Инструменты |
Фортепиано, Скрипка, Виолончель |
Состав исполнителей |
Трио |
Тип нот |
Партитура |
Издатель |
Michael Staff |
Время звучания |
6'0 |
One form in which Alkali borate is structured is trigonal planar, with one
boron atom (+) being bridged to 3 oxygen atoms (-). As a structure with SP2
hybridization, it possesses symmetrical properties that find a close analog in musical
composition, which were used in Permutations in a number of ways. First, the
molecular symmetry is mathematically-related to the harmonic and melodic shape of
the thematic material. Second, the overall thematic structure in the music follows a
similar symmetrical outline. Lastly, the mathematical group to which the symmetry
of BO3 belongs is also a cyclic group, as is the mathematical group to which musical
notes belong; as a cyclic group must contain at least one generator element, the
generator elements are extensively used to create an auditory illusion in the piece's
final permutation. In addition to these direct relationships the music has with the
aforementioned element in chemistry, a final yet more abstract approach is
suggested by the title of the work.
In the form of a 2-dimensional planar triangle, BO3 possesses a 120-degree
bond angle, and thus can be rotated or reflected while satisfying all the axioms
necessary to be classified as a mathematical group (also a permutation group due to
these operations). In addition to this fact, it is also a cyclic group since it possesses 2
generator elements, 1 and 2 (if we label each vertice with 0, 1, and 2, respectively)
since under addition both of these elements generate the entire group.
The 12 musical notes can be ordered chromatically into a dodecagon, and
under the operation of addition, these notes as intervals also satisfy the same group
axioms.
If we assign each integer (0-11) to a corresponding interval, our first interval,
P1(The Perfect first, which is the interval between the base note and itself),
corresponds to 0. Similarly, the next interval, m2, corresponds to 1, and represents
the interval from the base tone to the next tone up, or a chromatic half-step (from c
to c# as an example). The last interval then, the M7 (from c to b for example),
corresponds to 11.
The 12 intervals are closed under addition (group axiom 1). The P1 interval is
the identity element (group axiom 2), since when added to any other element a, the
result is that element a. The intervals are also associative (group axiom 3), since
changing the order of a series of harmonic transformations will not change the result.
Most interestingly, each element also has an inverse (group axiom 4), so that when
the two are combined, the identity element is the result.
However, if we form a subgroup using 0, 4, and 8 from our Z12 group, we not
only end up with the augmented triad, but we end up with the same symmetrical
properties as Alkali borate. Both the rotations and reflections in the chemistry
application provide us with all the possible chord inversions for this chord in the
musical context. Interestingly, our 3-element subgroup is also cyclic.
The main thematic material from which each permutation is derived is an
augmented harmony overlapped by a chromatic figure, construced as such because
the augmented triad is the subgroup analagous to Alkali borate, and the minor 2nd
(the interval used in chromatic passages) is one of the generator elements of the
cyclic group Z12.
The work's conclusion contains an even tighter construction of these concepts
in order to create an auditory illusion, an effect that makes the music sound as if it's
constantly rising even though it's staying within the same small range. This is
accomplished by using a variant of the thematic material and rotating/reflecting it as
a series of permutations; and since the group is cyclic and satisfies group closure,
continuing the group permutations (that would normally extend outside the octave)
but as their counterparts within the octave, it creates an anxious and confusing
effect. This climax represents the point of maximum entropy.