Introduction

An interval in music is usually understood as the difference in pitch between two notes.  In other words, it is a measurement of distance between to points on the “vertical” axis in musical space.  This concept is more complex than it seems at first.  For example, think about the following questions:

• How do we understand distance in musical space?  Is it the difference between the fundamental frequency of two notes? Is it the vertical distance that separates two notes on a staff, measured in lines and spaces, or scale degrees?
• How do we, as listeners, experience this difference between two notes?

Before we get into the basics, consider the first question in relation to two notes chosen at random, let’s say C4 and G4.  We would normally think, in musical terms, of the interval between C4 and G4 as a “fifth.”  This is because we usually measure intervals by counting lines and spaces (or steps) starting on the lower note and finishing on the upper one.  Thus, for C4 to G4, we would count five steps: C4—D4—E4—F4—G4.

What about C3 and G3?  That will also be five steps: C3–D3–E3–F3–G3.  So, the distance, measured this way, is the same as the distance from C4 to G4.  Of course, counting lines and spaces leaves some notes out (all the black-key notes). Moreover, not all the intervals between adjacent pairs of notes in the series C–D–E–F–G are the same—most pairs form tones, but one pair (E–F) forms a semitone. So, we could make this counting more precise by counting all the notes, and thus all the semitones between C4 and G4, or C3 and G3.  If we count semitones, the result will still be the same—seven semitones.

So, either way, C4 and G4 or C3 and G3 are five steps or seven semitones apart.

Now consider the fundamental frequencies of these notes (in equal temperament at A4=440Hz):

Ihe difference between C3 and G3 is 65.1849Hz while that between C4 and G4 is 130.3698Hz. In other words, the distance between C4 and G4, measured this way, is twice the distance between C3 and G3.

The reason one measurement yields the same result while the other yields a result that is different by a factor of two is that the measurement in semitones is a linear function while the measurement in frequency is a logarithmic function. This is illustrated by the red line and the blue curve in Example 00 below.

The intriguing thing about all this is that we “hear” these two intervals  (C3–G3 and C4–G4), or any perfect fifth for that matter, as being the same “size.” Keep in mind, as we go through the music theory around intervals, that we are describing distance on a scale that is different to the physical reality, yet we are, or have become, capable of hearing it the way we describe it

Basic considerations

Melodic and harmonic intervals

A interval can be either melodic or harmonic:

• Melodic intervals involve a pair of notes in succession.
• Harmonic intervals involve a pair of notes simultaneously.

With melodic intervals, the direction from the first note to the second can be either ascending or descending.  Below are three representations of random intervals (don’t worry about the names yet) according to these configurations (Example 00).

Interval Quantity and Quality

In describing intervals, we refer to two related concepts: interval quantity and interval quality.

Interval quantity is the basic measurement of interval size (aka the “distance” between two notes on a staff).  It is measured according to the lines and spaces on the staff and always includes the two notes themselves in the counting.

Interval quantity is a more refined modification that we apply to intervals, and is most precisely measured in the exact number of semitones (inclusive of the notes themselves).

For instance, as shown below, the intervals C–E and D–F^[1] are both intervals of a third, because we count three staff positions (or letter-names) inclusively: C–D–E and D–E–F.  However, there is one more semitone between C and E than there is between D and F, meaning that they are different kinds of third. C–E is a major third; D–F is a minor third.  We will talk about quality of intervals more once we have covered measuring quantity.

As the example above makes clear, there are five semitones in the interval from C to E while D to F has four.  Both are thirds according to how we measure them on a staff or as notes on a keyboard, but they are qualitatively different.

Interval Quantity

In the following discussion, it does not matter whether we are talking about melodic or harmonic intervals.

Measuring interval quantity—counting Lines and Spaces

Interval quantity is measured by counting the lines and spaces on the staff.  In practice, it is useful to get accustomed to the appearance of interval sizes at sight without need to always count.  This takes a little practice but it is worth the effort to facilitate your reading.

