Exercise 4

Readings: Please read Chapter 4 from Miller's book.

Exercises:

1. In the Western tempered scale, if A is tuned to 440 Hz., what is the frequency of the D below it?

2. What is the frequency of the same D as in problem 1, using the Just Intoned scale in D instead of the tempered scale?

3. A major triad may be formed by the frequencies 100, 125, and 150 Hz. Another may be formed of frequencies two octaves up: 400, 500, 600. Which triad is likely to sound "sweeter"? Why?

4. How many equal-tempered tritones does it take to span the normal frequency range of the human ear?

5. Imagine a tuning system not based on equal divisions of an octave (R = 2), but rather a tripling of value (R=3). Assuming we keep the familiar 12 equal divisions of this "tritave", what would the equivalent of a major 3rd (ie, 4 "half-steps" up) be as a ratio?

Home Lab:

How much detuning makes an interval sound sour? This project is a test of the Helmholz theory of consonance and dissonance. The interval we'll work on is the fourth below 440 Hz. (and later, 220 Hz.)

First, using "sinusoid" objects, make a perfect fourth using the frequencies 440 and 330. You can connect them to the same "output" object so that they have the same amplitude as each other. Now drag the 330 Hz. tone down in frequency until, to your ears, the result starts to sound ``sour". How many Hz. did you have to decrease the 330-Hz. tone to make it sour? (If it never sounds sour to you at all, just report that.)

Now do the same things with pulse trains. You'll need the "pulse" object. Make two of them, frequencies 440 and 330, with "BW" (bandwidth) set to 2000, and connect them to an "output" object as you did with the sinusoids. Now reduce the 330-Hz. one to 329. What do you hear?

Now reduce it further until it sounds sour. How many Hz. less than 330 did you have to go? Was it further away than the tempered fourth (329.628)?

One could think that the number of Hz. you have to mis-tune an interval to get sourness might be a constant or else that it might be a constant proportion (i.e., interval). To find out, repeat the experiment for 220 Hz. and 165 Hz. Again, decrease the lower frequency (165) until you think it sounds sour. How many Hz. did it take and is it more nearly the same frequency difference or the same proportion?