Row patterns are first presented, modeled on the row set of Webern's Konzert that exhibit higher degrees of symmetry. Start with the motive
(E): Bb | A / B
It really does take 3 pitches to create the first and most elementary pattern; an interval is not a pattern, unless it has another, possibly rhythmic structure imposed on it; e.g., the opening decending major third of Beethoven's fifth symphony, (GGG Eb).
thinking perhaps of the Chopin B minor etude. Using appropriate transpositions of its (R), (I) and (C), rows of the forms
(E) (I) (R) (C) (E) (R) (I) (C) (C) (I) (R) (E) (C) (R) (I) (E) (I) (C) (E) (R) (I) (E) (C) (R) (R) (C) (E) (I) (R) (E) (C) (I)
(8 permutations of four symbols out of a total of 24 total possible permutations) rows can be constructed so that their two halfs are crabs of each other:
(E) (I) (R) (C) Bb | A / B | G / Ab | F# | F | D# / E | C / D | C# (E) (R) (I) (C) Bb | A / B | F | D# / E / G / Ab | F# | C / D | C# (C) (I) (R) (E) (C) (R) (I) (E) (I) (C) (E) (R) (I) (E) (C) (R) (R) (C) (E) (I) (R) (E) (C) (I)
Also available are the more obvious rows with less global symmetry constructed by motivic transpositions:
(E) (E) (E) (E) (I) (I) (I) (I) (R) (R) (R) (R) (C) (C) (C) (C) (E) (E) (E) (I) (E) (E) (I) (I) (E) (I) (E) (I) (I) (E) (E) (I) (I) (E) (I) (E) ...
Each of the 4 positions in the row can be filled by 4 intervallically translated permutations of a 3 note motive giving a total of 4^4 = 2^8 = 256 possible 12 tone rows derivable from a 3 note motive. Some have inbuilt symmetries while others do not; all, however, have discernable and audible inbuilt structure clearly related to the generating motive.
The 3 note motive itself, as set of 3 pitches has 3! = 6 possible permutations:
(cyclic) 1 2 3 2 3 1 3 1 2 (alternating) 2 1 3 3 2 1 1 3 2 Including repetitions, there are 3^3 = 27 possibilities: 111 112 121 113 131 122 123 132 133 211 212 221 213 231 222 223 232 233 311 312 321 313 331 322 323 332 333
The 6 previous permutations are clearly contained within these possibilities. There is then in the fundamental context of any 3 note motive a vast amount of mathematical structure - and more importantly here, a vast amount of variational, growth (systematic, audible and sensible accretion of pitches) and death processes (systematic, audible and sensible elimination of pitches) for musical material and structure in a wider context of dodecaphonic and pitch set oriented musical presentation and development.
Now consider the intervallic evolution of the fundamental 3 note motive, by intervallic expansion:
(Bb|A/B) -> (Bb|A/C) -> (Bb|A/C#) -> (Bb|Ab/C) -> (Bb|A/D) -> (Bb|Ab/Db) -> (Bb/|A/Eb) -> (Bb|A|E) -> (Bb|Ab/Eb) -> (Bb|G/D) -> (Bb/|Ab/E) -> (Bb|Gb/D)
Putting this evolution of motive together with the first row form, for which the retograde is transpositionally equivalent to the inversion, generate a sequence of 12 evolving rows all with the same symmetry.
(E) (I) (R) (C) Bb | A / B | G / Ab | F# | F | D# / E | C / D | C# Bb | A / C | G / Ab | F / F# | D# / E | B / D | C# Bb | A / C# | B / C | Ab | G | D# / E | D / Gb| F Bb | Ab / C / D#/ F | C# / F# | D / E / G / B | A Bb | A / D / F / F# | C# / E | B / C / Eb / Ab| G Bb | Ab / Db | A / B | F# / G | D / E | C / F | Eb Bb | A / Eb / E / F | B / C | F# / G / Ab / D | C# Bb | A / E | D / Eb | Ab / C# | F# / G | F / C | B Bb | Ab / Eb | D / E | A / C | F / G | F# / C#| B Bb | G / D / D#/ F# | B / E | A / C / C# / G#| E# Bb | Ab / E | C / D | F# / G | B / C# | A / F | Eb Bb | Gb / D / E / G# | C / F | A / C# / Eb / B | G
Even further, consider any sequence of specific pitches translated in a well tempered environment by the intervals in the sequence. For example,
(E) (I) (R) (C) Bb | A / B | G / Ab | F# | F | D# / E | C / D | C# (down a minor 2nd) A | Ab/ Bb | Gb/ G | F | E | D / Eb | B / Db| C (up a major 2nd) B | Bb/ C | Ab/ A | G | F#| E / F | C#/ Eb| D (down a major 3rd) G | Gb/ Ab | E / F | Eb | D | C / C# | A / B | Bb (up a minor 2nd) Ab | G / A | F / F# | E | Eb| C# / D | Bb/ C | B (down a major 2nd) F# | F / G | Eb/ E | D | Db| B / C | Ab/ Bb| A (down a minor 2nd) F | E / F# | D / Eb | Db | C | Bb / B | G / A | Ab (down a major 2nd) D# | D / E | C / C# | B | Bb| Ab / A | F / G | F# (up a minor 2nd) E | D#/ F | C#/ D | C | B | A / Bb | F#/ G#| G (down a major 3rd) C | B / C# | A / Bb | Ab | G | F / F# | D / E | Eb (up a major 2nd) D | C#/ D# | B / C | Bb | A | G / G# | E / F#| F (down a minor 2nd) C# | C / D | Bb/ B | A | Ab| Gb / G | Eb/ F | E
This is, of course, NOT music, but simply a systematic way of using notions of sets, groups, permutations and symmetries to generate self-similar material (think fractals) for pitch sequences that may be created, and that make sense to the human ear.
One can also use the same or similar techniques on other musical parameters and densities: tempo, rythm, event activity rate, pitch extension rate, intervallic extension, intervallic extension rate, pitch density, dynamics, color-timbre, as well as row transposition, crossrelations betwee any of these and application of group theoretic transformations to complex activities of any time length. E I E, E I R I E, e.g. Think of how useful the classical ABA form and its extensions have been for so long.
This is one way of loosening of the bonds of a monochromatic serialism of pitch, and of all and total serialism generally. "Serialism without serialism" a language that gives methods of musical discipline while leaving all the actual controls in the composer's hands, where it should be.
Email me, Bill Hammel at