Fundamentals

# 16 Intervals

Chelsey Hamm and Bryn Hughes

Key Takeaways

- Two pitches form an interval, which is usually defined as the distance between two notes.
- Melodic intervals are played or sung separately, while harmonic intervals are played or sung together.
- Every interval has a size and a quality. An interval’s size is the distance between two notes on a staff—i.e. it is a measurement of the number of lines and spaces between two notes.
- Size is considered generic. In other words, it doesn’t matter what accidentals you apply to the notes, the size is always the same.
- A quality makes an interval specific when used in combination with a size. Unisons, fourths, fifths, and octaves form perfect intervals, while seconds, thirds, sixths, and sevenths form major and minor intervals.
- Any interval can be augmented or diminished. Augmented intervals are one half-step larger than a perfect or major interval. Diminished intervals are one half-step smaller than a perfect or minor interval.
- Intervals between a unison and an octave are called simple intervals . Any interval larger than an octave is a compound interval.
- Intervallic inversion occurs when two notes are “flipped.” Inverting intervals can be useful when you do not want to work in the key signature of the note that is the original lower note.
- Consonant intervals are intervals that are considered more stable, as if they do not need to resolve, while dissonant intervals are considered less stable, as if they do need to resolve.

Two pitches form an interval, which is usually defined as the distance between two notes. But what does an interval measure? Physical distance, on the staff? Difference in wavelength between pitches? Something else? Music theorists have had contradictory ideas on the definition of “interval,” and these definitions have varied greatly with milieu. This chapter will focus on intervals as a measure of two things: written distance between two notes on a staff, and an aural “distance” (or space) between two sounding pitches. It will be important to keep in mind at all times that intervals are both written and aural, so that you are thinking of them musically (and not simply as an abstract concept that you are writing and reading).

# Size

You might encounter melodic intervals or harmonic intervals. Melodic intervals are played or sung separately, while harmonic intervals are played or sung together. shows the difference:

As you can see and hear in

, the notes in the first measure sound together (harmonically), while in the second measure they sound separately (melodically).Every interval has a size and a quality. A size is the distance between two notes on a staff—i.e. it is a measurement of the number of lines and spaces between two notes. Sizes are written with Arabic numbers (2, 3, 4, etc.); however, they are spoken with ordinal numbers (second, third, fourth, fifth, sixth, seventh, etc.). Always count a note to itself as one when counting size. shows the first 8 sizes within an C major scale:

As you can see in

, a note to itself is not said to be a “first;” instead, it is a unison. Likewise, notes eight lines and spaces apart are not said to be an “eighth” but instead they are an “octave.”Size is considered generic. In other words, it doesn’t matter what accidentals you apply to the notes, the size is always the same. demonstrates this:

As you can see in

, each of these intervals is a third because there are three lines/spaces between the two notes. Accidentals do not matter in the determination of generic size. We would say that each of these intervals is a “generic third.”# Perfect, Major, and Minor Qualities

A quality makes an interval specific when used in combination with a size. Quality more precisely measures written distance between notes, and—in combination with an interval’s size—it describes the aural sound of an interval.

There are five possible interval qualities:

- Augmented (designated as A or +)
- Major (M)
- Perfect (P)
- Minor (m)
- Diminished (d or
^{o})

When speaking about or writing intervals, one says or writes the quality first and then the size. For example, an interval could be described as a “perfect fourth” (abbreviated P4), a “minor third” (abbreviated m3), or an “augmented second” (abbreviated +2 or A2).

For now, we will only discuss three qualities: perfect, major, and minor. Different theorists (in different locations and time periods) have applied these qualities to different sizes of intervals, depending on milieu. shows how these qualities are applied today:

[table id=61 /]

As you can see in

, unisons, fourths, fifths, and octaves are perfect intervals (shown in the right column), while seconds, thirds, sixths, and sevenths are major and/or minor (shown in the left column).# The “Major Scale” Method for Determining Quality

There are several different methods for learning to write and identify qualities of intervals. One method you may have heard of is counting half-steps. This is not a recommended method, because it is time consuming and often inaccurate. Instead, we recommend the “Major Scale” method.

