Math: The Invisible Hand Behind The Music
From NCTM News Bulletin July/August 1999
Want a mathematical challenge? Try writing, reading, and
playing music. Not only does it take an ear for music, it
requires an appreciation for the principles of mathematics.
Because Jimmy Buffett started his career on raw talent,
some of the mathematical aspects of music (counting,
forming chords, and so forth) came to him quite naturally.
But he realized how important understanding certain
mathematical concepts were when he decided to write a
musical called "Don't Stop The Carnival" with Pulitzer
Prize-winning author Herman Wouk. Composing music
required a knowledge of music theory, which has
mathematical underpinnings. "Of all the academic
subjects, math is most closely connected with music. Music
is all based on fractions and patterns," says Michele Adams,
a middle grades mathematics teacher, music teacher, and
piano player from The Woodlands, Texas. "Where
fractions are concerned, music focuses on divisions of
time for the rhythm and space for dealing with intervals
such as octaves or fifths." Adams points to the Gregorian
chants. "They are based on strict rules of mathematics,"
she notes. Adams points out some mathematical concepts
underpinning music:
* Counting: It's fundamental to playing music. One must
count beats per measure and count how long to hold notes.
* Patterns: Music is full of patterns -- patterns of notes,
chords, and key changes. Musicians learn to recognize
these quickly. Patterns, and being able to invert them
(known as counterpoint), help musicians form harmonies.
* Geometry: Music students use geometric shapes to help
them remember the correct finger positions for notes or
chords (more than one note played simultaneously). For
instance, guitar players' fingers often form triangular shapes
on the neck of the guitar.
* Ratios and proportions/equivalent fractions: Reading
music requires an understanding of ratios and proportions.
For instance, a whole note needs to be played for twice as
long as a half note, four times as long as a quarter note, and
so forth. In addition, since the amount of time allotted to
one beat in a given time signature is a mathematical
constant, the durations of all the notes in that piece are
all relative to one another and are played on the basis of
that constant. Finally, different frameworks of time with
which musicians work are based on an understanding of
fractions and multiples -- for example, understanding the
rhythmic difference between 3/4 and 4/4 time signatures.
* Sequences: Music and mathematics are also related
through sequences, particularly intervals. Teacher Eli
Maor expounded on this relationship further in the
September 1979 Mathematics Teacher. "Although a
mathematical interval corresponds to the difference
between two numbers, a musical interval corresponds to
the ratio of the frequencies of the tones." He goes on to
say, "Here, then, is a single principle that underlines all
musicomathematical relations: Arithmetic progressions
in music correspond to geometric progressions in
mathematics; that is, the relation between the two is
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