# 2.1: Intervals

- Page ID
- 186173

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# 11.1 Introduction

The preceding chapters have dealt primarily with musical events happening one at a time. The rhythms, pitches, and scales discussed so far, in other words, might all be performed by a single individual. In tonal Western art music, however, voices rarely sound alone. Soloists are provided accompaniment, melodies converse with other melodies, and orchestras full of unique sounds contribute to massive symphonic textures. For many practitioners, this joining of individual voices is the essence of the art.

This chapter will consider the structure and effect of intervals, combinations of pitches heard either together or in close proximity. Of course, this is not a new topic. In Chapter 5 we defined two types of intervals: octaves and semitones. In Chapter 6 we discussed the intervals found between successive major scale degrees and noticed how these steps are equal in size to either semitones and whole tones. The present chapter will expand this discussion. Our main focus will be to describe a system for identifying, classifying, and labeling intervals of any size and type. Along the way, we will explore how various intervals relate to each other and to broader contexts such as scales and keys. Finally, we will discuss how intervals may be manipulated—expanded, contracted, and inverted—to create new intervals. All of these skills and ideas will provide a solid foundation for the chapters that follow.

# 11.2 Interval size

An *interval* is the distance a listener perceives between two pitches. When the two pitches sound simultaneously, we refer to it as a *harmonic interval*:

When they sound one after the other, we refer to it as a *melodic interval*:

Some intervals, such as semitones, whole tones, and octaves, have special names related to their acoustic properties or relationship to a scale. Useful though these names are, they do not tell us much about how such intervals relate to one another. (What does a whole tone sound like? The name tells us nothing in this regard.) In the widely-used system described here, all intervals are classified—and subsequently labeled—according to their *size* and *quality*, or aesthetic effect. We will begin our discussion with interval size, the more straightforward of these two attributes.

Imagine two voices singing different steps of a C-major scale, say, C and D. If the voice singing the D raised the pitch to the next scale degree, E, while the other stayed on C, we would say that the interval heard between the two voices grew larger in size. When we speak of an interval’s size, we are concerned with the distance between the notes as they appear in a scale, on a staff, or in the cycle of pitch letter names A through G.

The size of an interval specifies the number of staff lines and spaces—or pitch letter names—spanned by the two notes. C and D, a whole tone apart, are adjacent to one another on the staff. Together, they span two staff positions and therefore the size of this interval is called a *second*. C and E have either a line or space between them and therefore span three staff positions. They are said to be a *third* apart. The following example shows all of the interval sizes up to an octave:

(Despite not technically being an interval, the unison is included in Example 11–3 for reasons that will become clear momentarily.)

\[\hat1\]

## Activity 11-1

Activity 11–1

Identify the size of each of the following intervals.

### Exercise 11–1a:

### Question

What is the size (unison, second, third, etc.) of this interval?

## Hint

To determine the size of an interval, count the number of lines and spaces it spans on the staff. (Don’t forget to include the lines and spaces occupied by the notes themselves!)

## Answer

fourth

### Exercise 11–1b:

### Question

What is the size (unison, second, third, etc.) of this interval?

## Hint

To determine the size of an interval, count the number of lines and spaces it spans on the staff. (Don’t forget to include the lines and spaces occupied by the notes themselves!)

## Answer

seventh

### Exercise 11–1c:

### Question

What is the size (unison, second, third, etc.) of this interval?

## Hint

To determine the size of an interval, count the number of lines and spaces it spans on the staff. (Don’t forget to include the lines and spaces occupied by the notes themselves!)

## Answer

second

### Exercise 11–1d:

### Question

What is the size (unison, second, third, etc.) of this interval?

## Hint

## Answer

fifth

### Exercise 11–1e:

### Question

What is the size (unison, second, third, etc.) of this interval?

## Hint

## Answer

third

### Exercise 11–1f:

### Question

What is the size (unison, second, third, etc.) of this interval?

## Hint

## Answer

octave

By definition, an interval cannot be smaller than a unison. It can, however, be larger than an octave. When it comes to naming larger intervals, there are two options. One might, as suggested above, simply count the span of staff positions or scale steps between the two notes. The interval from a C up *ten* scale steps to E would then be called a *tenth*.

