Solutions to Exercises in Chapter 4

Notice that although they look completely different, the scales of F sharp major and G flat major (numbers 5 and 6) sound exactly the same when played, on a piano as shown in the following Figure 4.59, or on any other instrument using equal temperament tuning. If this surprises you, please read more about enharmonic scales.

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Figure 4.59 Enharmonic Scales

Using this figure of a keyboard, or the fingerings from your own instrument, notice that the notes for the F sharp major scale and the G flat major scale in Figure 4.58, although spelled differently, will sound the same. 

Solution to Exercise 4.6:

Figure 4.60 Solution to Exercise 4.6

Solution to Exercise 4.7:

1. A minor: C major
2. G minor: B flat major
3. B flat minor: D flat major
4. E minor: G major
5. F minor: A flat major
6. F sharp minor: A major

Solution to Exercise 4.8:

Figure 4.61 Solution to Exercise 4.8

Solution to Exercise 4.9:

Figure 4.62 Solution to Exercise 4.9

Solution to Exercise 4.10:

Figure 4.63 Solution to Exercise 4.10

Solution to Exercise 4.11:

Figure 4.64 Solution to Exercise 4.11

Solution to Exercise 4.12:

Figure 4.65 Solution to Exercise 4.12

Solution to Exercise 4.13:

Figure 4.66 Solution to Exercise 4.13

Solution to Exercise 4.14:

Figure 4.67 Solution to Exercise 4.14

Solution to Exercise 4.15:

1. Diminished sixth
2. Perfect fourth
3. Augmented fourth
4. Minor second
5. Major third

Solution to Exercise 4.16:

1. The ratio 4:6 reduced to lowest terms is 2:3. (In other words, they are two ways of writing the same mathematical 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 fifteen; 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 4.17:

Opening both first and second valves gives the harmonic series one-and-a-half steps lower than "no valves".

Figure 4.68 Solution to Exercise 4.17

Solution to Exercise 4.18:

• E flat major (3 flats):
• B flat major (2 flats)
• A flat major (4 flats)
• C minor (3 flats)
• G minor (2 flats)
• F minor (4 flats)

A minor (no sharps or flats):

• E minor (1 sharp)
• D minor (1 flat)
• C major (no sharps or flats)
• G major (1 sharp)
• F major (1 flat)

Solution to Exercise 4.19:

Figure 4.69 Soultion to Exercise 4.19

Solution to Exercise 4.20:

• A major adds G sharp
• E major adds D sharp
• B major adds A sharp
• F sharp major adds E sharp

Figure 4.70 Solution to Exercise 4.20

Solution to Exercise 4.21:

• B minor adds C sharp
• F sharp minor adds G sharp
• C sharp minor adds D sharp

Figure 4.71 Solution to Exercise 4.21

Solution to Exercise 4.22:

• E flat major adds A flat
• A flat major adds D flat
• D flat major adds G flat
• G flat major adds C flat

Figure 4.72 Solution to Exercise 4.22

Solution to Exercise 4.23:

Figure 4.73 Solution to Exercise 4.23

This whole tone scale contains the notes that are not in the whole tone scale in

Solution to Exercise 4.24:

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Figure 4.74 Solution to Exercise 4.24

The flats in one scale are the enharmonic (Section 1.1.5) 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.

Solution to Exercise 4.25:

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

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