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 two points on the vertical axis in musical space.  This concept is more complex than it seems at first.  Think about the following questions:

• How do we define “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 distance 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.

Now consider C3 and G3  They are also five steps apart: C3–D3–E3–F3–G3.  So, the distance, measured this way, is the same as the distance from C4 to G4—a fifth.  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.  None-the-less, when we count semitones, the result will still be the same—seven semitones (eight notes in all).

So, measured this way, the distance from C4 to G4 and C3 and G3 are is uniform—five steps or seven semitones.

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

The 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 musical concepts around intervals, that we are describing distance on a scale (linear) that is different to the physical reality (logarithmic), 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 harmonic or melodic:

• Harmonic intervals involve a pair of notes simultaneously.
• Melodic intervals involve a pair of notes in succession.
• For 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 numerical measurement of interval size; that is, the “distance” between two notes on a staff).  It is measured according to the number of lines and spaces on the staff and always includes the two notes themselves in the counting.  We sometimes call these intervals “generic” intervals.

Interval quantity is a more refined modification that we apply to intervals of the same quantity,  and is most precisely measured in the exact number of semitones (inclusive of the notes themselves).  To distinguish quality among intervals of the same quantity, we add modifiers—perfect, major, minor, diminished, augmented.

For instance, as shown below in Example 00, 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.

Interval Quantity

Measuring interval quantity—counting Lines and Spaces

Interval quantity is measured by counting the lines and spaces on a 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 Example 00, 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.

Naming Interval Quantity

In naming intervals according to quantity, we use ordinal numbers for the number of steps and treat these as nouns. That is, we say or write “a second,” “a fifth,” and so on.  (This is short for saying “the interval of a second,” and so on).

There are two exceptions to this, and they involve cases where the interval involves the same note letter name.

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.” For this interval, we always say or write “a unison” (not “a first”).

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 always 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 (unisons, thirds, fifths, sevenths) will have both notes on either lines or spaces, while even numbered intervals (seconds, fourths, sixths, octaves) 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 (Example 00). Again, we can still show this without any particular clef or note names—we are still just considering the “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).

Naming Compound Intervals

The naming of compound interval quantity follows a couple of conventions:

• Ninths are always called ninths, never compound seconds. This is because seconds are usually understood in the context of steps (as in a scale), while ninths (as we will see later) typically come about by stacking a series of thirds: 1–3–5–7–9.
• It is also quite common to refer to intervals from the tenth to the thirteenth (inclusive) by their outright number but also acceptable to use the terms “compound third” through to “compound sixth.”
• For some reason, the fourteenth is usually just called a “compound seventh.”

Interval Inversion—Simple intervals

Inverting intervals is a fundamental part of the organisation of Western common-practice music.  Inversion 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” this octave 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, and vice-versa.

You might 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, again, that the pairs of numbers always add up to nine.

Inverting compound intervals

Compound intervals will invert within a compound octave. There are three ways to do this:

• Move the lower note up two octaves—see (a) and (d) in Example 00, below.
• Move the upper note down two octaves—see (b) and (e) in Example 00.
• Move the lower note up one octave and the upper note down one octave—see (c), (f) and (i) in Example 00.

The sum of the two intervals will now be sixteen and the total octave movement must be two octaves altogether—two down, two up, or one down and one up.

Note that in Example 00 neither (g) nor (h) represent inversions. There is only one octave movement—the lower note goes up one octave in (g) and the upper note goes down one octave in (h). Because the interval exceeds and octave, moving one note an octave is insufficient to change the relative position of the notes so that they are reversed.  It has only reduced a compound interval (ninth) to its simple equivalent (second).

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 be placed 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 (P)
• major (M)
• minor (m)
• augmented (A, or +)
• diminished (d, or °)

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.

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

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.

As shown in the third example above, compound intervals can be worked out by starting an octave (or more) higher than the lower note.   

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.^[2] We have already seen an example of inverting a perfect fifth in Example 00, above.

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).^[3] 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 increasing (augmenting) or decreasing (diminishing) the size a perfect, major, or minor interval.  This is done in such as way as the interval quantity is not changed.  For instance, a major sixth and an augmented sixth are 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 made 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 by a semitone, 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.
• Note that (although rare) any augmented interval can be further expanded by adding a semitone, creating a doubly augmented interval.  Correspondingly, any diminished interval can be further reduced, creating a double diminished interval.

