Building intervals

Knowing that a major third spans two tones is one thing. Writing, with no piano nearby, the major third above B flat, or the diminished sixth below D, is another. Counting semitones note by note works, but it is slow and full of traps, because accidentals change the count at every step.

Lacerda gives three chapters to a method that avoids that count. The idea is always the same: first find the natural interval, the one formed with no accidental at all, and only then settle the difference (Lacerda 1966, ch. XXXII).

Natural intervals, by heart

A natural interval is one where neither note carries an accidental (Lacerda 1966, ch. XXX). They are memorised with a single key: the two natural semitones, E–F and B–C. Lacerda arranges the rules around them:

Interval	Rule
seconds	major, except E–F and B–C, which are minor
thirds	major when they span neither semitone; minor when they span one
fourths	perfect, except F–B, which is augmented
fifths	perfect, except B–F, which is diminished
sixths and sevenths	major when they span only one semitone; minor when they span both
octaves	all perfect

"Span" here means the semitone lies between the interval's two notes. D–F passes through E–F: a minor third. C–E passes through neither: a major third. A–G passes through B–C and E–F: a minor seventh.

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Naming: reduce to natural and compare

To name an interval with accidentals, Lacerda has you do two things, in this order (Lacerda 1966, ch. XXXI):

1. Reduce the interval to natural: take the accidentals away and name what is left, by the rules above.
2. Compare the natural interval with the given one: each note that has moved away widens the interval by a semitone; each note that has moved closer narrows it.

An example. A flat up to G:

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Without the flat, A–G is a minor seventh, because it spans both natural semitones. In the given interval the lower note has fallen: the notes moved apart, and the interval grew by a semitone. A minor seventh plus a semitone is a major seventh.

Another, with both notes altered. F sharp up to D flat:

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F–D, natural, spans only B–C: a major sixth. In the given interval the lower note rose and the upper one fell. Both moved closer, and the interval shrank by two semitones: from major to minor, and from minor to diminished. It is a diminished sixth.

When both notes carry the same accidental, nothing changes: the given interval has the natural interval's quality. Blatter reaches the same result in steps, one accidental at a time (Blatter 2007, part 2, art. 1).

Building upwards

Building is the reverse. Given a note and an interval, find the other note. Lacerda uses the same process: take the natural interval as a base and introduce the alterations it needs (Lacerda 1966, ch. XXXII).

Asked for: the major third above B flat.

• Take the accidental off the given note: B. Count three names up: B, C, D.
• The natural B–D spans B–C: it is a minor third.
• But the given note is B flat, lower than B. That already moves the notes a semitone apart, and the minor third becomes major.

The answer is D, with no accidental.

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Another: the perfect fifth above B. The natural B–F is a diminished fifth; to be perfect it has to grow a semitone. The upper note rises: F sharp.

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Building downwards

The given note can be the upper one. Then you count downwards, and any adjustment falls on the lower note (Lacerda 1966, ch. XXXII).

Asked for: the minor third below G. Three names down: G, F, E. The natural E–G spans E–F: it is already a minor third. The answer is E.

Asked for: the major third below G. The same E–G is minor; to grow, the notes have to move apart, and the note free to move is the lower one, which falls: E flat.

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The usual rule holds both ways. To widen, the upper note rises or the lower note falls; to narrow, the opposite. All that changes is which of the two is free to take the accidental.

A shorter way, with the major scale

In an appendix to the chapter, Lacerda gives a second method. Think of the major scale that starts on the interval's lower note. From the first note of that scale to each of the others, the intervals are all major or perfect: major second, third, sixth and seventh; perfect fourth, fifth and octave (Lacerda 1966, ch. XXXII). You only need to compare the given note with the note of the scale.

Naming D–B flat: in D major the sixth note is B natural, and D–B is a major sixth. The B flat is a semitone lower: a minor sixth.

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Lacerda finds this process generally simpler than the first, and warns where it gets harder: when the given note is the upper one, because then you do not know in advance which scale to imagine, and when double sharps or double flats appear. Med teaches a procedure close to the first one: form the interval from the note without its accidental, name the natural interval, and only then apply the given note's accidental, adjusting the other (Med 1996, ch. X).

What comes next

With intervals built in either direction, you can see what happens when the two notes swap places, the lower one rising an octave. That is inversion, and it turns each interval into another, with rules of its own for number and quality.