In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates (all twelve tones).[1] Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."[1]
Combinatoriality is a side effect of derived rows, where the initial segment or set may be combined with its transformations (T,R,I,RI) to create an entire row. "Derivation refers to a process whereby, for instance, the initial trichord of a row can be used to arrive at a new, 'derived' row by employing the standard twelve-tone operations of transposition, inversion, retrograde, and retrograde-inversion."[2]
Combinatorial properties are not dependent on the order of the notes within a set, but only on the content of the set, and combinatoriality may exist between three tetrachordal and between four trichordal sets, as well as between pairs of hexachords,[3] and six dyads.[4] A complement in this context is half of a combinatorial pitch class set and most generally it is the "other half" of any pair including pitch class sets, textures, or pitch range.
Definition
Most generally complementation is the separation of pitch-class collections into two complementary sets, one containing the pitch classes not in the other.[1] More restrictively complementation is "the process of pairing entities on either side of a center of symmetry".[5]
The term, "'combinatorial' appears to have been first applied to twelve-tone music by Milton Babbitt" in 1950,[7] when he published a review of René Leibowitz's books Schoenberg et son école and Qu’est-ce que la musique de douze sons?[8] Babbitt also introduced the term derived row.[2]
Hexachordal combinatoriality
Combinatorial all-trichord hexachords from Elliott Carter's Piano Concerto, mm. 59–60
A 12-tone row has hexachordal combinatoriality with another 12-tone row if their respective first (as well as second, because a 12-tone row itself forms an aggregate by definition) hexachords form an aggregate.
There are four main types of combinatoriality. A hexachord may be:
Prime (transpositional) combinatoriality of a hexachord refers to the property of a hexachord whereby it forms an aggregate with one or more of its transpositions. Alternatively, transpositional combinatoriality is the lack of shared pitch classes between a hexachord and one or more of its transpositions. For example, 0 2 4 6 8 t, and its transposition up one semitone (+1): 1 3 5 7 9 e, have no notes in common.
Retrograde hexachordal combinatoriality is considered trivial, since any row has retrograde hexachordal combinatoriality with itself (all tone rows have retrograde combinatoriality).
Inversional combinatoriality is a relationship between two rows, a principal row and its inversion. The principal row's first half, or six notes, are the inversion's last six notes, though not necessarily in the same order. Thus, the first half of each row is the other's complement. The same conclusion applies to each row's second half as well. When combined, these rows still maintain a fully chromatic feeling and don't tend to reinforce certain pitches as tonal centers as might happen with freely combined rows. For example, the row from Schoenberg's Moses und Aron, above contains: 0 1 4 5 6 7, this inverts to: 0 e 8 7 6 5, add three = 2 3 8 9 t e.
Retrograde-inversional combinatoriality is a lack of shared pitches between the hexachords of a row and its retrograde-inversion.
Babbitt also described the semi-combinatorial row and the all-combinatorial row, the latter being a row which is combinatorial with any of its derivations and their transpositions. Semi-combinatorial sets are sets whose hexachords are capable of forming an aggregate with one of its basic transformations (R, I, RI) transposed. There are thirteen hexachords that are semi-combinatorial by inversion only.
(0) 0 1 2 3 4 6 // e t 9 8 7 5 (1) 0 1 2 3 5 7 // e t 9 8 6 4 (2) 0 1 2 3 6 7 // e t 9 8 5 4 (3) 0 1 2 4 5 8 // e t 9 7 6 3 (4) 0 1 2 4 6 8 // e t 9 7 5 3 (5) 0 1 2 5 7 8 // e t 9 6 4 3 (6) 0 1 3 4 6 9 // e t 8 7 5 2 (7) 0 1 3 5 7 9 // e t 8 6 4 2 (8) 0 1 3 5 8 9 // 7 6 4 2 e t (9) 0 1 3 6 7 9 // e t 8 5 4 2 (10) 0 1 4 5 6 8 // 3 2 e t 9 7 (11) 0 2 3 4 6 8 // 1 e t 9 7 5 (12) 0 2 3 5 7 9 // 1 e t 8 6 4
Any hexachord which contains a zero in its interval vector possesses transpositional combinatoriality (in other words: to achieve combinatoriality a hexachord cannot be transposed by an interval equaling a note it contains). For example, there is one hexachord which is combinatorial by transposition (T6):
(0) 0 1 3 4 5 8 // 6 7 9 t e 2
Neither hexachord contains tritones.
