We have discussed how processes
like harnessing randomness, or defining
precomposition structures, micro-
and macrostructures give rise
to a composition and we also looked at
three examples that show contemporary
musicians and how contemporary composers
use these tools, but there
is more to this.
Now, we will introduce this concept
of pattern and how patterns
and compositions are intrinsically
related.
So, patterns can operate on all levels
of composition. From a microstructural
point of view to a macrostructural
point of view. So from, say the deep
structural organisation of pitch
and rhythm to how these combine
in a macroscopic sense in the creation
of the piece of music. Patterns can
involve many different things, and there
is a very general concept. It can be just
a single song parameter or involve
a single song parameter, such as
the frequency for instance, fundamental
frequency of a sound, or multiple sound
attributes so the amplitude, the duration,
the pitch, and how this evolve over time.
And again, we operate on patterns with
most, or all, of the same operations that
we have introduced in the previous units,
but we can add more operations now
in a kind of broader palette
of possibilities for composition.
So patterns can be cycles, where I have
a sequence of elements that I repeat
in a loop; it can be just a line, so that
I go from one beginning to an end in a
particular sequence and that's a pattern;
I can do palindrome, where I go forward
and backward on a same sequence;
I can permute elements randomly in what
we call a heap; or I can use stochastic
elements to generate, to select elements
in different distributions,
so we encounter again
some of the operations that we
already defined in previous units, but
now we put them in a larger and broader
context. You can see here in this chart
that we can have rotations, we can have
Markov chains, we can use graphs
and networks as elemental structures
of a piece, or how we structure
the space of pitch, for instance,
or rhythm. We can have functions and we
can have rules, so we can define actually
a grammar that we can then follow in
the composition of the piece.
In the example that is discussed in
the Pierre Boulez video, that's actually
a form of grammar that we have defined
with some specific rules that are
associated with musical meaning.
Now, patterns are by definition, objects
that have a length, so different lengths
correspond to different chunks of
the pattern, or they have a period that
is repeated, so it kind of defines
a longer timeframe of the composition.
And again, this plays into this micro vs.
macro scale of the compositional process
and the actual result of the composition
that is the musical piece.
Patterns can contain other patterns, so
again we have another aspect of scale
economy in music, I have patterns that
are composed of different patterns, and
I can play this game to the microscopic
scale of the single pitches and single
durations. They can work in parallel
because I can add patterns for rhythm
and patterns for harmony, and patterns
for melody and so on. And the idea is that
we combine all of these patterns
in a composite that is
the foundation of the composition,
the design, the framework and the content
of the piece.
Let's talk now about specific patterns
that we can use to compose,
and let's start with
mathematical patterns, so what I
call music from math. Now let's say that
beautiful mathematics doesn't necessarily
give rise to beautiful music, and vice
versa, but sometimes it does. And so we
can look at some classical examples
of number series of integers for instance,
like you know the Pascal's triangle, or
the Fibonacci's sequence. The Fibonacci's
sequence has been used by innumerable
numbers of composers over centuries,
it's all related with this golden ratio,
it has philosophical implications
and so on. So, we find Fibonacci sequences
in the structure of Mozart piano sonatas,
or Bach chorals, or you know, many
different examples. Now, here what is
really important, in order to translate a
mathematical function or a mathematical
sequence into music, is to define a
mapping strategy for both pitch duration
and so on. So again, we go back to
the idea that we use a deterministic
function in a way, that we want to
translate into some material.
Of course, the mapping depends on
the data that you have, so you can map
pitch durations or other parameters,
dynamics, orchestration,
macroparameters, like the length
relationships between different sections
of a single piece, for instance. And this
is where this Fibonacci sequence comes in
very prominently in the history of music.
Mapping pitches can be done on the
integer mapping that we have introduced
from 0 to 11, so the 12 pitches of the
equal tempered scale, or any other
scale for that matter. Or we can do it
with MIDI numbers, or we can map to a
specific scale, these are all possible
techniques, and there are probably many
others that one can utilize. Now one thing
that we need to do when we map from data
to pitch for instance, is that our data
need to be normalized. So even in
a sequence of numbers, if we are mapping
to a modulo 12 number system, we need to
normalize the data, or do something to
the data, so that we can eventually map
with the right correspondance.
So normalization is something that is, you
know, standard mathematical operations
that we can do on a series of data, and
here on this slide you see a way
of mapping, for instance, data points
to indices, indices that might be indices
of a scale, or indices of a sequence that
you predefine and then you map, you
use these indices to map your data points
to a particular sound
or pitch or duration.
You also need to map durations, so you
can use the same trick as in the previous
slide, in the normalization, you can
define an indexing sequence that maps to
a predefined set of durations. So you can
use intrinsic properties of the data
to extract some duration properties.
And here, I am showing you
an example in this formula, where I map
the duration to the pitch difference
between subsequent notes. So if
the notes are closer together, then
the duration is shorter; if they are
further apart, the duration is larger.
All these ideas are demonstrated in
the notebook that is associated to
this unit, and you will be able to play
with this and explore by yourself how
you can map specific sequences of integers
for instance into musical objects.
So another way that we can extract
patterns and then use them in a musical
context is, rather then using a
mathematical formula to generate
the data, it's to use the data itself,
data that is available, measurements
of any process that you can think of.
And here, I want to make an aside,
because using data to make music is not
what people call data sonification. It is
not like if I use data from, I don't know,
the sun activity over the past twenty
years, I am not giving you the sound
of the sun. I am giving you a musical
abstraction that uses the data
to generate parameters of a composition.
Data sonification is associated with the
utilitarian use of sound to convey
information. And so, there are instances
where sound can be used as an objective
representation of the data,
a perceptualization of the data, that is
complementary to the visual representation
for instance, and sound has been suggested
as one way, for instance, of making sense
of large data sets that are impossible
for a human to grasp just by visualizing
them. I will not be talking about this
aspect here, there is a lot of literature
that you are welcome to research
about data sonification and how this can
be used in a scientific setting. Here,
I want to use data with an aesthetic and
an artistic goal, that means I want to use
the data as pattern-generating objects
that I can then use in a composition,
a little bit like we did with random data,
in the previous unit.
Data needs to be read, and it comes in
many different formats. You have examples
of this in the notebook associated with
this unit, where we use actually existing
libraries that are very well known in the
data science community, like pandas
for instance for Python, that allow us
to read data and transform and manipulate
them and map them into musical parameters
within the context of this course.
You can find many different websites, like
the one that is indicated here in
this slide, where you can get data from,
and in the notebook there is at least
one example where we read and
manipulate data and then create
a mapping to a musical composition.
We have seen mathematical functions used
as generating sequences and patterns for
musical compositions, we have seen how
we can use data to do the same, but this
is not enough to create, to generate a
piece of music. Most of music is based on
the application of some processes
to ideas.
So, my pattern as I was saying earlier can
be manipulated and extended using
processes, and these processes are central
to the analysis and composition of music.
So, we can talk about combinations
of notes horizontally, so in a melodic
context, I can start from a motif,
or a pattern, and then changing
or applying a process to this pattern
in order to enrich the musical content.
And here, we have various examples
that have been present in music
for centuries, like the canon or the round
or any contrapuntal construction that we
can do on a pattern, or a motif.
Or, we can combine notes vertically, so
form chords and vertical combinations
of pitches that lead to chord sequences,
or what we call chord progressions, or
voice leadings.
All of this can actually be described at
the micro level as an algorithmic process.
Of course, these processes are rules that
one follows, and the genius
of the composer is to navigate
these rules and create something
that is original and compelling.
But the rules can be analyzed as elements
of a general grammar of music composition.