How Intervals Are Built

Every interval, once you know its quality, has a precise size. Danhauser expresses it in whole steps, diatonic half steps and chromatic half steps. Remember that a whole step is made of one diatonic half step plus one chromatic half step. A diatonic half step changes the letter name, as in E-F or C#-D, while a chromatic half step keeps it, as in C-C#.

How intervals are built: the table

Memory aid A: minor, major and perfect intervals

• The steps (whole and half) between the degrees of a minor, major or perfect interval add up to one less than the interval number. A fifth contains 4 steps, a seventh 6.
• Major and perfect intervals contain 1 diatonic half step.
• Minor intervals contain 2 diatonic half steps.
• There are a few exceptions: the major 2nd and the major 3rd contain no half step, the minor 2nd and the minor 3rd contain only one, and the perfect octave contains two.

Example 1. Let's try it. What is a perfect fifth made of? Take 1 away from 5 and you get 4 steps. Since the interval is perfect, one of them is a diatonic half step, and the remaining 3 are whole steps. So the answer is 3 whole steps and 1 diatonic half step.

Example 2. Which interval contains 4 whole steps and 2 diatonic half steps? 4 plus 2 makes 6 steps, so it's a seventh. With two diatonic half steps, it's minor. It is a minor seventh.

Memory aid B: augmented intervals (no exceptions)

An augmented interval is always one chromatic half step larger than the same interval in its major or perfect form. For example, an augmented fifth is a perfect fifth, that is, 3 whole steps and 1 diatonic half step, plus 1 chromatic half step. A doubly augmented interval adds one more chromatic half step.

Memory aid C: diminished intervals (no exceptions)

A diminished interval is always one chromatic half step smaller than the same interval in its minor or perfect form. For a diminished fifth, take one chromatic half step away from 3 whole steps and 1 diatonic half step. Since a whole step minus a chromatic half step leaves a diatonic half step, you get 2 whole steps and 2 diatonic half steps.

Counting half steps, the modern way

Today, on an equal-tempered keyboard, we simply count half steps, that is, consecutive keys, white and black. A whole step is 2 half steps, whatever their kind.

Identifying the interval between two natural notes

Among the natural notes, the white keys, there are only two diatonic half steps: E-F and B-C. Count the degrees first, then check whether the interval contains two of these half steps, one or none. Memory aid A then gives you the quality.

For example, take C up to B. That's 7 degrees, so a seventh. Between them lies a single half step, E-F, so it's a major seventh.

Identifying the interval when there are accidentals

Now let's bring in accidentals. Mentally remove them, find the natural interval as above, then put the accidentals back and see what they do:

• You enlarge the interval by one chromatic half step when you raise the upper note with a raising accidental, or lower the lower note with a lowering accidental. It takes the next quality up: major instead of minor, augmented instead of perfect or major. Do both at once and it grows by 2 chromatic half steps.
• You shrink the interval by one chromatic half step when you lower the upper note or raise the lower note. It takes the next quality down: minor instead of major, diminished instead of perfect or minor. Do both at once and it shrinks by 2 chromatic half steps.
• If both notes carry accidentals of the same kind, such as two sharps or two flats, the distance does not change and the quality stays the same.

Finding the note that forms a given interval

Finding a note works the same way, in reverse. What is the major third above B? The third above B is D. But B-D contains a half step, B-C, so it's a minor third. To make it major, raise the upper note: the major third above B is D#.

Now that you can measure any interval, let's see what happens when you turn one upside down. That's inversion, the subject of the next lesson.