We already learned about half steps and whole steps, which are types of intervals. In this chapter, we will explore the most basic types of intervals.

8.1   Interval Numbers

An interval is the distance between two notes. We already learned about half steps and whole steps, which are types of intervals. When we first learned how to write scales, we used a sequence of half steps and whole steps. Although half steps and whole steps are technically intervals, we use very specific labels when identifying intervals. Interval labels consist of two parts.

• Interval quality: major, minor, augmented, or diminished
• Interval number: 1–8

We will start by focusing on the interval number. Being able to quickly recognize and write the interval number will save you a great deal of time.

For the interval number, always count the bottom note as “1” and then count upward, alternating between lines and spaces. For example, both a diatonic half step and a diatonic whole step have the interval number of “2,” which we call a “second” (Example 8.1.1).

Example 8.1.1. Seconds

The intervals in Example 8.1.1 are seconds because there are two notes, and the distance between them is a second. Examples 8.1.1A, B, and C are correctly written, but Examples 8.1.1D and E are not.

• Example 8.1.1A: The interval from C₅ up to D₆ is an ascending melodic second.
  • This is a melodic interval because the notes are played one after another.
  • This is an ascending interval because the notes go up when reading from left to right.
• Example 8.1.1B: The interval from D₆ down to C₅ is a descending melodic second.
  • Even though D₆ appears before C₅, C₅ is still labeled as “1,” while D₆ is labeled as “2.” This is because intervals are always determined by the lowest note.
• Example 8.1.1C: This interval is a harmonic second.
  • A harmonic interval occurs when two notes are played at the same time. Another name for a two-note harmonic interval is a dyad.
  • This time, since B4 is the lowest note, B4 is labeled as “1,” while C5 is labeled as “2.”
  • We never use the words “ascending” or “descending” for harmonic intervals.
  • To avoid confusing melodic intervals with harmonic ones, make sure to leave enough space between melodic intervals.
  • Notice how the harmonic second is written. Because both noteheads cannot fit directly above one another, they must be staggered as shown. The lower note always goes on the left. Think: Lower Left.
• Example 8.1.1D: This harmonic second is incorrectly written because the notes are not staggered. Seconds are very difficult to distinguish when they overlap.
• Example 8.1.1E: This harmonic second is incorrectly notated because the lower note is on the right.
  • Remember that the lower note of the dyad must be on the left.^[1]

As previously mentioned, it does not matter if the second is a half step or a whole step. As long as the letter names are a second apart, the interval is called a second. Observe the following examples, which are all seconds (Example 8.1.2).

Example 8.1.2. More seconds

• Example 8.1.2A: C♭B flat is enharmonically equivalent to C. However, because the note names B and C are a second apart, this interval is a second.
• Example 8.1.2B: Although this example differs from Ex. 8.1.2A because of the C♯C sharp, it is also a second.
• Example 8.1.2C: This example is extreme. The distance from B𝄫B double flat to C♯C sharp is four half steps. However, since the note names B and C are only a second apart, this interval is a second.

Play the three seconds in Example 8.1.2 on the piano and listen to how different they all sound. The top line of Rothschild’s “Si j’étais Rayon” illustrates several seconds (Example 8.1.3).

Example 8.1.3. Seconds in Rothschild,^[2] “Si j’étais Rayon”

There is a mix of melodic and harmonic seconds, and some intervals are not seconds.

• Blue square brackets represent melodic seconds (see Examples 8.1.3A, C, D, E, and G).
  • Most seconds in this example are half steps (Examples 8.1.3A, C, E, and G), but one second is a whole step (Example 8.1.3D).
• Red boxes represent intervals that are not seconds (Examples 8.1.3B and F).
  • Although Examples 8.1.3B and F are half steps, they are chromatic half steps (and not diatonic half steps). Because the note names are the same, they cannot be seconds.
  • For instance, Example 8.1.3B connects G♯G sharp to G♮G natural. Since the distance from G to G is not a second, the interval from G♯G sharp to G♮G natural is also not a second.
• There is one harmonic second identified in Example 8.1.3H. Notice how the second is notated, with the lower note on the left.

Example 8.1.4 shows all the different interval numbers from 1 to 8 written as harmonic intervals.

Example 8.1.4. Interval numbers

The first row of Arabic numerals is how we label the interval, and the second row shows how we say the interval. Most intervals are named after their ordinal numbers (second, third, fourth), with two exceptions: the unison and the octave. They are never called the first and the eighth. Observe the following characteristics in Example 8.1.4:

• The third to the octave is stacked neatly above one another. However, the unison and second are written differently.
• Harmonic unisons with whole notes must have two noteheads next to each other to tell that there are two notes. If there is only one notehead, it is assumed that there is only one note.
• Harmonic seconds must be written side by side because the top note doesn’t fit neatly above. The lower note is always placed to the left.

While you may need to count every line and space at this point, it is important to quickly identify the interval number. We can use the following shortcuts.

• When the bottom note is in a space, all the odd-numbered intervals are in spaces of the staff.
  • The interval from F up to C is an odd number (fifth) because both notes are in spaces.
• When the bottom note is on a line, all odd-numbered intervals are on lines of the staff.
• When the bottom note and the top note are in a space and on a line (or vice versa), the interval number is even.
  • The interval from F up to B is an even number (fourth) because F is in a space and B is on a line.

Sometimes composers choose to use a variety of intervals in the melody, as in Berg’s Lyric Suite (Example 8.1.5). Cover the interval numbers and see if you can quickly identify them using the shortcut we just learned.

