### 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) is 3 semitones:

- PI(0,3) = 3 - 0

While C4 to D♯5 ) is 15 semitones:

- PI(0,15) = 15 - 0

However, under #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:

- $\backslash operatorname\{ip\}\backslash langle\; x,y\backslash rangle\; =\; y-x$

and:

- $\backslash operatorname\{ip\}\backslash langle\; y,x\backslash rangle\; =\; x-y$

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:

- $\backslash operatorname\{ip\}(x,y)\; =\; |y-x|$
^{[4]}

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:

- $\backslash operatorname\{i\}\backslash langle\; x,y\backslash rangle\; =\; y-x$ (in modular 12 arithmetic)

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

## See also

## Sources