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 measure the number of semitones between pitches, and indicate direction with a plus or minus sign (+/−).
- Unordered pitch intervals measure the number of semitones between pitches, but do not indicate direction (no +/−).
- Ordered pitch class intervals measure the distance between two pitches in pitch class space (the clock face), always moving clockwise (ascending). This is similar to measuring time on a 12-hour clock: 9–5 is 8 hours, never 4.
- Interval classes (unordered pitch class intervals) represent the smallest possible distance in semitones between two pitch classes; therefore, an interval class is always ≤6.
- 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 in these different tonal contexts, even though they are the same keys on a piano keyboard ( ).
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.
| pitch intervals | pitch-class intervals | |
|---|---|---|
| ordered intervals | ordered pitch intervals | ordered pitch-class intervals |
| unordered intervals | unordered pitch intervals | unordered pitch-class intervals / interval classes (IC) |
Pitch Intervals (ordered and unordered)
Pitch intervals are the distance between pitches as measured in semitones, which is to say that octave is taken into consideration. Thus, the interval C4–E4 is 4: four semitones are between these notes. But if that E is moved up an octave (C4–E5), the interval becomes 16: four semitones between C and E, plus an octave (twelve semitones) between th