Solutions to Exercises in Chapter 4 (Understanding Basic Music Theory)

Solutions to Exercises

Solution to Exercise 4.1.1

Solution to Exercise 4.2.1

Solution to Exercise 4.2.2

Figure 4.59: If your answer is di erent, check to see if you have written a di erent enharmonic spelling (Section 1.1.5) of the note in the answer. For example, the B at could be written as an A sharp.

Solution to Exercise 4.3.1

1. Major

2. Major

3. Minor

4. Major

5. Minor

Solution to Exercise 4.3.2

Notice that although they look completely di erent, the scales of F sharp major and G at major (numbers 5 and 6) sound exactly the same when played, on a piano as shown in Figure 4.61 (Enharmonic Scales), or on any other instrument using equal temperament (Section 6.2.3.2: Equal Temperament) tuning. If this surprises you, please read more about enharmonic (Section 1.1.5) scales.
Enharmonic Scales

Figure 4.61: Using this gure of a keyboard, or the ngerings from your own instrument, notice that the notes for the F sharp major scale and the G at major scale in Figure 4.60, although spelled di erently, will sound the same.

Solution to Exercise 4.4.1

Solution to Exercise 4.4.2

1. A minor: C major

2. G minor: B at major

3. B at minor: D at major

4. E minor: G major

5. F minor: A at major

6. F sharp minor: A major

Solution to Exercise 4.4.3

Solution to Exercise 4.4.4

Solution to Exercise 4.5.1

Solution to Exercise 4.5.2

Solution to Exercise 4.5.3

Solution to Exercise 4.5.4

Solution to Exercise 4.5.5

Solution to Exercise 4.5.6

1. Diminished sixth

2. Perfect fourth

3. Augmented fourth

4. Minor second

5. Major third

Solution to Exercise 4.6.1

1. The ratio 4:6 reduced to lowest terms is 2:3. (In other words, they are two ways of writing the samemathematical relationship. If you are more comfortable with fractions than with ratios, think of all the ratios as fractions instead. 2:3 is just two-thirds, and 4:6 is four-sixths. Four-sixths reduces to two-thirds.)

2. Six and nine (6:9 also reduces to 2:3); eight and twelve; ten and fteen; and any other combination that can be reduced to 2:3 (12:18, 14:21 and so on).

3. Harmonics three and four; six and eight; nine and twelve; twelve and sixteen; and so on.

4. 3:4

Solution to Exercise 4.6.2

Opening both rst and second valves gives the harmonic series one-and-a-half steps lower than “no valves”.

Solution to Exercise 4.7.1E at major (3 ats):

• B at major (2 ats)

• A at major (4 ats)

• C minor (3 ats)

• G minor (2 ats)

• F minor (4 ats)

A minor (no sharps or ats):

• E minor (1 sharp)

• D minor (1 at)

• C major (no sharps or ats)

• G major (1 sharp)

• F major (1 at)

Solution to Exercise 4.7.2

Solution to Exercise 4.7.3

• A major adds G sharp

• B major adds A sharp

• E major adds D sharp

• F sharp major adds E sharp

Solution to Exercise 4.7.4

• B minor adds C sharp

• F sharp minor adds G sharp

• C sharp minor adds D sharp

Solution to Exercise 4.7.5

• E at major adds A at

• A at major adds D at

• D at major adds G at

• G at major adds C at

Solution to Exercise 4.8.1

Figure 4.75: This whole tone scale contains the notes that are not in the whole tone scale in Figure 4.48 (A Whole Tone Scale).

Solution to Exercise 4.8.2

Figure 4.76: The ats in one scale are the enharmonic equivalents of the sharps in the other scale.

Assuming that octaves don’t matter – as they usually don’t in Western music theory, this scale shares all of its possible pitches with the scale in Figure 4.48 (A Whole Tone Scale).

Solution to Exercise 4.8.3

If you can, have your teacher listen to your compositions.

Basic Music Theory

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