__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."