# hwk07B - in which the piece is played so different keys can...

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1 Chapter 9 Solution RQ14. 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 A b 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. RQ15. Semitone of equal temperament gives an interval that is exactly 1/12 of an octave. The frequency ratio is 2 1 12 = 1.05946. .. RQ17. An octave is 1200 cents. RQ20. In equal temperament, the circle of fifths is a closed circle. QTD3. In equal temperament, all keys should sound alike because the interval relationships between notes are unchanged in moving from one key to another. However, in other temperaments, such as the just temperament, this is not true. The frequency ratios vary depending on the key
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Unformatted text preview: in which the piece is played, so different keys can be associated with different characteristics. In this day of tuning to equal temperament, the key in which a piece is played still changes the frequency range of the piece. Therefore, even though the frequency ratios are the same, there will be very small changes in consonance and dissonance because of the variation in the width of the critical bands. EX1. Since the semitone of equal temperament has the ratio of 05946 . 1 2 12 / 1 = , we can calculate other ratios by direct multiplication. For a major third: 260 . 1 ) 05946 . 1 )( 05946 . 1 )( 05946 . 1 )( 05946 . 1 ( = For a minor third: 189 . 1 ) 05946 . 1 )( 05946 . 1 )( 05946 . 1 ( = EX2. We know that twelve equal semitones make up an octave. The ratio of a semitone is 05946 . 1 2 12 / 1 = 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 12 / 1 2 059 . 1 ! , the investment approximately doubles in 12 years....
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## This document was uploaded on 02/11/2012.

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