To identify an interval (size and quality) using this method, complete the following steps:

- Determine size (by counting lines and spaces between the notes).
- Determine if the top note is in the major scale of the bottom note.
- If it is: the interval is perfect (if it is a unison, fourth, fifth, or octave) or it is major (if it is a second, third, sixth, or seventh). If it is not: then, for now, the interval is minor (a lowered second, third, sixth, or seventh).

shows two intervals. Try identifying their size and quality:

For the first interval: the notes are F and C in treble clef. Here is the process in more detail:

- First, this interval is a generic fifth (F to itself is 1; to G is 2; to A is 3; to B is 4; to C is 5).
- Second, C is within the key of F major (which has one flat, B♭).
- The interval is a perfect fifth because fifths are perfect (not major/minor), and the notes are unaltered by accidentals.

Let’s now use this process for the second example. The notes in this example are E♭ and C♭ in treble clef. Let’s go through the process in more detail:

- First, this interval is a generic sixth (E♭ to itself is 1; to F is 2; to G is 3; to A is 4; to B is 5; to C is 6).
- Second, C♭ is NOT in the key of E♭ major (which has three flats, B♭, E♭, and A♭).
- Therefore, this is a minor sixth. If it were a major sixth, then the C would have to be C♮ instead of C♭, because C♮ is in the key of E♭ major.

# Augmented and Diminished Qualities

To review, there are five possible interval qualities, of which we have covered major, minor, and perfect:

- Augmented (designated as A or +)
- Major (M)
- Perfect (P)
- Minor (m)
- Diminished (d or
^{o})

Augmented intervals are one half-step larger than a perfect or major interval. shows this:

As you can see in the first measure of

, the notes F and C form a perfect fifth (because C is in the key of F major). The top note of that interval has been raised by a half-step to a C♯, and so is one half-step larger; consequently, the interval F to C♯ is an augmented fifth (abbreviated either A5 or +5). In the second measure of , a major sixth is shown with the notes G and E (because E is in the key of G major). The top note of that interval has been raised by a half-step to E♯, and so the interval is one half-step larger and is now an augmented sixth.Note that it is not always the top note that is altered.

shows two augmented intervals in which the bottom notes have been altered:

In the first measure of

, F and C again form a perfect fifth. However, the bottom note has now been lowered by a half-step to an F♭, creating an augmented fifth (because the interval is one half-step larger than a perfect fifth). In the second measure of , G and E once again form a major sixth. The bottom note, G, has been lowered a half-step to G♭, creating an augmented sixth because the interval is now one half-step larger than a major sixth.Diminished intervals are one half-step smaller than a perfect or minor interval. shows this:

In the first measure of ^{o}5). In the second measure of , G and E form a major sixth which becomes a minor sixth when the top note is lowered by a half-step (making the entire interval one half-step smaller). The minor sixth then becomes a diminished sixth when it is again contracted by a half-step from G to E𝄫.

It is very important to note that major intervals do not become diminished intervals directly; a major interval becomes minor when contracted by a half-step. It is only a minor interval that becomes diminished when further contracted by a half-step.

Again, it is not always the top note that is altered.

shows two diminished intervals in which the bottom notes have been altered:

In the first measure of

, F to C form a perfect fifth. This interval becomes diminished when it is made a half-step smaller by the bottom note moving up a half-step from F to F♯. In the second measure of , G to E form a major sixth. This interval is made into a minor sixth when the G moves up a half-step to G♯, making the interval a half-step smaller (or contracted). Furthermore, this minor interval becomes diminished when the G moves to G♯♯, making the minor interval a further half-step contracted.and again demonstrate and summarize the relative size of intervals. Each bracket in these examples is one half-step larger or smaller than the brackets to their right and left. shows intervals with the top note altered by accidentals:

As you can see in

, intervals one half-step larger than perfect intervals are augmented, while intervals one half-step smaller than perfect intervals are diminished. Likewise, in , intervals one half-step larger than major intervals are augmented, while intervals one half-step smaller than major are minor and intervals one half-step smaller than minor are diminished.shows intervals with the bottom note altered by accidentals:

outlines the same qualities as ; the only difference between the examples is which note is altered by accidentals. In it is the top note, while in it is the bottom note.

# Doubly and Triply Augmented and Diminished Intervals

Intervals can be further contracted or expanded outside of the augmented and diminished qualities. An interval a half-step larger than an augmented interval is a doubly augmented interval, while an interval a half-step larger than a doubly augmented interval is a triply augmented interval.

An interval a half-step smaller than a diminished interval is a doubly diminished interval, while an interval a half-step smaller than a doubly diminished interval is a triply diminished interval.

