Pitch interval
Encyclopedia
In musical set theory, a pitch interval (PI or ip) is the number of semitone
s that separates one pitch
from another, upward or downward.
They are notated as follows:
For example C4 to D4 is 3 semitones:
While C4 to D5 is 15 semitones:
However, under octave equivalence these are the same pitches (D4 & D5), thus the #Pitch-interval class may be used.
.
The PIC is notated and related to the PI thus:
12, ordered pitch interval, ip, may be defined, for any two pitches x and y, as:
and:
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:
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.
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
s that separates one pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
from another, upward or downward.
They are notated as follows:
- PI(a,b) = b - a
For example C4 to D4 is 3 semitones:
- PI(0,3) = 3 - 0
While C4 to D5 is 15 semitones:
- PI(0,15) = 15 - 0
However, under octave equivalence these are the same pitches (D4 & D5), 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 twelveModular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
.
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 moduloModular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
12, ordered pitch interval, ip, may be defined, for any two pitches x and y, as:
and:
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:
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.