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 notes separate, while are played or sung notes together.
  • Every interval has a size and a quality. A is the physical measurement between two notes on a staff–i.e. it is a measurement of the number of lines and spaces between two notes. Always be sure to count each line and each space, including the line or space that the first note is on.
  • 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–those which are perfect and those which are major/minor–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 with the sizes of a unison through an octave are . 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,” which has 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 notes separate, while harmonic intervals are played or sung notes together. Example 1 shows the difference:

Two notes are shown in treble clef, first played together (or as a harmonic interval), and second played separately (or as a melodic interval)
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 separate (melodically).

Every interval has a size and a quality. A is the physical measurement between two notes on a staff–i.e. it is a measurement of the number of lines and spaces between two notes (be sure to count each line and each space, including the line or space that the first note is on). 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 F major scale:

A unison, second, third, fourth, fifth, sixth, seventh, and octave are shown (all harmonic) in G major in bass clef
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:

Many thirds are shown in G major (G to B), with different accidentals applied to the different notes. No matter the accidentals, each of the intervals shown is still a "generic third"
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 not only more precisely measure written distance between notes, but it also–in combination with an interval’s size–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 talking out loud about intervals, one says the quality first and then the size. For example, an interval could be described as a “perfect fourth,” a “minor third,” or an “augmented second.”

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:

***Example 4 is under construction and will be added ASAP***

As you can see in Example 4, unisons, fourths, fifths, and octaves are perfect intervals, while seconds, thirds, sixths, and sevenths are major and/or minor. You will under NO circumstances want to cross the impenetrable barrier of fire and death. For example, never say or write “major fourth” or “perfect sixth.”

The “Major Scale” Method for Determining Quality

There are several different methods for learning to write and identify the qualities of intervals. One method you may have heard of is counting half-steps. This is NOT a good method and Open Music Theory does not recommend utilizing it. 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)
  4. If it is not: then, for now, the interval is minor (a lowered second, third, sixth, or seventh)

Example 5 shows two intervals:

F and C are shown in treble clef in the first example; in the second example, e-flat and c-flat are shown in treble clef
Example 5: Two intervals

Let’s use this process for each of these examples in turn. 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-flat)
  3. The interval is a perfect fifth, because fifths are perfect

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

  1. First, this interval is a generic sixth (E-flat 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, Cb is NOT in the key of E-flat major (which has three flats, B-flat, E-flat, and A-flat)
  3. Therefore, this is a minor sixth. If it were a major sixth, then the C would have to be C-natural instead of C-flat, because C-natural is in the key of E-flat 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)

Any interval–those which are perfect and those which are major/minor–can be augmented or diminished. are one half-step larger than a perfect or major interval. Example 6 shows this:

A perfect fifth, F to C is shown, then an augmented fifth, F to C# is shown. Likewise, a major sixth, G to E is shown, and then an augmented sixth, G to E sharp is shown. All of the examples are in treble clef.
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:

Two augmented intervals are shown with their counterparts that are unaltered. F and C become F flat and C; G and E become G flat and E
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:

Two diminished intervals are shown; F and C become F and C flat, while G and E (a major sixth) first become G and E flat, a minor sixth, and then become G and E double flat, a diminished sixth
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:

Two diminished intervals are shown with the bottom notes altered. F to C, a perfect fifth, becomes diminished when the F moves to F sharp. G to E, a major sixth, becomes minor when the bottom note moves from G to G sharp, and then becomes diminished when the bottom note moves from G sharp to G double sharp.
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 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 a conceptualization of intervals with the top note altered by accidentals:

***Example 10 is under construction and will be added ASAP***

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 a conceptualization of intervals with the bottom note altered by accidentals:

***Example 11 is under construction and will be added ASAP***

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 triply augmented interval..

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:

The notes A and C are shown, first as a simple interval (a minor third) and then as a compound interval (a minor tenth)
Example 12: A simple and compound interval

The notes A to C form a minor third (C♯ is in the major key of A, and C is one half-step lower than C♯, making the interval one half-step contracted). 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 m3 and m10 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, C (on bottom) with E (above) is an inversion of E (on bottom) with C (above), as can be seen in Example 13:

A pair of notes, C and E (c on bottom) is flipped (E on bottom) in bass clef
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 of notes 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.

