Intervals

In music theory, an interval is the relationship between two pitches (notes) that are played simultaneously or in sequence. If the distinction is important, the former is defined as a harmonic interval, whereas the latter is a melodic interval.

There are many ways to calculate the relationship between pitches. In acoustics, their frequency ratio is important, such as the relationship between middle A at 440 Hz, and the next higher A at 880 Hz (a ratio of 1:2). When discussing various tuning systems, intervals are frequently described in cents where an octave is exactly 1200 cents and an equally tempered semitone is 100 cents.

Most of the time in music theory, however, we measure intervals by how many scale degrees and semitones separate the notes.

Interval Names

The names of intervals are derived from how many scale steps (or degrees) they encompass:

Note that most interval names are simply ordinals—the exceptions being the unison and the octave. (The latter of which anyway means "eighth".)

There are also intervals larger than the octave, commonly referred to as compound intervals:

Counting and Adding Intervals

When determining the interval name, the first step is always counted. Hence, the interval C–E is a third, because C=1, D=2 and E=3. Because of this, in interval theory, you always wind up with "one less" then you would do in arithmetic. If you stack two seconds on top of each other, you get a third (2 + 2 = 3). Two thirds equal a fifth, and so on.

Interval Quality

Having determined the name of an interval, its quality is decided by how many semitone steps the interval contains:

When determining interval quality, do not count the starting note. The interval C–E is a third because C=1, D=2 and E=3 scale steps. It is a major third because C=0, C♯=1, D=2, D♯=3 and E=4 semitones.

There are two distinct classes of intervals: perfect and major/minor intervals. Perfect intervals include the unison, fourth and fifth (as well as their compound brethren: the octave, eleventh and twelfth). These only appear in one basic form. The major/minor intervals include the second, third, sixth and seventh (plus the compound equivalents) and appear in two basic forms, neither of which is more prevalent than the other. A perfect interval can never become major or minor, and vice versa.

The perfect intervals are so called because they have the simplest frequency ratios in the overtone series: 1:1, 1:2, 2:3 and 3:4.

Altering Intervals

When either or both notes are modified with an accidental (or through a key signature), the semitone span of the interval changes. Such an interval is said to be altered.

A perfect interval becomes diminished when it shrinks by one semitone and augmented when it grows by the same amount:

A minor interval that is raised by a semitone becomes a major interval. If it is further raised, it becomes augmented.

A major interval that is flattened by a semitone becomes a minor interval. If flattened further, it becomes diminished.

This is an example of how a minor/major interval grows from diminished to augmented:

Note how subtracting a sharp is the same thing as adding a flat, and the other way around. Intervals are grown by raising the upper note and/or flattening the lower note. Conversely, they are shrunk by flattening the upper note and/or raising the lower note.

Diminished and augmented intervals that are further altered become double diminished/augmented, then triple diminished/augmented, and so on. Still, a major/minor interval can never become perfect, and vice versa.

Intervals and Enharmonicity

When you add altered intervals to the mix, you have several ways of notating what on an instrument with equal temperament would sound exactly the same. We can therefore expand the interval list with a couple of new variants:

The list is by no means exhaustive. It merely contains most of the intervals that actually tend to occur in written music. It could theoretically be expanded almost infinitely. Just add all the weird edge cases like augmented thirds, double-augmented fourths, triple-diminished fifths, etc.

Even though such exotic intervals are rarely seen, they do serve to illustrate that it's the underlying natural notes that determine the name of the interval. No matter how many accidentals you add to the notes C and E, and no matter how much that shrinks and grows the interval, it remains some sort of a third. It might be triple-diminished or quadruple-augmented, but it's still just a third. (In case you're curious, a triple-dimished third is enharmonically equivalent to a perfect unison, and a quadruple-augmented third equivalent to an augmented fifth.)

On the other hand, if you play the notes C and E on a musical instrument, those notes are simply two notes at the interval of four semitones. Until you place them within the context of a chord or a key, C could just as easily be a B♯; E could equally well be an F♭. It is more likely that the notes are C and E, but absent some sort of a context, no one can claim this, that or the other thing.

Interval Inversion

Inverting intervals means that you transpose either of the notes by an octave so that the note that was formerly at the bottom is now on top or vice versa:

Interval Class

As you see, intervals tend to appear in discrete pairs with similar properties. Unisons and octaves can be thought of as aspects of each other, likewise with fourths and fifths, seconds and sevenths and thirds and sixths. For this reason, theorists sometimes discuss interval classes rather than intervals proper. Just as with pitch classes, register is ignored and the shortest possible distance between notes is calculated.

Inversion of Quality

A major interval that is inverted always becomes a minor interval. Conversely, a minor interval becomes major upon inversion. Altered intervals work exactly the same: an augmented interval inverts to a diminished interval, and the other way around:

The exceptions are the perfect intervals. These remain perfect even in inversion.

The Tritone

The diminished fifth/augmented fourth between B and F is the only interval between natural notes that isn't perfect, major or minor. This interval is known as the tritone, since it spans three [whole] tones.

The tritone has many interesting properties. It consists of six semitone steps, which means that it is a subdivision of the octave into two parts of equal size. It is also the only interval that remains identical when inverted. Since it is midway through the octave, it is also the largest possible interval class—the perfect fifth (7 semitones) inverts to a perfect fourth (5 semitones).

Learning the Intervals

Constructing intervals as well as naming them is a skill that is developed with experience and practice. It is not dissimilar to the multiplication table: we actually don't calculate 9 times 7 whenever we're confronted with it. We just know the product is 63. It is very much the same with intervals: much in music theory becomes that much easier if you just know that there is a major sixth between a C and an A, a major seventh between a D and a C♯, and so on.

Therefore, I have decided to provide the musical equivalent of the said multiplication table:

These are just the natural notes, but with the knowledge you gained by reading this article, you know how to make a minor interval major, or how to make a perfect interval diminished. Just remember that it's the underlying natural notes that determine how many scale steps that separate the notes, therefore whether it's a second, third and so on. Staff notation makes this considerably easier, since all intervals have their own distinct "look"—thirds occupy adjacent lines or spaces, fifths every other line or space, and so on. (Refer back to the illustration near the top of the page.) If you know that the notes B and F constitute a diminished fifth, then it will be easy to make a valid perfect fifth—just flatten the lower note (B♭–F) or sharpen the upper note (B♮–F♯).