I. Fundamentals

Intervals

Chelsey Hamm and Bryn Hughes

Key Takeaways

  • Two pitches form an , which is usually defined as the distance between two notes.
  • are played or sung separately, while are played or sung together.
  • Every interval has a size and a quality. An interval’s 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 . In other words, it doesn’t matter what accidentals you apply to the notes, the size is always the same.
  • A makes an interval 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. are one half-step larger than a perfect or major interval. are one half-step smaller than a perfect or minor interval.
  • Intervals between a unison and an octave are called . Any interval larger than an octave is a .
  • 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.
  • intervals are intervals that are considered more stable, as if they do not need to resolve, while intervals are considered less stable, as if they do need to resolve.

Two pitches form an , 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 . 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 or . Melodic intervals are played or sung separately, while harmonic intervals are played or sung together. Example 1 shows the difference:

Example 1. A harmonic and a melodic interval.

As you can see and hear in Example 1, 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 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. Example 2 shows the first 8 sizes within an C major scale:

Example 2. Sizes of intervals.

As you can see in Example 2, 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 . In other words, it doesn’t matter what accidentals you apply to the notes, the size is always the same. Example 3 demonstrates this:

Example 3. Generic size is demonstrated.

As you can see in Example 3, 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 makes an interval 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 . Example 4 shows how these qualities are applied today:

Major/Minor Perfect
2nds Unisons
3rds 4ths
6ths 5ths
7ths Octaves

Example 4. Interval qualities.

As you can see in Example 4, 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:

  1. Determine size (by counting lines and spaces between the notes).
  2. Determine if the top note is in the major scale of the bottom note.
  3. 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).

Example 5 shows two intervals. Try identifying their size and quality:

Example 5. Two intervals.

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

  1. 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).
  2. Second, C is within the key of F major (which has one flat, B♭).
  3. 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:

  1. 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).
  2. Second, C♭ is NOT in the key of E♭ major (which has three flats, B♭, E♭, and A♭).
  3. 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)

are one half-step larger than a perfect or major interval. Example 6 shows this:

Example 6. Two augmented intervals.

As you can see in the first measure of Example 6, 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 Example 6, 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. Example 7 shows two augmented intervals in which the bottom notes have been altered:

Example 7. Two more augmented intervals.

In the first measure of Example 7, 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 Example 7, 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.

are one half-step smaller than a perfect or minor interval. Example 8 shows this:

Example 8. Diminished Intervals.

In the first measure of Example 8, the perfect fifth F and C has been made a half-step smaller, since the top note has been lowered by a half-step. Consequently, F to C♭ is a diminished fifth (abbreviated usually as a d5 or o5). In the second measure of Example 8, 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. Example 9 shows two diminished intervals in which the bottom notes have been altered:

Example 9. Diminished intervals with the bottom notes altered.

In the first measure of Example 9, 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 Example 9, 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.

Examples 10 and 11 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. Example 10 shows intervals with the top note altered by accidentals:

Example 10. Relative size of intervals with top note altered.

As you can see in Example 10, intervals one half-step larger than perfect intervals are augmented, while intervals one half-step smaller than perfect intervals are diminished. Likewise, in Example 10, 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.

Example 11 shows intervals with the bottom note altered by accidentals:

Example 11. Relative size of intervals with bottom note altered.

Example 11 outlines the same qualities as Example 10; the only difference between the examples is which note is altered by accidentals. In Example 10 it is the top note, while in Example 11 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 , while an interval a half-step larger than a doubly augmented interval is a .

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

Compound Intervals

The intervals discussed above, from unison to octave, are , which have a size an octave or smaller. Any interval larger than an octave is a . Example 12 shows the notes A and C, first as a simple interval and then as a compound interval:

Example 12. A simple and 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

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 Example 13:

Example 13. Intervallic inversion.

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. Example 14 shows one such instance of this:

Example 14. An interval in which the key of the bottom note is imaginary.

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 Example 15:

Example 15. The interval from Example 14 has been inverted.

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 or . 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 . Example 16 shows a table of melodically consonant and dissonant intervals:

Melodically Consonant Melodically Dissonant
Perfect Intervals Augmented Intervals
M2, m2 Diminished Intervals
M3, m3 M7, m7
M6, m6

Example 16. Melodically consonant and dissonant intervals.

Example 17 shows harmonically consonant and dissonant intervals:

Harmonically Consonant Harmonically Dissonant
M3, m3 M2, m2
M6, m6 Augmented Intervals
PU, P8, P5 Diminished Intervals
M7, m7
P4

Example 17. Harmonically consonant and dissonant intervals.

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 Example 18:

Example 18. White-key seconds.

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 Example 19:

Example 19. White-key thirds.

All of the fourths are perfect except for one: F and B, which is augmented, as seen in Example 20:


Example 20. White-key fourths.

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 Example 21:

Example 21. White-key sevenths.

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

Example 22 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

Example 22. Enharmonic equivalence of intervals.

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. Example 23 reproduces the interval from Example 14:

Example 23. An interval in which the key of the bottom note is imaginary.

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 Example 24:


Example 24. An interval that is enharmonically equivalent to Example 23.

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

Online Resources
Assignments from the Internet
  1. Interval Identification (.pdf.pdf.pdf), in Major Keys (.pdf), in Minor Keys (.pdf)
  2. Interval Identification and Construction, pp. 18–19 (.pdf)
  3. Interval Construction (.pdf, .pdf, .pdf)
  4. Compound Intervals, pp. 15–17 (.pdf)
Assignments
  1. Writing and Identifying Intervals Assignment #1 (.pdf, .mcsz)
  2. Writing and Identifying Intervals Assignment #2 (.pdf, .mcsz)
  3. Writing and Identifying Intervals Assignment #3 (.pdf, .mcsz)

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OPEN MUSIC THEORY by Chelsey Hamm and Bryn Hughes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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