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Solutions to Exercises in Chapter 4

22 July, 2019 - 10:18
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Solution to Exercise 4.1:

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Figure 4.55 Solution to Exercise 4.1

Solution to Exercise 4.2:

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Figure 4.56 Solution to Exercise 4.2

Solution to Exercise 4.3:

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Figure 4.57 If your answer is different, check to see if you have written a different enharmonic spelling (Section 1.1.5) of the note in the answer. For example, the B flat could be written as an A sharp. 

Solution to Exercise 4.4:

  1. Major
  2. Major
  3. Minor
  4. Major
  5. Minor

Solution to Exercise 4.5:

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Figure 4.58 Solution to Exercise 4.5

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:

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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:

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Figure 4.61 Solution to Exercise 4.8

Solution to Exercise 4.9:

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Figure 4.62 Solution to Exercise 4.9

Solution to Exercise 4.10:

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Figure 4.63 Solution to Exercise 4.10

Solution to Exercise 4.11:

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Figure 4.64 Solution to Exercise 4.11

Solution to Exercise 4.12:

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Figure 4.65 Solution to Exercise 4.12

Solution to Exercise 4.13:

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Figure 4.66 Solution to Exercise 4.13

Solution to Exercise 4.14:

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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".

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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:

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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
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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
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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
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Figure 4.72 Solution to Exercise 4.22

Solution to Exercise 4.23:

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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.