| THE MATHEMATICS OF MUSIC|THE MUSIC OF MATHEMATICS | ||
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Music by Chance? Die Musikalisches Wurfelspiel Mozart wrote a sixteen measure minuet with eleven different possibilities to choose from for each measure. The performer chooses one of the eleven possibilities for each measure by rolling two dice. The amazing thing is that Mozart wrote each of these measures so that no matter which combination is chosen, they fit together to make a pleasant piece of music!
Here are the measures for a simple four measure melody with which you can play the musical dice game. There is only one choice for the first measure. To choose each of the other three measures, roll the die. If you roll 1 or 4, choose choice 1 for that measure. If you roll 2 or 5, choose choice 2 and if you roll 3 or 6, chooose choice 3. You can play your composition on the xylophone.
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"Music is a secret arithmetical exercise and the person who indulges in it does not realize
that he is manipulating numbers." -Gottfried Liebniz 1646-1716, an inventor of the calculus and a contemporary of J. S. Bach
Many musical compositions show symmetry in their structure. In many musical forms a theme (short melody) is repeated many times, often with small changes, throughout the piece. When the theme is repeated it may simply start later than the original (a slide or translation) or it may be played backwards (a flip or reflection), or played upside-down (another type of reflection). The theme may be stretched so that it goes twice as slowly or shrunk so that it goes twice as fast. If you have ever sung :"Row, Row, Row Your Boat" as a round you have used symmetry in your music. J. S. Bach's Musical Offering is probably the most famous example of mathematical symmetry in music.
Can you figure out what symmetry he used in writing this piece of music?
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Counterpoint Counterpoint is the art of writing two or more melodies which sound good when played together. Often one melody is chosen first and then symmetry tricks are used to make the second melody from the first melody. Write your own counterpoint 1. Write the melody backwards and play it with the original melody. 2. Write the melody upside-down and play it with the original. Choose C or G as your starting note, and whenever the melody goes up make your part go down the same distance, and vice versa. 3. Play the melody as a round. 4. Write the melody twice as fast-use eighth notes wherever there are quarters and quarters wherever there are halves.
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