Wolf in the Fold

The nun Yíwen was tuning a guqin in her quarters when her roommate Hwídah returned and slumped into a chair, visibly defeated.

“Ten thousand pardons for my curiosity,” said Yíwen, “but your brow is more deeply-furrowed than usual. Have you again been investigating failures in our production systems?”

“It is all I seem to do these days,” sighed Hwídah. “Although the cause of this evening’s trouble was particularly irksome. Every choice the developer made is optimal when taken by itself, yet the combination taken as a whole is disastrous. Thus the best approach to fixing the problem remains elusive.”

“How unfortunate!” said Yíwen. “Have you informed the developer of this conundrum, that they might meditate upon it?”

“There was no need,” said Hwídah. “This miserable soul is she.”

For a while the two sat in silence, as Hwídah took solace in sweet fragments of melody being plucked out by Yíwen.

“The act of coding shares much with the composition of music,” observed Hwídah. “Each note may be perfect in isolation, yet it cannot be judged by itself. It must be weighed against the notes that come before, and the ones that come after, like steps in an algorithm.”

“Yet each must also be weighed against the notes sounded alongside it, would you not agree?” asked Yíwen. “Melody and harmony are the warp and weft of music; for the staff is two-dimensional, and time is but one axis. Even in my humble guqin, the vibration of one string causes its fellows to vibrate in sympathy, as surely as multiple threads may affect each other in an application.”

“Would that the harmonies of software were as simple as those of sound!” said Hwídah. “For I did ‘play’ my application many times in private, and never a note seemed out of place!”

“Simple?” asked Yíwen. “Have you never heard of the Twelfth Root?”

“Long ago,” said Yiwen, “it was observed that if a string is plucked along with another string exactly one-half its length, the two notes sound so similar as to be judged identical. It was also observed that if a string is plucked alongside another which is two-thirds its length, the resulting harmony is quite pleasing. The same held true for many simple ratios, like three-fourths and four-fifths, and two-fifths, and so on. Thus did a musical scale take shape around simple fractions.

“Yet all attempts to produce a perfect scale for the keyboard were thwarted. For there are many such fractions, and many ways to arrange them, yet no matter which tuning was tried it would always contain some intolerable imperfection. An interval that sounded exquisite in one key would howl like a wolf in another.

“Every choice is optimal when taken by itself, yet the combination is disastrous,” murmured Hwídah. “But then how is your keyboard tuned?”

“By the best bad compromise of all,” said Yíwen. “There are twelve notes per octave, so the ratio of each note to the preceding is the twelfth root of two. For although that is a most irrational ratio, it ensures that twelve identical steps upward will bring us exactly one octave above our starting place.

“In this temperament, no interval is an integer ratio except the octave: they are all the slightest bit sharp or flat. Yet it allows us to transpose songs into any key without distortion or dissonance. In this way perfection yields to pragmatism... for though the harmony of the pasture is disturbed when we gird it with a fence, to do otherwise would be to let the wolf in.”