In the previous unit, we talked

about networks and their representation

through graphs: nodes, links,

that represent the connection

between the different data points,

but we didn't talk about music,

and how music fits in this picture.

So, in this unit, I will introduce you

to a way of representing musical

spaces using networks.

And when I talk about musical spaces,

I talk about melodies, or harmonies,

or rhythms, timbres, orchestrations,

so the arrangements of the different

instruments in a particular song,

for instance, and how this can all

be represented through network

and giving us this view of music

through the complexity of the graphs.

In what follows, we'll concentrate on the

Western classical music tradition, but

this approach is completely general, you

can apply it to any musical tradition,

being Western or not, and one can see

how there are some universality structures

that emerge from the representation of

these musical spaces through networks.

The individual elements, or units, that we

will use in defining these musical

networks, are actually the pitches,

the individual notes, that form the basis

for this classical music tradition and

not only classical.

A pitch is a perceptual representation

of a sound through its frequency

and timbre. It's what we hear when we hear

a note played by an instrument. In Western

classical music tradition, there are

12 pitches that are defined, or that are

tuned to a particular temperament that

we call equal temperament. This is not

very important in the following discussion

but it is something that you should keep

in mind. There are musical traditions

where there are more than 12 pitches,

for instance, where the tuning is

different and it depends on the particular

instrument, the particular genre, or style

of music.

So, with these 12 pitches, we can create

all the possible melodies, all the

possible harmonies and so they provide

us with an enormous space that you can

explore by combining them.

So, in a way, one of the most famous

novels by Jorge Luis Borges,

The Library of Babel, he hypothesized

a potentially infinite library, where each

book contains all possible combinations

of the 26 characters of the alphabet and

so eventually, out of all these possible

combinations, you might have a book

that is Moby-Dick, for instance, or

The Odyssey, in this infinite number

of combinatorial possibilities of letters

arranged in all possible ways.

In a way, what I want to push is the idea

that we could do the same with pitches,

with the notes of our music, and basically

construct all the possible spaces that

connect notes with each other,

or that can connect chords,

so combinations of notes with each other.

So, in the following, I will talk mostly

about the concept of harmony space,

that means all the ways I can construct

vertical combinations of pitches in chords

so, you now, typical chords in any song

where you have a succession of chords

that provide the backbone of the harmony,

of the progressions of the song.

And we need to start doing some

mathematical abstractions, here, so as

you will see in this slide, I can

construct all the possible

combinations of these 12 pitches

in this harmonic space

by basically calculating all the

combinatorial possibilities excluding

all pitch repetitions, all permutations

within the same chord, so that I reduce

this space to a minimal dimension.

And if I do this, I can still combine the

12 pitches in a large number of

effective chords. So in this redaction

if I use 12 pitches, I can construct

4'096 chords, with different numbers of

notes per chords, but if I extend this to

a 24-note space, or a different

temperament, a different tuning system,

for different cultures, musical cultures,

these combinations become enormous.

So, in a 24-note space, we go higher than

16 million possible chords and if we look

at the whole keyboard of a piano, as you

have here in this slide, and we think

about all the possible chords that we

can combine notes with, then we get to

an astronomically large number of

10 to the 26 different chords.

So, out of all these possible

combinations, what we want to do is

we want to construct a geometrical model

of this harmonic space. And remember that

here, what we are looking for is how we

can represent this harmonic space through

a network representation, through nodes

and links. So, in this approach here, we

can see that chords as order pitches,

combinations of pitches, that

mathematically can be seen as vectors.

So, a vector of three components is,

for instance, the first chord in this

slide, in the left corner, is a C Major

chord, is a vector of 3 different pitches.

On the other hand, since we are defining

a vector space, we can associate to

this vector space, a concept of metric,

of distance: how far our chords are

within this space, and if we can

quantify the distance. And indeed

we can, and it's actually relatively

simple, we just generalize the

Pythagoras's theorem, it's the

Euclidean metric that gives us a

distance between chords. By doing this,

we basically define our nodes of the

network, as the vectors, so the chords,

and we hypothesize links between chords

based on their distance. So, if I decide

I want to see all the chords, and the

connection between chords that are

distanced, say 1, in the space of chords,

then I will have a network representation

that corresponds to this metric.

In this slide, I am showing a very complex

network of chords. This is a network

that is obtained using the 24-pitch space,

so it's this larger pitch space that I was

talking about earlier, and I am only

connecting nodes that are distanced 1

from each other in this metric space,

and you see that the representation

of this network becomes extremely

complex, actually even artistically

pleasing if you wish, and we will talk

about what the different colors mean

in this image in the next slides.

So, in order to make things easier and

kind of better follow the relationship

between all these geometrical elements,

let's focus on a subset of chords, of

the 12-pitch space, that are the chords

made of 3 pitches, the triads.

And what I'm showing in this slide, are

the networks that are formed building

links at different distances.

So, the first one on the left is the

network where I am connecting triads

that are distanced 1 through each other,

that means that I can go from one chord

to the next chord by changing only one

note by a semitone.

If I do a similar representation but for a

distance of, say, square root of 2, like

in the right top corner, then the topology

of this network changes. Instead of having

one single network, I have two networks

that are separated, it means that the

movement from one chord to another

is constrained by this topology, and if

I want to move by a square root of 2,

in this space, I can either stay on one

network or on the other.

And so on and so forth, I mean,

the square root of 3 is also bipartite,

it's a kind of 2 different networks -

one is very small, that contains

chords that are very dissonant,

and another one that contains

all the chords that are more

harmonious to our perception.

So, not only do we have a representation

of a space, that we can navigate

as composers, for instance, choosing

paths through these networks, but we also

have some sort of classification of

different chord categories, that emerges

from these representation itself, so from

the topology of this representation.

So, what I'm showing you here, is some

sort of map, of a space that we can

navigate, and that composers navigate

when they write music that is based on

these particular chords. So, the classical

chord progression of a rock song, for

instance, might belong to one or more

of these networks that I am showing you

here, and this kind of spatial

representation, or topological

representation is very much linked

to what we learned by talking with

Pr. Dmitri Tymoczko about the geometry

of chords, of chord spaces, in one of the

previous units.

Now, this is not a composition, this is

just a map. So, how do we do a

composition starting from this mapping?

Well, the composer here becomes the

agent that takes from this deterministic

representation of the pitch space,

the complexity of a work of art, or

a piece of music, as an emergent

property of these systems.