**An animated Mandelbrot fractal **showing a large magnification
zoom.

(To review this fractal animation after it is fully loaded (which may take a few minutes), back out (Alt) and immediately return (Alt) to this page or right-click the image and choose "View Image".)

The topic of this essay is the manifestation of fractal-like structures in existing music, rather than the creation of new music. For those interested in the use of fractals to create music, a list of of websites is provided at the end of this exposition.

The word "fractal" means "broken" and originally referred to fractional dimensions and irregular (hence "broken") surfaces. Fractal geometry is a new branch of experiential mathematics that recognizes that objects in the real, or natural, world are not purely square, triangular, circular, spherical, cylindrical, etc., but, rather that the latter are are only idealized geometric models. Planet Earth is described in traditional science as a sphere or ellipsoid, although its surface is not smooth, but rough, with mountains and valleys. The coastline and boundary of India may be described in Euclidean geometry as a triangle, but it is really very complex and irregular.

One of the defining aspects of fractals is their *fractional dimension*.
We are used to dealing with whole dimensions. We say, for instance, that
a line is one dimensional, a plane is two dimensional, a sphere is three
dimensional. But what does it mean to have a fractional dimension? Taking
India's coastline as an example, we can ask how we should measure its perimeter.
On the surface this seems to be straightforward, but before we can actually
measure it we must determine the unit, or scale, of measurement. Zooming
into the details of a coastline results in ever increasing intricacy and
complexity of the line that defines it. Should we measure the line around
boulders, pebbles, grains of sand? As the unit of measurement becomes smaller
and smaller the perimeter approaches infinity, and the larger our measuring
unit, or scale, the shorter the perimeter seems to be. Before long we realize
that such a measurement can only be an approximation. So, how long is the
coastline? There is no one definitive answer, but it depends upon the unit
of measurement and how much approximation one is willing to settle for.

The fractal dimension is a measure of the space-filling ability. The coastline seems to be a line of one dimension, but if it were to continue infinitely it could fill a plane of two dimensions. Therefore, it is said to be somewhere between one and two dimensions. This fractional dimension can actually be mathematically determined by the degree of roughness or space-filling capability of the object. (Mandelbrot showed that the Hausdorff-Besicovitch dimension of a coastline is 1.26.)

Perhaps the most important defining property of fractals is self similarity
on many different scales; i.e., they have self-iterating geometric structures
that repeat in different sizes. "People use the word fractal in different
ways, but all agree that fractal objects contain structures nested within
one another like Chinese boxes or Russian dolls" [Kanadoff, *Physics
Today, *1986]. There are two kinds of self similarity: 1. exact similarity
(linear fractals), and statistical similarity (non-linear fractals). In
linear fractals, each scaling factor reproduces exactly the same shape,
as in the Sierpinski triangle and Cantor fractal cited later. In nonlinear
types, ocurring most often in natural objects, the resemblance at differing
scales is not exact, but is close enough to be recognized as statistically
self-similar.

Mathematical fractals have, theoretically, an infinite number of scales,
but fractals are also said to occur in nature, where the scaling factor
is finite. One example of this is the natural fern. This is a picture of
the natural fern, *Dicksonia culcita*:

And, here is a fractal computer modeling of one of the fern's fronds:

The shape of one of the whole branches of the fern is triangular, but it is not a true triangle. It is curved and has irregular edges. Let's call this fractal shape a "triferngle". Each of the fronds that make up the branch is also a statistical triferngle. Each leaf cluster is arranged in a triferngular shape, and each leaf is triferngular. Each leaf is also divided into triferngular shapes. Thus, the fern has a fractal structure of repeating self-similar triferngles on several diminishing scales.

From this remarkable replication we can see that fractal modeling can simulate the forms of nature. They are now used to create landscapes in movies and for computer modeling. Just as fractal geometries are found in nature, similar self-iterating structures can be found in music, architecture, and other arts.

As an example, one of the first and simplest fractals is called the Cantor fractal, which is based on the Cantor set or function.

The definition of the Cantor set is as follows:

A _{0} = [0, 1]

and define, for each *n*, the sets A* _{n} *recursively
as

A* _{ n}* = A

Then the Cantor set is given as:

C = A _{n}

A graphic representation of this set is as follows:

This fractal may be formed by taking a line of length A_{0}
and deleting the middle third, forming the second line in the above diagram.
By the iterative process the middle third of each of the two segments is
deleted, producing the third line. This process continues indefinitely,
deleting the middle third of each of the resulting segments, forming a
"Cantor comb". Thus, each of the steps is a binary division of
the result of the previous step, and the whole process generates the number
series: 1, 2, 4, 8, 16, etc. Looking at the process in reverse, the smaller
units are added in binary groups to form the larger units. As the set continues
to fractionalize it approaches a series of points having no dimension,
or "Cantor dust". The Cantor fractal has a fractional dimension
of 0.63, i.e., lying between a point and a line.

