Bach and My Musical Advent Calendar

 

·                                                                                                                                                  ILLUSTRATION: TOMASZ WALENTA

By 

Eugenia Cheng

Dec. 23, 2020 5:22 pm ET

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A keyboard masterpiece depends on the complex relationship between sound waves and musical notes.

Along with my usual chocolate Advent calendar, this year I decided to make myself a kind of musical Advent calendar by playing through a book of J.S. Bach’s “The Well-Tempered Keyboard,” a monumental collection of 24 preludes and fugues. Playing one a day made a perfect fit with my chocolates and the run-up to Christmas Eve.

Bach didn’t choose the number 24 arbitrarily. In Western music there are 12 notes in an octave, and for each note there is a major and a minor key, making 24 in total. To understand why there are 12 notes in an octave, rather than 10 or 14, we have to look at the math of sound waves.

The pitch of a note is determined by the frequency of its sound wave—that is, how many identical segments of the wave, or oscillations, pass a fixed point in a given period of time. The higher the frequency of a sound wave, the higher the pitch it produces. Harmony in music involves combining sound waves of different frequencies, making use of the fact that some waves reinforce one another while others interfere. 

For any musical note, the note that goes with it most harmoniously is the same note one octave higher. That’s because the higher note has exactly twice the frequency, leading the two sound waves to oscillate together neatly. The next most harmonious note will be the one with three times the frequency, which in musical terms is a perfect fifth higher—for instance, the jump from C to G, which you can hear at the beginning of “Twinkle, Twinkle Little Star.”

If we keep going up by perfect fifths, we get a sequence of harmonious intervals known as the cycle of fifths. The first five notes in the cycle form the pentatonic scale, which is used in the music of many ancient cultures. If we keep going up to seven notes, we get the notes of the major scale prevalent in Western classical music. Taking 12 notes of the cycle of fifths produces the 12 notes of the piano keyboard.

After 12 steps, something fortuitous happens: We get to a note that is almost the same as the one we started with, but seven octaves higher. The 12 notes of the keyboard essentially come from this numerical coincidence between the frequencies for seven octaves and 12 fifths. But there is still a slight mismatch in the frequencies—less than a quarter of the distance between two adjacent notes. 

Before Bach’s time, the typical approach was to “hide” this error in notes that the music was unlikely to use, like when you stuff your mess in a bedroom before guests come around. For example, if a piece of music is in the key of C major, it’s important for F and G to sound perfectly in tune, since they’re the fourth and fifth notes of the scale; but it’s less important for F sharp to sound right, since it will barely be used.

Musicians eventually had the idea of spreading the mess out more evenly so that no note would sound quite so wrong. This type of tuning of a keyboard is called “well-tempered,” hence the title Bach gave to his work. For the first time it was possible to play music in all the keys without some of them sounding out of tune, and Bach wrote a prelude and fugue in every key to explore and perhaps celebrate this fact.

People celebrate different things at this time of year, and many people have little to celebrate in 2020. But I will at least celebrate the beautiful effect that math can have on unexpected parts of life.