Example 8.1.5. Variety of intervals: Berg,^[3] Lyric Suite, i – Allegretto gioviale

In Example 8.1.5, the interval numbers are shown above the staff between notes. What do you observe above the interval numbers? Berg utilizes every interval number from 1 (the unison) to 8 (the octave). Compare Berg’s range of interval numbers to Rothchild’s use of steps (mostly seconds) in Example 8.1.3. Neither approach is better or more effective—each musical piece will have its own distinct characteristics.

Accidentals

When writing harmonic seconds, the notes had to be staggered because there was not enough room to fit both notes. Similarly, when writing harmonic intervals where both notes have accidentals, the accidentals must be staggered in a specific order when space is limited (Example 8.1.6). For melodic intervals, this is not an issue since the accidentals are simply written before each note.

Example 8.1.6. Order of accidentals

• Example 8.1.6A: For harmonic intervals of a seventh or larger, align the accidentals directly above each other. This is possible because larger intervals prevent the accidentals from overlapping.
• Example 8.1.6B: For harmonic intervals of a sixth or smaller, stagger the accidentals. Do not align the accidentals directly above each other, as they will overlap.
  • Write the accidental closer to the higher note.
  • Write the accidental farther away from the lower note.

Example 8.1.7 illustrates the correct way—and several incorrect ways—to write accidentals.

Example 8.1.7. Writing accidentals

• Example 8.1.7A: This demonstrates the proper way to stagger accidentals. Notice that the sharp is placed closer to the higher note, B, and that the sharps are not aligned.
• Example 8.1.7B: The accidentals are incorrectly placed, as the bottom note has the closer accidental.
• Example 8.1.7C: The accidentals are misplaced because they are aligned, but the interval is only a fourth. Notice how difficult it is to read the accidentals when they overlap.
• Example 8.1.7D: The accidentals seem to be properly aligned, but are too tiny. Have mercy on your theory professor’s eyes.
• Example 8.1.7E: Accidentals for melodic intervals don’t require special rules. These accidentals are correctly written.

Example 8.1.8 shows how accidentals should be written for dyads within an octave.

Example 8.1.8. Accidentals

• For the unison, although there are two whole notes to show that there are two simultaneous F♭F flats, there is only one accidental.
  • A second flat placed next to the flat would look like a double flat.
  • A second flat placed in between the two notes would look like a melodic interval.
• For intervals from seconds to sixths, the accidentals are arranged in a staggered manner.
  • The top note always gets the closer accidental, while the accidental for the bottom note sits farther away.
• For intervals of a seventh or larger, the accidentals are aligned because they can fit without overlapping.

Some accidentals, like the double sharp (𝄪 double sharp ), are smaller, allowing two double sharps to be placed directly above each other for smaller intervals. For now, make sure accidentals do not overlap, as they can be hard to read.

Wieck Schumann’s Caprice illustrates how accidentals are placed before harmonic dyads of different sizes (Example 8.1.9).

Example 8.1.9. Placement of accidentals in Wieck Schumann,^[4] Caprice, Op. 2, No. 6

Notice that accidentals are either aligned or staggered.

• Example 8.1.9A: Because the interval is an octave, there is enough space to comfortably align the two flats directly above each other.
• Examples 8.1.9B and C: This time, the intervals are only a third. For smaller intervals, the higher note gets the closer accidental, and the lower note’s accidental is placed farther away. The accidentals are not aligned.

8.2   Intervals in the Major Scale

In the last section, we learned that interval labels consist of two parts: interval quality and interval number. To understand the interval qualities better, it helps to think of them in relation to the major scale and natural minor scale. This approach will also assist you in aural training, especially when you are asked to sing and identify intervals.

Example 8.2.1 displays all the intervals relative to the tonic within a major scale.

Example 8.2.1. Intervals in the major scale

The first row shows how we label the intervals, and the bottom two rows show how we say the intervals.

• The intervals in blue boxes (unison, fourth, fifth, and octave) are perfect intervals (P).
  • The unison, fifth, and octave are called perfect because they produce the purest sounds. Historically, in Western art music, two-voice compositions could only start with a unison, fifth, or octave and must end with a unison or octave.
  • Labeling the fourth as perfect is just a convenience, as we will see later.
• The other intervals (second, third, sixth, and seventh) are major intervals (M).
  • Major intervals are simple to recall since they all fit within the major key.

Perfect intervals and major intervals are never interchangeable. In other words, the third can never be perfect, and the fourth can never be major.

When a major key signature is provided, identifying intervals is straightforward if you have the interval qualities memorized. For example, the third will always be major, and the fourth will always be perfect. Notice that in Example 8.2.2, which is in E♭E flat major, the interval qualities are the same as in Example 8.2.1. Again, this is because the interval qualities are always consistent in any major scale.

Example 8.2.2. Intervals in the E♭E flat major scale

We can rewrite Example 8.2.2 with accidentals and without a key signature (Example 8.2.3).

Example 8.2.3. Intervals in the E♭E flat major scale without a key signature

Notice that the notes and intervals in Example 8.2.3 are exactly the same as in Example 8.2.2. Since the accidentals are identical, the intervals are also the same. Observe the intervals in Beach’s “Peace I Leave With You,” which is also in E♭E flat major (Example 8.2.4).

Example 8.2.4. Intervals in Beach,^[5] “Peace I Leave With You”

Remember that intervals are always based on the lowest note. In Example 8.2.4, the bass has the lowest note, E♭E flat₃. Because the other voices (soprano, alto, and tenor^[6]) are higher than the bass, the intervals are measured from the bass’s E♭E flat₃. Since the key is E♭E flat major and E♭E flat is the lowest note, all the intervals will belong to the E♭E flat major scale.

