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List of tone rows and series

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List of tone rows and series

This is a list of tone rows and series. For a list of unordered collections, see set (music), Forte number, list of pitch class sets, and trope (music).


Twelve tone rows

Row form Integers Notation Composer: Composition Year Hexa-chord I Hexa-chord II Derivation/ Combina-toriality
P: 0 1 2 e t 9
5 4 3 6 7 8
Anton Webern:
Op. 32, later row[1]
un-
finished
6–1 6–1
I:
P: Igor Stravinsky:
Abraham & Isaac[2][3]
1963 6-Z3 6-Z36
I: 0 e 9 7 6 t
8 4 5 3 1 2
P: 0 1 3 6 7 9
8 t e 2 4 5
Arnold Schoenberg:
Serenade, op. 24, mov. 5[4]
1920–23 6–30 6–30 derived: 3–3
I:
P: 0 1 3 8 4 9
t 7 6 5 e 2
About this sound   
Alban Berg:
Kammerkonzert[5]
1923–25 6-Z19 6-Z44 cryptogram: "Schoenberg"
I: 0 e 9 4 8 3
2 5 6 7 1 t
P: 0 1 3 9 2 e
4 t 7 8 5 6
Arnold Schoenberg:
Suite for Piano, op. 25[6][7][8][9]
1921–23 6–2 6–2
I: 0 e 9 3 t 1
8 2 5 4 7 6
P: 0 1 4 2 9 5
e 3 8 t 7 6
Pohlman Mallalieu[10][11]

Robert Morris:
Not Lilacs[10]

n.a.