Now consider a simple piece of music by Beethoven, the first *Ecossaisen*.

**Link to a Key for symbols of formal analysis**
** **(Use
your "Go Back" or [Alt ]
key to return here)

This shows the score with a formal analysis. There are a total of 32
measures, which divide into 2 sections, **A** and **B**, of 16 measures
apiece (measure numbers are in boxes): 1-16 and 17-32. This represents
a symmetrical balancing of the 2 parts. Each of these sections divides
into 2 eight-bar periods, shown in slurs with the large numbers 1-4 above.
In turn, these eight-bar periods divide into binary 4-bar phrases, labeled
1-8. with smaller numbers. The 4-bar phrases divide into two sub-phrases
of two-bar lengths: labeled a1, b1, c1, d1, etc. The two-bar units subdivide
into binary one-bar motives: **m** and its transformations. Finally,
the one-bar motives are composed of a binary group of 2-eighths plus a
quarter, of which the eighths themselves are a binary unit. Binary one-bar
units are also contained in the bass part, labeled **n**. This figure
is a time-dilated (augmented) version of ** x**. In several places
it is also interval-dilated. Interval dilation is also common in the treble
part. Finally, the meter is a simple duple (binary).

Even the appearance of the hierarchical slurs in the above analysis mirrors the Cantor comb. Each successive subdivision of the 32 measures is a binary unit, and thus, each is a smaller replica of the larger unit containing it. The structure, therefore, is the same as that of the Cantor fractal.

The structure of Beethoven's *Eccosaisen *is not exceptional for musical
compositions. In fact, most compositions are structured in a similar manner,
with paired 4-bar and 2-bar phrases units being the most common. Their subdivisions
form motives and other small units that echo self-similar binaries. "Periods"
and "sections" are then constructed by the accumulation of these smaller
units into increasingly larger binary groups. The binary form is probably the
most common form in music, and there are several varieties of this form, including
the *rounded binary* and *sonata forms*. Even the sections of ternary
forms (**ABA**) are constructed with binary units. Since symphonies and concerti
use sonata form, they also have the same type of hierarchical structure. The
type of structuring where the smallest unit is echoed similarly into larger
and larger encompassing units is often called **architectonic**
in music and is virtually the definition of a fractal.

Another common fractal is called a Sierpinski triangle, in "two dimensions", or a Sierpinski pyramid in "three dimensions". Actually, the Sierpinski triangle has a fractional dimension of 1.584.

Sierpinski's triangle can be formed by what is called the "chaos
game". In this game any three points are chosen on a plane. They may
be labeled (1,2), (3,4), and (5,6) as above, which are the vertices of
the large triangle. The points need not be equidistant. Any point *x*
may be chosen in the same plane, inside or outside the figure. With the
roll of dice (1 die), a point is made halfway between *x* and the
vertex indicated by the die. This process is repeated from the last point
a large number of times, i.e., each new point generated from the last point.
Although the expected result is a randomly dispersed field of points, a
Sierpinski triangle is invariably the result.

**Link:
See the Chaos game generate a Sierpinski triangle. **(Use
your "Go Back" or [Alt ]
key to return here)

Sierpinski's triangle may also be created by starting with a solid triangle and bisecting each of its sides. Lines connecting these points of bisection then form a triangular hole with 3 solid triangles bordering the hole. The same operation of bisection is then done on each of the remaining solid triangles, opening up new holes similar to the first. Each of the new triangles has dimensions that are exactly half that of the original and a quarter of the area, representing binary divisions. This operation is repeated any number of times. Therefore, the structure may be seen as ternary and binary units reiterating on diminishing scales. (Use your "View Reload" key [Ctrl-R] if the following example is blank.)

If the operation is repeated indefinitely the bizarre result is a triangular figure having an infinite dimension (of one-dimensional length) but zero area (two dimensions). Although it appears two-dimensional, it hovers between one and two dimensions; i.e., of fractional dimension. A finite version, such as seen above, has a musical analogue in a ternary form (ABA) with repeated ternary and binary subdivisions (the line bisections and areas). Although such forms exist in music literature, they are somewhat less common than the plain binary "Cantor" types.

One example occurs in Beethoven's piano Sonata no. 15, op. 28, third movement (Scherzo), which is a combination of binary and ternary schemes similar to the Sierpinski structure.

This is only the first 32 measures of the scherzo, i.e., the first part
of the **A** section, or **A1**. **A1** here refers to the first
part of **A**, and **A2** is the second part. "**A**"
represents the whole section, which combines **A1** and **A2**. Similarly,
**B1** and **B2** are the first and second parts of the **B**
section, called the "Trio".