The processes described are summarised in Example 00.

The next example shows some instances of augmenting and diminished perfect, major and minor intervals.

In the example above, note how the location of the notes of the intervals on the staff does not change, only the accidentals applied to them.

Inverting Augmented and Diminished Intervals

Similar to the way major and minor intervals are reciprocal in inversion, augmented intervals become diminished when inverted, and diminished intervals become augmented.  The more common scenarios of inversion of augmented and diminished intervals are as follows:

• augmented fourths invert to diminished fifths; diminished fifths invert to augmented fourths
• augmented sixths invert to diminished thirds; diminished thirds invert to augmented sixths
• augmented seconds invert to diminished sevenths; diminished sevenths invert to augmented seconds

There are other possibilities, but they are very infrequent in common practice and not useful to consider here. Some examples are given below (Example 00), showing the counting of semitones. (If you compare this to Example 00, above, you can see the same principle at work.)

Interval spelling

It is important to be consistent when spelling intervals as regards both quantity and quality. By now, you may have noticed in some of the examples above that it would be possible to spell intervals in more than one way using enharmonic equivalents. In some cases, you might even think that using enharmonic notation is expedient because it sometimes seems simpler.  For example, why spell an augmented sixth above G as E when you could use it enharmonic equivalent, F, seemingly a more straightforward and familiar notation?  Both involve ten semitones, after all, and, if you play them on a piano, they use the same keys and sound the same.

The reasons will not be obvious now, but being aware that each interval has a unique and correct way of spelling it is important, and you should avoid making short cuts and learn the proper spelling according to quantity (number) and quality.  Example 00 gives some instances of enharmonically equivalent intervals that we often encounter.  Which one of each pair is “correct” will depend on rules and contexts that will be explained in other parts of this book. Note that each pair is aurally indistinguishable when played on a piano or keyboard.

Notating harmonic intervals with more than one accidental

Just as we avoid a “collision” of note heads when we write harmonic seconds (by placing the lower note head to the left), we also need to avoid colliding accidentals that might be applied to an interval There are some basic rules to follow:

For intervals less than a seventh, place the lower accidental sufficiently to the left that it would sit in its own virtual column.  If the accidentals apply to notes separated by a seventh or more, they should be in the same column.  Here are some examples of what and what not to do.

Intervals and Diatonic Scales

In discussing major and minor scales, we consider intervals already quite extensively in as much as we talk about the tones and semitones that separate adjacent degrees.  These steps in the diatonic scale are intervals—quantitatively, they are all seconds, and qualitatively:

• tones are major seconds
• semitones are minor seconds

We can also give a name to the tone+semitone (T+S) interval between the submediant () and leading note () in the harmonic minor scale—it is an augmented second.

We can also consider how other intervals relate to the diatonic major and minor scales.

Thirds and the diatonic scale

If we skip every other note in a scale, then we progress in thirds.  As is explained elsewhere, the third is a very important interval in common-practice tonality.  In fact, the style of harmony (chords) that is used in common-practice music is called “tertian harmony” (harmony of thirds).

If we take the major and natural minor scale, we will find that counting in thirds produces either major thirds (T–T progressions) or minor thirds (T–S or S–T progressions).  This is shown below using two octaves of (a) the major scale on D and (b) the natural minor scale on F (see Example 00). We use two octaves because this allows to a complete a full rotation counting in thirds.  When we introduce the harmonic minor, the pattern of major and minor thirds is changed slightly because of the swapping out of the subtonic () for the leading note (), as shown in (c) the harmonic minor scale on E.

Because thirds are so important in the organisation of common-practice music, it is worth practicing them in various ways.  For example, a “two steps forwards one step back” approach using sol-fa syllables.  This can be on any tonic pitch for the major (starting of do) or minor (starting on la), going up and down.

Fifths and the diatonic scale

The diatonic collection will mainly create perfect fifths.  There is one exception to this.  This can be shown by taking a Major scale and constructing a fifth (from other notes of the scale) above each degree.  It will be found that all degrees except the leading note () have a perfect fifth above them.  However, when we construct a fifth above the leading note the result is a diminished fifth (–), or ti–fa.

While the perfect fifths have a rather stable, or neutral sound, the effect of the diminished fifth is very different—it sounds quite unstable and active.  This is because, unlike the very simple ratio of fundamental frequencies for the perfect fifth, the diminished fifth has a much more complex frequency ratio and produces a harsher effect when its notes are heard in combination with each other.