1973

6–14 6–14 all-interval, RT6-invariant
I: 0 e 8 t 3 7
1 9 4 2 5 6
P: 0 1 4 3 2 5
6 9 8 7 t e
About this sound   
Anton Webern:
Variations for Orchestra, Op. 30[12][3][13]
1940 6–1 6–1 all-
combinatorial
I: 0 e 8 9 t 7
6 3 4 5 2 1
P: 0 1 4 3 2 5
6 9 8 e t 7
Igor Stravinsky:
Agon, "Pas de deux", "Four trios"[2][14]
1957 6–1 6–1 all-
combinatorial
I: 0 e 8 9 t 7
6 3 4 1 2 5
P: 0 1 4 3 7 6
e 2 t 9 5 8
Arnold Schoenberg:
Die Jakobsleiter[15]
un-
finished
6-Z13 6-Z42
I: 0 e 8 9 5 6
1 t 2 3 7 4
P: 0 1 4 5 8 7
6 3 2 7 e t
Walter Piston:
Symphony No. 8 (first movement)[16]
1965 6-Z19 6-Z44
I: 0 e 8 7 4 5
6 9 t 5 1 2
P: 0 1 4 5 7 6
8 9 t e 2 3
Roger Reynolds:
Ambages[17]
1965 6-5 6-5
I: 0 1 4 5 7 6
8 9 t e 2 3
P: 0 1 4 6 8 t
5 7 9 e 2 3
Ernst Krenek:
Lamentatio Jeremiae Prophetae, Op. 93, row 3[18]
1940–41 6–34 6–34 RI-symmetry
I: 0 e 8 6 4 2
7 5 3 1 t 9
P: 0 1 4 9 5 8
3 t 2 e 6 7
Milton Babbitt:
Composition for Twelve Instruments[3][19]
1948 6–20 6–20 dyadic, tri-, tetra-, and hexa- chordal combina-
toriality
I: 0 e 8 3 7 4
9 2 t 1 6 5
P: 0 1 5 2 6 t
3 8 9 4 7 e
Edison Denisov:
Fünf Geschichten von Herrn Keuner[20]
1966 6-Z43 6-Z17 all-trichord hexachord
I: 0 e 7 t 6 2
9 4 3 8 5 1
P: 0 1 5 8 t 4
3 7 9 2 e 6
Luigi Dallapiccola:
Quaderno musicale di Annalibera[21]
Variazioni per orchestra
Canti di Liberazione[22]
1952
1954
1951–55
6–31 6–31
I: 0 e 7 4 2 8
9 5 3 t 1 6
P: 0 1 6 2 7 9
3 4 t e 5 8
Arnold Schoenberg:
Violin Concerto, op. 36[23]
1936 6-Z50 6-Z29
I: 0 e 6 t 5 3
9 8 2 1 7 4
P: 0 1 6 2 t 9
4 7 3 8 e 5
Arnold Schoenberg:
A Survivor from Warsaw[24][25]
1947 6–15 6–15
I: 0 e 6 t 2 3
8 5 9 4 1 7
P: 0 1 6 5 7 e
2 t 3 9 8 4
Milton Babbitt:
All Set[26]
1957 6–7 6–7 second-order all-
combinatorial hexachord
I: 0 e 6 7 5 1
t 2 9 3 4 8
P: 0 1 7 2 3 9
t 8 6 5 e 4
Edison Denisov:
Viola concerto[27]
1986 6-Z41 6-Z12
I: 0 e 5 t 9 3
2 4 6 7 1 8
P: 0 1 7 5 6 4
t 8 9 e 2 3
About this sound   
Arnold Schoenberg:
Moses und Aron[24]
un-
finished
6–5 6–5 inversional
I: 0 e 5 7 6 8
2 4 3 1 t 9
P: 0 1 7 5 6 e
9 8 t 3 4 2
Igor Stravinsky:
Movements[2]
1958–59 6–7 6–7
I: 0 e 5 7 6 1
3 4 2 9 8 t
P: 0 1 7 6 8 9
3 2 4 5 e t
Alberto Ginastera:
Don Rodrigo, Act I, op. 31[28]
1963–64 6–5 6–5
I: 0 e 5 6 4 3
9 t 8 7 1 2
P: 0 1 7 9 8 6
3 t 2 5 e 4
Karlheinz Stockhausen:
Plus-Minus[29]
1963 (rev. 1974) 6–5 6–5
I: 0 e 5 3 4 6
9 2 t 7 1 8
P: 0 1 8 e 7 3
2 5 4 t 9 6
Karlheinz Stockhausen:
Zeitmaße[30][31]
1955–56 6–16 6–16
I: 0 e 4 1 5 9
t 7 8 2 3 6
P: 0 1 9 e 3 2
6 7 t 8 4 5
Anton Webern:
Op. 32, row 1[32]
un-
finished
6–2 6–2
I: 0 e 3 1 9 t
6 5 2 4 8 7
P: 0 1 9 e 3 2
7 6 t 8 4 5
Anton Webern:
Op. 32, row 2[32]
un-
finished
6–1 6–1 all-
combinatorial
I: 0 e 3 1 9 t
5 6 2 4 8 7
P: 0 1 9 e 8 t
4 5 6 2 3 7
About this sound   
Anton Webern:
Variations for piano, mov. II, op. 27[33][34]
1936 6–1 6–1 all-
combinatorial
I: 0 e 3 1 4 2
8 7 6 t 9 5
P: 0 1 9 e t 7
8 3 4 6 5 2
Igor Stravinsky:
A Sermon, a Narrative and a Prayer[2]
1961 6–2 6–2
