Set (music)

# Set (music)

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Encyclopedia
A set in music theory
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

, as in mathematics and general parlance, is a collection of objects. In musical contexts
Set theory (music)
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...

the term is traditionally applied most often to collections of pitches or pitch-classes
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations
Rhythm
Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...

or timbres, for example.

A time-point set is a duration set where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes .

A set by itself does not necessarily possess any additional structure, such as an ordering
Order (mathematics)
Order in mathematics may refer to:-In algebra:*Order , the cardinality of a group or period of an element*Order, or degree of a polynomial*Order, or dimension of a matrix*Order , an algebraic structure*Ordered group...

. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called segments); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.

## Serial

In the theory of serial music, however, some authors (notably Milton Babbitt
Milton Babbitt
Milton Byron Babbitt was an American composer, music theorist, and teacher. He is particularly noted for his serial and electronic music.-Biography:...

) use the term "set" where others would use "row" or "series", namely to denote an ordered collection (such as a twelve-tone row
Twelve-tone technique
Twelve-tone technique is a method of musical composition devised by Arnold Schoenberg...

) used to structure a work. These authors speak of "twelve tone sets", "time-point sets", "derived sets", etc. (See below.) This is a different usage of the term "set" from that described above (and referred to in the term "set theory
Set theory (music)
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...

").

For these authors, a set form (or row form) is a particular arrangement of such an ordered set: the prime form (original order), inverse
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...

(upside down), retrograde (backwards), and retrograde inverse (backwards and upside down) .

A derived set is one which is generated or derived from consistent operations on a subset, for example Webern
Anton Webern
Anton Webern was an Austrian composer and conductor. He was a member of the Second Viennese School. As a student and significant follower of Arnold Schoenberg, he became one of the best-known exponents of the twelve-tone technique; in addition, his innovations regarding schematic organization of...

's Concerto, Op.24, in which the last three sets are derived from the first :
B B D E G F G E F C C A
Represented numerically as the integers 0 to 11:
0 11 3 4 8 7 9 5 6 1 2 10

The first set (B B D) being:
0 11 3 prime-form, interval-string = <-1 +4>
The second set (E G F) being the retrograde-inverse of the first, transposed up one semitone:
3 11 0 retrograde, interval-string = <-4 +1> mod 12

3 7 6 inverse, interval-string = <+4 -1> mod 12
+ 1 1 1
------
= 4 8 7
The third set (G E F) being the retrograde of the first, transposed up (or down) six semitones:
+ 6 6 6
------
9 5 6
And the fourth set (C C A) being the inverse of the first, transposed up one semitone:
0 11 3 prime form, interval-vector = <-1 +4> mod 12

0 1 9 inverse, interval-string = <+1 -4> mod 12
+ 1 1 1
-------
1 2 10

Each of the four trichords (3-note sets) thus displays a relationship which can be made obvious by any of the four serial row operations, and thus creates certain invariances. These invariances in serial music are analogous to the use of common-tones and common-chords in tonal music.

## Non-serial

The fundamental concept of a non-serial set is that it is an unordered collection of pitch classes (Rahn 1980, 27). The normal form of a set is the most compact ordering of the pitches in a set. Tomlin defines the "most compact" ordering as the one where, "the largest of the intervals between any two consecutive pitches is between the first and last pitch listed". For example, the set (0,2) (a major second
Major second
In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...

) is in normal form while the set (0,10) (a minor seventh
Minor seventh
In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the...

, the inversion of a major second) is not, its normal form being (10,0).

Rather than the "original" (untransposed, uninverted) form of the set the prime form may be considered the either the normal form of the set or the normal form of its inversion, whichever is more tightly packed.

Though set theorists usually consider sets of equal-tempered pitch classes, it is possible to consider sets of pitches, non-equal-tempered pitch classes, rhythmic onsets, or "beat classes" (Warburton 1988, 148; Cohn 1992, 149).

In music, a dyad is a set of two notes or pitches. Although most chords have three or more notes, in certain contexts a dyad may be considered to be a chord. The most common two-note chord is made from the interval of a perfect fifth, which may be suggestive of music of the Medieval or Renaissance...

s, three-element sets trichord
Trichord
In music theory, a trichord is a group of three different pitch classes found within a larger group . For example a continguous three note set from a musical scale or twelve-tone row. The term is derived by analogy from the 20th-century use of the word "tetrachord"...

In music and music theory, a triad is a three-note chord that can be stacked in thirds. Its members, when actually stacked in thirds, from lowest pitched tone to highest, are called:* the Root...

). Sets of higher cardinalities are called tetrachord
Tetrachord
Traditionally, a tetrachord is a series of three intervals filling in the interval of a perfect fourth, a 4:3 frequency proportion. In modern usage a tetrachord is any four-note segment of a scale or tone row. The term tetrachord derives from ancient Greek music theory...

Pentachord
A pentachord in music theory may be either of two things. In pitch-class set theory, a pentachord is defined as any five pitch classes, regarded as an unordered collection . In other contexts, a pentachord may be any consecutive five-note section of a diatonic scale...

Hexachord
In music, a hexachord is a collection of six pitch classes including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.-Middle Ages:...

• Permutation (music)
Permutation (music)
In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...

• Pitch interval
Pitch interval
In musical set theory, a pitch interval is the number of semitones 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:...

• Similarity relation
Similarity relation
In music, a similarity relation or pitch-class similarity is a comparison between sets of the same cardinality , based upon shared pitch class and/or interval class content....

• Forte number
Forte number
In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in The Structure of Atonal Music...