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Functional Melodies
Finding Mathematical Relationships In Music

Sample Activity Demos

Functional Composer

Functional Composer is set in the context of a musical composer struggling with writer's block. Students solve the composer's problem by creating various mathematical transformations of a simple melody. The original melody is defined as f(x). Students create various transformations of f(x) to generate musical material for a composition. The audio CD demonstrates a musical compostion that can be made from the transformations. Students can then use this mathematical tool to create original compositions in subsequent projects. Teacher notes in the book suggest how the transformations can be used as a pure mathematical tool applicable to functions of all sorts; algebraic, trigonometric and logarithmic.

Shown below is the assortment of transformations studied in the activity and their graphs. Click here to hear a sample musical segment that utilizes the functions. (Note: You will need RealPlayer to hear the music. A free version of RealPlayer can be downloaded at http://www.real.com.) With the exception of a few small embellishments, each melody in the audio track is derived precisely from the agebraic functions (graphs presneted below). The harmonic accomaniment (background chords) suggests one way a composer might support the mathematically generated melodies.

It is important to note that the graph below is not an exact representation of what is heard on the audio. This representation is something of an interpretation, designed to highlight the shape of the function more clearly in the interest of teaching translation, stretching and reflection. This can be a good discussion to have with students, to point out that an exact mathematical graph of the sound would be a step function. The interpretation below might more accurately be described as vector resultants of the musical movements from note to note.

Copyright 2000 Key Curriculum Press

Inside Out

In this activity students use a polar coordinate system as an interface between the visual and sonic worlds. A graphic image is traced onto a polar coordinate grid and intersection points of the image with rays generate musical notes, a tone series that embodies the shape of the image, sonically.

Shown below is a sample from the book. Click here to hear a musical segment generated from it. The bass line is the fish contour while the high string line was created from the function, y=-f(x)+11 using the techniques of Functional Composer.

Copyright 2000 Key Curriculum Press

 

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