I started posting this as a comment to Basic Concepts for Atonal Theory, but decided it needed its own life.
Important additional comments/clarifications regarding interval terminology:
Unordered and ordered pitch intervals are the same except that ordered pitch intervals include a + or – sign.
Pitch-class intervals can never be larger than 11, since a pitch-class assumes octave equivalence.
Ordered pitch-class intervals can be counted in either direction (up/down, clockwise or counter-clockwise). The ordered pitch-class interval from C to A is both +9 and -3. Usually the positive number is chosen (by convention).
Unordered pitch-class intervals drop the + and – minus signs. Therefore, the unordered pitch-class interval between C and A is (again) both 9 and 3. However, we choose the smaller number to name an unordered pitch-class interval, so we would only call it a 3.
By choosing the shortest distance to label an unordered pitch-class interval, we can say that an unordered pitch-class interval is the same thing as an interval class. An interval class includes all the pitch intervals that can be expressed as the same unordered pitch-class interval. For instance, unordered pitch intervals 15, 9, 3, 21, and 27 all belong to the interval class 3. (15 – 12 = 3; 9 is the inversion of 3; 21 = 21 – 12 = 9, which is the inversion of 3; 27 = 27 – 12 = 15 – 12 = 3) In this way, you can think of an interval class as being a collection of equivalent pitch intervals.
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