During the day, David L. Wright, PhD, is chairman of the Department of Mathematics in Arts & Sciences at Washington University in St. Louis; at night, he is assistant director of Ambassadors of Harmony, a men’s a cappella chorus that has won international competitions.
Math and music might seem a strange combination to some. Certainly many famous performers are able to bring audiences to their feet without once thinking about ratios or anything else overtly mathematical. But Wright always has been gifted with an unusual, even eerie, ability to hear both the music and the math simultaneously.
He recalls, as a child, being brought up short one day by an unusual chord.
“When I heard that kind of blending, I simply had to stop and pay attention to what I was listening to and try to figure out what it was,” he says. What was the chord?
The 11th chord, in George Gershwin’s “Rhapsody in Blue.” “It’s that 11th harmonic of the E-flat fundamental that makes it sound so jazzy,” he says.
Wright would never say an understanding of math is needed to play music, but he would say that singers and musicians tend to seek out rhythms and pitch intervals based on integer primes such as 2, 3, 5 and 7 because they just sound right.
Sometimes when singers are really good, they get the math just right and magic happens. This YouTube video of Ambassadors of Harmony captures a moment when the director, Jim Henry, asks the singers with different vocal ranges to hit the same overtone simultaneously. He sings this high harmonic and then the choir sings and behind his voice the same tone emerges and floats above the massed voices.
“It is third harmonic of the low tone being sung and the second
harmonic of the upper tone being sung,” Wright says. “When harmonics are
reinforced in this way they often become audible if the voices are
really in tune.”
Singers say the chord “rings” when this happens.
Wright, who teaches a course for undergraduates titled “Mathematics and Music,” is the author of a book of the same title that serves as the course’s textbook.
In the keynote lecture he delivered this spring to the Mathematical Association of America district meeting at the University of Missouri-St. Louis, he began with counting.
“We prefer twos and multiples of two,” he said, “because we deal deftly with small integers.
“It’s very easy for us to take a beat and subdivide it so that we’re doing something twice as fast or four times as fast,” he says, demonstrating by snapping his fingers. “It’s not so easy to do something five times as fast.”
Similarly, time signatures, or the number of beats per measure, tend to be small integers over very small powers of two. Again, our most comfortable mode of counting is two, and most music comes in powers of two. A few pieces are in fives, but those are exceptional.
He demonstrates by playing short excerpts from two famous pieces of music.
The first mp3 is an excerpt from Scott Joplin’s “Maple Leaf Rag” and the second is Dave Brubeck’s “Take Five.”
Composers, Wright suggests, often generate an interesting musical pattern by cycling a melody made up of a small number of pitches through a rhythmic figure with a different number of notes.
For example, Glenn Miller’s “In the Mood” cycles three notes (C, E-flat and A-flat) through rhythmic figure that is four notes long, so that the entire pattern takes 12 notes to complete itself and return to its starting point.
“It’s the juxtaposition of three against four that makes the song fun to listen to,” Wright says.
Similarly, in Gershwin’s “Rhapsody in Blue,” three notes cycle through a five-note rhythmic pattern (two eight notes and three quarter notes), a juxtaposition that takes 15 notes to complete.
Staying on pitch
Just as single integer counts serve us well in the horizontal structure of music, Wright says, integer intervals serve us well in its vertical structure.
“We’re comfortable with two frequencies when one is twice as high as the other, just as we’re comfortable with two beats when one is twice as fast as the other,” he says. “The interval between one musical pitch and another that is double or half its frequency is the familiar octave.”
There’s even proof — in the form of an auditory illusion — that these intervals come naturally to us. “The feeling of an octave is so engrained in our minds that we have trouble distinguishing pitches that are exactly two octaves apart,” Wright says. “This is why the charge typically heard at baseball and hockey games seems to be continually going up and yet never gets anywhere.”
What you hear sounds like an ascending scale, but is in fact the same octave repeated. As the top note of the octave fades out, the note one step above the low note of the octave comes in. If you’re not listening closely, your brain hears the top note as the bottom note one octave below. And then, of course, the next note is higher, and so you seem always to be going up and never to be going down.
Wright says that a sung or played note is never a pure sinusoidal frequency — which would sound like a dull hum, the dull hum you hear when you hold a tuning fork up to your ear — but rather that frequency and some mixture of its harmonics (integer multiples of the fundamental frequency), called overtones.
This leads to one of the most startling vocal styles ever developed: throat singing, or overtone singing. This is an ancient singing style of the Tuva people who live in the far south of Siberia. The singer begins by producing a continuous, low pitch, like the drone of a bagpipe, and then by changing the shape of his vocal tract isolates the overtones so that they can be heard above the drone.
In this example, the melody is a changing series of overtones as the singer holds the same fundamental pitch.
In a different singing style, the isolated overtone is so high it sounds like a whistle above the drone.
For instruments that are not fretted, pitch is a continuum that can be endlessly varied. A good example is the rising clarinet note at the beginning of Gershwin’s “Rhapsody in Blue.”
Here singers in the group Ambiance mimic the clarinet in an arrangement of “Rhapsody” Wright wrote for them.
But on fretted and keyboard instruments, the available pitches are discrete instead of continuous, and this leads to a world of trouble because nothing quite works out musically or mathematically.
One way to tune a keyboard is to hit middle C and the G above it and adjust the G until it’s exactly in tune with the C and “beats,” — a wobbling or tremolo that occurs when two notes close in frequency are interfering with one another, — are no longer audible. Then the F can be tuned to the C in a similar manner.
This is called just intonation, or pure intonation, meaning the frequencies of notes are related by ratios of small whole numbers.
This is fine if the keyboard is only used to play music in the key of C and with limited harmonic variety. But music in A-flat or E-flat will sound awful on a keyboard tuned in this way, Wright says.
The solution is to space the intervals between keys equally so that they are all equally out of tune. This is called equal temperament and all keyboard instruments have been tuned this way since Wagner.
As it turns out, fifths (notes separated by five positions on the musical staff) sound pretty much the same in just intonation or equal temperament. Thirds are a bit more out and sevenths differ enough that even non-musicians can hear the difference.
“The chord tuned to just temperament may sound a bit annoying because the seventh at the top seems flat,” Wright says, “but this is a fantastic jazz chord.”
In fact, this is the chord in Gershwin’s “Rhapsody in Blue” that intrigued and puzzled him as a kid.
Singing by ear
“A keyboard musician, fine as that musician may be, can’t do a thing about the pitches of the keys. But singers are able to tune as their ear tells them to tune,” Wright says. “They tend to go to a slightly different pitch than the keyboard pitch simply because the mathematics is taking them to a different pitch.”
One example is an African-American gospel quartet singing the spiritual “De blind man stood on the road and cried” in the key of F.
“This is definitely not keyboard tuning,” Wright says. “The seventh is really low, so it sounds out of tune, and yet there’s a real consonance to it.”
It should be clear by now why Wright sings in a cappella choirs. He ends his talk with a piece he arranged for Vocal Spectrum, a local male quartet. It is Thelonious Monk’s classic jazz ballad “’Round Midnight.”
“The opening strain uses the 13th harmonic in a chord, which again is not well approximated in the tempered scale,” Wright says. “But a cappella singers don’t care. They go for what their ear tells them, and so this is very much in tune, just not the tuning you’d hear from a keyboard.”