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Multiplication of Whole Numbers by Fractions

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Multiplying Fractions Musically
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Credit: Laura Guerin
Source: CK-12 Foundation
License: CC BY-NC 3.0

Did you know that fractions can help create different musical tones? First, let's think about sound. Sound travels through air in waves with unique frequencies, which are measured in Hertz (Hz). Human ears generally can hear sounds in a frequency range from 20 Hz to 20,000 Hz. Each musical note has a unique pitch, which is determined by the frequency of its sound wave. Tones can be made up of one frequency or a combination of frequencies.

Why It Matters

Each note in a musical scale can be determined by multiplying fractions. When each pitch is multiplied by a fraction, the next tone in the scale is created. To create a major scale, start at any note. Then, multiply the frequency of that note by \frac{9}{8} to get the frequency of the second note. Multiply the frequency of the second note by \frac{10}{9} to get the frequency of the third note. Multiply the frequency of the third note by \frac{16}{15} to get the frequency of the fourth note. Then repeat this pattern. So, to get the frequency of the fifth note, you would again multiply by \frac{9}{8}.

Credit: Laura Guerin
Source: CK-12 Foundation
License: CC BY-NC 3.0

The C major scale is pictured above. "Middle C" on the left has a frequency of 264 Hz. How can we use fractions to build the C major scale?

Given that 264 \ \mathrm{Hz}=\mathrm{C}, the next note, D, is created by multiplying 264 by \frac{9}{8}:

264 \ \mathrm{Hz} \times \frac{9}{8}=297 \ \mathrm{Hz}=\mathrm{D}

Multiplying 297 by \frac{10}{9} gives the frequency of the next note, E:

297 \ \mathrm{Hz} \times \frac{10}{9}=330 \ \mathrm{Hz}=\mathrm{E}

And so on—we could keep multiplying and multiplying until all of the notes of the scale have been created!

Hear the C major scale for yourself: http://en.wikipedia.org/wiki/C_major

Explore More

Follow the instructions at the following link to create your very own "shoebox guitar" to help you learn more about sound and music.

http://scienceforkids.kidipede.com/physics/sound/doing/sound.htm

Image Attributions

  1. [1]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0
  2. [2]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0

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