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.