*Arch Carrellage* is a computer music composition for
organ sounds and modeled sound space. Atonal and micro tonal
sources are diffused using an Ambisonics six speaker
arrangement in a hexagon pattern. This composition is a
triptych with subsections in each part. Aside from
differences in size (length in time and duration), tempo
and timbre, each section is structured as a sequence of
tiles resulting from combinations of symmetric sets in the
form of dyads, trichords and tetrachords, as well as
threading of horizontal textures. Symmetric sets are
obtained by combining two different (perhaps orthogonal)
tone rows (more about this below). Besides geometrical
patterns, this piece is a search for clusters of diffused
sound sources, such as those heard on Philadelphia's
*Saints Peter and Paul Cathedral.* Reverberation on an
enclosed space such as this, is also a symmetrical feature,
in spite of differences in perspectives, formant sections,
and intersections of thousands of reflections that come from
sound sources and walls in the cathedral. We can pinpoint
that these reflections are like those images induced when a
hexagon is divided in twelve equal parts, to get another
circumscribed hexagon and further joining all points and
edges thereby creating equilateral triangles inside each
hexagon, in addition to a bunch of other circumscribed
geometric forms (provided perception is exercised). But on
its conception, this piece focuses on symmetrical structures
related to pitches that construct each element on each
part.

Sound wise, this piece experiments on sonority and textures resulting from blends on a myriad of timbres that hold the Organ adjective. It also touches upon atonal and micro tonal combinations of musical events in contrast to traditional tempered organ sounds. Sources come from own recordings, samples and modeled acoustics of the organ. Signal processing on some of these sources enhance gestures which cannot be achieved by any other means. Sequences of notes product of algorithmic output and computer aided composition triggered every part on this composition. For this purpose, a set of tools was developed in Lisp and Scheme by the composer, for assisting in processing of tone-rows that give symmetrical combinations, in addition to Rick Taube's Common Music (CM). Bill Shottstaedt's S7, Snd, and Common Lisp Music (CLM), as well as Joe Anderson's Ambisonics Toolkit (ATK) and Fernando Lopez-Lezcano's Dlocsig, still are extremely useful programs for synthesis and signal processing.

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**Figure No.1: Basic tile (or template) from the
***Herat Pattern.*

*Herat Pattern.*

In this figure two hexagons can be perceived inside a circle. They result from dividing the circle with diameters to get twelve equal parts. Further in, there are four circumscribed triangles that in turn form rhomboids and other polygons traced inside the outer hexagons. Since outer hexagons are symmetric, it is expected that inner divisions give more symmetries. Same with the four equilateral triangles that share a common center. Note that shapes on these divisions provide further symmetry. Consequently and by their symmetric properties these figures are also, rotations, tilts, mirror and transpositions, among others. Operations like those outline above can manipulate sequences of notes in the form of symmetric dyads, trichords and tetrachords that come form a tone-row.

#### Working with symmetries:

For the structure of *Herat
Pattern,* named after the Herat Mosque in Afghanistan. At
first sight patterns of hexagons, crosses, rhomboids,
diamonds and stars can be perceived and recognized. Further
into the *Herat pattern,* irregular pentagons as well
as rhomboids standout. Interesting enough, six smaller
triangles inside a hexagon also come into perception (see
figure No,1). These triangles are also centered on a smaller
hexagon and consequently, six little hexagons with
equilateral triangles at their center cast this
figure. Thus, from a visual perspective the Herat pattern
demonstrates perception at various levels.

Worth to point out, hexagons are an outstanding example of symmetry. By joining, or flapping, several hexagons, we get honeycomb patterns, known as one of the most basic of tilings, but also a template for more symmetric patterns. To get a Herat pattern we need two hexagons plus four inscribed equilateral triangles. Outer hexagon is divided into twelve equal parts. The second inner hexagon is drawn by joining the middle points of the outer hexagon. The bases of the first inner triangles are drawn by joining points of opposite sides of the inner hexagon. These lines are parallel sides of each triangle and go from the endpoints of these parallels to the opposite equilateral triangles which overlap each other (see figure again). The parallel bases for the second equilateral triangles are drawn from midpoints of adjacent sides of the inner hexagon. Recall we had divided the outer hexagon into twelve equal parts. The sides of the second triangle are drawn from the bases to the midpoints of the upper and lower adjacent sides of the inner hexagons. Twelve divisions of the outer hexagon help find these midpoints.