We do not need to refer to specific notes or even a clef to show the counting of intervals.  In the example below, we show the harmonic interval, then the melodic interval in open note heads.  Intervening lines and spaces are indicated by smaller closed note heads.  Note, as shown in the various examples, leger lines continue the counting of lines and spaces beyond the staff.

An interval smaller than a second is called a unison—it means that two pitches are occupying the same line or space.  Sometimes this is a purely theoretical matter, but in other cases it is a real interval, as, for instance, if two voices or instruments are singing or playing the same note—that is, “in unison.”

The next interval above the seventh is the octave.  Technically it is an “eighth” because we would count eight notes (or, lines and spaces), but the term octave is generally used.  As noted elsewhere, the octave involves notes with the same name and sharing a special quality based on the simple relationship of their fundamental frequencies (1:2:4:8 and so on).

When you read intervals on a score, remember that odd-numbered intervals will have both notes on either lines or spaces, while even numbered intervals will have notes on a line and space, or vice-versa. Thirds will occupy adjacent lines and spaces, fifths will leave one line or space between, and sevenths two lines or spaces.

Simple and Compound Intervals

Intervals up to and including the octave are called simple intervals.  Once an interval exceeds an octave, it is called a compound interval—because it is a compound of an octave and a simple interval.  We can think of compound intervals either in terms of the total number of lines and spaces (just as we do for simple intervals) or we can think of them as octave+simple interval.  As the intervals get larger, the octave+simple interval is an easier approach. Some examples are shown below. (Again, we can still show this without any particular clef or note names—we are just considering “distance” of lines and spaces.)

The table below shows the calculation for compound intervals.  To work out a compound interval, we add seven to the simple interval (because the count starts on one, not zero).

As example 00 suggests, intervals beyond a compound octave are just called by their simple name plus the prefix “compound,” irrespective of how many octaves lie between.  You will not often encounter such large intervals. ^[2]

Interval Inversion

Inverting intervals is a fundamental part of the organisation of Western common-practice music.  It involves a reciprocal relationship of intervals within an octave. It is easier to show than explain—for the moment we will focus only on interval quantity.

Let’s consider the interval of a fifth from G4 to D5 (Example 00) we can invert this interval in two ways: by taking G4 to the next octave above, G5—creating a fourth, D5–G5 or by taking the D5 down to the next octave below, also creating a fourth, now D4–G4.

Another way of looking at this is if we take an octave (in this case G4–G5, or D4–D5) and “bisect” with a note (in this case D5, or G4) that note creates a pair intervals inside the octave, one of which is the inversion of the other.

You might also notice some quirky “maths” going on in this process.  Obviously, 5 + 4 = 4 + 5, but 5 + 4 ≠ 8 (nor does 4 + 5)! Because we count intervals by including both notes, with the lower note as “1,” we count the internal note in an interval inversion twice, hence the mathematical fallacy.  Interval inversions within an octave (8) always add up to nine!  The example below shows all the possibilities, including the inversion of the unison and the octave, with reference to C. The arrows at either end reminds us that this relationship is always reciprocal.  And note that the pairs of numbers always add up to nine.

In summary:

Notating harmonic Intervals

You might have noticed some conventions around the way we write harmonic intervals.  In general, follow the following simple rules (referring to Example 00):

1. When notating seconds, the note heads sit either side of each other, with the lower note always to the left. In particular, do not stack one directly above the other! When there is a stem, it should sit be places between the two notes and its direction should be towards the centre of the staff with respect to the note furthest from the centre line.
2. For seconds that sit on leger lines, use a slightly elongated set of leger lines for the pair, to cover the width of each note sitting either side.
3. For intervals greater than a second, the notes are stacked vertically, where there is a stem it should be directed towards the centre of the staff with respect to the note furthest from the centre line. As for single notes, upward stems are placed right of the note head, downward stems to the left

Interval Quality

Interval quality is a further refinement of how we understand and perceive intervals. There are five different types of interval quality:

• perfect
• major
• minor
• augmented
• diminished

These apply to simple and compound intervals alike.

Perfect intervals can be:

• unisons
• fourths
• fifths
• octaves

Major or minor intervals can be

• seconds
• thirds
• sixths
• seventh

Augmented and diminished are modifications applied to perfect, major or minor intervals, and so any interval can be augmented or diminished.

