# 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 C4 to DTemplate:Music4 Play (help·info) is 3 semitones:

- PI(0,3) = 3 - 0

While C4 to DTemplate:Music5 Play (help·info) is 15 semitones:

- PI(0,15) = 15 - 0

However, under octave equivalence these are the same pitches (DTemplate:Music4 & DTemplate:Music5, Play (help·info)), thus the #Pitch-interval class may be used.

## 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:

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

## See also

## Sources

- ↑
^{1.0}^{1.1}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.
- ↑
^{3.0}^{3.1}John Rahn,*Basic Atonal Theory*(New York: Longman, 1980), 21. ISBN 9780028731605. - ↑ John Rahn,
*Basic Atonal Theory*(New York: Longman, 1980), 22.