Bach and The Tuning of the World. Jens Johler

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СКАЧАТЬ 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.’

      ‘Imperfect?’ asked the teacher, crossing his arms.

      ‘Well,’ said Erdmann, rising from his seat, ‘after all, the world is indeed anything but perfect! At least it’s in dire need of improvement – all progressive scholars are agreed on that.’

      ‘So God has created the world in an imperfect manner?’ enquired the teacher. ‘That’s how His Lordship meant it – right, Erdmann? So God created the world – well, what now, Erdmann? Give me a hand here. Did He do so sloppily? In a slipshod manner?’

      ‘Well …’

      Bach saw beads of perspiration on Erdmann’s upper lip.

      ‘But we just heard it from Bach,’ Erdmann said hesitantly. ‘Everything doesn’t fit together quite right here. It’s not as it ought to be. If you tune to perfect octaves, you get to a different note than you do when with tuning to perfect fifths. Such a difference would not exist in a perfect world. In a perfect world, the circle of fifths would be closed.’

      He folded his arms across his chest now, so they stood facing each other, the teacher and the student, both with crossed arms.

      ‘So Your Highness intends to improve upon God’s creation?’ the teacher said ironically, unfolding his arms. ‘It’s not good enough for His Lordship: His Lordship knows better, and His Lordship will show us. His Lordship will show GOD, am I right? Answer me!’

      Bach would have liked to help Erdmann, but how? Erdmann was his friend. He admired his courage. He admired his brilliance. But was it permitted to set oneself up as a judge of Creation?

      All the colour had drained from Erdmann’s face. Beads of cold sweat covered his forehead.

      ‘Just at the moment, I can’t answer that,’ said Erdmann evasively. ‘I have to think it over first.’

      ‘Well,’ the teacher said, smiling, ‘get on with it, Erdmann. So now His Lordship has three days to think over how he wants to improve God’s work. Three days in detention – get out!’

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