In Example 8.2.4, the intervals between the alto and bass and between the tenor and bass are shown. Notice that all the intervals are either major or perfect, just like in Examples 8.2.1 to 8.2.3. The intervals between the soprano and bass are not shown because the distance is greater than an octave. Intervals larger than an octave will be discussed in Chapter 9 (Intervals II: Further Concepts).

Identifying Intervals

Here are the steps to follow when asked to identify intervals, such as in Example 8.2.5.

Example 8.2.5. Identify the given intervals:

Step one: Write in the interval number (Example 8.2.6).

Example 8.2.6. Step one

Step two: Ask yourself whether the top note belongs to the major key of the bottom note. If it does, the quality is either major or perfect, depending on the interval number (Example 8.2.7).

Example 8.2.7. Step two

• Example 8.2.7A:
  • The bottom note is G.
  • The key signature of G major has one sharp (F♯F sharp), so E belongs to the key of G major.
  • Therefore, E is a major sixth (M6) above G because the interval is a sixth and E belongs to the key of G major.
• Example 8.2.7B:
  • The bottom note is F.
  • The key signature of F major has one flat (B♭B flat), so B♭B flat belongs to the key of F major.
  • Therefore, B♭B flat is a perfect fourth (P4) above F because the interval is a fourth and B♭B flat belongs to the key of F major.
• Example 8.2.7C:
  • The bottom note is F♯F sharp.
  • The key signature of F♯F sharp major has six sharps (F♯F sharp, C♯C sharp, G♯G sharp, D♯D sharp, A♯A sharp, E♯E sharp), so A♯A sharp belongs to the key of F♯F sharp major.
  • Therefore, A♯A sharp is a major third (M3) above F♯F sharp because the interval is a third and A♯A sharp belongs to the key of F♯F sharp major.

Example 8.2.8 shows the melodic interval between each note. All the intervals are from the major scale except for the one marked as “X.”

Example 8.2.8. Melodic Intervals in Abrams,^[7] “A Smile and a Tear”

Remember that intervals are always based on the lower note. For melodic intervals, analyze each pair of notes and think in the key of the lower note. Identify the interval number, then determine if the higher note belongs in the major key of the lower note.

• Example 8.2.8A: This is a perfect unison (P1) because the note repeats.
• Example 8.2.8B: The lower note is F, and C is a fifth above F. Because F major only has one flat (B♭B flat), C belongs to F major. Therefore, the interval from F to C is a perfect fifth (P5).
  • Even though the first note was C, it was not the lower note.
• Example 8.2.8C: The interval from F up to A is a major third (M3).
• Example 8.2.8D: The lower note is G, and A is a second above G. Because G major has only one sharp (F♯F sharp), A exists in G major. Therefore, the interval from A to G is a major second (M2).
  • Do not think in the key of A major because it appears first! A major has three sharps, including G♯G sharp. Always derive intervals from the lower note.
• Example 8.2.8E: The interval from G down to F is a major second (M2).
• Example 8.2.8F: The interval from F up to B♭B flat is a perfect fourth (P4) because F major has one flat (B♭B flat).
• Example 8.2.8G: The lower note is B♭B flat and D is a third above B♭B flat. Because the key of B♭B flat major has two flats (B♭B flat and E♭E flat), D does not have a flat and belongs to the B♭B flat major scale. Therefore, the interval from B♭B flat to D is a major third (M3).
• Example 8.2.8H: The lower note is D, and F is a third above D. The key of D major has two sharps (F♯F sharp and C♯C sharp). However, the next note is not F♯F sharp, but F♮F natural. Therefore, the interval between D and F does not belong to the D major scale and is marked with an “X.” We will learn about this interval later in the chapter.
• Example 8.2.8I: The lower note is C, and F is a fourth above C. Since C major has no sharps or flats, F is a perfect fourth (P4) above C.

Writing Intervals

Follow these steps when you’re asked to write the specified intervals (see Example 8.2.9).

Example 8.2.9. Write the following harmonic intervals above the given notes.

Step one: Write the pitch using only the interval number (no accidentals) (Example 8.2.10).

Example 8.2.10. Step one

• Example 8.2.10A: A seventh above A is G. Since A is in a space, the seventh above (an odd number) will also be in a space.
• Example 8.2.10B: A fifth above E♭E flat is B. Since E♭B flat is on a line, the fifth above (an odd number) will also be on a line.
• Example 8.2.10C: A second above B. For seconds, remember to position the notes side by side, with the lower note on the left.

Step two: Add any necessary accidentals to the top note so it fits within the bottom note’s major key (Example 8.2.11).

Example 8.2.11. Step two

• Example 8.2.11A:
  • Since the key signature of A major has three sharps (F♯F sharp, C♯C sharp, G♯G sharp), add a sharp to G.
  • G♯G sharp is a major seventh (M7) above A because it belongs to the key of A major.
• Example 8.2.11B:
  • Since the key signature of E♭E flat major has three flats (B♭B flat, E♭E flat, A♭A flat), add a flat to B.
  • B♭B flat is a perfect fifth (P5) above E♭E flat because it belongs to the key of E♭E flat
  • Notice that the flats are staggered and not aligned directly above each other. If the flats can fit directly over one another, it means that the flats are too small.
• Example 8.2.11C:
  • Because the key signature of B major has five sharps (F♯F sharp, C♯C sharp, G♯G sharp, D♯D sharp, A♯A sharp), add a sharp to C.
  • C♯C sharp is a major second (M2) above B because it belongs to the key of B major.

