Question

a multiple of the logarithm of the ratio of those two quantities. Une instance was the decibel as a measure of the loudness of sound. The other was the seismic moment magnitude scale as a measure of the intensity of earthquakes. Here is a third one. This one measures how different are the frequencies of two musical tones. The relevant unit is called a cent. If one knows the frequencies fi and f2 musical interval from fi to fo is given by the following formula: of two notes, then the number of cents measuring the d(fi, f2)-1200 log2 fi I've mentioned in class that the precise frequencies of notes (and, more to the point, the precise frequency ratios of the notes) generally depend on the tuning system. For example, in the tuning system that's called 'just intona- tion', the frequencies of the note C4 and of the note A4 form the ratio of 5/3. On the other hand, in the tuning system which is called 'equal temperament', these two notes will be nine semitones apart, where each semitone corresponds to a frequency ratio of a twelveth root of 2, 21/12 (a) In equal temperament tuning, one semitone corresponds to a frequency ratio of 21/12 How many cents is that? (b) What's the relationship between d(fi, f2) and d(f2, fi)? (c) The average hearing range of adult human population is normaly taken to be from 20 Hz to 20000 Hz. How many octaves does this correspond to? How many equal-temperament semitones? How many cents? (d) Assume that the frequency of the A4 note is the same in all tuning systems. How many cents apart are the C4 in just intonation and the C4 in equal temperament? Which one has higher frequency? (e) Suppose two tones are 340 cents apart, and that the higher one has a frequency of 517 Hz. What is the frequency of the lower tone? Vanja Dunjko Bonnie Miller

Answer 1

Hent 2 2 :2 5o, 12 CD 1 2 12ο 2): or(1-6.644 1200,,()の 12 GD l 2. 6' 1 4 4 x ,2. 79.728 y 80 172.8

5 4 12 au 2. 120x1318 887.76 A 4 (c 340 2. ) : 0.283 2 0 283 283 51

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