the former. That is why it seems to Philistines like me to be incomprehensible. The mind is needed to set the plan, to solve the problems, to order the creation into a challenging whole.

And what is such a fusion. Let us look at the greatest composer of music who ever lived: Johann Sebastian Bach (1685-1750). Let us look further at one tiny part of his oeuvre: The Well-Tempered Clavier. Listen to two of its preludes for just an idea of the man's artistic genius (Prelude number 8 in E flat minor, and Prelude number 10 in E minor).

In Bach's day there was an argument in the musical world about the preferred method of tuning musical instruments. The two leading contenders were Just intonation and Well tempering. Bach was pushing for the latter.

And what are these anyway? To understand, you need to look at just a little of the mathematics of music and tuning scales. Consider a piano keyboard. The bottom note on a concert grand piano (not counting the Bösendorffer which has a few extra bass notes) is an A. Its fundamental frequency is exactly 27.5 hertz. Each octave is a doubling of the fundamental frequency. Thus three octaves above this A is the A below middle C at a frequency of 220 hertz (27.5 times 2 is 55, times 2 is 110, times 2 is 220). Middle C itself is 261.63 hertz, approximately.

Why this odd number? Well, there are twelve notes in an octave (not eight -- don't forget the black keys on the keyboard and don't double count the end notes). The frequency of any note is the frequency of the preceding note, multiplied by the twelfth root of two. (220 times 2(1/12) is 233.08, the frequency of A#, times 2(1/12) is 246.94, the frequency of B, times 2(1/12) is 261.63 hertz.)

That is today. But it wasn't necessarily so in Bach's day. Well-tempering is very close to the system we use these days (which is called Equal Temperament) and it was for this that Bach was arguing. Just intonation was based on a system of perfect harmonies.

So how do harmonies work? What is it that makes two notes sound good together and two others rather teeth-grating?

Well, your typical piano note is not a pure single frequency. If it were, believe me, you wouldn't want to listen to it. Get a sine wave generator some time and have a listen. What you hear when you strike a piano note is a complex wave form. We know from Jean Baptiste Fourier (who was born some 18 years after Bach died, blinded by the same eye surgeon who saved Handel) that any repetitive wave form, no matter how complex, is constituted by a fundamental frequency (which may be of zero amplitude) and a number of additional frequencies which are whole multiples of the fundamental (the harmonics). Thus a square wave consists of a fundamental frequency, no second harmonic, a third harmonic that is one third the amplitude of the fundamental, no fourth harmonic, a fifty harmonic that is one fifth the amplitude, and so on).

Now the really strange thing about all this is that what we hear is the tone reconstituted into its individual frequencies. It is that balance of these frequencies (along with other elements such as the rates of attack and decay, variations of frequency balance through the sounding of the note, and spurious sounds) that give each instrument its distinctive character.

Now the mathematics comes in here. Take the musical interval of a fifth, for example: the notes C and G. On the modern scale, the third harmonic of middle C is 784.88 hertz. The second harmonic of G above middle C is 783.99 hertz. A remarkable coincidence that causes these notes to sound most pleasant when played simultaneously.

But they can sound even more pleasant. By tuning the instrument so that C's third harmonic and G's second harmonic are exactly the same frequency, rather than most of a hertz apart. This was what the system of Just intonation sought to do.

There were two problems with this system. The first was that notes such as F sharp and G flat were not identical. One can imagine the complexity of a keyboard designed to provide all the possible variations. The second problem was that if an instrument with fixed notes (this did not apply a fully variable instrument such as a violin or trombone) were tuned to provide Just intonation in a particular key, several other keys were thereby rendered useless because the harmonic coincidences were lost.

And J S Bach? What was his response?

The Well Tempered Clavier is a cycle of 24 Preludes and Fugues. Why 24? Because the first Prelude and Fugue are in the key of C Major. The second in C minor. The third in C sharp Major. And so on. For all 24 keys. The Well Tempered Clavier is unplayable on an instrument tuned with Just intonation. (That is Book 1. Some years later Bach produced Book 2, also covering all 24 keys.)

Bach's motivation and conception was in his conscious intellectual mind. It was his creative spirit that filled in the details with such moving works as Prelude number 8 in E flat minor and pretty pieces of virtuosity as Prelude number 10 in E minor.

The Preludes provided here were keyed by me (using mouse and computer keyboard that is, not musical keyboard) and while I retain all rights, I encourage people to download and distribute them. I have done little with them as far as tempo and dynamic variations, so I would welcome people using these as a template for better performances. If you do so, please leave in my authorship information (as well as adding your own), and send me a copy so that I too may enjoy the best possible version of these works.

© 2000 by Stephen Dawson