8.3   Shortcut: P4/P5

While it would be helpful to quickly identify and write all the intervals, some intervals are used more frequently. Being able to quickly recognize and write the perfect intervals will be useful. What do you notice about the following perfect intervals in Example 8.3.1?

Example 8.3.1. Perfect intervals

All note pairs in Example 8.3.1 share the same accidentals.

• Example 8.3.1A: Obviously, perfect unisons will always have identical accidentals, as they are the same pitch.
• Example 8.3.1B: Perfect octaves will always share the same accidentals because they are the same note, just an octave apart.
• Example 8.3.1C: Perfect fourths will always share the same accidentals, except for the notes F to B. The notes here are B♭B flat and E♭E flat. The lower note is B♭B flat, and B♭B flat’s key signature has two flats (B♭B flat and E♭E flat). Therefore, E has a flat.
• Example 8.3.1D: Perfect fifths will always share the same accidentals, except for the notes B to F. The notes here are C♯C sharp and G♯G sharp. The lower note is C♯C sharp, and C♯C sharp’s key signature contains all seven sharps. Therefore, G has a sharp.

Since the perfect unison and perfect octave always share the same accidentals, we will focus on the perfect fourth and perfect fifth (Example 8.3.2.).

Example 8.3.2. Perfect fourth and perfect fifth

• Example 8.3.2A:
  • The interval number is 4.
  • The notes are not F and B.
  • The accidentals match because neither note has an accidental.
  • Therefore, this is a perfect fourth.
  • Check: The key signature of A major has three sharps (F♯F sharp, C♯C sharp, and G♯G sharp); D does not have a sharp.
• Example 8.3.2B:
  • The interval number is 4.
  • The notes are not F and B.
  • The accidentals match because both notes are flat.
  • Therefore, this is a perfect fourth.
  • Check: The key signature of A♭A flat major has four flats (B♭B flat, E♭E flat, A♭A flat, and D♭D flat); D has a flat.
• Example 8.3.2C:
  • The interval number is 5.
  • The notes are not B and F.
  • The accidentals match because neither note has an accidental.
  • Therefore, this is a perfect fifth.
  • Check: The key signature of F major has one flat (B♭B flat); C does not have a flat.
• Example 8.3.2D:
  • The interval number is 4.
  • The notes are not F and B.
  • The accidentals match because both notes are sharp.
  • Therefore, this is a perfect fifth.
  • Check: The key signature of F♯F sharp major has six sharps (F♯F sharp, C♯C sharp, G♯G sharp, D♯D sharp, A♯A sharp, and E♯E sharp); C has a sharp.

For proof, you can literally write out all fifteen major scales and all fifteen minor scales to see that the perfect fourth and the perfect fifth will always share the same accidental as the tonic, except for F and B, or B and F. Example 8.3.3. shows how to make these intervals perfect.

Example 8.3.3. F and B (or B and F)

The accidentals for perfect fourths (between F and B) and perfect fifths (between B and F) will not match. You will need to rely on the key signatures for these intervals.

• Example 8.3.3A:
  • The interval number is 4.
  • The notes are F and B.
  • The key signature of F major has one flat (B♭B flat).
  • A perfect fourth above F is B♭B flat.
• Example 8.3.3B:
  • The interval number is 4.
  • The notes are F and B.
  • The key signature of F♯F sharp major has six sharps (F♯F sharp, C♯C sharp, G♯G sharp, D♯D sharp, A♯A sharp, and E♯E sharp).
  • A perfect fourth above F♯F sharp is B.
• Example 8.3.3C:
  • The interval number is 5.
  • The notes are B and F.
  • The key signature of B major has five sharps (F♯F sharp, C♯C sharp, G♯G sharp, D♯D sharp, and A♯A sharp).
  • A perfect fifth above B is F♯F sharp.
• Example 8.3.3D:
  • The interval number is 5.
  • The notes are B and F.
  • The key signature of B♭B flat major has two flats (B♭B flat and E♭E flat).
  • A perfect fifth above B♭B flat is F.

It is easy to remember which notes have accidentals when the notes are F and B.

• If there is one sharp in the key signature, that sharp is F♯F sharp.
• If there is one flat in the key signature, that flat is B♭B flat.
• This means that if you see F and B, or B and F, the pair will have either F♯F sharp or B♭B flat.
  • F and B♭B flat
  • F♯F sharp and B
  • B and F♯F sharp
  • B♭B flat and F

This shortcut can help you quickly identify perfect intervals. In Example 8.3.4, Bosmans utilizes harmonic perfect intervals.

Example 8.3.4. Harmonic perfect intervals in Bosmans,^[9] Impressions for Cello and Piano, I. Cortége

• Example 8.3.4A: The interval between G and C is a fourth. Since the notes are not F and B, and both notes do not have accidentals, this is a perfect fourth.
• Example 8.3.4B: The interval between F and B♭B flat is a fourth. Because the notes are F and B, one note must have an accidental (B♭B flat). Therefore, this interval is also a perfect fourth.
  • Do not forget to look at the key signature for accidentals.
• Example 8.3.4C: The interval between E♭E flat and A♭A flat is a fourth. Since the notes are not F and B, and the accidentals match, this is a perfect fourth.
• Example 8.3.4D: The interval between D and G is a fourth. Since the notes are not F and B, and both notes do not have accidentals, this is a perfect fourth.