# Compound Intervals

The intervals discussed above, from unison to octave, are simple intervals , which have a size an octave or smaller. Any interval larger than an octave is a compound interval. shows the notes A and C, first as a simple interval and then as a compound interval:

The notes A to C form a minor third in the first pair of notes; in the second pair of notes, the C has been brought up an octave. Quality remains the same for simple and compound intervals, which is why a minor third and minor tenth both have the same quality.

If you want to make a simple interval a compound interval, add 7 to its size. Consequently:

- Unisons (which get the number “1”) become octaves (“8s”)
- 2nds become 9ths
- 3rds become 10ths
- 4ths become 11ths
- 5ths become 12ths

These are the most common compound intervals that you will encounter in your music studies. Remember that octaves, 11ths, and 12ths are perfect like their simple counterparts, while 9ths and 10ths are major/minor.

# Interval Inversion

Intervallic inversion occurs when two notes are “flipped.” For example, the notes C (on bottom) with E (above) is an inversion of E (on bottom) with C (above), as can be seen in :

You might be wondering: why is this important? There are two reasons: first, because inverted pairs of notes share many interesting properties (which are sometimes exploited by composers), and second, because inverting a pair of notes can help you to identify or write an interval when you do not want to work from the given bottom note.

Let’s start with the first point: the interesting properties. First, the size of inverted pairs always add up to 9:

- Unisons (“1s”) invert to octaves (“8s”) (1 + 8 = 9) and octaves invert to unisons.
- Seconds invert to sevenths (2 + 7 = 9) and sevenths invert to seconds.
- Thirds invert to sixths (3 + 6 = 9) and sixths invert to thirds.
- Fourths invert to fifths (4 + 5 = 9) and fifths invert to fourths.

Qualities of inverted pairs of notes are also very consistent:

- Perfect intervals invert to perfect intervals.
- Major intervals invert to minor intervals (and minor intervals to major intervals).
- Augmented intervals invert to diminished intervals (and diminished intervals to augmented intervals).

With that information you can now calculate the inversions of intervals without even looking at staff paper. For example; a major seventh inverts to a minor second; an augmented sixth inverts to a diminished third; and a perfect fourth inverts to a perfect fifth.

Now for the second point: sometimes you will come across an interval that you do not want to calculate or identify from the bottom note.

shows one such instance of this:

The bottom note is E𝄫, and there is no key signature for this note (its key signature is “imaginary”). So, if you were given this interval to identify you might consider inverting the interval, as shown in

:

Now the inversion of the interval can be calculated from the non-imaginary key of A♭ major. The key of A♭ major has four flats (B, E, A, and D flat). An E♭ above A♭ would therefore be a perfect fifth; however, this interval has been contracted (made a half-step smaller) because the E♭ has been lowered to E𝄫. That means this interval is a d5 (diminished fifth).

Now that we know the inversion of the first interval is a d5, we can calculate the original interval from this inversion. A diminished fifth inverts to an augmented fourth (because diminished intervals invert to augmented intervals and because five plus four equals nine). Thus, the first interval is an augmented fourth (A4).

# Consonance and Dissonance

Intervals are categorized as consonant or dissonant . Consonant intervals are intervals that are considered more stable, as if they do not need to resolve, while dissonant intervals are considered less stable, as if they do need to resolve. These categorizations have varied with milieu. shows a table of melodically consonant and dissonant intervals:

[table id=63 /]

Melodically consonant and dissonant intervals.

shows harmonically consonant and dissonant intervals:

[table id=64 /]

# Another Method for Intervals: the White-Key Method

Ultimately, intervals need to be committed to memory, both aurally and visually. There are, however, a few tricks to learning how to do this quickly. One such trick is the so-called “white-key method,” which refers to the piano keyboard.

This method requires you to memorize all of the intervals found between the white keys on the piano (or simply all of the intervals in the key of C major). Once you’ve learned these, any interval can be calculated as an alteration of a white-key interval. For example, we can figure out the interval for the notes D and F♯ if we know that the interval D to F is a minor third, and this interval has been made one semitone larger: a major third.

Conveniently, there is a lot of repetition of interval size and quality among white-key intervals. Memorize the most frequent type, and the exceptions.

All of the seconds are major except for two: E and F, and B and C, which are minor*,* as seen in :

All of the thirds are minor except for three: C and E, F and A, and G and B, which are major, as shown in

:

All of the fourths are perfect except for one: F and B, which is augmented, as seen in

:

Believe it or not, you now know all of the white-key intervals, as long as you understand the concept of intervallic inversion, which was previously explained. For example, if you know that all seconds are major except for E and F and B and C (which are minor), then you know that all sevenths are minor except for F and E and C and B (which are major), as seen in

:

Once you’ve mastered the white-key intervals, you can figure out any other interval by taking into account the interval’s accidental or accidentals.