Gruppen's main first-order all-combinatorial tone row, though this property is not exploited compositionally in that work."Ode-to-Napoleon" hexachord in prime form One of Babbitt's six all-combinatorial hexachord "source sets".
All-combinatorial sets are sets whose hexachords are capable of forming an aggregate with any of its basic transformations transposed. There are six source sets, or basic hexachordally all-combinatorial sets, each hexachord of which may be reordered within itself:
(A) 0 1 2 3 4 5 // 6 7 8 9 t e (B) 0 2 3 4 5 7 // 6 8 9 t e 1 (C) 0 2 4 5 7 9 // 6 8 t e 1 3 (D) 0 1 2 6 7 8 // 3 4 5 9 t e (E) 0 1 4 5 8 9 // 2 3 6 7 t e (F) 0 2 4 6 8 t // 1 3 5 7 9 e
Note: t = 10, e = 11.
Because the first three sets (A, B, and C) each satisfy all four criteria for just one transpositional value, set D satisfies them for two transpositional values, E for three values, and F, for six transpositions, Babbitt designates these four groups as "first-order", "second-order", "third-order", and "sixth-order" all-combinatorial hexachords, respectively.[13] Notice that the first set, set "A," is the first six notes of an ascending chromatic scale, and that the last set, set "F," is a whole-tone scale.[14]
Combinatoriality may be used to create an aggregate of all twelve tones, though the term often refers simply to combinatorial rows stated together.
Hexachordal combinatoriality is a concept in post-tonal theory that describes the combination of hexachords, often used in reference to the music of the Second Viennese school. In music that consistently utilizes all twelve chromatic tones (particularly twelve-tone and serial music), the aggregate (collection of all 12 pitch classes) may be divided into two hexachords (collections of 6 pitches). This breaks the aggregate into two smaller pieces, thus making it easier to sequence notes, progress between rows or aggregates, and combine notes and aggregates.
The principal forms, P1 and I6, of Schoenberg's Piano Piece, op.33a, tone row feature hexachordal combinatoriality and contains three perfect fifths each, which is the relation between P1 and I6.
Occasionally a hexachord may be combined with an inverted or transposed version of itself in a special case which will then result in the aggregate (complete set of 12 chromatic pitches).
A row (B♭=0: 0 6 8 5 7 e 4 3 9 t 1 2) used by Schoenberg may be divided into two hexachords:
B♭ E F♯ E♭ F A // D C♯ G G♯ B C
When you invert the first hexachord and transpose it, the following hexachord, a reordering of the second hexachord, results:
G C♯ B D C G♯ = D C♯ G G♯ B C
Thus, when you superimpose the original hexachord 1 (P0) over the transposed inversion of hexachord 1 (I9 in this case), the entire collection of 12 pitches results. If you continued the rest of the transposed, inverted row (I9) and superimposed original hexachord 2, you would again have the full complement of 12 chromatic pitches.
In Schoenberg's Variations for Orchestra, Op. 31, tone row form P1's second half has the same notes, in a different order, as the first half of I10: "Thus it is possible to employ P1 and I10 simultaneously and in parallel motion without causing note doubling."