Note that in the explanations above, we did not connect every dyad to the key of the bottom note. Using the shortcut helped us identify these intervals more quickly.

8.4   Minor Intervals

When we lower a major interval by a half step while keeping its interval number, the quality changes to a minor interval. Perfect intervals can never become minor; only major intervals can turn into minor ones. In other words, only the second, third, sixth, and seventh can be minor intervals: minor second, minor third, minor sixth, and minor seventh.

Minor is abbreviated with a lowercase m. Although the uppercase M for major intervals (M) and the lowercase m for minor intervals (m) look different when typed, they can be hard to tell apart when written by hand. To clearly distinguish them, you can draw a line above a lowercase m to specify minor intervals when writing by hand.

There are two strategies you can use to quickly identify and write minor intervals: 1) start from major intervals, or 2) start from the minor scale.

From the Major Intervals

We learned that to transform a major scale into its parallel natural minor scale, we lower scale degree 3^, scale degree 6^, and scale degree 7^. Doing this changes the major third, major sixth, and major seventh into minor thirds, minor sixths, and minor sevenths. Likewise, the same process applies to the major second: lower it by a half step to create a minor second.

When using this method, remember that you can never alter the bottom note. You can only add an accidental to the top note.

Let’s practice writing a few minor intervals starting from the major intervals.

Step one: Write the pitch as it appears in the major scale (see Example 8.4.1).

Example 8.4.1. Step one

• Example 8.4.1A: Since the key signature of B major has five sharps, a major second (M2) above B is C♯C sharp.
• Example 8.4.1B: Since the key signature of C major has no sharps or flats, a major third (M3) above C is E.
• Example 8.4.1C: Since the key signature of A♭A flat major has four flats, a major sixth (M6) above A♭A flat is F.
• Example 8.4.1D: Since the key signature of C♭C flat major has all seven flats, a major seventh (M7) above C♭C flat is B♭B flat.

Step two: Lower the top note by half a step by adding, changing, or removing an accidental. Do not alter the note (letter) name or adjust the bottom note (Example 8.4.2).

Example 8.4.2. Step two

• Example 8.4.2A: To change the major second (M2) to a minor second (m2), lower the C♯C sharp to C(♮ natural ). The natural sign is not needed because there is no key signature; just remove the sharp.
• Example 8.4.2B: To change the major third (M3) to a minor third (m3), lower the E to E♭E flat.
• Example 8.4.2C: To change the major sixth (M6) to a minor sixth (m6), lower the F to F♭F flat.
  • Although E and F♭F flat are enharmonically equivalent, A to E is a fifth, not a sixth. The top note must be written as F♭F flat.
• Example 8.4.2D: To change the major seventh (M7) to a minor seventh (m7), lower the B♭B flat to B𝄫B double flat.
  • As in Example 8.4.2C, you must write B𝄫B double flat and not A. A is only a sixth above C, and the instructions ask for a minor seventh above C, so the answer must be B-something.

Notice that the top note can range from no accidental to a flat, or even a double flat. As we will later learn, you can also have minor intervals with a sharp or double sharp on the top note. In other words, there is no consistently reliable way to write a minor interval. Sometimes students will simply add a flat to the top note, but that is not always correct.

There is a well-known example of lowering the top note of a major interval to become a minor interval in Strauss’s Also sprach Zarathustra (Example 8.4.3).

Example 8.4.3. Major to minor interval: Strauss,^[11],Also sprach Zarathustra (Thus Spoke Zarathustra), Op. 30 (Orchestral reduction)

The first bracket shows a major third (M3) from C to E. Across the bar line, the top note E is lowered to E♭E flat, creating a minor third (m3). Listen to the distinct qualities of the major third and the minor third, and how simply changing one note creates such a different sound.

From the Minor Key Signature

Another method you can use to find minor intervals is to base it on the natural minor scale. However, if you use this technique, you cannot apply it to minor seconds, as the second in the natural minor scale is a major second. Example 8.4.4. shows the minor intervals found in the natural minor scale.

Example 8.4.4. Intervals in the natural minor scale

The first row shows how we label the interval, and the bottom two rows show how we say the intervals.

• Notice that the intervals in the blue boxes (third, sixth, and seventh) are minor (m).
• All the other intervals are exactly the same as in the major scale.
  • Perfect intervals are the same in both major and minor keys. This is why the shortcut for perfect intervals works.
  • The second is major, not minor.

Some students prefer the minor key signature method because it only requires one step to find the minor intervals. Using the major key signature requires two steps: finding the major interval then lowering the top note with an accidental. However, other students struggle with minor key signatures, so they prefer starting with the major key signature. Let’s identify some minor intervals starting from the minor key (Example 8.4.5).

Example 8.4.5. Minor intervals

• Example 8.4.5A: The key signature of A♭A flat minor has all seven flats. Therefore, a minor third (m3) above A♭A flat is C♭C flat.
• Example 8.4.5B: The key signature of F♯F sharp minor has three sharps (F♯F sharp, C♯C sharp, and G♯G sharp). Therefore, a minor sixth (m6) above F♯F sharp is D.
• Example 8.4.5C: The key signature of B♭B flat minor has five flats (B♭B flat, E♭E flat, A♭A flat, D♭D flat, and G♭G flat). Therefore, a minor seventh (m7) above B♭B flat is A♭A flat.

All the notes and intervals in Example 8.4.6 belong to the key of A minor, except for one (marked with an asterisk).

