VIII. 20th- and 21st-Century Techniques
Intervals in Integer Notation
Brian Moseley and Megan Lavengood
Key Takeaways
- When analyzing atonal music, intervals may be better understood as a number of semitones, rather than using tonal interval names.
- There are four types of interval: ordered pitch intervals, unordered pitch intervals, ordered pitch class intervals, and interval classes (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 created 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, then, the two intervals are different even though they are the same in isolation ( ).
Contrast this with atonal music. Because atonal music has no tonality, the distinction between B♭ and A♯ no longer matters. In this context, 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
. Each of these interval types is explained below.[table id=18 /]
Pitch Intervals (ordered and unordered)
Pitch intervals are the distance between pitches 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 ordered pitch interval, simply add a plus or minus sign to indicate whether the interval is ascending or descending. Unordered pitch intervals, 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 .
Ordered Pitch-Class Intervals
Pitch-class intervals are the distance between pitch classes as measured in semitones in pitch class space—that is, around the clock face. Returning to our C4–E5 interval, we are now interested just in the pitch classes C and E, without reference to a specific octave
Ordered pitch-class intervals measure the di