On the other hand, despite the large distance between the two pitches, this interval sounds a lot like the third shown in Example 11–3:

The difference between these two intervals is that in the second case the upper note, E, has been displaced by an octave. Intervals that are smaller than an octave are called *simple intervals*. Intervals that are greater than an octave are called *compound intervals* since they consist of a simple interval plus one or more octave displacements. Both of the intervals in Example 11–4, then, are thirds. The first one is a simple third and the second one is a compound third.

For the sake of highlighting the relationship between corresponding compound and simple intervals, we often refer to large intervals as though the two pitches were an octave or less apart.

The intervals in Example 11–5 are, in turn, an octave, a ninth (or compound second), tenth (or compound third), and eleventh (or compound fourth).

## Activity 11-2

Activity 11–2

Several intervals have been put in boxes in the score below. Identify each interval as either simple or compound:

### Exercise 11–2a:

### Question

Is the interval in box “a” simple or compound?

## Hint

Intervals up to and including an octave are considered simple.

## Answer

compound

### Exercise 11–2b:

### Question

Is the interval in box “b” simple or compound?

## Hint

Intervals up to and including an octave are considered simple.

## Answer

simple

### Exercise 11–2c:

### Question

Is the interval in box “c” simple or compound?

## Hint

Intervals up to and including an octave are considered simple.

## Answer

simple

### Exercise 11–2d:

### Question

Is the interval in box “d” simple or compound?

## Hint

Intervals up to and including an octave are considered simple.

## Answer

simple

### Exercise 11–2e:

### Question

Is the interval in box “e” simple or compound?

## Hint

Intervals up to and including an octave are considered simple.

## Answer

compound

### Exercise 11–2f:

### Question

Is the interval in box “f” simple or compound?

## Hint

Intervals up to and including an octave are considered simple.

## Answer

simple

Interval size is unaffected by accidentals.

All of the intervals in the example above are sixths, even though they sound very different. The presence of the accidentals does not change the fact that each of these intervals spans six staff positions.

As Example 11–6 makes clear, interval size is directly related to the spelling of the individual pitches. There is always, however, more than one enharmonically equivalent way to spell a pitch. Since intervals are made of pitches, it follows that there are multiple ways of enharmonically spelling an interval. If the Abb in Example 11–6 were spelled as G and if the Cx were spelled as D, the interval would still sound exactly the same. Written this way, though, it would be considered a fourth instead of a sixth:

Enharmonic equivalence allows for some counter-intuitive scenarios. In the following example, the interval shown between the two staves is a unison, since both voices are on middle C. If one of these voices were changed to C#, the interval would still be a type of unison, even though there are two distinct pitches:

# 11.3 Interval quality

Consider the following two intervals:

Both of these intervals have the same size; they are both thirds. Despite the similarity in notation, they sound quite different. We address such difference by identifying the *quality *of an interval. Interval size is a generic classification; multiple (different) intervals can have the same size. The four intervals in Example 11–6 were all of the same size, but had different qualities. Combining interval size with interval quality allows us to specify the exact distance between—and spelling of—two notes.

In terms of naming intervals, quality is related—though not entirely tied—to how blended or stable the two pitches sound together. Pitches that sound stable and harmonious together are said to be *consonant*. An octave is an example of a consonant interval. The two pitches in an octave blend together so well, in fact, that it can be difficult for some listeners to distinguish them. On the other hand, intervals that sound unstable and agitated—as though the pitches are rubbing against one another—are said to be *dissonant*. A semitone is an example of a dissonant interval.

The following example presents three dissonant intervals followed by three consonant intervals above middle C:

a. three dissonant intervals

b. three consonant intervals

Most listeners will hear the first three intervals as dissonant and the next three as consonant. Consonance and dissonance, however, are relative terms. An interval might sound consonant in comparison to one interval but dissonant in comparison to another. Furthermore, the application of these terms is dependent on a listener’s subjective experience. What one listener hears as dissonant, another might hear as consonant and vice versa.

**Note:** As you can see, the terms *consonant* and *dissonant* are difficult to pin down. The way a listener hears a musical sound is subjective and very much tied to cultural preference and prior listening experience. It should come as no surprise, then, that throughout history music theorists have disagreed when classifying intervals into categories based on these criteria. In this book, we will use the terms “dissonant” and “dissonance” to refer to sonorities that sound unstable or foreign to the immediate musical context. Correspondingly, “consonant” and “consonance” will here refer to sonorities that sound stable and harmonious with the immediate musical context.