I: 0 e 3 1 2 5
4 9 8 6 7 t
P: 0 1 t 2 6 4
5 3 7 e 8 9
Luigi Dallapiccola:
Dialoghi[35]
6–21 6–21 RI-invariant:
P0 = RI9
I: 0 e 2 t 6 8
7 9 5 1 4 3
P: 0 1 t 3 4 6
8 9 e 5 7 2
Chen Yi:
Near Distance[3][36]
1988 6-Z23 6-Z45
I: 0 e 2 9 8 6
4 3 1 7 5 t
P: 0 1 e 2 t 3
9 4 8 5 7 6
About this sound   
Luigi Nono:
Canti per tredeci
Il canto sospeso[15][37][38]
Composizione per orchestra n. 2: Diario polacco ’58
1955
1955–56
1959
6–1 6–1 symmetrical all-interval
row
; all-
combinatorial
I: 0 e 1 t 2 9
3 8 4 7 5 6
About this sound   
Nicolas Slonimsky: Grandmother chord[39] 1938
P: 0 2 1 3 8 7
5 6 4 e 9 t
Joonas Kokkonen:
Symphonic Sketches, mov. 3, row A[40]
1968 6-Z38 6-Z6
I: 0 t e 9 4 5
7 6 8 1 3 2
P: 0 2 1 3 4 6
7 9 8 t 5 e
Igor Stravinsky:
Agon, second coda[41][42]
1957 6–2 6–2
I: 0 t e 9 8 6
5 3 4 2 7 1
Igor Stravinsky:
coda[43]
P: 0 2 1 7 t 8
9 e 3 4 6 5
Igor Stravinsky:
The Flood[2]
1962 6-Z12 6-Z41
I: 0 t e 5 2 4
3 1 9 8 6 7
P: 0 2 1 7 e 8
9 6 t 4 3 5
Luigi Dallapiccola:
Requiescant[35]
6–5 6–5 RI-invariant:
P0 = RI5
I: 0 t e 5 1 4
3 6 2 8 9 7
P: 0 2 1 t 3 e
5 9 4 7 8 6
Alfred Schnittke:
String Quartet No. 1[3]
1966 6–1 6–1 all-
combinatorial
I: 0 t e 2 9 1
7 3 8 5 4 6
P: 0 2 3 1 4 5
6 7 8 9 e t
Alban Berg:
Lulu, Schigolch[24][44]
1934 6–1 6–1 all-
combinatorial
, derived: 4–1
I: 0 t 9 e 8 7
6 5 4 3 1 2
P: 0 2 3 5 4 9
7 8 t e 1 6
Igor Stravinsky:
Agon, "Bransles"[45]
1957 6-Z40 6-Z11
I: 0 t 9 7 8 3
5 4 2 1 e 6
P: 0 2 3 5 7 8
e 4 6 t 8 1
About this sound   
Alban Berg:
"Der Wein"[46]
1929 6–16 6–16 D harmonic minor & in the inverted row-form added-sixth chord with both major and minor third
I: 0 t 9 7 5 4
1 8 6 2 4 e
P: 0 2 3 6 8 t
5 7 9 e 1 4
Ernst Krenek:
Lamentatio Jeremiae Prophetae, Op. 93, row 2[18]
1940–41 6–34 6–34 RI-symmetry
I: 0 t 9 6 4 2
7 5 3 1 e 8
P: 0 2 4 5 8 t
6 7 9 e 1 3
Ernst Krenek:
Lamentatio Jeremiae Prophetae, Op. 93, row 1[18]
1940–41 6–34 6–34 RI-symmetry
I: 0 t 8 7 4 2
6 5 3 1 e 9
P: 0 2 4 6 7 t
5 8 9 e 1 3
Ernst Krenek:
Lamentatio Jeremiae Prophetae, Op. 93, row 6[18]
1940–41 6–34 6–34 RI-symmetry
I: 0 t 8 6 5 2
7 4 3 1 e 9
P: 0 2 4 6 8 9
5 7 t e 1 3
Ernst Krenek:
Lamentatio Jeremiae Prophetae, Op. 93, row 5[18]
1940–41 6–34 6–34 RI-symmetry
I: 0 t 8 6 4 3
7 5 2 1 e 9
P: 0 2 4 6 t 8
e 7 9 1 3 5
Alban Berg:
Lulu[24]
1934 6–35 6–35
I: 0 t 8 6 2 4
1 5 3 e 9 7
P: 0 2 5 3 t 4
6 1 e 7 8 9
Igor Stravinsky:
Variations[47][2][3][48][49][50][51][52]
1963–64 6–9 6–9
I: 0 t 7 9 2 8
6 e 1 5 4 3
P: 0 2 6 4 t 9
7 e 8 5 1 3
Arnold Schoenberg:
Piano Piece [Klavierstück], op. 33b[24]
1931 6–34 6–34
I: 0 t 6 8 2 3
5 1 4 7 e 9
P: 0 2 6 t 4 7
8 3 5 9 e 1
About this sound   
Alban Berg:
Lulu, Schoolboy[44][53][54]
1934 6–34 6–34
I: 0 t 6 2 8 5
4 9 7 3 1 e
P: 0 2 9 e 8 6
5 7 t 1 4 3
Igor Stravinsky:
The Owl and the Pussy Cat[2]
1966 6-Z23 6-Z45
I: 0 t 3 1 4 6
7 5 2 e 8 9
P: 0 7 6 5 8 9
4 3 t e 2 1
Karel Goeyvaerts:
Nummer 2[55]
1951 6Z36 6-Z3
I: 0 5 6 7 4 3
8 9 2 1 t e
P: 0 2 t e 1 8
6 7 9 4 3 5
