Moving slowly into non-serial atonality, let’s start with the basics.
Octave equivalence is really a hold over from basic music theory. We hear pitches in octaves as being functionally the same. C in one register is the same analytically as C in another register. What we’re really doing is still pointing out the difference between a pitch and a pitch class. A pitch refers to a specific, single note in a single register — i.e., C4. Pitch class refers to all C’s, no matter what the register.
While we’ve always agreed that C# and Db sound the same, enharmonic spellings are used for certain tonal functions in tonal harmony. In D major, C# is the leading tone and implies movement up to D. In the same key, Db would imply downward motion from D through Db to C-natural. In atonal theory, C# and Db are exactly the same. Any choice of enharmonic spelling usually is based on local significance (readability).
Enharmonic equivalence means that it is most useful to think of pitches and intervals in terms of integer notation. You can find the review post here. Get used to writing out (or having a copy handy that you can write on) the pitch-integer clockface.
All intervals will be measured in half steps, using integers to denote number of half steps.
Pitch Intervals (ip)
Pitch intervals describe the actual distance between two pitches (not pitch classes). For example, C4 up to D5 is a pitch interval of 14 (14 half steps). C4 up to D4 is a pitch interval of 2 (2 half steps).
A pitch interval can be unordered or ordered. Ordered intervals use a + to indicate up and a – (minus) to indicate down. For our examples above, the unordered 14 changes to +14, and the unordered 2 changes to +2. If I move from F3 down to C#3 the ordered interval is a -4. The relationship of unordered to ordered is like the relationship of speed to velocity. Both measurements tell you how fast something is moving, but velocity tells you what direction it is moving in. Speed does not contain direction information.
Moving clockwise on the clockface is moving up (+) and counter-clockwise movement is moving down (-).
Pitch-class intervals describe the distance between two pitch classes, and will always reduce the interval to something less than an octave (11 or less) and use positive numbers. Remember that with octave equivalence one octave is the same as a unison. Also, with octave equivalence it doesn’t matter what register either of the pitches is in, so we have to have some consistent way of counting.
With octave equivalence we can use modulo math to reduce intervals larger than 11 down to the range of less than an octave. Modulo math simply divides by a base number and gives the remainder as the answer. For our purposes we will use Mod 12. The pitch-class interval for the pitch interval 14 becomes 14(Mod 12) = 2. (12 divides into 14 once, with a remainder of 2.)
Important: Since ordered pitch-class intervals will always be positive and between 0 and 11, the easiest way to determine an ordered pitch-class interval is to use the clockface and always count in the clockwise (positive) direction. If you want to just think in half-steps, you will always count up to the second pitch class.
Unordered pitch-class intervals will always be determined by counting around the clockface in the shortest direction.
Another term for an unordered pitch-class interval is an interval class.
- Because of octave equivalency, intervals larger than an octave are the same as the corresponding intervals less than an octave.
- The premise of unordered pitch-class intervals considers interval complements to belong to the same interval (you can describe the same interval by travelling in either direction around the clockface).
- If you travel the shortest distance between two pitch classes you will never have an interval larger than 6.
The concept of an interval class means that any octave-equivalent interval or interval complement is essentially the same interval. For example, pitch intervals of a 1, 11, and 13 (among others) all belong to the interval class 1. The interval class 3 includes pitch intervals of 3, 9, 18, and 21.
Interval classes are not ordered.