Based loosely on a scale of consonance and dissonance, there are two broad categories for different sizes of intervals: *perfect* intervals and *imperfect* intervals. Perfect intervals tend to sound more consonant. Unisons, fourths, fifths, and octaves (along with the corresponding compound intervals) are perfect intervals. All of the remaining interval sizes tend to sound less consonant. Seconds, thirds, sixths, and sevenths (along with the corresponding compound intervals) are imperfect intervals. The following table summarizes:

Perfect interval sizes: | Imperfect interval sizes: |
---|---|

unisons (U), fourths (4), fifths (5), octaves (8), and compound versions of the same |
seconds (2), thirds (3), sixths (6), sevenths (7), and compound versions of the same |

## Activity 11-3

Activity 11–3

Several intervals are surrounded by boxes in the score below. Identify each interval as either perfect or imperfect:

### Exercise 11–3a:

### Question

Is the interval in box “a” perfect or imperfect?

## Hint

The perfect intervals are unisons, fourths, fifths, octaves, and the corresponding compound intervals.

## Answer

perfect

### Exercise 11–3b:

### Question

Is the interval in box “b” perfect or imperfect?

## Hint

The perfect intervals are unisons, fourths, fifths, octaves, and the corresponding compound intervals.

## Answer

imperfect

### Exercise 11–3c:

### Question

Is the interval in box “c” perfect or imperfect?

## Hint

The perfect intervals are unisons, fourths, fifths, octaves, and the corresponding compound intervals.

## Answer

perfect

### Exercise 11–3d:

### Question

Is the interval in box “d” perfect or imperfect?

## Hint

## Answer

imperfect

### Exercise 11–3e:

### Question

Is the interval in box “e” perfect or imperfect?

## Hint

## Answer

imperfect

### Exercise 11–3f:

### Question

Is the interval in box “f” perfect or imperfect?

## Hint

## Answer

perfect

Within the *imperfect* category, intervals tend to be one of two qualities: *major* or *minor*. A minor interval is a semitone smaller than the corresponding major interval. Recall the two thirds from Example 11–9:

The first of these thirds is a major third. We abbreviate “major” with an uppercase “M,” so the interval is labeled “M3.” The second of the two thirds is a semitone smaller, a minor third. We abbreviate “minor” with a lowercase “m,” so that interval is labeled “m3.” A minor interval is always a semitone smaller than the major interval of the same size: a m6 is a semitone smaller than a M6, a m2 is a semitone smaller than a M2, and so on.

Minor intervals are often said to sound somewhat darker or more somber than the corresponding major intervals which are often said to sound brighter and more cheerful. Listen again to the two thirds in the example above and think about how you would describe the difference in quality. Keep in mind, though, that the current discussion is somewhat abstract. One’s subjective aesthetic experience of an interval is very much tied to the musical context. In other words, minor intervals can sound cheerful and major intervals somber under certain circumstances.

Within the *perfect *category, intervals tend to be *perfect* in quality. (Here, the quality name matches the category name.) The following example presents three perfect intervals above middle C:

Each of these intervals is perfect in quality. We abbreviate “perfect” with an uppercase “P,” so the intervals are labeled “P4,” “P5,” and “P8.” Compare the effect of these intervals with the effect of the thirds heard above. Perfect intervals are often described as sounding bold, stately, or serious. More importantly, they sound stable.

As with determining interval size, it is very helpful to think of a major scale when determining interval quality. All of the intervals formed by scale degrees above the keynote are either major or perfect in quality. In other words, the imperfect intervals—the second, third, sixth, and seventh above the keynote—are all major: M2, M3, M6, and M7. Meanwhile, the perfect intervals—the fourth, fifth, and octave above the keynote—are all perfect in quality: P4, P5, and P8:

Now consider the same intervals in a minor scale:

\[\hat3\]

**Note:** The fact that all of the intervals above the keynote in a major scale are either major or perfect is particularly handy when either identifying or writing intervals. When determining the size and quality of an interval, imagine a major scale built on the lower of the two notes. If the upper note belongs to that scale, the interval will be either major or perfect in quality depending on its size. When writing an interval above any given note, imagine a major scale with that note as the keynote and find the corresponding scale degree. Then, adjust the major or perfect interval as needed.