Hexachordal combinatoriality is closely related to the theory of the 44 tropes created by Josef Matthias Hauer in 1921, although it seems that Hauer had no influence on Babbitt at all. Furthermore, there is little proof suggesting that Hauer had extensive knowledge about the inversional properties of the tropes earlier than 1942 at least.[17] The earliest records on combinatorial relations of hexachords, however, can be found amongst the theoretical writings of the Austrian composer and music theorist Othmar Steinbauer.[lower-alpha 1] He undertook elaborate studies on the trope system in the early 1930s which are documented in an unpublished typescript Klang- und Meloslehre (1932). Steinbauer's materials dated between 1932 and 1934 contain comprehensive data on combinatorial trichords, tetrachords and hexachords including semi-combinatorial and all-combinatorial sets. They may therefore be the earliest records in music history.[18] A compilation of Steinbauer's morphological material has in parts become publicly available in 1960 with his script Lehrbuch der Klangreihenkomposition (author's edition) and was reprinted in 2001.[19]
Trichordal combinatoriality is a row's ability to form aggregates through the combination of trichords. "Trichordal combinatoriality involves the simultaneous presentation of four rows in parcels of three pcs."[20] The existence of trichordal combinatoriality, or any other form, in a row does not preclude the existence of other forms of combinatoriality (at the least trivial hexachordal combinatoriality exists between every row form and its retrograde). All trichordally derived rows possess trichordal combinatoriality.
1 2 Christensen, Thomas (2002). The Cambridge History of Western Music Theory, [unpaginated]. Cambridge. ISBN9781316025482.
↑ George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, fourth edition, revised (Berkeley, Los Angeles, London: University of California Press, 1977), 129–131. ISBN0-520-03395-7
↑ Kielian-Gilbert, Marianne (1982–83). "Relationships of Symmetrical Pitch-Class Sets and Stravinsky’s Metaphor of Polarity", Perspectives of New Music 21: 210. JSTOR832874.
↑ Mead, Andrew (2002). "Twelve-Tone Composition and the Music of Elliott Carter", Concert Music, Rock, and Jazz Since 1945: Essays and Analytical Studies, pp. 80–81. Elizabeth West Marvin, Richard Hermann; eds. University Rochester. ISBN9781580460965.
↑ Leeuw, Ton de (2005). Music of the Twentieth Century: A Study of Its Elements and Structure. Translated by Stephen Taylor. Amsterdam: Amsterdam University Press. pp.155–157. ISBN90-5356-765-8. Translation of Muziek van de twintigste eeuw: een onderzoek naar haar elementen en structuur. Utrecht: Oosthoek, 1964. Third impression, Utrecht: Bohn, Scheltema & Holkema, 1977. ISBN90-313-0244-9.
↑ Diederichs, Joachim. Fheodoroff, Nikolaus. Schwieger, Johannes (eds.). 2007. Josef Matthias Hauer: Schriften, Manifeste, Dokumente 428–440. Vienna: Verlag Lafite
↑ Sedivy, Dominik. 2011. Serial Composition and Tonality. An Introduction to the Music of Hauer and Steinbauer, p. 70. Vienna: edition mono/monochrom. ISBN978-3-902796-03-5. Sedivy, Dominik. 2012. Tropentechnik. Ihre Anwendung und ihre Möglichkeiten, 258–264. Salzburger Stier 5. Würzburg: Königshausen & Neumann. ISBN978-3-8260-4682-7
↑ Neumann, Helmut. 2001. Die Klangreihen-Kompositionslehre nach Othmar Steinbauer (1895–1962), 184–187, 201–213, 234–236. 2 vols.. Frankfurt et al.: Peter Lang
↑ Morris, Robert (1991). Class Notes for Atonal Music Theory, p. 82. Frog Peak Music. ASINB0006DHW9I[ISBN unspecified].
In music, a tone row or note row, also series or set, is 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.
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music.
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 theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.
In music theory, a trichord is a group of three different pitch classes found within a larger group. A trichord is a contiguous three-note set from a musical scale or a twelve-tone row.
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.
In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.
In music, a permutation (order) of a set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. Different permutations may be related by transformation, through the application of zero or more operations, such as transposition, inversion, retrogradation, circular permutation, or multiplicative operations. These may produce reorderings of the members of the set, or may simply map the set onto itself.
