Derived row
Encyclopedia
The term "partition" is also French for the sheet music
of a transcription
.
In music
using the twelve-tone technique
, derivation is the construction of a row through segments. A derived row is a tone row
whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton Webern
often used derived
rows in his pieces. A partition is a segment created from a set through partitioning.
of any number of pitch class
es that is a divisor
of 12, the most common being the first three pitches or a trichord
. This segment may then undergo transposition
, inversion
, retrograde
, or any combination to produce the other parts of the row (in this case, the other three segments).
One of the side effects of derived rows is invariance. For example, since a segment may be equivalent
to the generating segment inverted and transposed, say, 6 semitone
s, when the entire row is inverted and transposed six semitones the generating segment will now consist of the pitch classes of the derived segment.
Here is a row derived from a trichord
taken from Webern's Concerto
:
B, B, D, E, G, F, G, E, F, C, C, A
P represents the original clavichord, RI, retrograde and inversion, R retrograde, and I inversion.
The entire row, if B=0, is:
For instance, the third trichord:
is the first trichord:
backwards:
and transposed 6
difference.
In music
using the twelve-tone technique
a partition is, "a collection of disjunct, unordered pitch-class sets that comprise an aggregate
." It is a method of creating segments from sets
, most often through registral
difference. The opposite of derivation
used in derived rows.
More generally, in musical set theory partitioning is the division of the domain of pitch class sets into types, such as transpositional type, see equivalence class and cardinality.
Partition is also an old name for types of compositions in several parts; there is no fixed meaning, and in several cases the term was reportedly interchanged with various other terms.
A cross-partition is, "a two-dimensional configuration of pitch classes whose columns are realized as chords, and whose rows are differentiated from one another by registral, timbral, or other means." This allows, "slot-machine transformations that reorder the vertical trichords but keep the pitch classes in their columns."
A mosaic is, "a partition that divides the aggregate into segments of equal size," according to Martino (1961). "Kurth 1992 and Mead 1988 use mosaic and mosaic class in the way that I use partition and mosaic," are used here. However later, he says that, "the DS determines the number of distinct partitions in a mosaic, which is the set of partitions related by transposition and inversion."
of the constituent pitch class
sets
of a partition. For trichord
s and hexachord
s combined see Alegant 1993, Babbitt 1955, Dubiel 1990, Mead 1994, Morris and Alegant 1988, Morris 1987, and Rouse 1985; cited in.
Sheet music
Sheet music is a hand-written or printed form of music notation that uses modern musical symbols; like its analogs—books, pamphlets, etc.—the medium of sheet music typically is paper , although the access to musical notation in recent years includes also presentation on computer screens...
of a transcription
Transcription (music)
In music, transcription can mean notating a piece or a sound which was previously unnotated, as, for example, an improvised jazz solo. Further examples include ethnomusicological notation of oral traditions of folk music, such as Béla Bartók's and Ralph Vaughan Williams' collections of the national...
.
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
using the twelve-tone technique
Twelve-tone technique
Twelve-tone technique is a method of musical composition devised by Arnold Schoenberg...
, derivation is the construction of a row through segments. A derived row is a tone row
Tone row
In music, a tone row or note row , also series and set, refers to a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.-History and usage:Tone rows are the basis of...
whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton 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...
often used derived
Derivation
Derivation may refer to:* Derivation , a function on an algebra which generalizes certain features of the derivative operator* Derivation * Derivation in differential algebra, a unary function satisfying the Leibniz product law...
rows in his pieces. A partition is a segment created from a set through partitioning.
Derivation
Rows may be derived from a sub-setSet 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...
of any number of pitch class
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...
es that is a divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
of 12, the most common being the first three pitches or a 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"...
. This segment may then undergo transposition
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
, inversion
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
, retrograde
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...
, or any combination to produce the other parts of the row (in this case, the other three segments).
One of the side effects of derived rows is invariance. For example, since a segment may be equivalent
Equivalence
Equivalence or equivalent may refer to:*In chemistry:**Equivalent **Equivalence point**Equivalent weight*In computing:**Turing equivalence *In ethics:**Moral equivalence*In history:...
to the generating segment inverted and transposed, say, 6 semitone
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, when the entire row is inverted and transposed six semitones the generating segment will now consist of the pitch classes of the derived segment.
Here is a row derived from a 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"...
taken from Webern's Concerto
Concerto (Webern)
Anton Webern's Concerto for Nine Instruments, Op. 24 is a twelve-tone concerto for nine instruments: flute, oboe, clarinet, horn, trumpet, trombone, violin, viola, and piano; containing three movements: I. Etwas lebhaft, II. Sehr langsam, and III...
:
P represents the original clavichord, RI, retrograde and inversion, R retrograde, and I inversion.
The entire row, if B=0, is:
- 0, 11, 3, 4, 8, 7, 9, 5, 6, 1, 2, 10.
For instance, the third trichord:
- 9, 5, 6
is the first trichord:
- 0, 11, 3
backwards:
- 3, 11, 0
and transposed 6
- 3+6, 11+6, 0+6 = 9, 5, 6 mod 12Modular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
.
Partition and mosaic
The opposite is partitioning, the use of methods to create segments from entire sets, most often through registralRegister (music)
In music, a register is the relative "height" or range of a note, set of pitches or pitch classes, melody, part, instrument or group of instruments...
difference.
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
using the twelve-tone technique
Twelve-tone technique
Twelve-tone technique is a method of musical composition devised by Arnold Schoenberg...
a partition is, "a collection of disjunct, unordered pitch-class sets that comprise an aggregate
Chromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
." It is a method of creating segments from sets
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...
, most often through registral
Register (music)
In music, a register is the relative "height" or range of a note, set of pitches or pitch classes, melody, part, instrument or group of instruments...
difference. The opposite of derivation
Derivation
Derivation may refer to:* Derivation , a function on an algebra which generalizes certain features of the derivative operator* Derivation * Derivation in differential algebra, a unary function satisfying the Leibniz product law...
used in derived rows.
More generally, in musical set theory partitioning is the division of the domain of pitch class sets into types, such as transpositional type, see equivalence class and cardinality.
Partition is also an old name for types of compositions in several parts; there is no fixed meaning, and in several cases the term was reportedly interchanged with various other terms.
A cross-partition is, "a two-dimensional configuration of pitch classes whose columns are realized as chords, and whose rows are differentiated from one another by registral, timbral, or other means." This allows, "slot-machine transformations that reorder the vertical trichords but keep the pitch classes in their columns."
A mosaic is, "a partition that divides the aggregate into segments of equal size," according to Martino (1961). "Kurth 1992 and Mead 1988 use mosaic and mosaic class in the way that I use partition and mosaic," are used here. However later, he says that, "the DS determines the number of distinct partitions in a mosaic, which is the set of partitions related by transposition and inversion."
Inventory
The first useful characteristic of a partition, an inventory is the set classes produced by the unionUnion (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of the constituent pitch class
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...
sets
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...
of a partition. For 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"...
s and hexachord
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:...
s combined see Alegant 1993, Babbitt 1955, Dubiel 1990, Mead 1994, Morris and Alegant 1988, Morris 1987, and Rouse 1985; cited in.