Simple vs. Compound Intervals

Introduction

In our last session, we explored the elegant symmetry of interval inversion, seeing how intervals exist in complementary pairs within the confines of an octave. This was described through simple rules, the physics of frequency ratios, and the formal structure of modular arithmetic. Today, we will expand our view beyond the octave to understand how musical space is structured on a larger scale.

This lesson addresses the distinction between simple and compound intervals. You'll learn to identify and name intervals that span more than an octave and understand their relationship to the simple intervals you already know. For your work in electronic music, this is crucial for creating wide, spacious harmonic textures—think of ambient pads or sweeping arpeggios—and it lays the groundwork for understanding the extended chords (9ths, 11ths, 13ths) that add so much color to house, techno, and jazz-influenced electronic music.

Simple vs. Compound Intervals: The Formal Definition

First, let's establish a clear definition. The distinction is straightforward:

Simple Intervals: These are intervals that are less than or equal to an octave in size. All the intervals we have focused on so far—from the minor second to the perfect octave—are simple intervals.
Compound Intervals: These are intervals that are larger than an octave.

Essentially, a compound interval is simply a simple interval with one or more octaves added. The most common compound intervals are those within two octaves.

Interval (music)

The Wikipedia article on musical intervals provides a concise definition of this concept.

Please read the brief section under the heading "Simple and compound," starting from the definition. The accompanying image of a simple vs. compound major third is a very effective visual aid.

From Simple to Compound: The "Rule of 7"

Since a compound interval is just an extended version of a simple one, there's a predictable relationship between them. The quality of the interval (Perfect, Major, Minor, etc.) remains the same. A compound version of a Major third will also be Major. The number, however, changes.

To find the number of a compound interval, you simply add 7 to the number of the simple interval.

A 2nd becomes a 9th (2 + 7 = 9)
A 3rd becomes a 10th (3 + 7 = 10)
A 4th becomes an 11th (4 + 7 = 11)
A 5th becomes a 12th (5 + 7 = 12)
A 6th becomes a 13th (6 + 7 = 13)
A 7th becomes a 14th (7 + 7 = 14)
An octave (8) becomes a 15th (8 + 7 = 15), also known as a double octave.

This is why a Major 3rd becomes a Major 10th, and a Perfect 5th becomes a Perfect 12th. The image below shows how these are represented in standard notation. Note the alternative naming system, which makes the relationship explicit (e.g., "compound major 2nd" for a major 9th).

Compound intervals shown on a musical staff, illustrating how they are named as extensions of simple intervals beyond the octave.

Intervals – Open Music Theory

The "Open Music Theory" text offers a very clear explanation of this relationship.

Focus on the section titled "Compound Intervals." It directly states the "add 7" rule and lists the most common transformations.

The Mathematical Framework

You might wonder why we add 7 and not 8 (the number of notes in an octave). This is an artifact of inclusive counting in music. When we stack an interval on top of an octave, the top note of the octave also serves as the bottom note of the new interval, so it's only counted once.

Given your background, we can state this more formally. Let $D N_{s}$ be the diatonic number of a simple interval, and $D N_{o}$ be the number for an octave (8). The number of the resulting compound interval, $D N_{c}$ , is given by:

D N_{c} = (D N_{s} - 1) + (D N_{o} - 1) + 1 = D N_{s} + D N_{o} - 1

Since $D N_{o} = 8$ , this simplifies to:

D N_{c} = D N_{s} + 7

This formula neatly confirms the "add 7" rule.

We can also view this through two other lenses:

Semitones (Modular Arithmetic): A simple interval can be represented by a number of semitones, $x$ , where $0 \leq x \leq 12$ . Its compound equivalent is simply $x + 12 n$ , where $n$ is the number of added octaves. For example, a major third is 4 semitones. A major tenth is $4 + 12 = 16$ semitones.
Frequency Ratios (Physics): If a simple interval has a frequency ratio of $r$ , adding an octave corresponds to multiplication by 2. The compound interval's ratio is $r \times 2^{n}$ . For example, a perfect fifth (P5) has a ratio of approximately 3/2. A perfect twelfth (P12), which is a P5 plus an octave, has a ratio of $(3/2) \times 2 = 3/1$ .

The following chart visually connects the piano keyboard layout to frequency ranges. You can see how compound intervals naturally span across these ranges, from the bass fundamentals to the midrange and high-end harmonics, creating a sense of space and breadth in the sound.

This chart aligns the piano keyboard with frequency octaves and their common labels in audio production, such as "Sub Bass," "Midrange," and "High End."

Application: The Sound of Compound Intervals

Knowing the names is one thing; understanding their musical function is another. Compound intervals are essential for creating richer, more complex harmonies. Adding a 9th, 10th, or 11th on top of a basic triad completely changes its color and emotional weight.

Midi Music Theory 22 - Compound Intervals

The video "Midi Music Theory 22 - Compound Intervals" by Composing Gloves provides a great demonstration of this concept within a DAW environment.

First, watch from the introduction to understand the basic concept of extending intervals past the octave. Then, pay close attention to the segment from the chord demonstration, where the host plays a basic chord and then adds compound intervals (a 9th, a 10th, an 11th) on top. Listen carefully to how each addition alters the harmonic texture.

As you heard, these larger intervals form the basis of what are called "extended" or "upper-structure" chords, a topic we will explore in depth in a later module. They are fundamental to adding sophistication and space to your productions.

Practice

Let's solidify this concept. For each interval below, provide its corresponding simple or compound equivalent.

Simple Interval: Minor 6th
- What is its compound equivalent?
Simple Interval: Perfect 4th
- What is its compound equivalent?
Compound Interval: Major 9th
- What is its simple equivalent?
Compound Interval: Perfect 12th
- What is its simple equivalent?

Answers:

Minor 13th (m6 -> m13)
Perfect 11th (P4 -> P11)
Major 2nd (M9 -> M2)
Perfect 5th (P12 -> P5)

Conclusion

In this lesson, we have extended our understanding of intervals to encompass distances greater than an octave. You have seen that this seemingly complex topic is governed by simple, logical rules rooted in the same mathematical and physical principles we have discussed before.

Key Takeaways:

Simple vs. Compound: Simple intervals are an octave or smaller; compound intervals are larger.
Relating Intervals: A compound interval is a simple interval plus one or more octaves. Its quality remains the same, and its number is found by adding 7 to the simple interval's number.
The "Why": This relationship can be formally described by the formula $D N_{c} = D N_{s} + 7$ , by adding 12 semitones, or by multiplying the frequency ratio by 2.
The Sound: Compound intervals are used to create wider, more spacious voicings and form the upper extensions of complex chords, adding harmonic richness.

Now that you can identify and distinguish all types of intervals, both simple and compound, our next step will be to put them into direct practice. In the following lesson, we will move into the DAW to program simple two-voice melodic and harmonic phrases, training your ear to connect the theoretical names with their actual sound in a musical context.

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Introduction