#### Composition with the above:

*Arch Carrellage* is seeded on two tonerows are
borrowed from Luigi Dallapiccola, lyric compositions
*Preguiere* and *Parole di San Paolo.* Two rows
because they resemble a cross, although not necessarily
horizontal or vertical but almost orthogonal so that they
have one point (note) in common. Nevertheless intersecting
points slide from top to bottom, or from right to left. This,
in order to achieve symmetries on their tetrachordal or
trichordal combinations. While working with tone rows, basic
symmetries are obtained by mirroring the row, by inverting
its intervals, or by cycling the position of each note in
the row. With these operations we get cycles, rotations and
palindromic musical shapes. For compositional purposes these
figures work as they do on geometrical analogies such as
symmetric figures as in basic honeycombs.

From a musical perspective, mapping of the above geometry, takes into account that size translates into length and duration, but regardless of patterns, translations, rotations, shrinking or enlarging as repetition processes, we need to isolate or constraint parameters so that we can map visual shapes into music events. Since analogies of these shapes can establish counterparts in the sound domain, the process of generating geometric patterns and compositions with these figures on the plane or space, gives a parallel for generating musical ideas. Visual mapping of these figures don't have exact correspondence on the musical domain and vice versa. However, operations to generate other symmetries and more shapes are analogous in both domains. (i.e by rotating a square, we get a diamond. In a diamond there are two equilateral triangles tat can also be tilted. Similarly a tetrachord can be rotated so that we have a palindrome of its pitches. Dyads in this tetrachord can also be tilted so that we get other variants of the original symmetry -see Figure No.2-.)

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**Figure No.2: Tesselations and tilings using the
***Herat Pattern.*

*Herat Pattern.*

In a further dimension, rotations, translations,
transpositions and folding of the basic *Herat
Pattern* create more symmetrical shapes which now can
tease the mind. By the same token in the musical domain,
these operations generate variations on a tone row,
thereby creating symmetries on left and right
hexachords, left, center, right terachords, and the four
triads. Mind constructions of this type produce
generative grammars that help on the cognitive features
of perception while trying to make sense of shape an
structure on either visual domain or sound domain.

Intuitively a hexagon can be used as a template to get combinations of notes for tetrachords, a triangle for triads, and so on. But knowing that a hexagon casts such a symmetric structure, why not go further?. By transposition or rotations, a symmetric shape creates more symmetries. With this axiom we can assume that two symmetrical combinations of tetrachords can generate more structural shapes, closely related to the original by means of symmetry (i.e. Two opposed triangles can be perceived as a diamond. Two opposed triads can generate a symmetric tetrachord, though an axis note needs to be found so that the triads unfold similarly top or bottom, left and right.)

Prime forms of each row are labeled as T_{0} and
P_{0}, respectively. Subscripts give the number of
transposition of the prime form. Palindromic and inverted
versions of these primes are labeled RIT_{0} et
RIP_{0}, respectively. By arithmetic operations
(or rather permutations) and by listening, we found that
combinations T_{0}, P_{0}; T_{1},
P_{1}; T_{6}, P_{6}; in addition
to, RIT_{10}, RIP_{10}; RIT_{11},
RIP_{11}; RIT_{6}, RIP_{6}; are
primes and variations, suitable for creating
symmetries. By assigning six of these variations to the
edges of the hexagon, we create paths to each of the other
five edges to find combinations of rows an
variations. Hexachords for horizontal sequences are the
result of combinations of these edges; hexachords can also
be used to get dyads and triads vertically. If these
hexagons are used, we can also build a square symmetric
matrix with combinations of the patterns found in each
hexagon. In this fashion we end up with a six-by-six
matrix with no horizontal or vertical contiguous same kind
hexachords.

#### ✇ Listen 🗧

... Compressed UHJ Ambisonics down mix of *``Arch Carrellage''*