Intervals

Image Description. An employee of the US Bureau of Land Management in Oregon using a theodolite to survey land. Like surveyors, music theorists measure distances between pitches to help analyze music.
(US Bureau of Land Management | CC BY-2.0 Domain)

When analyzing melodies and chords in 12TET, music theorists often find it useful to measure the distance between two pitches by using scale degrees or half steps rather than basic frequencies.

Diatonic Intervals

The distance between two pitches is called an interval. Music theorists will sometimes differentiate between melodic intervals, in which notes are played consecutively, and harmonic intervals, which describes two notes played at the same time.

The notes E4 and C5, shown in sequence and then as a single chord.
Figure 1: A melodic interval and a harmonic interval.

The most basic type of interval is a diatonic interval, which measures scale degrees without regard for accidentals. Diatonic intervals describe the interval distance — the total number of scale degrees or staff positions the two notes encompass — and are described using ordinal numbers.

Scale Degrees Spanned Interval
1 unison
2 second
3 third
4 fourth
5 fifth
6 sixth
7 seventh
8 octave
Five harmonic intervals: a third from E4 to G4, a seventh from C4 to B flat 4, a fifth from G sharp 4 to D5, an octave from F sharp 4 to F natural 5, and a second from B4 to C flat5.
Figure 2: Five different intervals. In each case, the diatonic interval describes the number of diatonic steps included in the interval; accidentals are not included in these measurements.

Compound Intervals

Intervals larger than an octave are called compound intervals and share many aural characteristics with the corresponding interval within the octave; for example, the interval of a tenth is generally heard as having the same consonance as a third. Because of this, music theorists usually analyze intervals as though the notes were in the same octave.

Chromatic Intervals

Adding accidentals to the notes in an interval can change the interval's size. For example, the distances between C and F and between C and F# are both fourths, but C to F# is a half step larger.

The intervals C4 to F4 and C4 to F sharp 4 on the staff, and illustrated on the piano keyboard. The first interval is shown to encompass 5 half steps, and the second interval is shown to encompass 6 half steps.
Figure 3: Despite both being fourths, C and F and C and F# are different intervals.

To specify this difference, we describe the interval's quality — major, minor, perfect, augmented or diminished — in addition to the diatonic distance. Intervals measured this way are called chromatic intervals.

It is important to remember that adding, changing or removing accidentals does not change the diatonic measurement of an interval; for example, C to Fx is a type of fourth, but C to G — which sounds identical — is a type of fifth.

Perfect Intervals

Unisons, fourths, fifths and octaves are called perfect intervals.

A perfect octave is the distance between two notes which have the same name in adjacent octaves, such as from E3 to E4.

A perfect octave from E3 to E4.
Figure 4: A perfect octave.

A perfect unison is the distance between two notes of the same name and octave, such as from F4 to F4 or from Bb2 to Bb2.

A perfect unison from F4 to F4.
Figure 5: A perfect unison.

A perfect fourth is a diatonic fourth that spans 5 half steps. All fourths that can be created without accidentals are perfect fourths, except for F to B, which spans 6 half steps.

Perfect fourths from C4 to F4, D4 to G4, E4 to A4, F4 to B4, G4 to C5, A4 to D5, and B4 to E5. All are labeled as P4 except for F4 to B4.
Figure 6: All fourths created using notes without accidentals are perfect fourths except for F to B.

A perfect fifth is a diatonic fifth that spans 7 half steps. All fifths that can be created without accidentals are perfect fifths, except for B to F, which spans 6 half steps.

Perfect fifths from C4 to G4, D4 to A4, E4 to B4, F4 to C5, G4 to D5, A4 to E5, and B4 to F5. All are labeled as P5 except for B4 to F5.
Figure 7: All fifths created using notes without accidentals are perfect fifths except for B to F.

When a perfect interval is made a half step larger by adding or removing an accidental, it becomes an augmented interval.

A perfect octave from G4 to G5, an augmented octave from G4 to G sharp 5, a perfect fifth from G4 to D5, an augmented fifth from G flat 4 to D5, a perfect fourth from G4 to C5, an augmented fourth from G4 to C sharp 5, a perfect unison from G4 to G4, and an augmented unison from G4 to G sharp 4.
Figure 8: When accidentals are used to increase the size of a perfect interval by a half step, the interval becomes augmented.

When a perfect interval is made a half step smaller by adding or removing an accidental, it becomes a diminished interval.

A perfect octave from G4 to G5, a diminished octave from G4 to G flat 5, a perfect fifth from G4 to D5, a diminished fifth from G sharp 4 to D5, a perfect fourth from G4 to C5, a diminished fourth from G4 to C flat 5, a perfect unison from G4 to G4, and an augmented unison from G flat 4 to G4.
Figure 9: When accidentals are used to decrease the size of a perfect interval by a half step, the interval becomes diminished. Note that diminished unisons do not exist, since a perfect unison represents zero distance.

When perfect intervals are inverted by moving the bottom note up and octave, the result is another perfect interval, according to the following chart.

