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Solutions for the Crash Course

Exercise 1

  1. A1, d5, d4.
  2. 32768/19683, 6561/4096, 65536/59049.
  3. 9/8, 32/27, 9/8, 32/27, 9/8.
  4. D–F♯–G–A–C♯–D
  5. 531441/524288. (Or its reciprocal; this is a bit tricky. 531441/524288 is the interval from C to the B♯ below it, which cannot be represented easily as an interval: augmented zeroth?! Anyway, by size, it’s a diminished second.)
  6. AAAAAAA3. It’s a septuple augmented third, minus an octave. Yes. Above C, it would be written as a major third with seven sharps. Yikes.

Exercise 2

  1. −1, −4.
  2. 513/512.
  3. +1.
  4. E–F♯–G♯5–A–B–D7–E
  5. A–A–E–A–C♯5–E–G7–A–B–C♯5–D11–E–F13–G7
  6. A–A–D–A–F-5–D–B-7–A

Exercise 3

  1. M65, m35, m37, m75.
  2. 10/9, 9/7, 11/6.
  3. C–D–E5–F–G–A5–B5–C
  4. No. It would be the sixth mode if the fourth degree of the scale was F5 instead of F. The two scales are actually built on tonic, subdominant, and dominant triads of 4:5:6 (5/4 major) and 10:12:15 (6/5 minor) respectively. The scale with 3 major triads contains only 2 minor triads, and the scale with 3 minor triads contains only 2 major triads.

Exercise 4

  1. m27, A2513, d41125.
  2. 12/7, 13/10, 243/200.
  3. C5–D5–E♭–G5–B♭, F–C1–E♭7–F–A5, B♭–C–D5–F–A5
  4. The following five chords:
    • I chord: 1/1, 3/2, 5/4, 1/1
    • vi chord: 5/3, 5/3, 5/4, 1/1
    • ii chord: 4/3, 5/3, 4/3, 10/9
    • V chord: 3/2, 3/2, 9/8, 15/8
    • I chord: 1/1, 3/2, 5/4, 1/1
  5. A–B♭11-5, A–B5, A–B, A–B-7
  6. P155. The interval is 440/432 = 55/54. Please don’t buy into 432 Hz stuff, thanks.
  7. P1657.