Note that unisons, fourths, fifths or octaves are never major or minor; seconds, thirds, sixths or sevenths are never perfect.

Compound intervals can be worked out by starting an octave (or two, or three, etc.) higher than the lower note.  E.g. for the interval C4–A3, count semitones from C3 to A3 (inclusive), for the interval G2–E♭4, count semitones from G4 to E♭4 (inclusive). 

Perfect Intervals

Perfect intervals derive from the simplest frequency ratios on the harmonic series—1:1 (unison); 2:1 (octave); 3:2 (perfect fifth); 4:3 (perfect fourth).  Because it is relatively rare to make unisons or octaves diminished or augmented, in general usage we can take “unison” to mean “perfect unison” and “octave” to mean “perfect octave,” unless context demands clarification.  However, because the effects of augmenting and diminishing are more commonly applied even in quite simple music to fourths and fifths, it is best to be clear and write/say “perfect fourth” or “perfect fifth,” even if it is relatively obvious.

The following shows the number of semitones (inclusive of both notes in the interval) in each perfect interval:

Here are some examples, showing the interval with open notes and the intervening semitone steps with filled note heads.

Inverting perfect intervals

When you invert a perfect interval, you get another perfect interval. Clearly, inverting a unison creates octave, and vice versa. Also, considering the number of semitones in the perfect fourth and fifth, we see that they sum to twelve, meaning that a perfect fourth inverts to a perfect fifth, and vice versa.^[3]

Major and Minor Intervals

Major and minor intervals derive from the increasingly complex higher frequency ratios of the harmonic series—5:4 (major third); 6:5 (minor third); 8:7 (major second); 16:15 (minor second).^[4] The counting of semitones for these intervals (again, inclusive of the two notes forming the interval) is as follows:

Inverting Major and Minor Intervals

When you invert a major or a minor interval, the quality is reciprocated:

• major inverts to minor
• minor inverts to major

Again, if we consider the semitone counts for these intervals, you can see why this is the case.  For instance, a major third has four semitones (inclusive), if you take four semitones from twelve, you are left with eight semitones, a minor sixth.

The follow table summarises the inversions for major and minor intervals:

In the example below, three cases are shown, including the counting of semitones and the interval inversion with reciprocal counting.

Counting shortcuts

With practice, recognising perfect, major and minor intervals, both on paper and by ear, becomes easier and, ultimately, instantaneous.  However, at first, counting semitones in large intervals can be laborious.  Remember that sixths invert to thirds and sevenths invert to seconds and that major inverts to minor, and vice versa. So, for example, if you come across E♭–D♭, instead of counting this interval, invert it so that you have D♭–E♭. D♭–E♭  is a major second, so E♭–D♭ is a minor seventh.

Augmented and diminished intervals.

Augmented and diminished intervals result from a process of (obviously) augmenting (increasing) or diminishing (decreasing) the size a perfect, major, or minor interval.  This is done in such as way as the interval quantity is not changed.  That is, for instance, a major sixth and an augmented sixth as still both sixths. If we count lines and spaces between note heads, we will count six for both intervals, but the augmented sixth will have one extra semitone, because we will have either raised (sharpened) the upper note by a semitone, or lowered (flattened) the lower note.

The process of augmenting or diminishing is slightly different depending on whether it is applied to a perfect interval, or to a major or minor interval.

• If we increase a perfect fourth or a perfect fifth by a semitone, the result is an augmented fourth or an augmented fifth, respectively.
• If we decrease a perfect fourth or perfect fifth by a semitone, the result is a diminished fourth or a diminished fifth, respectively.
• The same is true of unisons and octaves, although these are much less augmented or diminished in common practice.

Thus, perfect intervals,  can be either diminished or augmented.

Major and minor intervals can only be affected by one of these processes:

• Major intervals can only be augmented.  If you decrease a major interval, you get a minor interval.
• Likewise, minor intervals can only be diminished. If you increase a minor interval by a semitone, you get a major interval.