Example 8.4.6. Intervals in Beach, Theme and Variations for Flute and String Quartet, Op. 80 – Theme. Lento di molto, sempre espressivo

• Remember that intervals are always based on the lowest note. In Example 8.4.6, the cello has the lowest note, A₃.
  • Because the other instruments (violin I, violin II, and viola) are higher than the cello, the intervals are based on the cello’s A₃.
  • Since the key is A minor and A is the lowest note, all the intervals listed (except the one with an asterisk) belong to the A minor scale.
• The intervals between the viola and cello, as well as between the second violin and the cello, are shown. The intervals between the first violin and the cello are not shown because the distance exceeds an octave. Intervals larger than an octave will be addressed in Chapter 9 (Intervals II: Further Concepts).
• The last interval between the viola and cello (marked with an asterisk) does not belong to the minor scale. However, C♯C sharp does belong to A major, making the interval a major third (M3).

For the interval marked with an asterisk, we used the parallel major. Using the parallel relationship can be helpful, especially when a key does not exist in major. As you may recall, there are three minor keys that lack a parallel major: G♯G sharp minor, D♯D sharp minor, and A♯A sharp minor. When writing major intervals above G♯G sharp, D♯D sharp, or A♯A sharp, use the minor key signature, then raise the top note (Example 8.4.7).

Example 8.4.7. Major intervals from the minor key signature

• Example 8.4.7A: There is no key signature of G♯G sharp major, but G♯G sharp minor has five sharps.
  • Therefore, a minor third (m3) above G♯G sharp is B.
  • Raising B to B♯B sharp makes a major third (M3).
• Example 8.4.7B: There is no such key signature of D♯D sharp major, but D♯D sharp minor has six sharps.
  • Therefore, a minor sixth (m6) above D♯D sharp is B.
  • Raising B to B# results in a major sixth (M6).
• Example 8.4.7C: There is no such key signature of A♯A sharp major, but A♯A sharp minor has all seven sharps.
  • Therefore, a minor seventh (m7) above A♯A sharp is G♯G sharp.
  • Raising G♯G sharp to G𝄪G double sharp results in a major seventh (M7).

You can apply a similar method when building or identifying minor thirds above notes that do not have a minor key, such as D♭D flat minor, G♭G flat minor, and C♭C flat minor. When writing minor intervals above D♭D flat, G♭G flat, or C♭C flat, start with the major key and then lower the third.

It is best to be able to use both techniques when identifying and writing minor intervals. In the next section, we will learn how to handle other trickier intervals.

8.5   Shortcut: m2/M2

As we learned, the minor second is not found in the natural minor scale. Because of this, we cannot use the minor key signature to identify the minor second. We can use the first method—starting with a major second and then lowering the top note by a half step—but there is a quicker shortcut.

• A minor second is a diatonic half step.

Always write the second first, because a chromatic half step would be incorrect. Example 8.5.1 shows a variety of minor seconds.

Example 8.5.1. Minor seconds

For each of the examples above, you could think of the major key or minor key, then lower the second by a half step, but it is much quicker to simply think of a diatonic half step above.

• Example 8.5.1A: B to C is a half step and a minor second, as shown on the keyboard.
• Example 8.5.1B: D to E♭E flat is also a half step and a minor second, as shown on the keyboard.
• Example 8.5.1C: D to D♯D sharp is a half step but not a minor second. D to D♯D sharp is a type of unison, so although it is a half step, it is not a minor second. It is a type of unison because both notes have the letter name “D.”
• Example 8.5.1D: G♯G sharp to A is a half step and a minor second, as shown on the keyboard.
• Example 8.5.1E: A♭A flat and B𝄫B double flat is a half step and a minor second, as shown on the keyboard. To identify this interval, simply locate the pitches on the keyboard. If the note (letter) names form a second and are a half step apart, the interval is a minor second.

You can also apply a similar approach to quickly identify and write major seconds.

• A major second is a diatonic whole step.

Again, always write the second first. Example 8.5.2 shows a variety of major seconds.

Example 8.5.2. Major seconds

For each of the examples above, you could think of scale degree 2^ in the major key or minor key, but it is sometimes quicker to just think of a diatonic half step above.

• Example 8.5.2A: B to C♯C sharp is a whole step and a major second, as shown on the keyboard.
• Example 8.5.2B: D to E is also a whole step and a major second, as shown on the keyboard.
• Example 8.5.2C: G♯G sharp to A♯A sharp is a whole step and a major second, as shown on the keyboard.
• Example 8.5.2D: G♯G sharp to B♭B flat is a whole step, but is not a major second. G to B is a third, so although it is a whole step on the keyboard, it is not a major second.

In Chapter 4, we learned that a major scale is built from a pattern of whole steps (W) and half steps (H): WWHWWWH. We can now replace these half steps and whole steps with minor seconds and major seconds (Example 8.5.3).

Example 8.5.3. Steps in a major scale

Using diatonic half steps (m2 minor seconds) and diatonic whole steps (M2 major seconds), see if you can quickly identify the seconds in Example 8.5.4.

Example 8.5.4. Seconds in Farrenc^[12], Impromptu pour Piano

• For the notes with stems pointing up, all the intervals are minor seconds and major seconds.
  • If you base the interval of the first two notes on the key signature of the lower note (E♯E sharp), it would be very difficult to determine whether the interval from E♯E sharp to F♯F sharp is a minor second or a major second. However, when considering half steps and whole steps, the distance between E♯E sharp and F♯F sharp is a diatonic half step. Therefore, the interval is a minor second (m2).
• For the notes with stems pointing down, all intervals are minor or major seconds, except for the last interval, which is marked with an X.
  • Although E♮E natural to E♯E sharp is a half step apart, it is a chromatic half step (same letter name). Therefore, this interval is a type of unison, not a second.