## Activity 11-4

Activity 11–4

Identify the quality of each of the following imperfect intervals.

### Exercise 11–4a:

### Question

What is the quality of this third?

## Hint

To determine if an imperfect interval is major, imagine a major scale using the lower note as its tonic. If the upper note is a member of this scale, then the quality of this interval is major.

## Answer

minor (m)

### Exercise 11–4b:

### Question

What is the quality of this sixth?

## Hint

To determine if an imperfect interval is major, imagine a major scale using the lower note as its tonic. If the upper note is a member of this scale, then the quality of this interval is major.

## Answer

minor (m)

### Exercise 11–4c:

### Question

What is the quality of this seventh?

## Hint

To determine if an imperfect interval is major, imagine a major scale using the lower note as its tonic. If the upper note is a member of this scale, then the quality of this interval is major.

## Answer

major (M)

### Exercise 11–4d:

### Question

What is the quality of this second?

## Hint

## Answer

major (M)

### Exercise 11–4e:

### Question

What is the quality of this compound third?

## Hint

## Answer

minor (m)

### Exercise 11–4f:

### Question

What is the quality of this seventh?

## Hint

## Answer

major (M)

Intervals in the perfect category are normally perfect in quality while intervals in the imperfect category are normally either major or minor in quality. Any of these intervals, however, may also be made larger or smaller, typically by adding accidentals. The term *diminished* is used to identify an interval that is a semitone smaller than normal and the term *augmented* is used to identify an interval that is a semitone larger than normal.

Take, for example, the four fifths shown below:

The first fifth is the same perfect fifth shown in the examples above (C and G). The next fifth, with its Gb instead of G§, is a semitone smaller than the more normal perfect fifth and is therefore said to be diminished. We abbreviate “diminished” with a lowercase “d,” so the interval is labeled “d5.” The next fifth raises the C to C# and restores the G§. It, too, is a semitone smaller than the perfect fifth and so is also labeled “d5.” Finally, the last fifth—C to G#—is a semitone larger than a perfect fifth and is therefore said to be augmented. We abbreviate “augmented” with a capital “A,” so the interval is labeled “A5.”

The same principle applies to imperfect intervals, though here the matter is complicated by the fact that there are two normal qualities: major and minor. An imperfect interval that is a semitone smaller than the corresponding minor interval is diminished. An interval that is a semitone larger than the corresponding major interval is augmented. The following example shows four sixths in order of increasing size:

The minor sixth (A and F) is a semitone smaller than the major sixth (A and F#). The diminished sixth (A# and F) is a semitone smaller still, while the augmented sixth (Ab and F#) is a semitone larger than the major sixth.

The following example shows the semitone differences between the different interval qualities:

**Note:** You may occasionally encounter interval qualities other than those listed here. An interval that is a semitone smaller than the diminished interval of the same size, for example, is said to be *doubly diminished*. (The same principle also allows for *doubly augmented* intervals.) Such interval qualities usually require uncommon accidentals and appear far less frequently than the ones discussed here.

## Activity 11-5

Activity 11–5

Identify each of the following intervals by size and quality.

### Exercise 11–5a:

### Question

What is the size and quality of the following interval?

## Hint

Think of a major scale with the lower note as its keynote. Find the corresponding scale degree and know that the interval it forms with the keynote is either major (if its size is imperfect) or perfect (if its size is perfect). Compare this known interval to the interval shown here.

## Answer

M6

### Exercise 11–5b:

### Question

What is the size and quality of the following interval?

## Hint

Think of a major scale with the lower note as its keynote. Find the corresponding scale degree and know that the interval it forms with the keynote is either major (if it is imperfect) or perfect (if it is perfect). Compare this known interval to the interval shown here.

## Answer

P4

### Exercise 11–5c:

### Question

What is the size and quality of the following interval?

## Hint

Think of a major scale with the lower note as its keynote. Find the corresponding scale degree and know that the interval it forms with the keynote is either major (if it is imperfect) or perfect (if it is perfect). Compare this known interval to the interval shown here.

## Answer

P8

### Exercise 11–5d:

### Question

What is the size and quality of the following interval?

