Read my post from last year. It has a good description of all the interval counting concepts for atonal theory. What is included in this post is a brief review.
You can download a sheet of “clock faces.” ( clockfacePage )
Interval size is always determined by counting half steps, and using integers to notate.
Inverting an interval is the same as subtracting the interval size (in half steps) from 12.
Pitch intervals describe the distance between specific pitches (which includes register). C4 to D4 is 2. C4 to D5 is 14.
Ordered pitch intervals also describe the direction between specific pitches. + is up; – is down. C4 to D5 is +14. D5 to C4 is -14.
Pitch class intervals remove register from the description, so they will always be 11 or less in size.
Ordered pitch class intervals don’t use a + or -. They will always be determined by counting up from the first pitch class to the second pitch class. C4 to D5 is a pitch class interval of C to D. Since C comes first, the ordered pitch class interval starts on C and counts up to D (2). D5 to C4 is D to C. Start on D and count up to C (10). You can also count clockwise around the clock face.
Unordered pitch class intervals are always counted in the closest direction (around the clock). In unordered pitch class interval terms, C to D and D to C are the same interval. Unordered pitch class intervals are always 6 or less. Anything above 6 can be inverted to something less than 6.
Interval classes are the same as unordered pitch class intervals.
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