hw16_solutions

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Physics 198
Homework 16
Due October 27
1. What is the main problem with the just scale?
Solution:
In the just scale, a few musical intervals are perfect but many other intervals are imperfect. For
example, if the fifths in the just scale are examined, five are perfect but D:A is imperfect. The
same is true of the fourths in the scale; five are perfect but A:D is not. Another problem is that
Ab is a little higher in pitch than G# in the just scale; i.e., enharmonic notes have different pitch.
Finally, on must retune to modulate to a different key in order to keep the intervals as consonant.
2. What frequency ratio gives an interval that is exactly 1/12 of an octave?
Solution:
Semitone of equal temperament gives an interval that is exactly 1/12 of an octave. The frequency
ratio is 21 12  1.05946
3. How many cents are in an octave?
Solution: An octave is 1200 cents.
4. At what temperament is the circle of fifth a closed circle?
Solution: In equal temperament, the circle of fifths is a closed circle.
5. Verify by direct calculations that a major third in equal temperament has the ratio of 1.26 and
a minor third the ratio 1.19.
Solution:
Since the semitone of equal temperament has the ratio of 21 12  1.05946 we can calculate other
ratios by direct multiplication.
   2  1.260 ,
For a minor third: (1. 2   2  1.189
For a major third: 21 12
4
1 12 3
13
14
6. From your knowledge of equal temperament, show that if you invest money at an interest
rate 5.9% compounded annually, your investment doubles in 12 years.
Solution:
We know that twelve equal semitones make up an octave. The ratio of a semitone is 21 12  1.059
and the ratio of an octave is 2. Hence if we invest money at the rate of 5.9%, the money will
become (1+5.9%) =1.059 times next year. Since 1.059  21 12 , the investment approximately
doubles in 12 years.
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