Michael Clarke uses numbers and notes. Mr. Clarke, chairman of mathematics and science at Duke Ellington School for the Arts in Northwest, who holds a doctorate in physical chemistry, says many links exist between math and music, such as the use of number relationships, patterns, symbols, graphs and fractions. As a concert pianist, he knows this from experience. "The skills that you develop in the creative mode in the arts are the skills that you need to be successful in all the concepts that you will come across in math and science," Mr. Clarke says.

Music is connected to math on two levels. Primarily, mathematical principles are used to notate musical pieces. However, in higher mathematics, a deeper correlation exists. Both advanced mathematicians and musicians use the right side of the brain the side inclined toward the intuitive rather than the analytical when they are thinking.

Notating music is based on mathematical proportions, Mr. Clarke says. Every musical piece uses a unit to measure what is to be played. For instance, a whole note has four beats, while a half note has two beats, and a quarter note has one beat. As the proportions become smaller, an eighth note receives one-half beat, and a 16th note receives one-fourth beat. One can clap to the downbeat, which is the accent of the music that reveals the time signature of the songs. Some pieces have duple meter, such as 2/4 and 4/4, while others have triple meter, such as 3/4.

Chords also are formed through specific numeric intervals. For instance, there are two whole steps on a musical staff between the first two notes of a major chord, with 1 steps between the next two notes. There are 1 steps on a musical staff between the first two notes of a minor chord, while there are two whole steps between the second two notes. These notes are placed on the musical staff in somewhat the same way as coordinates are plotted on a mathematical graph.

Symbols are interspersed throughout the musical text. For instance, rests show when silence is to occur and for how long. These markings communicate an instruction the same way a plus or minus sign would in a mathematical equation. Crescendo and decrescendo symbols, used for increasing and descreasing volume in musical passages, are almost the same as "greater than" and "less than" signs in a math equation.

David Cope, professor of music at the University of California in Santa Cruz, says many of Johann Sebastian Bach's works relied upon intrinsic mathematical principles. Mr. Cope cites Bach's fugues, a type of composition with a repeated melodic theme that must reoccur at an interval of a fifth during the beginning of the piece. Mr. Cope says Bach is believed to have said, "Music is combination of math and magic."

Bach is also famous for the numerical symbolism of his music. In the "St. John Passion," he repeats the melodic theme in the piece 10 times to allude to the Ten Commandments in the Bible.

"Unless someone points it out to you, you might not realize it, but there is more math beneath music than we realize," Mr. Cope says. "The math occurs by sensory and intuitive methods."

• • •

Beyond basic notation, higher mathematics and music involve abstract thinking, says Dr. Samuel Potolicchio, a neurologist at the Neurological Institute at George Washington University in Northwest. According to studies done by Gordon Shaw, a neuroscientist from the University of California at Irvine, someone working at an advanced mathematical level is using the right side of the brain to reason conceptually and the right side is the part of the mind musicians use when composing and playing.

The right side of the brain is the more creative and intuitive side, whereas the left side of the brain is the side that tends toward the realism and logic used in simple math or in notating music.

"If you look at functional imaging of the brain and you ask a highly trained mathematician to perform an exercise in mathematics, his thoughts will approach that of a musician more so than you would think," Dr. Potolicchio says.

John Conway, von Neumann Professor of Mathematics at Princeton University in Princeton, N.J., who holds a doctorate in mathematics from Cambridge University in England, says his mind travels far beyond the simple grade school calculations. The right side of his brain allows him to think beyond the realistic world.

About 1970, he discovered "surreal numbers," which include real numbers, such as positive and negative whole numbers, along with zero, integral fractions, and irrational numbers, such as the square root of 2. "Surreal numbers" also provide a way to represent numbers bigger than infinity or smaller than the smallest fraction.

"I'm not just saying, 'One plus one equals two,'" he says. "I'm exploring combinations and seeing patterns, wondering whether something connects to something else, noticing something common to several situations."

Kathryn Judd, director of the Washington Conservatory of Music, says creativity is important in both math and music. Pythagoras, a Greek philosopher who is considered the father of geometry, is also considered the father of music theory. Not only did he invent the Pythagorean theory used in mathematics, but he also researched the overtone series found in music. This series has exact multiples of the frequency of the lowest note in the series, which is called the fundamental. The pitches that result from the first five multiples form a major triad.

When Albert Einstein arrived at his theories, he formulated them in his mind first and then used math to prove them, which is the same way many musicians compose, Ms. Judd says. Einstein also was an amateur violinist.

"The 'eureka moment' is a poetic or musical moment," Ms. Judd says, referring to the point in time when Einstein encountered his profound insights.

Similar to Einstein, Wolfgang Amadeus Mozart constructed musical passages in his mind and then wrote them on paper using the mathematical principles designated for musical notation. Often, Mozart did this without altering the text at a later time, Ms. Judd says.

"Composing just comes to you, and you use math to write it out," she says. "The emotional and spiritual component of math probably comes from where music comes from, wherever that is."

Diana Dabby, who holds a doctorate in electrical engineering from the Massachusetts Institute of Technology in Cambridge, teaches a course called "Music, Art, Literature: Intersections With Science" at the Juilliard School in New York City. She says mathematicians and musicians both attempt to find graceful answers to the problems set before them, whether it's proving a theorem or changing key signatures. Ms. Dabby is a concert pianist and composer.

"I'm always searching for elegance in whatever I do," she says. "I've experienced it in both worlds an elegant math solution moves the math community, and an elegant turn of phrase moves the musicians in an audience."

• • •

Dan Ullman, chair of mathematics at George Washington University in Northwest, says many times people believe math is entirely uncreative. Mathematics is often mistaken as being simply memorizing multiplication tables or balancing a checkbook. Mr. Ullman, who also is a pianist, says the same abstract element in music appears in math.

"Math problems don't feel like the real world," he says. "Music is like that, too. You have to close your eyes and think in the imaginary world."

Robert Sirota, director of the Peabody Institute of the Johns Hopkins University in Baltimore, says music is one of the ultimate abstractions. He says in a concrete sense it does not exist in the same way as a building or a painting.

"These are things you can reach out and touch," he says. "Music is totally invisible in a sense. It relies completely on memory and the ability to predict the future. It's always in the moment."

Pattern recognition in sound is similar to the study of mathematical puzzles, Mr. Sirota says. In both instances, one needs to imagine the completed phrase or problem and recognize when the pattern is broken. Training the mind this way sharpens thought processes for all fields.

"Music instruction is an important way that many students are able to improve their academic skills," Mr. Sirota says. "It's not the answer for every student, but a doorway for many students to improve their academic studies."

Meanwhile, Mr. Clarke at the Duke Ellington School says he is excited to know that many of his math and science students study music. He says music is one of the subjects the Greek philosopher Plato suggested that people study in order to become a "philosopher king," Plato's term for the most highly educated type of person. Because of his theories, a "quadrivium" of mathematical arts existed, including geometry, arithmetic, astronomy and music.

"Music is a form of math," Mr. Clarke says. "Studying music can only help you understand the other areas of math."

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