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AllCombinatorial Hexachords
explained by Dr. Jody Nagel
See Milton Babbitt's "Some Aspects of TwelveTone Composition"
from The Score and I.M.A Magazine 12 (1955) pp.5361.
"AllCombinatorial Hexachords" have the following property:
Call the first hexachord (six pitchclasses) of some specified 12tone row "M" and the second hexachord "N".
Require "M" and "N" to have the same Prime Form settype.
The specified row will have the order "M, N."
Then look for rows (out of the possible 48) that have the order "N, M."
If at least one transposition of each of (1) type "T" (2) type "I" (3) type "RT" and (4) type "RI" can be found that have the order "N, M," then the hexachord is said to be an "AllCombinatorial Hexachord," and the row is an "AllCombinatorial HexachordallyGenerated 12Tone Row."
A row, generated from an "AllCombinatorial Hexachord," with pitchclass order "M, N" could participate in a firstspecies counterpoint "duet" with one of the rows of order "N, M" so that the following takes place:
 Voice1 will play all 12 notes.
 Voice2 will play all 12 notes.
 The first half of Voice1 and the first half of Voice2 will together constitute all 12 notes (an aggregate).
 The second half of Voice1 and the second half of Voice2 will together constitute all 12 notes.
Voice 1: M, N
Voice 2: N, M
The two voices are in a "hexachordal combinatorial" relationship.
Only six primeform hexachords have the property of being "allcombinatorial."
Milton Babbitt named them Hexachord "A," "B," "C," "D," "E," and "F."
These will be considered in detail in the following charts.


Hexachord

Degree of Symmetry

Possible Trichord Generators

A

[0,1,2,3,4,5]

first order (one tritone axis)

(012), (013), (014), (024)

B

[0,2,3,4,5,7]

first order (one tritone axis)

(013), (015), (024), (025)

C

[0,2,4,5,7,9]

first order (one tritone axis)

(024), (025), (027), (037)

D

[0,1,2,6,7,8]

second order (two tritone axes)

(012), (015)^{2} *, (016), (027)

E

[0,1,4,5,8,9]

third order (three tritone axes)

(014)^{3}, (015)^{3}, (037)^{3}, (048)

F

[0,2,4,6,8,10]

fourth order (six tritone axes)

(024)^{3}, (026)^{6}, (048)


Trichordal Generators (that can be used to construct allcombinatorial rowforms.)


Trichords

Hexachords
(formable by the given trichord and an equivalent trichord at some T_{n} or T_{n}I)

31

(012)

A,D

32

(013)

A,B

33

(014)

A,E^{3} *

34

(015)

B,D^{2},E^{3}

35

(016)

D

36

(024)

A,B,C,F^{3}

37

(025)

B,C

38

(026)

F^{6}

39

(027)

C,D

310

(036)

none

311

(037)

C,E^{3}

312

(048)

E,F


AllCombinatorial Hexachordally Generated TwelveTone Rows :
Setforms of the original row (T_{0}) include T_{n}, I_{n}, R(T_{n} ), and R(I_{n} ).
Each allcombinatorial hexachord, shown above in Prime Form, can be used along with its compliment (which will be of the same hexachordal type  nonZrelated), to form a 12tone row such that at least one T_{n}, one I_{n}, one R(T_{n} ), and one R(I_{n} ) can unfold simultaneously with T_{0}, partitioned 6+6, resulting in two 6^{2} vertical partition types. ("n" is an integer between 0 and 11, representing some transposition of the original row [T], the inversion [I], or the retrograde [R] of either of these.)
 Hexachords A, B, and C are "firstorder allcombinatorial hexachords" in that only one of each setform type (T, I, R, RI) can combine with T_{0} and produce combinatorial 6+6 partitioning. (Also, A, B, and C have only one tritoneaxis of symmetry.)
 Hexachord D is "secondorder" because there are two of each setform type combinable with T_{0}. (D has two axes of symmetry.)
 Hexachord E is "thirdorder" because there are three of each setform type combinable with T_{0}. (E has three axes of symmetry.)
 Hexachord F is "fourthorder" and there are six of each setform type combinable with T_{0}. (F has six axes of symmetry.)
* Note: When a trichordal generator can be used in more than one way to construct an allcombinatorial hexachord, a superscripted number shows how many possibilities exist.
terms to know:
aggregate, derived set, secondary set, partition, compliment, zrelated hexachord
______________________________________________________________________________
Tranpositions of Hexachordal Prime Forms and Inversions Necessary for Generating
AllCombinatorial TwelveTone Rows.
Inversion is calculated in the following manner:
If T_{0} = [0, 1, 2, 3, 4, 5], then I_{0} = [0, 11, 10, 9, 8, 7] = [7, 8, 9, 10, 11, 0] Starting with the first (lowest) note of the ascending Prime Form, the Inversion consists of the same set of intervals, though now descending. (The inversion may then be presented in ascending order.) In this example, I_{11} = the 11th transposition of I_{0}. I_{11} = [6, 7, 8, 9, 10, 11].
Notice that the necessary transposition(s), T_{x}, consists of those intervals missing from the hexachord's interval vector. The interval vector, usually expressed within triangle brackets, is a list of the number of semitones, whole tones, minor thirds, major thirds, perfect fourths and tritones found between all possible pairs of pitch classes found within the set. The six smallest traditional intervals, along with their "inversions," are thought of now as interval class 1, 2, 3, 4, 5 and 6. So, in the case of A [0,1,2,3,4,5], the interval vector is < 5, 4, 3, 2, 1, 0 >, and the missing interval (there are no "interval class 6's") is the interval used to obtain the compliment of T_{0}, namely T_{6}.