## Hint

Think of a major scale with the lower note as its keynote. Find the corresponding scale degree and know that the interval it forms with the keynote is either major (if it is imperfect) or perfect (if it is perfect). Compare this known interval to the interval shown here.

## Answer

A4

### Exercise 11–5e:

### Question

What is the size and quality of the following interval?

## Hint

## Answer

M3

### Exercise 11–5f:

### Question

What is the size and quality of the following interval?

## Hint

## Answer

d5

## Activity 11-6

Activity 11–6

In each of the following exercises, you are presented with a single note and directions for creating an interval. Add the appropriate note to complete the interval.

### Exercise 11–6a:

### Question

Write a note that is a P5 above the given note.

## Hint

Begin by adding the appropriate notehead. Check the quality formed by the natural notehead and adjust the new note with accidentals as necessary.

## Answer

### Exercise 11–6b:

### Question

Write a note that is a m6 above the given note.

## Hint

Begin by adding the appropriate notehead. Check the quality formed by the natural notehead and adjust the new note with accidentals as necessary.

## Answer

### Exercise 11–6c:

### Question

Write a note that is a d5 above the given note.

## Hint

Begin by adding the appropriate notehead. Check the quality formed by the natural notehead and adjust the new note with accidentals as necessary.

## Answer

### Exercise 11–6d:

### Question

Write a note that is a P4 below the given note.

## Hint

## Answer

### Exercise 11–6e:

### Question

Write a note that is a M2 below the given note.

## Hint

## Answer

### Exercise 11–6f:

### Question

Write a note that is a A6 below the given note.

## Hint

## Answer

# 11.4 Interval size in semitones

Thinking about intervals as they relate to scales will strengthen your understanding of how different combinations of pitches function within a given key. (The convention for labeling interval size is, after all, directly related to the relationships found between various scale degrees and the keynote.) It will also be helpful to familiarize yourself with the exact size of common intervals in terms of how many semitones they span.

The following table shows all of the most common intervals along with their size in semitones:

Interval: | Number of semitones: |
---|---|

PU | 0 |

m2 (semitone) | 1 |

M2 (whole tone) | 2 |

m3 | 3 |

M3 | 4 |

P4 | 5 |

A4, d5 (tritone) | 6 |

P5 | 7 |

m6 | 8 |

M6 | 9 |

m7 | 10 |

M7 | 11 |

P8 (octave) | 12 |

Note that this table does not show the less common enharmonic equivalents of these intervals. A diminished third (d3), for example, is enharmonically equivalent to a major second (M2)—both are two semitones in size.

Notice, too, that some intervals are listed along with their common nicknames: semitone, whole tone, etc. There is one nickname in Table 11–2, however, that we have not yet discussed. All of the intervals from zero to twelve semitones may be written as minor, major, or perfect in quality with the exception of one. Two pitches, six semitones apart, cannot be written as a minor, major, or perfect interval. This interval, commonly known as a *tritone* typically appears as either an augmented fourth (A4) or a diminished fifth (d5):

\[\hat1\]

**Note:** The name “tritone” is derived from the size of the interval. Table 11–2 shows that this interval is equal in size to 6 semitones. Since a whole tone (or “tone”) is equal in size to two semitones, it follows that a tritone is equal in size to three whole tones, hence, “*tri*-tone.”

## Activity 11-7

Activity 11–7

Answer the following questions regarding the size of various intervals.

### Exercise 11–7a:

### Question

How many semitones are in a minor sixth?

## Answer

8

### Exercise 11–7b:

### Question

A perfect fifth is how many semitones larger than a perfect fourth?

## Answer

2

### Exercise 11–7c:

### Question

\[\hat1\]

## Answer

4

### Exercise 11–7d:

### Question

What common interval is one semitone larger than a major seventh?

## Answer

P8

# 11.5 Interval inversion

When the two pitches in an interval switch position—that is, when the lower note is raised one or more octaves so that it sounds above the upper note—the resultant interval sounds different, but noticeably similar to the original. Take, for example, the major third between C and E. If we move the C up an octave, the result is a minor sixth:

We call such a rearrangement of notes an *inversion*. Because of the similarity in quality and effect, we may think of these two intervals as forming a kind of pair. The major third between C and E inverts, in other words, to a minor sixth. Invert the interval again and we see that this works both ways: if we were to take the lower note of the minor sixth (E) and raise *it* an octave, the interval would invert back to a major third. This is true of any major third or minor sixth.