A perfect fifth from E4 to B4 and a perfect fourth from B4 to E5, then a perfect fourth from E4 to A4 and a perfect fifth from A4 to E5, a perfect octave from E4 to E5 and a perfect unison from E5 to E5, and a perfect unison from E4 to E4 and a perfect octave from E4 to E5.
Figure 11: When an interval is inverted by moving its bottom note up one octave, the resulting interval is also perfect.

Imperfect Intervals

Seconds, thirds, sixths and sevenths are called imperfect intervals. Like the name suggests, imperfect intervals cannot be perfect; instead, they can be major or minor.

Interval Name Number of Half Steps
m2 minor second 1
M2 major second 2
m3 minor third 3
M3 major third 4
m6 minor sixth 8
M6 major sixth 9
m7 minor seventh 10
M7 major seventh 11

While imperfect intervals can be measured using half steps as shown in table above, it is generally easier to understand them in relation to the major scale. All intervals in the major scale measured up from the tonic are either perfect or major.

A perfect unison from C4 to C4, a major second from C4 to D4, a major third from C4 to E4, a perfect fourth from C4 to F4, a perfect fifth from C4 to G4, a major sixth from C4 to A4, a major seventh from C4 to B4, and a perfect octave from C4 to C5.
Figure 12: Intervals derived from the ascending major scale are all major or perfect.

Likewise, all intervals in the major scale measured down from the tonic are either perfect or minor.

A perfect unison from C5 to C5, a minor second from B4 to C5 a minor third from A4 to C5, a perfect fourth from G4 to C5, a perfect fifth from F4 to C5, a major sixth from E4 to C5, a major seventh from D4 to C5, and a perfect octave from C4 to C5.
Figure 12: Intervals derived from the descending major scale are all minor or perfect.

Like perfect intervals, when an imperfect interval is made larger or smaller with accidentals, it can become augmented or diminished, respectively.

A diminished third from A4 to C flat 4, a minor third from A4 to C natural 4, a major third from A4 to C sharp 4 and an augmented third from A4 to C double sharp 4. Then, a diminished sixth from B flat 3 to A double flat 4, a minor sixth from B flat 3 to A flat 4, a major sixth from B flat 3 to A natural 4, and an augmented sixth from B flat 3 to A sharp 4.
Figure 14: Increasing the size of a major interval creates an augmented interval, and decreasing the size of a minor interval creates a diminished interval.

Identifying Intervals

While some level of memorization will come with practice, any interval within the octave can be identified by using the following steps.

  • Determine the diatonic interval by counting the letter names covered by the interval.
  • Temporarily ignore any accidentals and follow the appropriate steps below to identify the resulting interval:
    • For unisons and octaves, the resulting interval will be perfect.
    • For fourths, if the resulting interval is F to B, it is augmented; otherwise, it is perfect.
    • For fifths, if the resulting interval is B to F, it is diminished; otherwise it is perfect.
    • For all other intervals, if the top note belongs in the major key starting with the bottom note, the resulting interval is major; otherwise it is minor.
  • Lastly, replace any accidentals one at a time and track how the quality of the interval is affected.
Figure 4: An flowchart for labelling an interval. First, reduce the interval to an octave or smaller and remove any accidentals. If the resulting interval is a unison or octave, it is perfect. If it is a fourth or fifth, it is perfect unless it involves the notes F and B. If it is a second, third, sixth or seventh, the resulting interval is major if the top note is in the major key of the bottom note; otherwise, it is minor. Finally, add any accidentals back in one at a time and track how the inflection of the interval changes.

Intervals: Summary

  • The distance between two pitches is called an interval.
    • A melodic interval is the distance between two consecutive pitches.
    • A harmonic interval is the distance between two pitches played simultaneously.
    • A compound interval is an interval greater than an octave.
  • A diatonic interval indicates the number of scale degrees between two notes, without regard for accidentals.
  • A chromatic interval specifies both the diatonic interval and the inflection created by the presence or absence of accidentals.
    • Unisons, fourths, fifths and octaves are perfect intervals.
      • When no accidentals are present, unisons and octaves are always perfect.
      • When no accidentals are present, all fourths are perfect fourths, except for F to B, which is an augmented fourth.
      • When no accidentals are present, all fifths are perfect fifths, except for B to F, which is a diminished fifth.
    • Seconds, thirds, sixths and sevenths are imperfect intervals.
      • When no accidentals are present and the top note is in the major key of the bottom note, imperfect intervals are major.
      • When no accidentals are present and the bottom note is in the major key of the top note, imperfect intervals are minor.
    • To identify an interval with accidentals, add accidentals one at a time and track how the distance changes.
      • For perfect intervals, a diminished interval is a half step smaller than a perfect interval and an augmented interval is a half step larger than a perfect interval.
      • For imperfect intervals, a minor interval is a half step smaller than a major interval, a diminished interval is a half step smaller than a minor interval, and an augmented interval is a half step larger than a major interval.

Exercises

Exercise 1: Description of Exercise

Exercise 2: Description of Exercise