Suggested Learning Resources

SET-CLASSES – contain all possible transpositions of a pitch-class-set and all transpositions
of its inversion. All pitch-class-sets within a Set-Class share the same Interval-Class-Vector.

To find all pitch-class-sets within a set-class...

- Write out all 11 transpositions

- Find the inversion of original pitch-class-set and reverse its order

- Write out all eleven transpositions of the inverted set

- Example, let's determine the set-class of {9,t,e,0,2}

- First: write out all the twelve transpositions of this set {9 t e 0 2}
, {t e 0 1 3} {e 0 1 2 4} {0 1 2 3 5} {1 2 3 4 6} {2 3 4 5 7} {3 4 5 6 8} {4 5 6
7 9} {5 6 7 8 t} {6 7 8 9 e} {7 8 9 t 0} {8 9 t e 1}

- Second, determine the inverse of the original set {9 t e 0 2} becomes {3
2 1 0 t} then reverse the order to obtain {t 0 1 2 3}

- Third, write all the transpositions of that {t 0 1 2 3} {e 1 2 3 4} {0
2 3 4 5} {1 3 4 5 6} { 2 4 5 6 7} {3 5 6 7 8} {4 6 7 8 9} {5 7 8 9 t} {6 8 9 t e
0} {7 9 t e 0} {8 t e 0 1} {9 e 0 1 2}

NOTE: Most Set-Classes have 24 members (all transpositions and inversion transpositions),
but if the content of a pitch class set is repeated, it is only counted once…. Some
items have fewer members (like augmented triads have 4 members… diminished seventh
chords have 3 members.. octatonic has 3 members and whole tone has 2 members. The
total number of sets that reproduce themselves is called the degree of symmetry. If
Set-Class has three pitch-class-sets that are the same in transposition and 3 pitch-class-sets
that are the same in the inversion transposition, the degree of symmetry is 6.

https://youtu.be/3LvUqWIxquw

Objective 54.1: Define set-classes, and identify all possible pitch-class-sets contained
within a set-class.