# Intervallic Enharmonic Equivalence

may be useful when thinking about enharmonic equivalence of intervals:

unis. | 2nd | 3rd | 4th | 5th | 6th | 7th | oct. | |
---|---|---|---|---|---|---|---|---|

0 | P1 | d2 | ||||||

1 | A1 | m2 | ||||||

2 | M2 | d3 | ||||||

3 | A2 | m3 | ||||||

4 | M3 | d4 | ||||||

5 | A3 | P4 | ||||||

6 | A4 | d5 | ||||||

7 | P5 | d6 | ||||||

8 | A5 | m6 | ||||||

9 | M6 | d7 | ||||||

10 | A6 | m7 | ||||||

11 | M7 | d8 | ||||||

12 | A7 | P8 |

In this chart, the columns are different intervallic sizes, while the rows present intervals based on the number of half-steps they contain. Each row in this chart is enharmonically equivalent. For example, a M2 and d3 are enharmonically equivalent (both are 2 half-steps). Likewise, an A4 and d5 are enharmonically equivalent—both are six half-steps in size.

Intervallic enharmonic equivalence is useful when you come across an interval that you do not want to calculate or identify from the bottom note. We have already discussed one method for this situation previously, which was intervallic inversion. You may prefer one method or the other, though both will yield the same result.

reproduces the interval from :

As you’ll recall, there is no key signature for the bottom note (E𝄫), making identification of this interval difficult. By using enharmonic equivalence, however, we can make identification of this interval easier. We can recognize that E𝄫 is enharmonically equivalent with D and that A♭ is enharmonically equivalent with G♯, as shown in

:

Now we can identify the interval as an A5 (augmented fourth), using the key signature of the enharmonically equivalent bottom note (D).

- Specific Intervals (musictheory.net)
- Interval Introduction (Robert Hutchinson)
- Intervals (Hello Music Theory)
- Diminished and Augmented Intervals (Open Textbooks)
- Diminished and Augmented Intervals (Robert Hutchinson)
- Compound Intervals (Hello Music Theory)
- Interval Inversion (musictheory.net)
- Interval Ear Training (musictheory.net)
- Interval Ear Training (Tone Dear)
- Interval Ear Training (teoria)
- Interval Identification (musictheory.net)
- Keyboard Interval Identification (musictheory.net)

- Interval Identification (.pdf, .pdf, .pdf), in Major Keys (.pdf), in Minor Keys (.pdf)
- Interval Identification and Construction, pp. 18–19 (.pdf)
- Interval Construction (.pdf, .pdf, .pdf)
- Compound Intervals, pp. 15–17 (.pdf)

- Writing and Identifying Intervals Assignment #1 (.pdf, .mcsz)
- Writing and Identifying Intervals Assignment #2 (.pdf, .mcsz)
- Writing and Identifying Intervals Assignment #3 (.pdf, .mcsz)

The distance between two notes

The interval is played or sung separately (one note after another)

The interval is played or sung together (both notes at the same time)

Interval size is written with Arabic numbers (2, 3, 4, etc.); it is the distance between two notes on a staff

The number of scale steps between notes of a collection or scale

When applied to an interval, the term "quality" modifies the size descriptor in order to be more specific about the exact number of semitones in the interval

When applied to triadic harmony, "quality" refers to the size of the different intervals that make up the harmony

A combination of size and quality to describe an interval

Intervals that are one half-step larger than a perfect or major interval

Intervals that are one half-step smaller than a perfect or minor interval

Intervals with a size an octave or smaller

An interval that is larger than an octave

Occurs when two notes (such as C and E) are flipped; C (on bottom) with E (above) is an inversion of E (on bottom) with C (above)

A quality in an interval or chord that, in a traditional tonal context, is stable; this stability is the result of its perceived independence from a need to resolve

A quality in an interval or chord that, in a traditional tonal context, is unstable; this instability is the result of its perceived dependence on a need to resolve

A physical and/or social setting

An interval a half-step larger than an augmented interval

An interval a half-step larger than a doubly augmented interval

An interval a half-step smaller than a diminished interval

An interval a half-step smaller than a doubly diminished interval