Other intervals can be inverted too. A minor second inverts to a major seventh (and vice versa):

A perfect fourth inverts to a perfect fifth (and vice versa)

And a tritone inverts to another tritone:

The following table shows the results that come from inverting common intervals. (The inversion of a unison or octave is negligible since these intervals consist of two members of the same pitch class. These have therefore been left off of the table.)

Interval: | Inversion: |
---|---|

m2 | M7 |

M2 | m7 |

m3 | M6 |

M3 | m6 |

P4 | P5 |

A4 | d5 |

d5 | A4 |

P5 | P4 |

m6 | M3 |

M6 | m3 |

m7 | M2 |

M7 | m2 |

**Note:** The result of inverting an interval is very predictable. The quality of an inverted perfect interval will also be perfect. The quality of an inverted minor interval will be major (and vice versa). And finally, the quality of a diminished interval will be augmented (and vice versa). Furthermore, the interval sizes will always add up to nine.

With a little arithmetic and by memorizing the following table, you will be able to quickly determine the size and quality of any inverted interval:

d ↔ A |

m ↔ M |

P ↔ P |

A M7, for example, inverts to a m2 since seven and two add up to nine and since major intervals invert to minor. Similarly, a diminished third will invert to an augmented sixth for the same reasons.

## Activity 11-8

Activity 11–8

Invert each of the following intervals.

### Exercise 11–8a:

### Question

Invert the following interval and identify its size and quality.

## Hint

To invert an interval, raise the lower note up an octave so that it appears above the other note.

## Answer

### Exercise 11–8b:

### Question

Invert the following interval and identify its size and quality.

## Hint

To invert an interval, raise the lower note up an octave so that it appears above the other note.

## Answer

### Exercise 11–8c:

### Question

Invert the following interval and identify its size and quality.

## Hint

To invert an interval, raise the lower note up an octave so that it appears above the other note.

## Answer

### Exercise 11–8d:

### Question

Invert the following interval and identify its size and quality.

## Hint

To invert an interval, raise the lower note up an octave so that it appears above the other note.

## Answer

### Exercise 11–8e:

### Question

Invert the following interval and identify its size and quality.

## Hint

To invert an interval, raise the lower note up an octave so that it appears above the other note.

## Answer

### Exercise 11–8f:

### Question

Invert the following interval and identify its size and quality.

## Hint

To invert an interval, raise the lower note up an octave so that it appears above the other note.

## Answer

# 11.6 Summary

An *interval* is the perceived distance between two pitches. Intervals are named/labeled according to two criteria: *size* and *quality*. The size of an interval is determined by the number of scale steps, staff positions, or pitch letter names spanned by the two notes. *Simple intervals* are those that are less than or equal to an octave in size. *Compound intervals* are greater than an octave in size. Compound intervals are usually named after their simple versions, as though the two notes were brought one or more octaves closer together.

The *quality* of an interval reflects its aesthetic effect. To determine the potential qualities for any given interval size, one must first know whether the interval size in question is *perfect* or *imperfect*. Perfect intervals include unisons, fourths, fifths, octaves, and their corresponding compound intervals. Imperfect intervals include seconds, thirds, sixths, sevenths, and their corresponding compound intervals. Perfect intervals are typically perfect in quality, but may also be diminished (a semitone smaller than perfect) or augmented (a semitone larger than perfect). Imperfect intervals are typically major or minor in quality—with minor being a semitone smaller than major—but may also be diminished (a semitone smaller than minor) or augmented (a semitone larger than major).

The major scale is a helpful tool since all of the intervals formed by scale degrees above the key note are predictably major or perfect. This knowledge, combined with an awareness of each common interval’s size in semitones, is very useful in identifying and writing intervals.

Intervals may be *inverted* (moving the lower note to the upper position or vice versa). The size of the resulting interval is very predictable since inversionally-related interval sizes always add up to nine. The resulting interval quality is likewise predictable since minor always inverts to major (and vice versa), augmented inverts to diminished (and vice versa), and perfect inverts to perfect.