He recalls, as a child, being brought up short one day by an
"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
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.
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.
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
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 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."