About this sound   
Igor Stravinsky:
Requiem Canticles, 1st series[2]
1966 6–2 6–2
I: 0 t 2 1 e 4
6 5 3 8 9 7
P: 0 2 e 9 t 1
3 6 4 5 7 8
Joonas Kokkonen:
Woodwind Quintet, mov. 1 & 2, row A & B[40]
1973 6–1 6–1 all-
combinatorial
I: 0 t 1 3 2 e
9 6 8 7 5 4
P: 0 2 e 9 t 4
3 7 5 6 8 1
Arnold Schoenberg:
Dreimal Tausend Jahre, Op. 50a[3]
1949 6–9 6–9
I: 0 t 1 3 2 8
9 5 7 6 4 e
P: 0 3 1 2 4 7
5 6 8 e 9 t
Arvo Pärt:
Symphony No. 2[56]
1966 6-Z36 6-Z3 derived: 4–1
I: 0 9 e t 8 5
7 6 4 1 3 2
P: 0 3 1 2 e t
8 9 5 7 6 4
Milton Babbitt:
Woodwind Quartet[57]
1953 6–1 6–1
I: 0 9 e t 1 2
4 3 7 5 6 8
P: 0 3 1 6 2 4
t 5 9 7 8 e
Ernst Krenek:
Sechs Vermessene, Op. 168[58]
1958 6–2 6–2
I: 0 9 e 6 t 8
2 7 3 5 4 1
P: 0 3 1 6 2 t
9 8 e 7 5 4
Ernst Krenek:
Karl V[59][60]
1938 6-Z39 6-Z10
I: 0 9 e 6 t 2
3 4 1 5 7 8
P: 0 3 1 t 2 8
e 5 6 9 7 4
Donald Martino:
Fantasy Variations[61]
1962 6–9 6–9
I: 0 9 e 2 t 4
1 7 6 3 5 8
P: 0 3 2 1 4 8
e 5 t 7 6 9
Benjamin Frankel:
The Curse of the Werewolf[62]
1961 6-Z37 6-Z4
I: 0 9 t e 8 4
1 7 2 5 6 3
P: 0 3 2 9 7 8
e 6 1 5 4 t
Karlheinz Stockhausen:
"Aries", from Tierkreis[63]
1974–75 6–18 6–18
I: 0 9 t 3 5 4
1 6 e 7 8 2
P: 0 3 2 1 5 4
t e 7 8 9 6
Anton Webern:
Symphony, Op. 21[4][64]
1928 6–1 6–1 all-
combinatorial
I: 0 9 t e 7 8
2 1 5 4 3 6
P: 0 3 4 6 e 8
7 t 9 5 1 2
Igor Stravinsky:
Agon, "Mystery" (sketches only)[65]
1957 6–31 6–31
I: 0 9 8 6 1 4
5 2 3 7 e t
P: 0 3 5 1 2 6
4 e 7 8 e 9
Karlheinz Stockhausen:
Klavierstück IV (second row of 48)[66]
1952 6-Z3 6-Z36
I: 0 9 7 e t 6
8 1 5 4 1 3
P: 0 3 5 4 e t
9 8 7 6 2 1
Karlheinz Stockhausen:
"Aquarius", from Tierkreis[63]
1974–75 6-Z6 6-Z38
I: 0 9 7 8 1 2
3 4 5 6 t e
P: 0 3 5 7 9 e
6 8 t 1 2 4
Ernst Krenek:
Lamentatio Jeremiae Prophetae, Op. 93, row 4[18]
1940–41 6–34 6–34 RI-symmetry
I: 0 9 7 5 3 1
6 4 2 e t 8
P: 0 3 7 e 2 5
9 1 4 6 8 t
About this sound   
Alban Berg:
Violin Concerto[4][24][67]
1935 6-Z24 6-Z46
I: 0 9 5 1 t 7
3 e 8 6 4 2
P: 0 3 e 2 t 1
9 5 8 4 7 6
Anton Webern:
"Erlösung" from Drei Lieder, Op. 18, No. 2[68]
1910–13 6–1 6–1 all-
combinatorial
I: 0 9 1 t 2 e
3 7 4 8 5 6
P: 0 3 e 4 2 1
7 8 t 5 9 6
Ernst Krenek:
Quaestio temporis, Op. 170[69]
1959 6–1 6–1 all-
combinatorial
retrograde-symmetry
I: 0 9 1 8 t e
5 4 2 7 3 6
P: 0 3 e t 2 9
1 5 4 8 7 6
Anton Webern:
Op. 31[12]
1941–43 6-Z3 6-Z36
I: 0 9 1 2 t 3
e 7 8 4 5 6
P: 0 4 1 8 5 9
3 e 2 7 t 6
Milton Babbitt:
Composition for Four Instruments, row iv[26]
1948 6–20 6–20 all-
combinatorial hexachord
I: 0 8 e 4 7 3
9 1 t 5 2 6
P: 0 4 1 e 3 2
8 6 5 t 7 9
Milton Babbitt:
Composition for Four Instruments[70]
1948 6–1 6–1 all-
combinatorial hexachord
I: 0 8 e 1 9 t
4 6 7 2 5 3
P: 0 4 2 5 7 8
t 9 e 1 6 3
Igor Stravinsky:
Anthem[2]
1962 6–33 6–33
I: 0 8 t 7 5 4
2 3 1 e 6 9
P: 0 4 3 2 5 6
t e 9 1 7 8
Karlheinz Stockhausen:
Licht, Eve formula[71][72]
1977–
2003
6–2 6–2
I: 0 8 9 t 7 6
2 1 3 e 5 4
P: 0 4 3 7 e 8
9 6 5 1 2 t
Arnold Schoenberg:
Ode to Napoleon Buonaparte
Suite Op. 29[73]