8.6   Tricky Intervals

We just learned how to write minor and major seconds starting on all notes—including those without a key signature, like E♯E sharp. We will now learn how to identify and write tricky intervals that begin on other notes.

We found out that not every note has both major and minor keys (Example 8.6.1).

Example 8.6.1. Possible key signatures in major and/or minor

• For notes that have both major and minor keys, such as C major and C minor, you can select which method to use for constructing and identifying intervals. For example, to write a minor sixth above C, you have two options.
  • Think of the key signature of C major. Write a major sixth above C, then lower the top note by a half step using an accidental.
  • Think of the key signature of C minor. Write a minor sixth above C based on the key signature of C minor.
• For notes that only have keys in major, like D♭D flat major, think of the key signature of D♭D flat major and lower the top note to create minor intervals.
• For notes that only have keys in minor, like D♯D sharp minor, think of the key signature of D♯D sharp minor and raise the top note for major intervals.

However, what happens when a note does not have a major or minor key? For example, what is a major sixth above F♭F flat? There is no such key as F♭F flat major or F♭F flat minor. Although less common, we do encounter tricky intervals like this.

Identifying Tricky Intervals

When tricky intervals share the same accidentals, you can ignore both accidentals because the quality remains the same (Example 8.6.2).

Example 8.6.2. Simplifying tricky intervals with the same accidentals

• Example 8.6.2A:
  • Example 8.6.2A1: The bottom note of the interval is F♭F flat, which seems intimidating.
  • Example 8.6.2A2: Since both accidentals are the same (both flats), simply remove both flats to easily determine the interval. The interval from F to E is a major seventh (M7).
  • Example 8.6.2A3: Returning to the original notes (F♭F flat and E♭E flat), we transfer our answer: M7 major seventh.
• Example 8.6.2B:
  • Example 8.6.2B1: The bottom note of the interval is B𝄪B double sharp, which appears daunting.
  • Example 8.6.2B2: Since both accidentals are double sharps, simply remove them to determine the interval easily. The interval from B up to G is a minor sixth (m6).
  • Example 8.6.2B3: Returning to the original notes (B𝄪B double sharp and G𝄪G double sharp), we transfer our answer: m6 minor sixth.
  • Note that although the interval was less than a seventh, the double sharps are aligned. This is because double sharps are smaller accidentals that do not overlap in this example.

To identify tricky intervals where the accidentals do not match, follow these steps.

Step one: Change the accidental of the bottom note to a natural sign and make the same change to the top note. You can only modify the notes by using accidentals; you cannot change the note names.

• If you raise a bottom note with a flat to a natural, then raise the top note by a half step.
• If you lower a bottom note with a sharp to a natural, then lower the top note by half a step.

Step two: Follow your usual process for identifying intervals, then write your answer for the original interval.

See Example 8.6.3 for these two steps in action.

Example 8.6.3. Simplifying tricky intervals with different accidentals

• Example 8.6.3A:
  • Example 8.6.3A1: There is no such key as F♭F flat major or F♭F flat minor.
  • Example 8.6.3A2: Raise F♭F flat a half step to become F♮F natural.
  • Example 8.6.3A3: Apply the same transformation to the top note: Raise the top note (D𝄫D double flat) by a half step to D♭D flat. The interval from F to D♭D flat is manageable; it is a minor sixth (m6).
  • Example 8.6.3A1: Returning to the original notes (F♭F flat and D𝄫D double flat), transfer your answer: m6 minor sixth.
• Example 8.6.3B:
  • Example 8.6.3B1: There is no such key as B♯B sharp major or B♯B sharp minor.
  • Example 8.6.3B2: Lower B♯B sharp a half step to become B♮B natural.
  • Example 8.6.3B3: Apply the same transformation to the top note: Lower the top note (F𝄪F double sharp) by a half step to F♯F sharp. The interval from B to F♯F sharp is manageable; it is a perfect fifth (P5).
  • Example 8.6.3B1: Returning to the original notes (B♯B sharp to F𝄪F double sharp), transfer your answer: P5 perfect fifth.
  • Alternatively, we could use the shortcut for perfect fifths to quickly identify this interval.

We can apply this technique to more extreme bottom notes, including double flats and double sharps. Follow the same steps, but instead of adjusting notes by a half step, adjust them by a whole step (Example 8.6.4). Remember that a whole step consists of two half steps, and you can never change the note (letter) names.

Example 8.6.4. Extreme bottom notes

• Example 8.6.4A:
  • Example 8.6.4A1: There is no such key as E𝄫E double flat major or E𝄫E double flat minor.
  • Example 8.6.4A2: Raise E𝄫E double flat a whole step to become E♮E natural.
  • Example 8.6.4A3: Apply the same transformation to the top note: Raise the top note (G♭G flat) by a whole step to G♯G sharp. You cannot raise the top note to A♭A flat because E to A would be a fourth, and the interval must be a third. The interval from E to G♯G sharp is manageable; it is a major third (M3).
  • Example 8.6.4A1: Returning back to the original notes (E𝄫E double flat to G♭G flat), we transfer our answer: M3 major third.
• Example 8.6.4B:
  • Example 8.6.4B1: There is no such key as F𝄪F double sharp major or F𝄪F double sharp minor.
  • Example 8.6.4B2: F𝄪F double sharp must be lowered a whole step to become F♮F natural.
  • Example 8.6.4B3: Apply the same transformation to the top note: Lower the top note (B♯B sharp) by a whole step to B♭B flat. Again, you cannot lower the top note to A♯A sharp because F to A would be a third and the interval must be a fourth. The interval from F to B♭B flat is manageable; it is a perfect fourth (P4).
  • Example 8.6.4B1: Returning to the original notes (F𝄪F double sharp to B♯B sharp), we transfer our answer: P4 perfect fourth.
  • Alternatively, we could use the shortcut for perfect fourths to quickly identify this interval.

