Discussion 2a - Identifying and Labeling Intervals

Class Discussion

How would you define interval size?

• By the spacing between intervals (How is this different than quality?)
• By how many notes you can fit between x note and y note (How would you define a note?): count up from the bottom of interval, counting the line/space occupied by the bottom note. Otherwise the interval ID will be incorrect
• If the bottom note is on a line and the next is also on a line, the interval number will be odd while opposite placements will be even

How would you define interval size without lines and spaces?

• Note names! Just like the lines and spaces you have to count the note you start on. Can be used with last lesson’s “there are only 7 note names” idea
• Even if there’s no clef, students with basic music knowledge will default to thinking in their main clef–this is something you can use

How would you define interval quality?

• Why is quality harder to define than interval?
  • Requires familiarity with tonal system
• Do enharmonically equivalent notes = the same interval?
  • No–they will be different distances apart from the other note

Chromatic vs diatonic intervals

• If the top note belongs to the scale of the bottom note the interval is diatonic. Otherwise it is chromatic
• Chromatic interval =/= “there’s an accidental”

What are our interval hierarchies?

• One built around P, one built around M/m
• Both had A/d
• P used on 1, 4, 5, 8 only
• M/m used for 2, 3, 6, 7 only

Difference between simple and compound?

• Range of the interval. Within an octave = simple interval, but anything exceeding an octave is a compound interval
• Is there a shortcut for connecting simple and compound intervals?
  • Subtracting 7 from the compound interval, not 8 or 9. 8 is an octave, and 9 is important for inversions. If this is breaking your brain, think of reducing an octave to a unison: P8 - 7 = P1
• Do simple and compound care about quality?
  • No, only size. Look solely at the number for simple vs compound

Harmonic vs melodic intervals

• Harmonic = two pitches that sound at the same time
• Melodic = two pitches sounding sequentially
• Is there a difference between the harmonic and melodic interval of a unison?
  • Yes! Depends on how they are written in your part
  • Augmented unisons also exist and that’s where things get a little confusing and very strict

Inversions

• How to invert?
  • Take either of the notes and drop them down?: Not quite, but you can take either note and move it an octave in the direction of the other note (ex. G4 and D5 to D5 and G5 OR D4 and G4)
• How do we get our inversions number?
  • Subtract the original interval number from 9
• How to invert quality?
  • M <-> m
  • d <-> A
  • P <-> P
• Must invert size and quality independently

Further Reading

From Open Music Theory

Defining an Interval

An interval is the distance between two pitches, usually measured as a number of steps on a scale.

A dyad is a pair of pitches sounding together (in other words, a two-note chord). Since a dyad is defined by the interval between the two pitches, dyads are often simply called intervals.

Thus, the term interval regularly refers both to the distance between two pitches on a scale and to a dyad whose pitches are separated by that distance.

Diatonic intervals

More commonly for tonal music, we are interested in the number of steps on the diatonic (major or minor) scale. This is a bit tricky—not because it’s difficult, but because it’s counter-intuitive. Unfortunately, the system is too old and well engrained to change it now! But once you get past the initial strangeness, diatonic intervals are manageable.

When identifying a diatonic interval, begin with the letter names only. That is, treat C, C-sharp, and C-flat all as C for the time being. Next, count the number of steps (different letters) between the two pitches in question, including both pitches in your count. This gives you the generic interval.

For example, from C4 to E4, counting both C and E, there are three diatonic steps (three letter names): C, D, E. Thus, the generic interval for C4–E4 is a third. The same is true for any C to any E: C#4 to E4, Cb4 to E#4, etc. They are all diatonic thirds.

Often more specificity is needed than generic intervals can provide. That specificity comes in the form of an interval’s quality. Combining quality with a generic interval name produces a specific interval.

There are five possible interval qualities:

• augmented (A)
• major (M)
• perfect (P)
• minor (m)
• diminished (d)

To obtain an interval’s quality, find both the generic interval and the chromatic interval. Then consult the following table to find the specific interval.

	unis.	2nd	3rd	4th	5th	6th	7th	oct.
i0	P1	d2
i1	A1	m2
i2		M2	d3
i3		A2	m3
i4			M3	d4
i5			A3	P4
i6				A4	d5
i7					P5	d6
i8					A5	m6
i9						M6	d7
i10						A6	m7
i11							M7	d8
i12							A7	P8

For example, C4–E4 is a generic third, and has a chromatic interval of i4. A third that encompasses four semitones is a major third (M3). Note that both generic interval and chromatic interval are necessary to find the specific interval, since there are multiple specific diatonic intervals for each generic interval and for each chromatic interval.

Note that some generic intervals can be augmented, perfect, or diminished, and other intervals can be augmented, major, minor, or diminished. There is no generic interval that can be both major/minor and perfect; if it can be major or minor, it cannot be perfect, and if it can be perfect, it cannot be major or minor. An augmented version of an interval is always one semitone wider than major or perfect; diminished is always one semitone smaller than minor or perfect.

Solfège can also help to determine the specific interval. Each pair of solfège syllables will have the same interval, no matter what the key, as long as it is clear which syllable is the lower pitch and which is the upper pitch. Memorizing the intervals between solfège pairs can help speed along your analysis of dyads as they appear in music. For example, knowing that do–mi, fa–la, and sol–ti are always major thirds and knowing that re–fa, mi–sol, la–do, and ti–re are always minor thirds will allow for faster analysis of dyads in major keys.

