In
musical set theory
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the ...
, a pitch interval (PI or ip) is the number of
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.
It is defined as the interval between two adjacent no ...
s that separates one
pitch from another, upward or downward.
[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. .]
They are notated as follows:
:PI(''a'',''b'') = ''b'' − ''a''
For example C
4 to D
4 is 3 semitones:
:PI(0,3) = 3 − 0
While C
4 to D
5 is 15 semitones:
:PI(0,15) = 15 − 0
However, under
octave equivalence
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
these are the same pitches (D
4 & D
5, ), 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.
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
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave posit ...
and
modulo 12, ordered pitch interval, ''ip'', may be defined, for any two pitches ''x'' and ''y'', as:
*
and:
*
the other way.
John Rahn
John Rahn, born on February 26, 1944, in New York City, is a music theorist, composer, bassoonist, and Professor of Music at the University of Washington School of Music, Seattle. A former student of Milton Babbitt and Benjamin Boretz, he was edito ...
, ''Basic Atonal Theory'' (New York: Longman, 1980), 21. .
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:
*
[John Rahn, ''Basic Atonal Theory'' (New York: Longman, 1980), 22.]
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, may be defined as:
*
(in modular 12 arithmetic)
Ascending intervals are indicated by a positive value, and descending intervals by a negative one.
See also
*
Interval class
In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'" (; ), is the shortest distance in pitch ...
References
{{Set theory (music)
Intervals (music)
Musical set theory