Luigi Nono:
Variazioni canoniche sulla serie dell'op. 41 di Arnold Schönberg[74]

1942
1926

1950

6–20 6–20 all-
combinatorial
I: 0 8 9 5 1 4
3 6 7 e t 2
P: 0 4 3 8 7 e
5 1 2 9 t 6
Milton Babbitt:
Composition for Four Instruments, row iii[26]
1948 6–20 6–20 all-
combinatorial hexachord
I: 0 8 9 4 5 1
7 e t 3 2 6
P: 0 4 3 e 7 8
t 9 2 5 1 6
Karlheinz Stockhausen:
"Scorpio", from Tierkreis[63]
1974–75 6–20 6–20
I: 0 8 9 1 5 4
2 3 t 7 e 6
P: 0 4 3 e t 9
1 7 8 6 2 5
Ernst Krenek:
Suite, Op. 84 for solo cello[75]
6–5 6–5
I: 0 8 9 1 2 3
e 5 4 6 t 7
P: 0 4 5 2 6 7
8 9 e t 3 1
Alban Berg:
Lulu[24]
1934 6–9 6–9
I: 0 8 7 t 6 5
4 3 1 2 9 e
P: 0 4 5 2 7 9
6 8 e t 3 1
About this sound   
Alban Berg:
Lulu, Basic[44][53][54]
1934 6–32 6–32 circle of fifths
I: 0 8 7 t 5 3
6 4 1 2 9 e
P: 0 4 5 e 6 t
7 3 1 2 9 8
Milton Babbitt:
All Set[76][77]
1957 6–7 6–7 all-interval row
I: 0 8 7 1 6 2
5 9 e t 3 4
P: 0 4 5 t 7 6
1 8 e 2 9 3
Karlheinz Stockhausen:
"Gemini", from Tierkreis[63]
1974–75 6-Z41 6-Z12
I: 0 8 7 2 5 6
e 4 1 t 3 9
P: 0 4 8 9 3 2
t e 7 6 5 1
Robert Moevs:
Three Symphonic Pieces, mov. 2[78]
1954–55 6-Z17 6-Z43 all-trichord hexachord
I: 0 8 4 3 9 t
2 1 5 6 7 e
P: 0 4 t 6 3 8
7 2 5 9 1 e
Juan Carlos Paz:
Canciones y Baladas, Balada II[79]
1936 6–34 6–34
I: 0 8 2 6 9 4
5 t 7 3 e 1
P: 0 5 1 2 3 9
8 t 4 e 6 7
Alban Berg:
Lulu, Chorale[44]
1934 6-Z39 6-Z10
I: 0 7 e t 9 3
4 2 8 1 6 5
P: 0 5 1 4 2 3
9 8 t 7 e 6
Michael Gielen:
Six Songs, for bass, violin, viola, clarinet, bass clarinet, and piano[80]
1958 6–1 6–1 retrograde-symmetrical all-interval
row
, all-
combinatorial hexachords
I: 0 7 e 8 t 9
3 4 2 5 1 6
P: 0 5 1 4 t 8
e 9 3 6 2 7
Alfred Schnittke:
Concerto Grosso No. 3[3]
1985 6–31 6–31
I: 0 7 e 8 2 4
1 3 9 6 t 5
P: 0 5 2 7 4 9
6 e 8 1 t 3
Benjamin Britten:
The Turn of the Screw[81]
1954 6–32 6–32
I: 0 7 t 5 8 3
6 1 4 e 2 9
P: 0 5 2 7 t 1
e 4 6 8 9 3
Dmitri Shostakovich:
Symphony No. 14, mov. III[82]
1969 6-Z47 6-Z25
I: 0 7 t 5 2 e
1 8 6 4 3 9
P: 0 5 2 t 1 7
8 6 e 4 9 3
Alban Berg:
Lulu, Countess Geschwitz[24][44][54]
1934 6-Z47 6-Z25
I: 0 7 t 2 e 5
4 6 1 8 3 9
P: 0 5 4 7 t 6
e 8 1 9 2 3
Igor Stravinsky:
Threni[2]
1958 6-Z41 6-Z12
I: 0 7 8 5 2 6
1 4 e 3 t 9
P: 0 5 7 4 2 3
9 1 8 e t 6
Milton Babbitt:
Three Compositions for Piano, no. I[83]
1947 6–8 6–8
I: 0 7 5 8 t 9
3 e 4 1 2 6
P: 0 5 7 6 e 3
4 2 9 8 t 1
Alban Berg:
Lulu, Acrobat/Athlete[24][44][54]
1934 6-Z17 6-Z43 all-trichord hexachord
I: 0 7 5 6 1 9
8 t 3 4 2 e
P: 0 5 9 t 4 7
6 e 1 2 8 3
About this sound   
Alban Berg:
Lulu, Dr. Schön[44][53][54]
1934 6-Z25 6-Z47
I: 0 7 3 2 8 5
6 1 e t 4 9
P: 0 5 e 6 3 2
8 7 9 t 4 1
Karlheinz Stockhausen:
"Leo", from Tierkreis[63]
1974–75 6Z-13 6-Z42
I: 0 7 1 6 9 t
4 5 3 2 8 e
P: 0 6 4 2 8 9
3 1 e t 7 5
Igor Stravinsky:
Elegy for J.F.K.[2]
1964 6–34 6–34
I: 0 6 8 t 4 3
9 e 1 2 5 7
P: 0 6 4 3 2 9
t 5 7 8 e 1
Karel Husa:
Mosaïques[84]
1960 6-Z45 6-Z23
I: 0 6 8 9 t 3
2 7 5 4 1 e
P: 0 6 5 1 e 7
4 8 9 3 2 t
Arnold Schoenberg:
"De Profundis", Op. 50b[3]
1950 6–7 6–7
I: 0 6 7 e 1 5
8 4 3 9 t 2
P: 0 6 5 2 3 4
7 8 9 t e 1
Karlheinz Stockhausen:
"Sagittarius", from Tierkreis[63]
1974–75 6–2 6–2
I: 0 6 7 t 9 8
5 4 3 2 1 e
P: 0 6 5 e t 4
3 9 8 2 1 7
Witold Lutosławski:
Musique funèbre (Funeral Music)[85]
1954–58 6–7 6–7
I: 0 6 7 1 2 8
9 3 4 t e 5
P: 0 6 7 8 9 1
2 5 e 4 t 3
Wolfgang Amadeus Mozart:
String Quartet in E major, K. 428 first movement, opening of development section[86]
6–5 6–5
I: 0 6 5 4 3 e
t 7 1 8 2 9
P: 0 6 8 5 7 e
4 3 9 t 1 2
About this sound   
Arnold Schoenberg:
Variations, Op. 31[79][87][88]
1926–28 6–5 6–5 Bach motif
I: 0 6 4 7 5 1
8 9 3 2 e t
P: 0 6 8 t 2 4
9 5 3 e 7 1
Arnold Schoenberg:
Three Songs, Op. 48[64]
1933 6–35 6–35
I: 0 6 4 2 t 8
3 7 9 1 5 e
P: 0 6 9 2 8 t
4 7 1 e 3 5
Josef Matthias Hauer:
Salambo, op. 60[89]
1929 6-21 6-21
I: 0 6 3 t 4 2
8 5 e 1 9 7
P: 0 6 t 7 9 8
3 2 4 1 5 e
Michel Fano: Sonata for Two Pianos[90] 1951 6-Z3 6-Z36 retrograde-symmetry
I: 0 6 2 5 3 4
9 t 8 e 7 1
P: 0 6 t 8 4 7
2 5 1 e 3 9
Ben Johnston:
String Quartet No. 7, mov. 2[91]
1984 6–21 6–21 just intonation: otonality & utonality
I: 0 6 2 4 8 5
t 7 e 1 9 3
P: 0 7 1 t e 9
5 8 4 3 2 6
Karlheinz Stockhausen:
Klavierstück IV (third row of 48)[66]
1952 6–2 6–2
I: 0 5 e 2 1 3
7 4 8 9 t 6
P: 0 7 2 1 e 8
3 5 9 t 4 6
About this sound   
Arnold Schoenberg:
Piano Piece, op. 33a[24][92]
1928 6–5 6–5
I: 0 5 t e 1 4
9 7 3 2 8 6
P: 0 7 3 t 2 6
9 1 4 8 e 5
Alberto Ginastera:
Quintet, op. 29[28]
1963 6–31 6–31
I: 0 5 9 2 t 6
3 e 8 4 1 7
P: 0 7 4 e 6 t
3 9 1 8 5 2
Elliott Carter:
Piano Concerto, Orchestra[93]
1964–65 6-Z43 6-Z17 all-trichord hexachord; ordered in register
I: 0 5 8 1 6 2
9 3 e 4 7 t
P: 0 7 6 4 5 9
8 t 3 1 e 2
Igor Stravinsky:
Requiem Canticles, 2nd series[2]
1966 6-Z40 6-Z11
I: 0 5 6 8 7 3
4 2 9 e 1 t
P: 0 7 6 e 9 1
4 t 8 5 2 3
Ernst Krenek:
String Quartet No. 7, op. 96[59]
1943–44 6-Z12 6-Z41
I: 0 5 6 1 3 e
8 2 4 7 t 9
P: 0 7 e 2 1 9
3 5 t 6 8 4
Arnold Schoenberg:
Piano Concerto, Op. 42[94][95]
1942 6–9 6–9 inversional
I: 0 5 1 t e 3
9 7 2 6 4 8
P: 0 7 e 8 2 t
6 9 5 4 1 3
Anton Webern:
Op. 23[96]
1933 6-Z10 6-Z39
I: 0 5 1 4 t 2
6 3 7 8 e 9
P: 0 7 e t 2 9
6 1 5 4 8 3
Ross Lee Finney:
Edge of Shadow[21]
1959 6–8 6–8
I: 0 5 1 2 t 3
6 e 7 8 4 9
P: 0 7 e t 9 8
3 6 4 1 5 2
Karlheinz Stockhausen:
"Cancer", from Tierkreis[63]
1974–75 6–1 6–1 all-
combinatorial
I: 0 5 1 2 3 4
9 6 8 e 7 t
P: 0 8 1 7 2 e
9 6 4 5 3 t
Arnold Schoenberg:
String Trio, Op. 45[97]
1946 6–5 6–5
I: 0 4 e 5 t 1
3 6 8 7 9 2
P: 0 8 1 t 9 e
5 3 4 7 2 6