The opposite of each of these points is also true; for example, if seconds invert to sevenths, then sevenths invert to seconds.

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 the interval for which you do not want to calculate or identify the interval from the bottom note. Example 14 shows one such instance of this:

An interval in which the key of the bottom note (e double flat) is imaginary; the top note is A flat, and both notes are written in bass clef
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:

The interval from the previous example has been inverted; now the A flat is on bottom and the E double flat is on top
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-flat major. The key of A-flat 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 . The following intervals are considered melodically consonant:

  • All perfect intervals (P4, P5, P8)
  • All diatonic steps (M2, m2)
  • Major and minor thirds
  • Major and minor sixths

While these intervals are considered melodically dissonant:

  • All augmented and diminished intervals (including those that are enharmonically equivalent to consonant intervals)
  • All sevenths

The following intervals are considered harmonically consonant:

  • Major and minor thirds
  • Major and minor sixths
  • All perfect intervals except the perfect fourth

While all other harmonic intervals are dissonant, including:

  • All diatonic steps (M2, m2)
  • All augmented and diminished intervals (including those that are enharmonically equivalent to consonant intervals)
  • All sevenths
  • Perfect fourths

Another Method for Learning 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 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 D4-F#4 if we know that the interval D4-F4 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-F, and B-C, which are minor, as seen in Example 16:

All of the white keys seconds in C major are shown. E/F and B/C are boxed because these are minor
Example 16: White key seconds

All of the thirds are minor except for three: C-E, F-A, and G-B, which are major, as shown in Example 17:

All white keys thirds are shown with boxes around C/E, F/A, and G/B because they are minor
Example 17: White key thirds

All of the fourths are perfect except for one: F-B, which is augmented, as seen in Example 18:

All white key fourths are shown with a box around the F/B dyad because it is augmented
Example 18: 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-F and B-C (which are minor), then you know that all sevenths are minor except for F-E and C-B (which are major), as seen in Example 19:

All white key sevenths are shown with boxes around F/E and C/B, because they are major
Example 19: 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

The following chart 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.

Online Resources
Assignments from the Internet
  1. Interval Identification (.pdf)
  2. Interval Identification in Major Keys (.pdf)
  3. Interval Identification in Minor Keys (.pdf)
  4. Interval Identification (.pdf)
  5. Interval Identification and Construction (pp. 18–19) (.pdf)
  6. Interval Identification (.pdf)
  7. Interval Construction (.pdf)
  8. Interval Construction (.pdf)
  9. Interval Construction (.pdf)
  10. Compound Intervals (pp. 15–17) (.pdf)
Assignments
  1. Writing and Identifying Intervals # 1 (.pdf)
  2. Writing and Identifying Intervals # 2 (.pdf)
  3. Writing and Identifying Intervals # 3 (.pdf)
  4. Interval Project (.pdf)

Media Attributions

  • Harmonic and Melodic Interval © Chelsey Hamm is licensed under a Public Domain license
  • Intervallic Size © Chelsey Hamm is licensed under a Public Domain license
  • Generic Size © Chelsey Hamm is licensed under a Public Domain license
  • Two Intervals © Chelsey Hamm is licensed under a Public Domain license
  • Two Augmented Intervals © Chelsey Hamm is licensed under a Public Domain license
  • Augmented Intervals Bottom Notes © Chelsey Hamm is licensed under a Public Domain license
  • Diminished Intervals © Chelsey Hamm is licensed under a Public Domain license
  • Diminished Intervals Bottom Note © Chelsey Hamm is licensed under a Public Domain license
  • Simple and Compound Intervals © Chelsey Hamm is licensed under a Public Domain license
  • Intervallic Inversion © Chelsey Hamm is licensed under a Public Domain license
  • Bottom Note is E double flat © Chelsey Hamm is licensed under a Public Domain license
  • Inverted Interval © Chelsey Hamm is licensed under a Public Domain license

License

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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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