In Example 8.6.4, both bottom notes did not have to move by whole step; they could have been moved by only a half step.

• Example 8.6.4A: Raising E𝄫E double flat by just a half step changes E𝄫E double flat to E♭E flat as the bottom note. In this case, you would only raise the top note a half step from G♭G flat to G♮G natural. The interval from E♭E flat to G is a major third (M3).
• Example 8.6.4B: Lowering F𝄪F double sharp by just a half step results in F♯F sharp as the bottom note (instead of F♮F natural). In this case, you would only lower the top note a half step from B♯B sharp to B♮B natural. The interval from F♯F sharp to B is a perfect fourth (P4).

As you can see, whether you move the bottom note by a half step or a whole step, you get the same result. Sometimes students prefer identifying intervals where the bottom note does not have an accidental. Other times, students prefer moving notes by only a half step instead of a whole step. As usual, it’s best to be able to use both techniques.

Although you may wonder why we need to learn about intervals with tricky bottom notes, they actually happen quite often in music. Example 8.6.5 shows an interval where the bottom note does not have a key.

Example 8.6.5. Tricky interval: Zybina,^[14] Mazurka

In Example 8.6.5, the circled dyad is a tricky interval with E♯E sharp on the bottom.

• There is no such key as E♯E sharp major or E♯E sharp minor.
• Because the accidentals match, we can temporarily remove both sharps. Now the notes are E and C.
• The key of E minor has one sharp (F♯F sharp). Therefore, E up to C is a minor sixth.
• This means that the circled interval is also a minor sixth.

Writing Tricky Intervals

Writing intervals with tricky bottom notes involves similar steps as identifying them. Temporarily change the bottom note to a note that’s easier for you (see Example 8.6.6).

Example 8.6.6. Writing tricky intervals

• Example 8.6.6A:
  • Example 8.6.6A1: Directions ask to write a major sixth (M6) above B♯B sharp.
  • Example 8.6.6A2: Because there is no such key as B♯B sharp major or B♯B sharp minor, temporarily lower B♯B sharp by a half step to B♮B natural.
  • Example 8.6.6A3: Write a major sixth above B: G♯G sharp.
  • Example 8.6.6A1: Return to the original question. Since B♯B sharp is a half step higher than B, also raise G♯G sharp by a half step to G𝄪G double sharp. Therefore, a major sixth above B♯B sharp is G𝄪G double sharp.
• Example 8.6.6B:
  • Example 8.6.6B1: Directions ask to write a minor seventh (m7) above A𝄫A double flat.
  • Example 8.6.6B2: Because there is no such key as A𝄫A double flat major or A𝄫A double flat minor, temporarily raise A𝄫A double flat by a whole step to A♮A natural.
  • Example 8.6.6B3: Write a minor seventh above A: G.
  • Example 8.6.6B1: Return to the original question. Since A𝄫A double flat is a whole step lower than A, also lower G by a whole step to G𝄫G double flat. Therefore, a minor seventh above A𝄫A double flat is G𝄫G double flat.
  • Alternatively, instead of raising A𝄫A double flat a whole step to A, you could raise A𝄫A double flat by a half step to A♭A flat. You would follow the same instructions but apply a half step instead of a whole step.

8.7   Shortcut: m3/M3

Half Steps

There is one final shortcut for minor thirds and major thirds. Once you have the thirds written down, you can count half steps. Although this method will eventually lead you to the correct answer, it is similar to using your fingers to solve addition problems. Because it is very time consuming, the author discourages using this method unless the intervals are extremely tricky.

• Minor third: three half steps
• Major third: four half steps

Again, you must remember to write in the thirds first.

8.8   Analysis: Delibes, Lakmé

Composers often harmonize a melody using thirds. Remember that a harmonic interval occurs when two notes are played at the same time. Harmonizing a melody involves sounding a different note or notes alongside the melody. Notice how Delibes harmonizes the melody in Example 8.8.2.

Example 8.8.2. Harmonized melody: Delibes, “Sous le dôme épai,” Lakmé (“Flower Duet”), Act 1, Scene 2^[17]

The two sopranos do not always change notes simultaneously. However, notice that whenever they do, not only are the lyrics the same, but they are always singing in thirds. In particular, every interval between the two voices in the last two measures is a third. Listen to the example and how beautiful the thirds sound together. It might seem that harmonizing a melody only with thirds would sound boring. But when we look more closely, we see why it is anything but dull (Example 8.8.3).

Example 8.8.3. Intervals in Delibes, “Sous le dôme épai,” Lakmé (“Flower Duet”), Act 1, Scene 2

Example 8.8.3 shows the precise intervals in measures 3-4. Notice that although they are all thirds, there is a good mix of both major thirds and minor thirds. The variety of interval qualities creates diversity within the same interval number.

Terms

• dyad
• harmonic interval
• harmonizing a melody
• interval
• interval number
• interval quality
  • perfect
  • major
  • minor
• melodic interval