[98][99]
About this sound   
Karlheinz Stockhausen:
Klavierstück VII Gruppen
Klavierstück IX
Klavierstück X
1955
1955–57
1961
1961
6–1 6–1 all-
combinatorial
hexachord, retrograde-symmetrical all-interval
I: 0 4 e 2 3 1
7 9 8 5 t 6
P: 0 8 1 e t 7
5 9 4 6 3 2
Karlheinz Stockhausen:
Klavierstück V (row of original 1954 version)[100]
Klavierstück VIII[101]
1954–55
1954
6-Z3 6-Z36
I: 0 4 e 1 2 5
7 3 8 6 9 t
P: 0 8 4 9 t 3
1 6 e 2 7 5
Elliott Carter:
Piano Concerto, Solo[ist][93]
1964–65 6-Z17 6-Z43 all-interval hexachord; ordered in register
I: 0 4 8 3 2 9
e 6 1 t 5 7
P: 0 8 5 t 7 1
6 4 e 2 3 9
About this sound   
Alban Berg:
Lulu, Alwa[44][53][54]
1934 6-Z25 6-Z47
I: 0 4 7 2 5 e
6 8 1 t 9 3
P: 0 8 7 t 4 5
2 9 e 3 1 6
Elliott Carter:
A Symphony of Three Orchestras[93]
1976 6-Z24 6-Z46 ordered in register
I: 0 4 5 2 8 7
t 3 1 9 e 6
P: 0 8 7 e t 9
3 1 4 2 6 5
About this sound   
Anton Webern:
Variations for piano, mov. III, op. 27[102][103]
1936 6–1 6–1 all-
combinatorial
I: 0 4 5 1 2 3
9 e 8 t 6 7
P: 0 8 e t 2 1
4 3 7 6 9 5
Anton Webern:
Op. 29[104][35]
1938 6–2 6–2 semi-combinatorial, RI-invariant:
P0 = RI5
I: 0 4 1 2 t e
8 9 5 6 3 7
P: 0 8 e t 4 6
7 1 5 3 9 2
Ernst Krenek:
Sestina, op. 161[59]
1957 6–22 6–22
I: 0 4 1 2 8 6
5 e 7 9 3 t
P: 0 9 1 3 4 e
2 8 7 5 t 6
Luigi Dallapiccola:
Piccola musica notturna[105]
1954 6–8 6–8
I: 0 3 e 9 8 1
t 4 5 7 2 6
P: 0 9 1 8 4 5
7 2 3 6 t e
Milton Babbitt:
Composition for Synthesizer[106]
1961 6–20 6–20
I: 0 3 e 4 8 7
5 t 9 6 2 1
P: 0 9 1 e t 2
8 7 5 4 6 3
About this sound   
Milton Babbitt:
Composition for Four Instruments[107][108]
1948 6–1 6–1 all-
combinatorial
I: 0 3 e 1 2 t
4 5 7 8 6 9
P: 0 9 2 3 e t
5 4 1 6 7 8
Igor Stravinsky:
Epitaphium[2]
1959 6-Z3 6-Z36
I: 0 3 t 9 1 2
7 8 e 6 5 4
P: 0 9 2 8 7 5
6 1 3 4 t e
Karlheinz Stockhausen:
"Capricorn", from Tierkreis[63]
1974–75 6-Z47 6-Z25
I: 0 3 t 4 5 7
6 e 9 8 2 1
P: 0 9 8 4 7 6
1 t 3 5 e 2
David Dzubay: Di/Con[ver(gence/sions)][3] 1988 6-Z39 6-Z10
I: 0 3 4 8 5 6
e 2 9 7 1 t
P: 0 9 8 e t 2
3 4 5 7 1 6
Anton Webern:
Op. 22[109]
1929–30 6–2 6–2 semi-combinatorial
I: 0 3 4 1 2 t
9 8 7 5 e 6
P: 0 9 e 2 4 1
7 t 8 5 3 6
Milton Babbitt:
Composition for Four Instruments, row ii[26]
1948 6–8 6–8
I: 0 3 1 t 8 e
5 2 4 7 9 6
P: 0 t 1 2 5 4
6 3 7 e 8 9
Pierre Boulez:
Le marteau sans maître[3][110]
1953–55 6-Z10 6-Z39
I: 0 2 e t 7 8
6 9 5 1 4 3
P: 0 t 3 4 2 7
6 9 8 e 1 5
Charles Ives:
Tone Roads No. 1[111]
1911 6-Z46 6-Z24
I: 0 2 9 8 t 5
6 3 4 1 e 7
P: 0 t 3 e 8 7
1 2 5 9 4 6
Elliott Carter:
Night Fantasies[112]
1982 6–14 6–14
I: 0 2 9 1 4 5
e t 7 3 8 6
P: 0 t 9 8 7 5
e 1 2 3 4 6
Milton Babbitt:
Composition for Four Instruments, row i[26]
1948 6–8 6–8
I: 0 2 3 4 5 7
1 e t 9 8 6
P: 0 t 9 e 7 2
8 1 4 6 5 3
Karlheinz Stockhausen:
Kreuzspiel[113]
1951 6-Z50 6-Z29
I: 0 2 3 1 5 t
4 e 8 6 7 9
P: 0 e 1 2 9 8
t 7 5 6 4 3
Igor Stravinsky:
Agon, "Double Pas de Quatre"[2][114]
1957 6-Z3 6-Z36
I: 0 1 e t 3 4
2 5 7 6 8 9
P: 0 e 1 3 4 2
5 7 6 8 t 9
Igor Stravinsky:
Fanfare[2]
1964 6–1 6–1 all-
combinatorial
I: 0 1 e 9 8 t
7 5 6 4 2 3
P: 0 e 1 3 4 2
7 6 9 5 8 t
Igor Stravinsky:
Canticum Sacrum, II & IV[2]
1955 6–1 6–1 all-
combinatorial
I: 0 1 e 9 8 t
5 6 3 7 4 2
P: 0 e 1 7 9 8
t 4 6 5 2 3
Chen Yi:
Woodwind Quintet[3][36]
1987 6-Z4 6-Z37
I: 0 1 e 5 3 4
2 8 6 7 t 9
P: 0 e 2 1 3 4
9 t 7 8 6 5
Arvo Pärt:
Diagrams[56]
1964 6–1 6–1 all-
combinatorial
I: 0 1 t e 9 8
3 2 5 4 6 7
P: 0 e 2 1 5 6
3 4 8 7 t 9
About this sound   
Anton Webern:
String Quartet, Op. 28:[104][67][102]
1936 6–5 6–5 derived: 4–1 (BACH motif), semi-combinatorial
I: 0 1 t e 7 6
9 8 4 5 2 3
P: 0 e 2 3 4 1
6 5 8 9 t 7
Alberto Ginastera:
Violin Concerto, op. 30[28]
1963 6–1 6–1 all-
combinatorial
I: 0 1 t 9 8 e
6 7 4 3 2 5
P: 0 e 2 9 5 3
4 8 t 7 1 6
Elliott Carter:
String Quartet No. 3[115]
1973 6-Z23 6-Z45
I: 0 1 t 3 7 9
8 4 2 5 e 6
P: 0 e 3 1 7 9
6 2 t 5 8 4
Arnold Schoenberg:
Violin Phantasy, Op. 47[116][117]
1949 6–21 6–21
I: 0 1 9 e 5 3
6 t 2 7 4 8
P: 0 e 3 2 1 8
6 9 t 7 5 4
Igor Stravinsky:
Double Canon[2]
1959 6-Z36 6-Z3
I: 0 1 9 t e 4
6 3 2 5 7 8
P: 0 e 3 4 8 7
9 5 6 1 2 t
About this sound   
Anton Webern:
Concerto for Nine Instruments, Op. 24[24][68][87][92][118]
1934 6–20 6–20 all-
combinatorial
, derived: 3–3
I: 0 1 9 8 4 5
3 7 6 e t 2
P: 0 e 4 5 7 3
2 6 8 1 t 9
David Lewin:
Just a Minute, Roger[119]
1978 6–16 6–16
I: 0 1 8 7 5 9
t 6 4 e 2 3
P: 0 e 5 2 1 6
3 8 7 4 t 9
Alfred Schnittke:
Viola Concerto[3]
1985 6–5 6–5
I: 0 1 7 t e 6
9 4 5 8 2 3
P: 0 e 5 9 6 2
7 3 1 4 t 9
Luigi Dallapiccola:
Cinque Canti[35]
6-Z50 6-Z29 RI-invariant:
P0 = RI9
I: 0 1 7 3 6 t
5 9 e 8 2 3
P: 0 e 6 5 4 3
1 t 9 7 2 8
Pierre Boulez:
Structures Ia[3][67]
1952 6–5 6–5
I: 0 1 6 7 8 9
e 2 3 5 t 4
P: 0 e 6 5 7 1
8 2 t 3 4 9
George Perle:
Three Inventions for Piano, no. 3[120]
6–7 6–7 all-
combinatorial
I: 0 1 6 7 5 e
4 t 2 9 8 3
P: 0 e 7 1 2 9
3 8 t 4 5 6
About this sound   
Alban Berg:
Lyric Suite, mov. III[121]
1925–26 6–9 6–9
I: 0 1 5 e t 3
9 4 2 8 7 6
P: 0 e 7 1 4 8
3 9 t 2 5 6
About this sound   
Alban Berg:
Lyric Suite, mov. VI, row 1[122]
1925–26 6-Z44 6-Z19
I: 0 1 5 e 8 4
9 3 2 t 7 6
P: 0 e 7 4 2 9
3 8 t 1 5 6
About this sound   
Fritz Heinrich Klein[123]