Hexachord

Interval Vector

T0 + T_{x} or I_{x} necessary to produce haxachordal complementation

A [0,1,2,3,4,5]

< 5 4 3 2 1 0 >

T_{0} + T_{6} or I_{11}

B [0,2,3,4,5,7]

< 3 4 3 2 3 0 >

T_{0} + T_{6} or I_{1}

C [0,2,4,5,7,9]

< 1 4 3 2 5 0 >

T_{0} + T_{6} or I_{3}

D [0,1,2,6,7,8]

< 4 2 0 2 4 3 >

T_{0} + T_{3} or T_{9} or I_{5} or I_{11}

E [0,1,4,5,8,9]

< 3 0 3 6 3 0 >

T_{0} + T_{2} or T_{6} or T_{10} or I_{3} or I_{7} or I_{11}

F [0,2,4,6,8,10]

< 0 6 0 6 0 3 >

T_{0} + T_{1} or T_{3} or T_{5} or T_{7} or T_{9} or T_{11} or
I_{1} or I_{3} or I_{5} or I_{7} or I_{9} or I_{11}


Tranpositions of Trichordal Prime Forms and Inversions Necessary for Generating an AllCombinatorial Hexachord.


AllCombinatorial Hexachord

Trichord used as generator

T_{x} or I_{x} necessary to produce hexachord

A [0,1,2,3,4,5]

(012)

[ T_{0} or I_{2} ] + [ T_{3} or I_{5 }]

A [0,1,2,3,4,5]

(013)

T_{0} + I_{5}

A [0,1,2,3,4,5]

(014)

I_{4} + T_{1}

A [0,1,2,3,4,5]

(024)

[ T_{0} or I_{4} ] + [ T_{1} or I_{5} ]

B [0,2,3,4,5,7]

(013)

I_{3} + T_{4}

B [0,2,3,4,5,7]

(015)

I_{5} + T_{2}

B [0,2,3,4,5,7]

(024)

[ T_{0} or I_{4} ] + [ T_{3} or I_{7} ]

B [0,2,3,4,5,7]

(025)

I_{5} + T_{2} ]

C [0,2,4,5,7,9]

(024)

[ T_{0} or I_{4} ] + [ T_{5} or I_{9} ]

C [0,2,4,5,7,9]

(025)

T_{0} + I_{9}

C [0,2,4,5,7,9]

(027)

[ T_{5} or I_{7} ] + [ T_{2} or I_{4} ]

C [0,2,4,5,7,9]

(037)

I_{7} + T_{2}

D [0,1,2,6,7,8]

(012)

[ T_{0} or I_{2} ] + [ T_{6} or I_{8} ]

D [0,1,2,6,7,8]

(015)_{1}

T_{7} + T_{1}

D [0,1,2,6,7,8]

(015)_{2}

I_{1} + I_{7}

D [0,1,2,6,7,8]

(016)

T_{0} + I_{8}

D [0,1,2,6,7,8]

(027)

[ T_{0} or I_{2} ] + [ T_{6} or I_{8} ]

E [0,1,4,5,8,9]

(014)_{1}

T_{0} + I_{9}

E [0,1,4,5,8,9]

(014)_{2}

I_{1} + T_{4}

E [0,1,4,5,8,9]

(014)_{3}

T_{8} + I_{5}

E [0,1,4,5,8,9]

(015)_{1}

T_{0} + I_{9}

E [0,1,4,5,8,9]

(015)_{2}

I_{1} + T_{4}

E [0,1,4,5,8,9]

(015)_{3}

I_{5} + T_{8}

E [0,1,4,5,8,9]

(037)_{1}

T_{9} + I_{8}

E [0,1,4,5,8,9]

(037)_{2}

I_{0} + T_{1}

E [0,1,4,5,8,9]

(037)_{3}

T_{5} + I_{4}

E [0,1,4,5,8,9]

(048)

[ T_{0} or T_{4} or T_{8} or I_{0} or I_{4} or I_{8 }] + [ T_{1} or T_{5} or T_{9} or I_{1} or I_{5} or I_{9} ]

F [0,2,4,6,8,10]

(024)_{1}

[ T_{0} or I_{4} ] + [ T_{6} or I_{10} ]

F [0,2,4,6,8,10]

(024)_{2}

[ T_{10} or I_{2} ] + [ T_{4} or I_{8} ]

F [0,2,4,6,8,10]

(024)_{3}

[ T_{8} or I_{0} ] + [ T_{2} or I_{6} ]

F [0,2,4,6,8,10]

(026)_{1}

T_{0} + I_{10}

F [0,2,4,6,8,10]

(026)_{2}

I_{6} + T_{8}

F [0,2,4,6,8,10]

(026)_{3}

T_{10} + I_{8}

F [0,2,4,6,8,10]

(026)_{4}

I_{2} + T_{4}

F [0,2,4,6,8,10]

(026)_{5}

T_{6} + I_{4}

F [0,2,4,6,8,10]

(026)_{6}

I_{0} + T_{2}

F [0,2,4,6,8,10]

(048)

[ T_{0} or T_{4} or T_{8} or I_{0} or I_{4} or I_{8 }] +
[ T_{2} or T_{6} or T_{10} or I_{2} or I_{6} or I_{10} ]



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