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All-Combinatorial Hexachords
 
 
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All-Combinatorial Hexachords
explained by Dr. Jody Nagel
 
See Milton Babbitt's "Some Aspects of Twelve-Tone Composition"
from The Score and I.M.A Magazine 12 (1955) pp.53-61.
 
 
"All-Combinatorial Hexachords" have the following property:
 
Call the first hexachord (six pitch-classes) of some specified 12-tone row "M" and the second hexachord "N".
Require "M" and "N" to have the same Prime Form set-type.
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 "All-Combinatorial Hexachord," and the row is an "All-Combinatorial Hexachordally-Generated 12-Tone Row."
 
A row, generated from an "All-Combinatorial Hexachord," with pitch-class order "M, N" could participate in a first-species counterpoint "duet" with one of the rows of order "N, M" so that the following takes place:
- Voice-1 will play all 12 notes.
- Voice-2 will play all 12 notes.
- The first half of Voice-1 and the first half of Voice-2 will together constitute all 12 notes (an aggregate).
- The second half of Voice-1 and the second half of Voice-2 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 prime-form hexachords have the property of being "all-combinatorial."
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 all-combinatorial row-forms.)

Trichords Hexachords
(formable by the given trichord and an equivalent trichord at some Tn or TnI)
3-1 (012) A,D
3-2 (013) A,B
3-3 (014) A,E3 *
3-4 (015) B,D2,E3
3-5 (016) D
3-6 (024) A,B,C,F3
3-7 (025) B,C
3-8 (026) F6
3-9 (027) C,D
3-10 (036) none
3-11 (037) C,E3
3-12 (048) E,F
 

All-Combinatorial Hexachordally Generated Twelve-Tone Rows :
Set-forms of the original row (T0) include Tn, In, R(Tn ), and R(In ).
 
Each all-combinatorial hexachord, shown above in Prime Form, can be used along with its compliment (which will be of the same hexachordal type - non-Z-related), to form a 12-tone row such that at least one Tn, one In, one R(Tn ), and one R(In ) can unfold simultaneously with T0, partitioned 6+6, resulting in two 62 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 "first-order all-combinatorial hexachords" in that only one of each set-form type (T, I, R, RI) can combine with T0 and produce combinatorial 6+6 partitioning. (Also, A, B, and C have only one tritone-axis of symmetry.)
- Hexachord D is "second-order" because there are two of each set-form type combinable with T0. (D has two axes of symmetry.)
- Hexachord E is "third-order" because there are three of each set-form type combinable with T0. (E has three axes of symmetry.)
- Hexachord F is "fourth-order" and there are six of each set-form type combinable with T0. (F has six axes of symmetry.)
 
* Note: When a tri-chordal generator can be used in more than one way to construct an all-combinatorial hexachord, a superscripted number shows how many possibilities exist.
 
terms to know:
aggregate, derived set, secondary set, partition, compliment, z-related hexachord
 
______________________________________________________________________________
 
Tranpositions of Hexachordal Prime Forms and Inversions Necessary for Generating
All-Combinatorial Twelve-Tone Rows.

 
Inversion is calculated in the following manner:
If T0 = [0, 1, 2, 3, 4, 5], then I0 = [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, I11 = the 11th transposition of I0. I11 = [6, 7, 8, 9, 10, 11].
 
Notice that the necessary transposition(s), Tx, 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 T0, namely T6.

Hexachord Interval Vector
T0  +   Tx or Ix necessary to produce haxachordal complementation
A   [0,1,2,3,4,5] < 5 4 3 2 1 0 > T0  +   T6  or  I11
B   [0,2,3,4,5,7] < 3 4 3 2 3 0 > T0  +   T6  or  I1
C   [0,2,4,5,7,9] < 1 4 3 2 5 0 > T0  +   T6  or  I3
D   [0,1,2,6,7,8] < 4 2 0 2 4 3 > T0  +   T3  or  T9  or  I5  or  I11
E   [0,1,4,5,8,9] < 3 0 3 6 3 0 > T0  +   T2  or  T6  or  T10  or  I3  or  I7  or  I11
F   [0,2,4,6,8,10] < 0 6 0 6 0 3 > T0  +   T1  or  T3  or  T5  or  T7  or  T9  or  T11  or
             I1  or  I3  or  I5  or  I7  or  I9  or  I11
 

Tranpositions of Trichordal Prime Forms and Inversions Necessary for Generating an All-Combinatorial Hexachord.

