Class notes are reminders of some of the main points discussed in class. They are not always presented in a rigorous format.
Symmetries and group structures.
Definition of a group:
a set
a binary operation
E


if for allelements of the set  the following are true


associativity
identity element 
existence of inverses

if also:
commutativity
(a 




the group is Abelian
In music pitch intervals form an abelian group with the unison as the identity element and inverses: P4 the inversion of P5,
m2 the inversion of M7, etc. adding up to the octave which, in turn is equivalent to the unison. The tone-row system uses
such symmetries; the dorian scale (2 1 2 2 2 1 2) or the “gypsy scale” (1 3 1 2 1 3 1) are symmetrical structures as well.
Meter can be represented as an abelian group with the down beat being the identity element especially in the case of 4/4,
3/4. 6/8, etc. Compound rhythms form also an abelian group and so are non-retrogradable rhythms.
The second movement of the Variations Op.27 by Webern utilizes a grouping of three dynamic markings, f-p-f and its
symmetrical inversion, p-f-p.
Similar symmetries can be found in the formal architecture of music by Guillaume de Machaut – La fin est mon
commencement, Anton Webern – Symphony Op. 21, Variations Op. 27 (especially the second movement), the tone-row of
Op 24, and many other works. A most compelling example is Nomos α by Xenakis, for cello solo, where a group of
permutations analogous to the rotations of a cube, are used to create a complex set of 18 out of a total of 24 variations. In
fact, Xenakis uses two cubes rotating inside each other; each vertex is the meeting of three edgesm each representing a
sound parameter. The other six sections are freely composed but use the same pitch sieves. A subgroup in Nomos α is the
Klein V4 group, the symmetry group of a rhombus:
| u a b c
__|__________
u | u a b c
a | a u c b
b | b c u a
c | c b a u
Xenakis has used the same technique in Nomos γ for orchestra.
An interesting parallel may be drawn between the concept of broken symmetry in physics (all four forces were united into
only one moments after the big bang but they differentiated as matter started cooling off) and the broken symmetry in a
piece like Schoenberg's Variations for orchestra Op 30. The theme contains a perfect symmetry between the cello solo and
the accompaniment of the orchestra:
P0
RI7
R2
I9
I9
R2
RI7
P0
P3
which is “broken” by the addition of P 3 in the violin part along with the last pair thus allowing the composer to continue
with the variations.