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.
Demonstration
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.
Demonstration
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.
Demonstration
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.