**Link: See
a complete analysis of this movement **(Use
your "Go Back" key to return here)

Scherzos are in triple meter and ternary form (**ABA**), establishing a
triad scheme as in a triangle. **A2** repeats and is 38 measures without
the repeat (32 measures plus a 4-measure extension). The Trio (**B**) is
32 measures with its repeat. The ternary scherzo is 210 measures including the
repeats. The tonal scheme is that of a typical musical ternary (**A:I B:vi A:I**)
.

|**| A1 (32) ||: A2
(38)[76] :||: B1 (8)[16] :|| B2
(16) || A1 (32) || A2 (38) ||**

A hierarchical binary occurs within the ternary. The **A** section is a
rounded binary, with a return of the first part** || A1 ||: A2 A1 :||.**
So, the second part of **A** (measures 33-48, not shown) actually contains
**A1** within it, forming a miniature **ABA**within the **A.** This is due to the return of **A1** at the end
of **A2**. The tonal scheme of **A** also conforms to that of the rounded
binary (I V I) (or an incipient ternary). A binary substructure
also pervades the **A **section, including the two-note repeating staccato
figure in each measure, and the falling octave motive at the beginning and at
every 8-measure interval. In **A2** this octave figure becomes the primary
motive repeated sequentially in a series of modulations in the bass, accompanied
by the binary staccato figure in the treble. These binary figures are always
coupled with a quarter rest to form a ternary, as can be seen in the **A1**
section above. All the sections are formally multiples of 16 (**A2** having
a 4-measure extension). These all break up into 8-measure binaries and each
of these 8-measure units divides into binary 4-measure phrases.

The trio, or **B** section, is also a binary of 32 measures in the
relative minor key. The **B** is divided into hierarchical binary units
of eight and four measures similar to that of the **A** section. Binary
octave figurations in the bass accompany the obligato 4-measure repeating
melody eight times before the **A **section returns.

In measure 5, a prime motive consists of two eighths plus a quarter (rhythmically), falling as a downward arpeggiated 3-note chord. This "tertian motive" is repeated throughout the scherzo. Thus, the scherzo has ternary and binary forms on several scales. Rather than being exceptions, these schemes are abundant throughout the literature of music. Iterating binaries are common in the dances of the Renassance through Classical periods. Iterating ternaries are more common in the Classical and Romantic periods.

Fractal forms may be generated by using a simple iterative rule (formula) and a "seed" or motive. This seed is the basic shape used to generate the fractal. Thus a tree can be made from a basic geometric line and a simple rule of transformation.

Note the mirroring around the central axis of this figure. Operations
of a symmetry group may be applied in various ways and degrees. The process
here may be described as the imitation, or translation, of a basic shape
with attendant symmetry operations. Analogous operations on a basic motive
are carried out in most contrapuntal music. Imitation is a ubiquitous operation
in canons and fugues. From the Renaissance through the time of Bach composers
constructed *enigma canons* (also called "puzzle canons"
or "riddle canons") in which a simple musical figure (seed) was
notated and a canon was to be played from it by performers who could figure
out the rules of the canon, such as the intervals of imitation and whether
or not to apply various symmetry operations. In music these operations
are commonly called transposition, inversion, retrograde, retrograde-inversion,
augmentation, and diminution. More generally they are the traditional symmetry-group
operations of translation, reflection, and rotation.

These operations have been defined mathematically in my dissertation, ** Symmetry
as a Compositional Determinant** (1973, rev 2002). A brief
summary is provided here:

Reflections around y and x axes respectively: *T(x,y) = (-x, y)*
and *T(x,y) = (x, -y)*

Common rotations are:

90^{o} rotation: *T(x,y) = (y, -x)
*180

Translations are defined as:

Horizontal: *T(x,y) = (x+a, y)
*Vertical:

The following is an example of one of these enigma canons for 8 voices,
called *Trias Harmonica* by J. S. Bach. It is shown here in Bach's
original notation:

Believe it or not, this is the entire 8-voiced canon. The notation gives all the necessary pitch and rhythmic information, but the operations of transposition, time and pitch intervals, and symmetry operations must be determined by the performers. Here is one possible solution:

The time interval for the voice entrances is one quarter-note. The pitch interval of imitation is a perfect fifth, cycling back to tonic. The imitation is inverted and transposed on alternating entries.

In such works as Bach's *Art of Fugue,* *A Musical Offering,
*and Johannes Ockghem's *Missa Prolationem,* composers repeat simple
motivic ideas, subjecting them to the variation operations of symmetry
groups. A form of motive on the large scale is echoed by diminution on
smaller and smaller scales, repeated, inverted and retrograded (mirrored),
and translated (transposed) at many points in time.

**The Sound of Mathematics**

**MusiNum**

**Fractal Music Lab**

The Music of José Oscar
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