Chromatic intervals

The simplest way to measure intervals, particularly at the keyboard, is to count the number of half-steps, or semitones, between two pitches. To determine the chromatic interval between C4 and E4, for example, start at C4 and ascend the chromatic scale to E4, counting steps along the way: C#4, D4, D#4, E4. E4 is four semitones higher than C4. Chromatic intervals are notated with a lower-case i followed by an Arabic numeral for the number of semitones. C4–E4 is four semitones, or i4.

Compound intervals

The intervals discussed above, from unison to octave, are called simple intervals. Any interval larger than an octave is considered a compound interval. Take the interval C4 to E5. The generic interval is a tenth. However, it functions the same as C4 to E4 in almost all musical circumstances. Thus, the tenth C4–E5 is also called a compound third. A compound interval takes the same quality as the corresponding simple interval. If C4–E4 is a major third, then C4–E5 is a major tenth.

Interval inversion

In addition to C4–E4 and C4–E5, E4–C5 also shares a similar sound and musical function. In fact, any dyad that keeps the same two pitch classes but changes register will have a similar sound and function. However, the fact that E4–C5 has E as its lowest pitch instead of C means that it has a different generic interval: E4–C5 is a sixth, not a third. Because of that difference, it will also play a different musical function in some circumstances. However, there is no escaping the relationship.

Dyads formed by the same two pitch classes, but with different pitch classes on bottom and on top, are said to be inversions of each other, because the pitch classes are inverted. Likewise, the intervals marked off by those inverted dyads are said to be inversions of each other.

Again, take C4–E4 (major third) and E4–C5 (minor sixth). These two dyads have the same two pitch classes, but one has C on bottom and E on top, while the other has E on bottom and C on top. Thus, they are inversions of each other.

Three relationships exhibited by these two dyads hold for all interval inversions.

First, the chromatic intervals add up to 12. (C4–E4 = i4; E4–C5 = i8; i4 + i8 = i12) This is because the two intervals add up to an octave (with an overlap on E4).

Second, the two generic interval values add up to nine (a third plus a sixth, or 3 + 6). This is because the two intervals add up to an octave (8), and one of the notes is counted twice when you add them together. (Remember the counterintuitive way of counting off diatonic intervals, where the number includes the starting and ending pitches, and when combining inverted intervals, there is always one note that gets counted twice—in this case, E4.)

Lastly, the major interval inverts into a minor, and vice versa. This always holds for interval inversion. Likewise, an augmented interval’s inversion is always diminished, and vice versa. A perfect interval’s inversion is always perfect.

major ↔ minor
augmented ↔ diminished
perfect ↔ perfect

Interval inversion may seem confusing and esoteric now, but it will be an incredibly important concept for the study of voice-leading and the study of harmony.

Methods for learning intervals

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.

White-Key Method

The white-key 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.

All of the thirds are minor except for three: C-E, F-A, and G-B, which are major.

All of the fourths are perfect except for one: F-B, which is augmented.

Believe it or not, you now know all of the white-key intervals, as long as you understand the concept of interval inversion. 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).

Once you’ve mastered the white-key intervals, you can figure out any other interval by adjusting the size accordingly.

Melodic and harmonic intervals

The last distinction between interval types to note is melodic v. harmonic intervals. This distinction is simple. If the two pitches of a dyad sound at the same time (a two-note chord), the interval between them is a harmonic interval. If the two pitches in question are sounded back-to-back (as in a melody), the interval between them is a melodic interval. This distinction is important in voice-leading, where different intervals are preferred or forbidden in harmonic contexts than in melodic contexts. The difference is also important for listening, as hearing melodic and harmonic intervals of the same quality requires different techniques.

Consonance and dissonance

Intervals are categorized as consonant or dissonant based on their sound (how stable, sweet, or harsh they sound), how easy they are to sing, and how they best function in a passage (beginning, middle, end; between certain other intervals; etc.). Different standards apply to melody and harmony. The following categories will be essential for your work in strict voice-leading, and they will be a helpful guide for free composition and arranging work, as well.

Melodic consonance and dissonance

The following melodic intervals are consonant, and can be used in strict voice-leading both for successive pitches and as boundaries of stepwise progressions in a single direction:

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

All other melodic intervals are dissonant, and must be avoided for successive pitches and as boundaries of stepwise progressions in a single direction, including:

• All augmented and diminished intervals (including those that are enharmonically equivalent to consonant intervals, such as A2 and A1)
• All sevenths

Harmonic consonance and dissonance

The following harmonic intervals are imperfect consonances, and can be used relatively freely in strict voice-leading (except for beginnings and endings):

• Major and minor thirds
• Major and minor sixths

The following harmonic intervals are perfect consonances, and must be used with care in limited circumstances in strict voice-leading:

• All perfect intervals except the perfect fourth (P1, P5, P8)

All other harmonic intervals are dissonant, and must be employed in very specific ways in strict voice-leading, including:

• All diatonic steps (M2, m2)
• All augmented and diminished intervals (including those that are enharmonically equivalent to consonant intervals, such as A2 and A1)
• All sevenths
• Perfect fourths