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Powers, Roots, and Equal Temperament

Module by: Catherine Schmidt-Jones

Summary: A review of roots and powers for the music student who wishes to understand frequency relationships in equal temperament.

You do not need to use powers and roots to discuss music unless you want to talk about frequency relationships. They are particularly useful when discussing equal temperament. (See Tuning Systems.)

Powers are simply a shorthand way to write "a certain number times itself so many times".

Example 1

Figure 1
Figure 1 (powers.png)

Roots are the opposite of powers. They are a quick way to write the idea "the number that, multiplied by itself so many times, will give this number".

Example 2

Figure 2
Figure 2 (roots.png)

Roots and powers are relevant to music because equal temperament divides the octave into twelve equal half steps. A note one octave higher than another note has a frequency that is two times higher. So if you divide the octave into twelve equal parts (half steps), the size of each half step is "the twelfth root of two". (Notice that it is not "2 divided by twelve" or "one twelfth". For more on this, see Equal Temperament.)

Example 3

Figure 3
Figure 3 (rootsandpowers.png)

Exercise 1

Using a scientific calculator, find

  1. The frequency ratio of a half step (the twelfth root of 2), to the nearest ten thousandth (four decimal places).
  2. The frequency ratio of a perfect fourth (five half steps, or the twelfth root of 2 raised to the fifth power), to the nearest ten thousandth.
  3. The frequency ratio of a major third (four half steps), to the nearest ten thousandth.
  4. The frequency ratio of an octave.

Solution 1

  1. 1.0595
  2. The twelfth root of 2, to the fifth power, is approximately 1.3348
  3. The twelfth root of 2, to the fourth power, is approximately 1.2599
  4. The twelfth root of 2, to the twelfth power, is 2

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