- 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
- logarithmic."
-
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