Alban Berg:
Lyric Suite, mov. I[46][124]


Alberto Ginastera:
Sonata for Guitar, mvt. III, op. 47[28]

1920–22[125]

1925–26


1976

6–32 6–32 Mother chord (all-interval)
Trichordally combinatorial: 0369[126]
I: 0 1 5 8 t 3
9 4 2 e 7 6
P: 0 e 7 8 3 1
2 t 6 5 4 9
Arnold Schoenberg:
String quartet No. 4[64][116]
1936 6–16 6–16
I: 0 1 5 4 9 e
t 2 6 7 8 3
P: 0 e 8 4 7 3
1 5 2 6 9 t
Arnold Schoenberg:
Modern Psalm, Op. 50c[3]
1950 6–20 6–20
I: 0 1 4 8 5 9
e 7 t 6 3 2
P: 0 e 8 9 7 6
1 5 4 3 2 t
About this sound   
Alban Berg:
Lyric Suite, row 2[122]
1925–26 6-Z3 6-Z36
I: 0 1 4 3 5 6
e 7 8 9 t 2
P: 0 e 9 2 t 7
1 4 8 3 5 6
Milton Babbitt:
"The Widow's Lament in Springtime"[127]
1951 6–8 6–8
I: 0 1 3 t 2 5
e 8 4 9 7 6
P: 0 e 9 4 3 t
8 7 5 6 1 2
Dmitri Shostakovich:
Symphony No. 14, mov. I[82]
1969 6–5 6–5
I: 0 1 3 8 9 2
4 5 7 6 e t
P: 0 e 9 6 t 8
7 5 2 4 3 1
Igor Stravinsky:
Canticum Sacrum, II[2]
1955 6–2 6–2
I: 0 1 3 6 2 4
5 7 t 8 9 e
P: 0 e t 1 3 2
4 9 8 6 5 7
Alfred Schnittke:
String Quartet No. 4, Mov. I[3]
1989 6–1 6–1 all-combinatorial
I: 0 1 2 e 9 t
8 3 4 6 7 5
P: 0 e t 5 6 9
7 1 2 8 3 4
Arvo Pärt:
Symphony No. 1
Perpetuum Mobile[56]
1963
1963
6–5 6–5
I: 0 1 2 7 6 3
5 e t 4 9 8

Other lengths

Less than twelve

Five
Prime form Notation Composer:
Composition
Year Set(s) Features
0 e 8 9 t Igor Stravinsky:
In memoriam Dylan Thomas[128]
1954 5–1
Six
Prime form Notation Composer:
Composition
Year Set(s) Features
0 6 9 8 1 e
[129][130][131][132]
Pierre Boulez:
Messagesquisse