All-Combinatorial
Hexachord
Trichord used
as generator

Tx or Ix necessary to produce hexachord
A  [0,1,2,3,4,5] (012) [ T0  or  I2 ]  +  [ T3  or  I5 ]
A  [0,1,2,3,4,5] (013) T0  +  I5
A  [0,1,2,3,4,5] (014) I4  +  T1
A  [0,1,2,3,4,5] (024) [ T0  or  I4 ]  +  [ T1  or  I5 ]
B  [0,2,3,4,5,7] (013) I3  +  T4
B  [0,2,3,4,5,7] (015) I5  +  T2
B  [0,2,3,4,5,7] (024) [ T0  or  I4 ]  +  [ T3  or  I7 ]
B  [0,2,3,4,5,7] (025) I5  +  T2 ]
C  [0,2,4,5,7,9] (024) [ T0  or  I4 ]  +  [ T5  or  I9 ]
C  [0,2,4,5,7,9] (025) T0  +  I9
C  [0,2,4,5,7,9] (027) [ T5  or  I7 ]  +  [ T2  or  I4 ]
C  [0,2,4,5,7,9] (037) I7  +  T2
D  [0,1,2,6,7,8] (012) [ T0  or  I2 ]  +  [ T6  or  I8 ]
D  [0,1,2,6,7,8] (015)1 T7  +  T1
D  [0,1,2,6,7,8] (015)2 I1  +  I7
D  [0,1,2,6,7,8] (016) T0  +  I8
D  [0,1,2,6,7,8] (027) [ T0  or  I2 ]  +  [ T6  or  I8 ]
E  [0,1,4,5,8,9] (014)1 T0  +  I9
E  [0,1,4,5,8,9] (014)2 I1  +  T4
E  [0,1,4,5,8,9] (014)3 T8  +  I5
E  [0,1,4,5,8,9] (015)1 T0  +  I9
E  [0,1,4,5,8,9] (015)2 I1  +  T4
E  [0,1,4,5,8,9] (015)3 I5  +  T8
E  [0,1,4,5,8,9] (037)1 T9  +  I8
E  [0,1,4,5,8,9] (037)2 I0  +  T1
E  [0,1,4,5,8,9] (037)3 T5  +  I4
E  [0,1,4,5,8,9] (048) [ T0  or  T4  or  T8  or  I0  or  I4  or  I8 ]  +  [ T1  or  T5  or  T9  or  I1  or  I5  or  I9 ]
F  [0,2,4,6,8,10] (024)1 [ T0  or  I4 ]  +  [ T6  or  I10 ]
F  [0,2,4,6,8,10] (024)2 [ T10  or  I2 ]  +  [ T4  or  I8 ]
F  [0,2,4,6,8,10] (024)3 [ T8  or  I0 ]  +  [ T2  or  I6 ]
F  [0,2,4,6,8,10] (026)1 T0  +  I10
F  [0,2,4,6,8,10] (026)2 I6  +  T8
F  [0,2,4,6,8,10] (026)3 T10  +  I8
F  [0,2,4,6,8,10] (026)4 I2  +  T4
F  [0,2,4,6,8,10] (026)5 T6  +  I4
F  [0,2,4,6,8,10] (026)6 I0  +  T2
F  [0,2,4,6,8,10] (048) [ T0  or  T4  or  T8  or  I0  or  I4  or  I8 ]  +   [ T2  or  T6  or  T10  or  I2  or  I6  or  I10 ]
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