· ILLUSTRATION: TOMASZ WALENTA
By
Eugenia Cheng
Dec. 23, 2020 5:22 pm
ET
·
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.
