Combinatoriality

Combinatorial Rows are those which can produce some inversion, transposition, and/or retrograde in which the first 6 pitches will be completely different from the first 6 pitches in the original row.

                          Pitch Content     Pitch Content
      Prime Form:         hexachord a       hexchord b
      Transformation:     hexachord b       hexachord a

All-combinatorial rows feature combinatoriality between Po and an I, RI, and possibly an R form and in one or more transpositions (note that Po and Ro will always be combinatorial, so it isn't worth mentioning).

Semi-combinatorial rows feature combinatoriality with only an inverted (I) form (no transposition or RI).

Here is a procedure to determine if a row is combinatorial (it's possible to simply look at the matrix and figure it out, but it's pretty slow).

1. Take the first six notes and arrange them into a pitch class set and determine the prime form. The second 6 notes of any row will have the same set as the first, except for Z-related sets.
2. Check to see if that pitch class set is in the list below.
3. All-combinatorial sets will transpose at intervals where there are zeros in the interval vector (remember, those numbers indicate the number of common tones at that transposition level).
4. To find inversion possibilities: leave the original 6 notes of the row in pitch class numbers (do not turn them into a pitch class set). NOTE: for this to work, you need to call the first note 0 and then calculate the others as the number of half steps above that first pitch. Follow these steps:
   • eliminate all even numbers (I2, I4, I6, etc.), they will never work.
   • eliminate all odd numbers which are in the first six numbers.
   • eliminate all odd numbers which can be formed by adding together any two numbers from those first six numbers.

Example:         Bb  E  Gb  Eb  F  A  D  C#  G  G#  B  C
First 6 numbers:  0   6  8   5   7  11

Eliminate all even numbers, eliminate 5, 7, and 11 (they are in the first six numbers), 8+5=13 (1) (eliminate), 8+7=15 (3) (eliminate).

No two numbers added together equal 9, so the row is combinatorial at I9.

All Combinatorial		Semi-combinatorial
Pitch-class Set	Interval Vector	(023468) (012468) (013579) (012346) (023579) (013679) (012458) (013589) (014568) (012357) (013469) (012367) (012578)
(02468A) (014589) (012345) (012678) (023457) (024579)	[060603] [303630] [543210] [420243] [343230] [143250]

(013458) is a special set: T6 produces a combinatorial row.