In music, transposition refers to the process or operation of moving a collection of notes up or down in pitch by a constant interval.
The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.
Retrograde inversion is a musical term that literally means "backwards and upside down": "The inverse of the series is sounded in reverse order." Retrograde reverses the order of the motif's pitches: what was the first pitch becomes the last, and vice versa. This is a technique used in music, specifically in twelve-tone technique, where the inversion and retrograde techniques are performed on the same tone row successively, "[t]he inversion of the prime series in reverse order from last pitch to first."
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals, it has been used in other ways for twelve-tone technique, and musical set theory. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect.
A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.
In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. Other names include: ic vector, PIC vector and APIC vector
A trope or tropus may refer to a variety of different concepts in medieval, 20th-, and 21st-century music.
Composition for Four Instruments (1948) is an early serial music composition written by American composer Milton Babbitt. It is Babbitt's first published ensemble work, following shortly after his Three Compositions for Piano (1947). In both these pieces, Babbitt expands upon the methods of twelve-tone composition developed by Arnold Schoenberg. He is notably innovative for his application of serial techniques to rhythm. Composition for Four Instruments is considered one of the early examples of “totally serialized” music. It is remarkable for a strong sense of integration and concentration on its particular premises—qualities that caused Elliott Carter, upon first hearing it in 1951, to persuade New Music Edition to publish it.
In music, the "Ode-to-Napoleon" hexachord is the hexachord named after its use in the twelve-tone piece Ode to Napoleon Buonaparte (1942) by Arnold Schoenberg. Containing the pitch-classes 014589 it is given Forte number 6–20 in Allen Forte's taxonomic system. The primary form of the tone row used in the Ode allows the triads of G minor, E♭ minor, and B minor to easily appear.
6-Z44 (012569), known as the Schoenberg hexachord, is Arnold Schoenberg's signature hexachord, as one transposition contains the pitches [A], Es, C, H, B, E, G, E♭, B, and B♭ being Es, H, and B in German.
In music theory, the chromatic hexachord is the hexachord consisting of a consecutive six-note segment of the chromatic scale. It is the first hexachord as ordered by Forte number, and its complement is the chromatic hexachord at the tritone. For example, zero through five and six through eleven. On C:
A Sermon, a Narrative and a Prayer is a cantata for alto and tenor singers, a narrator, chorus, and orchestra by Igor Stravinsky, composed in 1960–61. It belongs to the composer’s serial period, and lasts a little over a quarter of an hour in performance.
The Tone Clock, and its related compositional theory Tone-Clock Theory, is a post-tonal music composition technique, developed by composers Peter Schat and Jenny McLeod. Music written using tone-clock theory features a high economy of musical intervals within a generally chromatic musical language. This is because tone-clock theory encourages the composer to generate all their harmonic and melodic material from a limited number of intervallic configurations. Tone-clock theory is also concerned with the way that the three-note pitch-class sets can be shown to underlie larger sets, and considers these triads as a fundamental unit in the harmonic world of any piece. Because there are twelve possible triadic prime forms, Schat called them the 'hours', and imagined them arrayed in a clock face, with the smallest hour in the 1 o'clock position, and the largest hour in the 12 o'clock position. A notable feature of Tone-Clock Theory is 'tone-clock steering': transposing and/or inverting hours so that each note of the chromatic aggregate is generated once and once only.
The String Quartet No.1 is a piece for two violins, viola and cello, composed by Robert Gerhard between 1951 and 1955, premiered at Dartington in 1956. This work marks a turning point in Gerhard's style and composition processes, because in one hand, he recovers some old techniques such as the sonata form in the first movement, along with others not as old like the 12-tone technique. Gerhard brilliantly develops, combines and transforms these resources along with new systematic processes created by himself, so that it leads to a new and broad theoretical framework that will be essential to his music thereafter.
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