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."
**Thanks to Tom Donaldson, European Music Educators
Association and one of our e-mail directors, for forwarding
this one to us!
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