Properties of 12-tone Sets

Combinatoriality-the most efficient way of controlling the rate of unfolding of aggregates (harmonic pacing, rhythm); discovered by Schoenberg, exploited by Babbitt

3 Basic large groups:

1. semi-combinatorial

a. I-combinatorial-a given prime form tht will combine with one inversion

b. RI-combinatorial-P0 will combine with one RI form (R is all comb. w/ its I)

c. 2nd order I combinatorial-will combine with two inversions

d. P-combinatorial-will combine with a transpositin of itself (TT)-rare; only one

Schoenberg Variations for Orchestra

any set is combinatorial w/ its own retrograde

2. non-combinatorial

Berg & Webern didn’t use combinatorial sets (when they did they didn’t know or care)

Schoenberg uses combinatoriality as a means of avoiding octaves

3. all-combinatorial

any form which will combine with at least one transpositon of itself, one I, one R and one RI

all-combinatorial sets-2nd hexachord is always a literal transpositon of the 1st

a. first order-a form that combines with one P, I, R & RI

i) 0 1 2 3 4 5 (720 orderings possible; 7202 possible for whole source set)
ii) 0 2 3 4 5 7 | 6 8 9 10 11 1-most used to make an all-interval set

ex.
source set: {0 2 3 4 5 7 | 6 8 9 10 11 1}
ordered set: {2 5 3 4 0 7 | 1 11 6 8 9 10}

go to the number in the 2nd hexachord that corresponds to the order number in the 1st one that the set
begins on (go to D then G#): THIS IS THE COMBINATORIAL PRIME FORM

count back from the end of the source set to find I form (D -> B)

R-which transposition of the 1st hex. will maintain exactly the same pitch content as the 1st
hexachord? (only are for first order: itself)

RI which inversion of the prime will maintain the same pitch content?
(itself in first order)

iii) {0 2 4 5 7 9 | 6 8 10 11 1 3}
-Guidonian hexachord

has maximum possible linear intervallic variety, but the least vertical intervallic possibilities
(least amount of sets to combine with)

in any 2 1st order all-comb. sets: if hexachordal content is maintained one will get exactly the same
vertical diads (P & I, R & RI)

b. second order-will combine with two of each {0 1 2 6 7 8 | 3 4 5 9 10 11}

hexachords composed of two symmetrical trichords; space between each aggregate same as each trichord

c. third order-will combine with three of each {0 1 4 5 8 9 | 2 3 6 7 10 11}

if set begins on C, then it will combine w/ set starting on D, F#, A#

if set begins on C, then it will invert w/ sets on B, G, D#

d. sixth order (or fourth)- will combine with six of each (WT scale)

{0 2 4 6 8 10 | 1 3 5 7 9 11}