Getting Started
Building upon your knowledge of identifying simple intervals, this section explores how to manipulate those intervals. We will learn two fundamental processes: inversion, which flips an interval upside down, and compounding, which expands an interval beyond the space of an octave. These skills are essential for understanding how composers create variety in texture and harmony from the same basic materials.
What You Should Be Able to Do
Identify the inversion of any simple interval presented in notated music.
Describe the relationship between a simple interval and its compound equivalent.
Aurally distinguish between simple and compound intervals in performed music.
Convert a compound interval to its simple form, and a simple interval to its compound form.
Apply the rules of inversion to determine the precise size and quality of an inverted interval.
Key Concepts & Analysis
Our analysis of intervals is grounded in the principles of voice leading, which governs how individual melodic lines interact to create coherent music. Interval inversion and compounding are not just mathematical operations; they are fundamental voice-leading techniques used to alter the spacing and sonority between two notes.
Interval Inversion
An interval is the distance between two pitches. Interval inversion is the process of changing this distance by transposing the lower note up by one octave. This action effectively flips the interval, placing the formerly lower note on top. Every interval has a unique counterpart, its inversion, and the two have a complementary relationship.
The core principle of inversion is that an interval and its inversion always combine to equal one perfect octave. This creates a predictable, mathematical relationship that affects both the interval's numeric size and its quality.
Calculating Inversion:
Numeric Size (The "Rule of 9"): The numeric sizes of an interval and its inversion always sum to nine. For example, a 2nd inverts to a 7th (2+7=9), a 3rd inverts to a 6th (3+6=9), and a 4th inverts to a 5th (4+5=9). This rule also applies to the unison (1), which inverts to an octave (8).
Quality: The quality of the interval also changes in a predictable way:
Major (M) intervals invert to minor (m) intervals, and vice-versa.
Augmented (A) intervals invert to diminished (d) intervals, and vice-versa.
Perfect (P) intervals invert to Perfect (P) intervals.
From a voice-leading perspective, inversion is a way to change the vertical spacing between two parts without changing the pitch classes involved. For example, the interval C4 up to E4 is a Major 3rd. By inverting it (moving the C4 up to C5), the interval becomes E4 up to C5, a minor 6th. The notes are still C and E, but their relationship and the resulting sound have changed. This technique allows a composer to retain a harmonic flavor while creating a different melodic contour or fitting the notes into different vocal or instrumental ranges.
Compound Intervals
While the intervals discussed so far fit within an octave, music often uses much wider spacings.
A simple interval is an interval with a size of an octave or smaller (e.g., 2nd, 5th, 7th, octave).
A compound interval is an interval larger than an octave. It is formed by adding one or more octaves to a simple interval. For example, a Major 2nd (like C4 to D4) becomes a Major 9th (C4 to D5) when an octave is added to the upper note. The Major 9th is the compound version of a Major 2nd.
To identify a compound interval, you can find its simple equivalent by subtracting seven from its numeric size.
A 9th is a compound 2nd (9 - 7 = 2).
A 10th is a compound 3rd (10 - 7 = 3).
An 11th is a compound 4th (11 - 7 = 4).
A 12th is a compound 5th (12 - 7 = 5).
The quality of the compound interval is the same as its simple equivalent. A Major 10th is a compound Major 3rd; a Perfect 11th is a compound Perfect 4th.
In terms of voice leading, using compound intervals is a primary tool for creating textural variety. A melody accompanied by chords voiced with simple intervals (like 3rds and 5ths) will sound close and dense. The same melody accompanied by the same chords voiced with compound intervals (like 10ths and 12ths) will sound open, spacious, and resonant. Analyzing whether an interval is simple or compound is the first step in describing musical texture.
Data & Organization Tools
This table summarizes the predictable changes that occur during interval inversion. Use it as a quick reference for calculating both the size and quality of any inverted interval.
| Original Interval | Inverted Interval |
|---|---|
| Size (Rule of 9) | |
| Unison (1) | Octave (8) |
| Second (2) | Seventh (7) |
| Third (3) | Sixth (6) |
| Fourth (4) | Fifth (5) |
| Quality | |
| Perfect (P) | Perfect (P) |
| Major (M) | minor (m) |
| Augmented (A) | diminished (d) |
Evidence Bank
Interval Inversion: The procedure of moving the lower note of an interval up an octave to create a new interval. It is a fundamental tool for varying harmonic spacing.
Simple Interval: An interval of an octave or less. Simple intervals form the basis of close-position harmony.
Compound Interval: An interval larger than an octave. Compound intervals are used to create open, spacious textures.
"Rule of 9": A mnemonic for finding the numeric size of an inverted interval. The original size and the inverted size will always sum to 9.
Quality Inversion (Major/minor): The principle that Major intervals invert to minor intervals, and minor intervals invert to Major intervals. This preserves the diatonic nature of the interval within a key.
Quality Inversion (Perfect): The principle that Perfect intervals (unisons, 4ths, 5ths, octaves) always invert to other Perfect intervals. This reflects their unique acoustic stability.
Compound Equivalent: The simple interval that corresponds to a compound interval. To find it, subtract 7 from the compound interval's number (e.g., a 10th is a compound 3rd).
Skill Snapshots
Voice-Leading Rule: Inverting a consonant 3rd or 6th results in another consonant 6th or 3rd.
Effect: This allows a composer to change the spacing between two voices (e.g., from a M3 to a m6) while maintaining a stable, consonant harmony. It is a common way to achieve contrary motion.
Voice-Leading Rule: Expanding a simple interval to its compound equivalent displaces one voice by an octave.
Effect: This technique is used to change musical texture. Moving from a P5 to a P12 creates a more open and resonant sound, often used to move from a dense passage to a more majestic one.
Voice-Leading Rule: The inversion of a dissonant 2nd or 7th is another dissonant 7th or 2nd.
Effect: While the level of tension may change slightly, the fundamental dissonant character remains. This means that the rules of resolution for dissonances will apply to both an interval and its inversion.
Common Misconceptions & Clarifications
Confusing Size and Quality Calculation: Remember that finding an inversion is a two-step process. First, use the "Rule of 9" to find the new number. Second, apply the quality-change rule (M↔m, P↔P, A↔d). Do not try to do both at once.
Inverting from the Top Note: While moving the top note down an octave produces the same resulting interval, the standard definition is to move the bottom note up. Sticking to this single procedure prevents errors and confusion.
Identifying Compound Intervals: A compound interval is not just any large leap. It is specifically a simple interval plus an octave. When you hear or see a large interval like a 10th, your first analytical step should be to think of it as its simpler component: a 3rd.
The Unison and Octave: The unison (P1) and octave (P8) are inversions of each other. This fits both rules: 1+8=9, and Perfect inverts to Perfect.
Summary
Understanding interval inversion and compound intervals allows us to look beyond the basic name of an interval and see its function and potential. Inversion is a transformative process where an interval and its inversion combine to form a perfect octave, following predictable rules for both size (the "Rule of 9") and quality (Major becomes minor, Perfect remains Perfect). Compound intervals are expansions of simple intervals, created by adding an octave to create wider, more open spacing. Both are fundamental voice-leading techniques that composers use to manipulate musical texture, manage consonance and dissonance, and create melodic and harmonic variety.