In the previous units,

we have explored how in using,

in a way, mathematical rules and patterns,

we can construct compositions that

have an aesthetic unity and a musical

sense. Right now, we need to move on,

refining some of these concepts,

and introducing more the idea

of simultaneity of sounds that, in

the musical language, corresponds to

harmony. Harmony will be one, if not

the most important aspect of music

that we will be looking at from the point

of view of complexity

and complexity theory.

The basic concept in harmony is the

idea of chords. A chord is a combination

of notes that are assembled vertically,

that means they sound together, and we

can hear here a chord played on a keyboard

this is a C Major triad - it's a

combination of three notes - played one

on top of the other.

<i>Demonstration</i>

This is an example of a three-note chord

called a triad, but of course you can

combine as many notes as you want, in

a chord. But in the classical music

tradition, of Western music tradition,

the triad itself is a foundational concept

in the idea and the theory of harmony.

So triads can be constructed using

all the notes of a scale. So we introduced

the idea of a scale before, we looked at

how we can build scales also through

the notebooks, each scale, each note

of a scale

can be the foundation for a chord on

that particular note, and so you see in

this image for instance, you have all

the triads that can be constructed over

a C Major scale.

<i>Demonstration</i>

Now of course you can have more

three notes, you can have four notes, five

notes or more. On more complex

situations on more modern music,

you can have a stack of different notes

with higher dimensions.

We want to explore these harmony space.

Given at our disposal our notes,

our pitches of equal temperament,

the 12 pitches of

the equal temperament scale, we want

to understand, or measure, how many

of these chords we can construct,

for instance using permutations

and combinations of different dimensions.

So, we can construct a dictionary of all

the possible combinations of notes,

vertical combinations of notes between

a dimension of, say, 2. That is the

minimal superposition of 2 notes that

are playing simultaneously, or even

1, like the unison, and 12 would be

we use all the pitches that we have at

our disposal in the scale. So if we try

to build a dictionary, or a kind of

measure of how many of these

combinations we can construct,

then we have a measure of

of how many chords can exist given

the raw material, our notes, our pitches

in the scale. So if we consider a 12-note

space, we have a number of chords that

we can construct that is of the order

of 4'000. And this number I will explain

a little bit in a second how we can

actually reduce this number to

the minimal units, the minimal

combinations of pitches, but if we expand

this to a temperament where instead of

having 12 pitches we have 24 pitches, so

we introduce the idea for instance of

quarter sharps and quarter flats, then

we get 16 million different combinations

of chords and if we look at the keyboard

here, a keyboard with 88 keys, and we try

to evaluate how many combinations of keys

we can construct, then we have a number

that is larger than the number of atoms

that you have in a cubic centimetre

of material. I mean 10 to 26 different

chords.

Now, this very large combinatorial space

can be deduced by operations that take

into account the equivalence of pitches,

so that every C in the keyboard

is actually one single entry in this large

combinatorial space.

We can deduce chords by transpositions

and inversions and we can introduce all

these mathematical operations on chords

that we discussed in previous units.

This enormous space that we have

constructed with all these

combinatorial possibilities can be

deduced by using operations on the pitches

of our chords that we have introduced in

the past units. So here, for instance,

I am reintroducing or showing again all

the possible operations that you can do

on a pitch, I mean on a chord, so a

connection of pitches, or a pitch set as

is called in modern music theory, and all

these operations reduce this

large combinatorial space

to minimal units. But still, this

is an enormously large combinatorial space

anyways. So the four operations that are

foundational to contemporary music theory,

or at least pitch set theory, are

the octave equivalents, so the fact that

all the octaves are equivalent, it doesn't

matter if we are on a higher octave or

a lower octave, the C is always a C.

That is permutations, I can do a

permutation of chords, of pitches

within a chord, and that corresponds

to the same set of our space, although

in terms of music theory this might

be kind of questionable, but we are not

going to get into these details

at this stage.

Another very important one is

transposition: I can transpose

my pitches by a constant shift, a constant

unit and so I can shift from one chord

to another chord, just by translating all

the pitches, by a given unit.

And then there is inversion, that is a

very particular operation in a pitch set,

pitch class set, where in terms of our

perception of the musical element, an

inversion is the operation that

transforms a major triad into a minor

triad. And this is an example,

for instance, between

a C Major and a C minor chord.

Going from one to the other is an

operation of inversion.

<i>Demonstration</i>

In this slide, we are starting from a

collection of pitches here, this is just

a collection of seven pitches, and using

only the pitches of the scale, we can

construct all these sequences of chords,

and you see here different triads, where

the pitches are combined to form

the harmony of that scale. Now this is

an important concept in what we call

traditional harmony or tonal music,

where the scale gives us this kind of

framework that is used to define also

the harmonic progressions within

that collection of pitches.

Now, the library that we have been working

with, in particular the class Note, that

we have introduced in previous notebooks,

has a method that is "Harmonize".

Harmonize is a method that, given a scale,

and given a particular interval and

the number of pitches that we want to

have, produces a collection of triads.

And you can try this in your notebook

associated with this module. So, in this

particular class and method of the class

Note, the scale is a scale of notes, so we

can decide any kind of number of notes we

want to have in that particular scale,

the interval defines the quantity between

notes in the chord, so if you want to have

a regular triad, then you will have 2/3,

and we can construct any combination

that we want, and then the size is the

number of pitches in that chord. And you

can explore all of these functions with

the notebook that is associated with

this module.

There are concepts here that are kind of

foundational for modern music theory,

and one of them is the concept of

transposition when applied to this

harmonization of a scale. So, we can

transpose our chords, as I showed you

before, the operation of transposition as

moving the whole chords, all the pitches

of a chord by a constant quantity

in absolute terms. So I can define

transposition on all the chromatic pitches

and say I want to shift this chord, each

note of this chord, say one semiton.

But I can also do a transposition where

instead, my reference system is not

the chromatic scale, but is actually

the scale itself. So, if I do that, then

I transpose, or I move, all the pitches

by a particular amount, relative

to the scale and not to the chromatic

aggregate. So, these are the two modes

of transposition that are actually coded

into the class Note. One is the so-called

regular transposition, or I could call it

regular transposition, where I add a

constant number of semiton

to all the pitches in the set.

So, if we start from a C Major

chord, 0-4-7, C-E-G, I can add one semiton

and make it C sharp, F and A flat. So we

call this the transposition along

the aggregate. The tonal transposition

is instead the transposition along

the scale. That means if I'm on C Major,

the only pitches that I can go to in

transposing are pitches that belong

to that scale. So in this case, instead of

transposing say by one semiton, here

I transpose by one step on the scale and

so I go from C-E-G, to D-F-A and this is

a transposition along the scale.

There is a third way, the third mode

of transposition that is associated

with the chord itself, that can be framed

within this idea of transposition

along the collection. So you can

transpose along different combinations

of pitches, you can transpose along

the chromatic scale, or the regular

diatonic scale or the scale internal to

the chord. All these concepts are

central to, what my friend and colleague

Dmitri Tymoczko at Princeton University

will talk about in a contribution to

this course, in the next couple of

modules.