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
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 a total of 24 total 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)
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
This is, of course, NOT music, but simply a systematic way of using notions of groups, permutations and symmetries to generate self-similar material (think fractal) for music that may be created.
Email me, Bill Hammel at