Répons


Dérive 1


Incises


sur Incises

1976

1984


1984


1994


1996

6-Z11, (6-Z40) Musical cryptogram:
Paul Sacher
Conrad Beck:
Für Paul Sacher: Drei Epigramme für Violoncello solo
1976
Luciano Berio:
Les Mots sont allés
Benjamin Britten:
Tema "Sacher"
Henri Dutilleux:
Trois Strophes sur le nom de Sacher
Wolfgang Fortner:
Zum Spielen für den 70. Geburtstag: Thema und Variationen für Violoncello Solo
Alberto Ginastera :Puneña n° 2, op. 45
Cristóbal Halffter:
Variationen über das Thema eSACHERe
Hans Werner Henze:
Capriccio per Paul Sacher
Heinz Holliger:
Chaconne, für Violoncello Solo
Klaus Huber:
Transpositio ad infinitum
Witold Lutosławski:
Sacher-Variationen
0 9 8 7 1 4
About this sound   
Arnold Schoenberg:
Die glückliche Hand, Op. 18[133]

"Seraphita", op. 22, no. 1[133]

1910–13


1913–16

6-Z44 (6-Z19) Schoenberg hexachord
Eight
Prime form Notation Composer:
Composition
Year Set(s) Features
0 2 8 9
t 4 3 e
Alban Berg:
Lyric Suite[44]
1925–26 8–5
0 2 9 7
t 8 e 1
Joonas Kokkonen:
Symphonic Sketches, mov. 3, row B[40]
1968 8–1
0 7 1 6
8 2 3 4
Alban Berg:
Lyric Suite[44]
1925–26 8–5
0 8 2 3
t 4 9 e
Alban Berg:
Lyric Suite[44]
1925–26 8–5
0 t 2 4
5 7 3 1
Igor Stravinsky:
Agon, "Pas-de-Deux" refrain
(b. 463–72)[134]
1957 8–10
0 t 3 5
2 4 1 e
Joonas Kokkonen:
Symphonic Sketches, mov. 3, row C[40]
1968 8–1
Nine
Prime form Notation Composer:
Composition
Year Set(s) Features
0 2 e 4 9
7 8 6 1
Joonas Kokkonen:
Symphonic Sketches, mov. 3, row D[40]
1968 9–8
Ten
Prime form Notation Composer:
Composition
Year Set(s) Features
0 e 2 1 5
6 4 9 7 8
Arvo Pärt:
Collage sur B-A-C-H[56]
1964 BACH motif
Eleven
Prime form Notation Composer:
Composition
Year Set(s) Features
0 1 e 5 6 8
7 9 2 3 4
Karel Husa:
Music for Prague 1968[3]
1968
0 e t 4 6 2
1 9 8 5 3
Karlheinz Stockhausen:
Licht, Lucifer formula[71][135]
1977–
2003

More than twelve

Thirteen
Prime form Notation Composer:
Composition
Year Set(s) Features
0 1 4 5 t 8 7
t e 6 3 2 9
Karlheinz Stockhausen:
"Taurus", from Tierkreis[63]
1974–75
0 1 t 6 8 3 5
2 4 e 9 7 0
Karlheinz Stockhausen:
Choral[136]
1950
0 3 e 8 5 4
9 2 1 t 7 3 6
Luciano Berio:
Nones[137][138]
1953–54
0 4 1 5 7 t 2
8 e 9 3 6 9
Karlheinz Stockhausen:
"Pisces", from Tierkreis[63]
1974–75
0 7 8 6 e t 2
5 1 3 9 4 0
Karlheinz Stockhausen:
Licht, Michael formula[71]
1977–
2003
0 e 2 4 t 9
7 3 7 1 8 5 6
Karlheinz Stockhausen:
"Virgo", from Tierkreis[63]
1974–75
0 e 4 t 9 5
8 6 2 1 6 7 3
Karlheinz Stockhausen:
Klavierstück IV
(first row of 48)[66]
1952 6-Z38, 6-Z6
Fourteen
Prime form Notation Composer:
Composition
Year Set(s) Features
0 1 t 3 9 4 6
7 4 5 7 3 8 e
About this sound   
Karlheinz Stockhausen:
"Libra", from Tierkreis[63]
1974–75
0 1 t 9 2 8 e
7 3 5 6 4 0 2
Karlheinz Stockhausen:
Inori[139]
1973–74
0 e 2 4 1 8 t
e 9 4 7 5 3 6
Joonas Kokkonen:
Cello Concerto, mov. 3, row A[40]
1969
Seventeen
Prime form Notation Composer:
Composition
Year Set(s) Features
0 2 1 e t 8
9 e t 0 1 e
t 0 e 9 0
Igor Stravinsky:
Agon, b. 104[140]
1957
0 2 3 5 4 9
7 8 e 1 6 e
t 0 e 9 0
Igor Stravinsky:
Agon, "Bransles"[140]
1957 6-Z40, 6-Z11
Twenty-two
Prime form Notation Composer:
Composition
Year Set(s) Features
0 e 6 8 7
2 1 3 t 9 4
6 9 7 8 3
1 2 4 e t 0
Joonas Kokkonen:
Cello Concerto, mov. 3, row B[40]
1969
Twenty-four
Prime form Notation Composer:
Composition
Year Set(s) Features
 0  1 12 14 23 22
19 18 20 21  3  4
16 13  6 15  8  7
17 10  9 11  2  5
Pierre Boulez:
Polyphonie X, quarter-tone series[141]
1950–51
 0 20  1 10  9 11
17 15 16  7  2  6
23 21 22 13  8 12
18 14 19  4  3  5
Karlheinz Stockhausen:
Himmelfahrt, mod24 (no octave equivalency)[142]
2004–
2005

See also

Sources

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  141. ^ Pierre Boulez, "The System Exposed" in Orientations: Collected Writings by Pierre Boulez, edited by Jean-Jacques Nattiez, translated by Martin Cooper, 129–42 (Cambridge: Harvard University Press, 1986): p. 129. ISBN 0-674-64375-5.
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