Ordered pitch interval

In musical set theory, ordered pitch interval is the distance in semitones between two pitches upward or downward. For instance, the interval from C to G upward is 7, but the interval from G to C downward is −7. Using integer notation and modulo 12, ordered pitch interval, "ip", may be defined, for any two pitches "x" and "y", as:
*$operatorname\{ip\}langle\; x,y\; angle\; =\; y-x$and:
*$operatorname\{ip\}langle\; y,x\; angle\; =\; x-y$the other way.

One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, similar to the interval of tonal theory. This may be defined as:
*$operatorname\{ip\}(x,y)\; =\; |y-x|$

The interval between pitch classes may be measured with ordered and unordered pitch class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. Thus the ordered pitch class interval, i<"x", "y">, may be defined as:
*$operatorname\{i\}langle\; x,y\; angle\; =\; y-x$ (in modular 12 arithmetic)

For unordered pitch class interval, see interval class.