Music Theory: Intervals, and How to Derive Them

Joel Falconer wrote a great article on how to use your ear to detect intervals with song associations. In this article we’ll look at how you come up with those intervals in the first place.

An interval is the distance between two notes, with the first note counted as 1. From C up to E is a major 3rd. From D to up G is a perfect 4th, and so on.

There are two parts to an interval: a modifier and a number. First let’s see how to figure out the number of an interval.

Interval Number

You find the number by counting up the letters from your first note to your last. So let’s say we wanted to find the number of the interval from C to A. Starting on C (counted as 1), we count up six letters (C D E F G A) to get to A, making C up to A an interval of a 6th.

Sharps and flats are not used when figuring out the number of an interval, only the distance between the letters. So if we wanted to go from Db to G we ignore the flat and count the letters. Starting with D we have D E F G. Four letters, making Db up to G an interval of a 4th (we’ll see what kind of 4th further down in the tutorial).

The name for an interval of 1 is unison. The name for an interval of 8 is octave. So two identical Cs played at the same time are considered in unison. A C played with the C above it is an octave.

Here are the interval numbers based off of C, from unison up to 13th:

Now that we know how to count the number of an interval, let’s look at how to figure out it’s modifier.

Modifiers

There are 5 possible modifiers of an interval: perfect, major, minor, diminished and augmented.

Perfect Intervals

Perfect intervals are used for unisons, 4ths, 5ths, and octaves. The best way to begin learning how to derive the perfect intervals is to think of the first note in the context of it’s related major scale. For example if your first note was Eb and you wanted to go up a perfect 4th, you would start on Eb and count up the notes in the Eb major scale, Eb F G Ab. A P4 above Eb is Ab.

Deriving a perfect 5th is the same process. Let’s say we’re starting on D. Count up 5 notes (with D as 1) in the D major scale and you land on A.

Major and Minor Intervals

The major and minor modifiers refer to intervals of a 2nd, 3rd, 6th and 7th. Again using your first note as if it is the 1 of a major scale, the major intervals are those that you’ll find by going up a 2nd, 3rd, 6th or 7th above that note. In other words C to E is a major third. Ab to F is a major 6th.

To change an interval from major to minor, you start with the major interval and then lower it by one half step. C to Db is a minor 2nd, C to Eb is a minor 3rd, C to Ab is a minor 6th, and C to Bb is a minor 7th.

Diminished and Augmented Intervals

A diminished interval (marked with a little circle º) is a half step lower than a minor or perfect interval. In real usage this usually only refers to a diminished 5th, in other words taking a Perfect 5th down a half step such as from C to Gb.

An augmented interval (notated with a +) is the opposite of diminished. You augment an interval by raising a major or perfect interval by half a step. An augmented 4th would be C to F#, an augmented 6th is C to A#. The most common augmented intervals are the 4th and 5th.

So remembering our interval of a 4th from earlier of Db to G, we can now see that it is an augmented 4th. How do I know that? Because a perfect 4th above Db is Gb (the 4th note up in the key of Db major). By raising the Gb up to a G we take it from Perfect up to Augmented.

Also note that the diminished 5th and augmented 4th are more commonly referred to as the interval of a tritone.

Let’s now look at the common intervals from anyone note up to any other note, using C and E as our example starting pitches. Make sure you are perfectly clear on every one of these intervals before moving on.

Inversions (Intervals Going Down)

So far we’ve looked at how to figure out an interval from one note up to another, but how about from one note down to another? One of the easiest ways to do this is to “invert” the interval you already know. Moving in towards the center: an 2nd up becomes a 7th down, a 3rd up becomes a 6th down, and a 4th up becomes a 5th down. Of course the reverse is also true: a 5th up becomes a 4th down, a 6th up becomes a 3rd down and a 7th up becomes a 2nd down.

A Major interval going up becomes a minor going down, and a minor interval going up becomes a major interval going down. Augmented intervals become diminished and vice versa. Perfect intervals remain perfect in either direction.

OK so what does that all mean? In other words, going up from C to Bb is a minor 7th. To figure out the interval from C down to Bb, we switch minor to major and invert the 7th to a 2nd. Going C down to Bb is thus a major 2nd.

As another example, Ab up to Db is a perfect 4th. To figure out Ab down to Db we leave the modifier of perfect as is, then invert the 4th to a 5th. We can then figure out that while Ab up to Db is a perfect 4th, Ab down to Db is a perfect 5th.

Use these two charts as a reference:

Conclusion

Finally, it’s worth pointing out that although C to Ab (a minor 6th) and C to G# (an augmented 5th) may “sound” exactly the same, they should not be considered as identical. Their usage depends entirely on context. For now just remember that your number (2nd, 3rd, etc) comes from the number of letters from your first to your second note, and your modifier (major, minor, diminished, etc) comes from the actual distance between those two notes in relation to the major scale.

I highly recommend you visit Joel’s tutorial to begin to familiarize yourself with what the different intervals sound like, not just what they look like written out.

Understanding intervals is a basic but crucial step in your development as a musician. With a firm grasp of intervals under your belt you’ll be able to handle more advanced and rewarding topics such as chord tensions and counterpoint.

Although memorizing all of the intervals from any note to any other note may seem tedious and boring, the long term rewards of doing so are immeasurable.

Got a tip learning intervals or a question about how they’re derived? Leave a comment and open up the discussion.