Another way of understanding this is to place the degrees that form this interval (ti and fa) in the context of the degrees that they are closest to—do and mi, respectively.  Both pairs of notes lie a semitone apart: ti–do forms a semitone, as does fa–mi.  Both fa and mi are, in the context of tonality, tendency notes—they have a strong tendency to move toward do and mi.  Because the interval between do–mi (a major third) sounds more stable than the diminished fifth, it gives the effect of a satisfactory destination if we play the two intervals (ti–fa and do–mi) in succession.

Also, when we invert both intervals—making the diminished fifth, ti–fa, into the augmented fourth, fa–ti, and the major third, do–mi, into the minor sixth, mi–do, and play this succession of intervals, the effect is similar—a highly unstable sounding interval proceeding to a more stable sounding one. Example 00 shows this in the key of D major.

The interval involving the leading note (ti, or ) and subdominant (fa, or ) of the major scale, which ever way we invert it (as diminished fifth or augmented) has a special role in tonality.  It is often referred to as the tritone, because as an augmented-fourth, we literally progress through three tones (fa–so–la–ti) to create it.  When we have interval successions as shown in Example 00, we call this “resolving” the tritone.

Tritones and other augmented/diminished intervals in the harmonic minor.

In the natural minor scale, the tritone ti–fa now lies between the supertonic () and submediant ().  However, when we use the harmonic minor scale, we create a second tritone, between the leading note, si (), and the subdominant, re ().

Additionally, we create the diminished seventh between the si and fa and, by inversion, the augmented second between fa and si.

All these can be resolved to more stable intervals in the context of the harmonic minor as shown below (with reference to G minor, for Example 00).  Note that, sonically, neither the diminished seventh or augmented second sound as harsh as the tritone when heard out of context—enharmonically, can sound as a major sixth or minor third, respectively.  However, in the context of the harmonic minor scale (and minor keys), the leading note (si) and submediant (fa) express a strong tendency to move (aka resolve) to the tonic (la) and dominant (mi).

Consonance and dissonance.

So far we have “measured” intervals according to both quantity (numerical size) and quality (perfect, major, minor, diminished augmented).  However, as hinted at in the discussion of the tritone in the previous section, there is another “qualitative” aspect to intervals which is somewhat less abstract, and this is how we experience them.

These terms are subjective and depend on a range of factors, including differences in musical style, individual perception, social consensus and so on.  In the context of Western common-practice music, however, we can start to narrow down the range of subjectivity and, in general, make broad categorisations of intervals in relation to whether they are “consonant” or “dissonant.”

• Consonant intervals tend to sound stable, at rest, “pleasing”
• Dissonant intervals tend to sound unstable, active, “harsh”

However, two things should be kept in mind:

1. As mentioned above, these are subjective terms. A “minor sixth,” for instance, can only ever be a minor sixth (an interval with notes separated by six lines and spaces and eight semitones).  Whether you, or I, experience it as, say, “stable” or “unstable,” will depend on any number of subjective factors. And this relates to the second point…
2. Consonance and dissonance should not be regarded as binary opposites. They are better thought of as different sides of a continuum.

Consonant intervals are traditionally divided into two categories: perfect consonances and imperfect consonances.

• Perfect consonances are those already described above as perfect—perfect unisons, fourths, fifths and octaves.
• Imperfect consonances are the major and minor thirds and sixths

Keeping in mind the subjective nature of this whole topic, as Western music evolved from the over centuries from the Middle Ages to the late renaissance period, the idea of the “pleasingness” of consonant intervals gradually shifted from favouring the perfect to the imperfect consonances.  While certainly stable, the perfect consonances are also often perceived as somewhat bland, or empty.  The imperfect consonances have a characteristically richer, colourful effect—due to the more complex relationships of there frequency ratios.

Dissonant intervals comprise all seconds and sevenths, as well as the tritone.

Also, within the category of dissonant intervals, there are a range of subjective levels of “harshness.”  Typically, minor seconds (semitones) are heard as harsher in effect than major seconds (tones), and likewise there inversions —major seventh (harsher) and minor seventh (less harsh).  The tritone might lie somewhere between.

Referring back to the idea of interval perception as a kind of continuum, we might try to represent this as follows, along with some subjective descriptions (Example 00).