Interval Quality: Major, Minor, Perfect, Diminished, Augmented

Every interval has two parts: a number, found by counting letter names (C to E is a third, counting C-D-E), and a quality, found by counting half steps. Unisons, fourths, fifths, and octaves get called perfect because in the major scale they appear in only one size against the tonic, with no major or minor version to confuse them with. Seconds, thirds, sixths, and sevenths come in two everyday flavors, major and minor, because the major scale produces a larger and a smaller version depending on which scale degree you measure to. Minor is always exactly one half step smaller than major, same letter-name distance, one key closer together.

The mechanism is just arithmetic once you fix the number. Figure out the generic size from the letters, then count half steps between the two actual notes. A major third spans 4 half steps, a minor third spans 3. A perfect fifth spans 7 half steps; shrink it by one half step and it becomes diminished, stretch a major or perfect interval by one half step and it becomes augmented. So C to E flat: letters give a third, half steps give 3, that is a minor third. F to B: letters give a fourth, half steps give 6, one more than perfect, so augmented fourth. A to E flat: letters give a fifth, half steps give 6, one less than perfect, so diminished fifth.

To build an augmented sixth above D flat, start from the major sixth, which is B flat, spanning 9 half steps. Raising B flat by a half step gives B, spanning 10 half steps above D flat, an augmented sixth. This topic sets up every later interval judgment, from spelling chords correctly to hearing why an augmented sixth chord sounds so different from an ordinary major sixth.