# Pitch interval

In musical set theory, a **pitch interval** (**PI** or **ip**) is the number of semitones that separates one pitch from another, upward or downward.[1]

They are notated as follows:[1]

- PI(
*a*,*b*) =*b*−*a*

For example C_{4} to D♯_{4}

- PI(0,3) = 3 − 0

While C_{4} to D♯_{5}

- PI(0,15) = 15 − 0

However, under octave equivalence these are the same pitches (D♯_{4} & D♯_{5},

## Pitch-interval class

In musical set theory, a **pitch-interval class** (**PIC**, also **ordered pitch class interval** and **directed pitch class interval**) is a pitch interval modulo twelve.[2]

The PIC is notated and related to the PI thus:

- PIC(0,15) = PI(0,15) mod 12 = (15 − 0) mod 12 = 15 mod 12 = 3

## Equations

Using integer notation and modulo 12, ordered pitch interval, *ip*, may be defined, for any two pitches *x* and *y*, as:

and:

the other way.[3]

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:

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:

- (in modular 12 arithmetic)

Ascending intervals are indicated by a positive value, and descending intervals by a negative one.[3]

## See also

## Sources

- Schuijer, Michiel (2008).
*Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts*, Eastman Studies in Music 60 (Rochester, NY: University of Rochester Press, 2008), p. 35. ISBN 978-1-58046-270-9. - Schuijer (2008), p.36.
- John Rahn,
*Basic Atonal Theory*(New York: Longman, 1980), 21. ISBN 9780028731605. - John Rahn,
*Basic Atonal Theory*(New York: Longman, 1980), 22.