Bach and The Tuning of the World. Jens Johler
asked.
‘Fifteen.’
‘And how many are on scholarships?’
‘The same number.’
‘Then, one of these days, we should organize a contest,’ said Bach. ‘Not in bowing and scraping, of course, but perhaps …’
‘In philosophizing,’ Erdmann suggested.
‘Or in singing,’ said Bach. ‘We can certainly do that much better than they do.
A singing contest never came about, though, and would anyway have been meaningless. They often sang together with the knightly students, and there was nobody who could deny that the choir students were more musical. The aristocratic gentlemen didn’t much care. They looked down on the scholarship students like they would on poor chirping birds who were born to warble, who had to do so out of necessity. The only one among them to whom they looked with something approaching respect after a while was Erdmann, because he spoke so well and got a kick out of styling his language to courtly etiquette.
‘I’ve thought it all over,’ he said after some time had passed. ‘I don’t want to become a philosopher after all, but a diplomat.’
This surprised Bach. Not so much because Erdmann all of a sudden wanted something different than what he’d wanted only a couple weeks ago but, rather because he had actually made such a decision. For him, Bach, the question didn’t exist. It had been clear from the onset he would be a musician. He came from a family of musicians, so what was there to think about? At most, the question was: What kind of musician? Town musician like his father? Organist like his uncle and his brother? Cantor like Elias Herda? Or kapellmeister at one court or another? And there was another question he asked himself sometimes before he fell asleep: With whom would he vie in the future? With the greatest musicians of his craft, with Reincken and Buxtehude, Corelli and Lully?
The discipline at the school was very strict. Every little thing was planned, and any deviation from the rules was strictly punished – when you were a scholarship student anyway.
But while Erdmann clandestinely rebelled against the unnaturalness of the unyielding rules, Bach acquiesced to the strictly disciplined system.
Along with the others, he got up at five in the morning, washed, combed his hair, dressed and, right where he was, got down on his knees for their first prayer, whether on a stone floor or scrubbed floorboards, as soon as the first quarter struck. During meals, he heard the chapter of the Bible that was read to them, refraining from speaking or any mischief, exactly as prescribed by the school’s set of rules. He kept his clothes, shoes, stockings and underwear clean; he swept the rooms when his turn came. During classes, he was attentive, made notes and memorized as much as he could, which required little effort since his memory had always been excellent.
The curriculum consisted of Latin and Greek, Religion and Logic, History and Geography, Mathematics, Physics and German Literature.
Bach had already found a special knack for mathematics when he went to school in Eisenach and Ohrdruf and so in this subject he could shine. During the first week, he had the chance to prove the theorem of Pythagoras and, when the teacher asked him what else he knew about Pythagoras, he answered that Pythagoras was one of the great sages of antiquity. Not least, he explained, Pythagoras was famous for finding the mathematical proportions of the harmony. The teacher asked whether he also knew how Pythagoras came to his discovery.
‘Certainly,’ Bach replied, stealing a quick look at Erdmann. ‘Lost in thought, Pythagoras was walking by a smithy, where several journeymen were hammering the iron on an anvil and suddenly he noticed how they created harmonic sounds; to wit, the fourth, the fifth and the octave. Astonished, he walked into the smithy to look for the cause of this array of sounds and ultimately discovered that the harmonic proportions of the notes have whole number ratios. He then demonstrated it on the monochord, which the Greeks called the kanón.’
‘How would you describe a monochord?’ the teacher asked, doing so because some of the students looked puzzled.
‘Well,’ said Bach, ‘it’s a board or, rather, a sound box over which a single string is stretched, let’s say with a length of four cubits. When strumming this string, you hear a note you could call the tonic. If the string is divided up into two equal halves by positioning it over a wooden bridge and the half-string is hit, the octave will sound. Hence the proportion: whole string to half string, or 2:1. If you now divide off two-thirds of the string and strum the longer part, you get the fifth. So the fifth has the ratio: three-thirds to two-thirds, i.e. 3:2. The fourth, in turn, is ruled by the ratio of 4:3, the major third by the ratio 5:4, and so forth. And, as mentioned before: all harmonic intervals are governed by whole number ratios.’
‘Excellent,’ said the teacher. ‘Then you also probably know what the Pythagorean Comma is?’
‘Oh, yes,’ Bach said eagerly, without noticing how the others’ eyes by now were turned on him with envy.
‘Well?’ asked the teacher.
‘A comma,’ said Bach, ‘if you translate it literally from Greek, is nothing but a section, and in this case – well, it’s not so easy to explain. Do I have permission to go to the blackboard and draw a sketch?’
‘Please do so,’ said the teacher.
Bach got up from his desk and walked to the blackboard. ‘Here is how it is,’ he said, turning to the class. ‘If you tune perfect fifths on an instrument, namely exactly in a ratio of 3:2, and go up higher from fifth to fifth, from C to G, from G to D, from D to A and so forth, you’ll return to the C after exactly twelve steps, only seven octaves higher. It’s called the circle of fifths.’
He turned his back to the class and drew the circle of fifths on the blackboard:
There you could see it. It began with C and ended with C, only seven octaves higher. It was simple.
‘And where is the Pythagorean Comma?’ enquired the teacher.
‘Yes,’ Bach said, ‘that’s the real problem. If you tune perfect octaves, namely from C to C’ and so forth, you’ll have a different note than by tuning to perfect fifths.’
‘Why?’ the teacher asked. ‘Why is that?’
‘Well,’ said Bach. ‘It’s a problem that hitherto no science has been able to resolve. The fact is, twelve perfect fifths result in a different note than seven perfectly tuned octaves.’ Bach turned to the blackboard again, wiping away a section of the chalk circle at the upper C and added a small spike. Then he drew an arrow pointing straight to the spike and said: ‘There. Here you can see it. The circle of fifths doesn’t close. The beginning and the end do not match. God has presented us with a riddle here.’
‘Thank you, Bach,’ said the teacher, ‘that was an excellent lecture.’
Bach put down the piece of chalk and strode back to his place.
‘But,’ queried an apothecary’s son after the teacher had allowed him to speak, ‘what does all this actually mean?’
‘What it means,’ said Bach, ‘is that you cannot play in all keys on the organ or the clavichord. If the instrument has been tuned in C, you can get barely to E major, and after that the wolf howls.
The howling of the wolf was an expression musicians used to describe a fifth that was so far out of tune that it only sounded miserable. It was called the wolf fifth.
‘All right,’ said the apothecary’s son, ‘but what does it all signify?’
‘It primarily signifies,’ Erdmann interjected, in the arrogant tone he had learned from listening to the aristocratic students, ‘that the order of the world is highly imperfect.’