VIII. 20th- and 21st-Century Techniques

# Intervals in Integer Notation

Brian Moseley and Megan Lavengood

Key Takeaways

• In atonal music, intervals are usually measured in , rather than using tonal interval names.
• There are four types of interval: , , , and (unordered pitch class intervals).
• Ordered pitch intervals are as specific as possible: they measure specific pitches (in specific octaves) and represent the directionality of the interval.
• Interval classes are the most abstract type of interval: they represent the smallest possible distance between two pitch classes.

In tonal music, because intervals are dependent upon the pitches that create them, the consonance and dissonance of intervals is determined by tonality itself. Imagine the interval create by G and B♭, a minor third. In the context of G minor, this is a consonant interval. Respelled as G and A♯, perhaps in the context of B minor, it creates a dissonant augmented second. From a tonal perspective, the two intervals are different even though they are the same in isolation (Example 1).

Example 1. Even though B♭ and A♯ are the same key on the keyboard, the intervals of a minor third and an augmented second are distinct in tonal theory.

Contrast this with atonal music. Because atonal music has no tonality, the distinction between B♭ and A♯ no longer matters. For us, the intervals G–B♭ and G–A♯ are the same. For this reason, we will not use tonal interval names like “minor third.” Instead, we will measure the intervals by the number of semitones between the pitches or pitch classes.

We can describe intervals according to two types of information: pitches vs. pitch classes, and ordered vs. unordered intervals. Combined, this makes four types of intervals, summarized in Example 2. Each of these interval types is explained below.

pitch intervals (pi) pitch class intervals
ordered intervals ordered pitch intervals ordered pitch class intervals
unordered intervals unordered pitch intervals unordered pitch class intervals / interval classes (IC)

Example 2. Four interval types in atonal theory.

# Pitch intervals (pi) Example 3. The top row of examples shows ordered pitch intervals: the number of semitones between the pitches, with a plus or minus sign to indicate direction. The bottom row shows unordered intervals, which simply disregards the direction, and thus drops the plus or minus sign.

Pitch intervals are the distance between as measured in half steps, which is to say that octave is taken into consideration. Thus, the interval C4–E4 is 4: four half steps are between these notes. But if that E is moved up an octave (C4–E5), the interval becomes 16: four half steps between C and E, plus an octave (twelve half steps) between the lower E and the higher E.

Within pitch intervals, there are ordered and unordered variants. To create an , simply add a plus or minus sign, to indicate whether the interval is ascending or descending. , by contrast, do not indicate which direction the pitches move in—they are thus more suitable for harmonic intervals. The differences between ordered and unordered pitch intervals are summarized in Example 3.

# Ordered pitch-class intervals

Pitch-class intervals are the distance between as measured in semitones in pitch class space—that is, around the clockface. Returning to our C4–E5 interval, we are now interested just in the pitch classes C and E, without reference to a specific octave

measure the distance between pitch classes, always ascending. This is visualized most easily by picturing the twelve tones around a clock face, and measuring the interval by going around the circle clockwise. Thus, from C to E = 4, but E to C = 8 (Example 4).

# Interval classes (IC)

Unordered pitch-class intervals are usually called interval classes. is the smallest possible distance between two pitch classes. On the clock face, this means traveling either clockwise or counter-clockwise, whichever is shortest. Interval class is a useful concept because it relates intervals, their inversions, and any compound versions of those intervals. You should be able to connect this concept to the concept of pitch vs. pitch class: a pitch class is a pitch, its enharmonic respelling(s), and any octave displacements of those spellings.

This means that there are only six interval classes: 1, 2, 3, 4, 5, and 6. If you reach the pitch class interval of 7, it becomes shorter to move counter-clockwise, and 7 becomes 5. For the same reason, 8 becomes 4, 9 becomes 3, and so on. Both C–E and E–C are interval class 4 (Example 5). Example 5. Unordered pitch class intervals are also known as interval classes. Interval classes measure the smallest possible distance between pitch classes.

# Summary

Using various combinations of pitch interval, pitch-class interval, ordered, and unordered, we arrive at four different conceptions of interval.To wrap your mind around each of these and begin to understand their various analytical uses, think of them on a sliding scale of most concrete—the ordered pitch interval—to most abstract—the unordered pitch-class interval. You can find this related to other concepts in the Set Theory Quick Reference Sheet.

As you analyze atonal music, you will find that different types of interval are useful for describing different types of phenomena. This is exactly like how, in tonal music, it’s useful to distinguish between a 13th and a 6th in some situations, but not others.

• Straus, Joseph Nathan. 2016. Introduction to Post-Tonal Theory. 4th ed. Upper Saddle River, NJ: Prentice Hall.
Resources
Assignments
1. Intervals (.pdf, .docx). Asks students to identify interval types (integer notation) within pieces of music. 