In the previous lecture on harmony,
I introduce this idea, this concept,
of transposition along a particular
collection. So we talk about transposition
in terms of transposition on a chromatic
scale, or a transposition on a diatonic
scale and in this particular lecture,
we will dig deeper into this concept of
transposition along a collection that
turns out to be foundational for gaining
a geometric understanding on how
the pitch space works in the
context of many different musical
traditions. And of course, here,
we will focus mostly on the tonal harmony
traditional Western classical music.
We can transpose along different
collections, and transposing along the
scale, or the intrinsic scale that a chord
belongs to, allows us to find what is
the most efficient movement between
a chord and another.
And this is called, in musical theory,
voice leading, or most efficient voice
leading. This has to do with the way in
which we actually move on a keyboard,
for instance, with our fingers, so we
always try to find the most efficient way
of going from one chord to another,
moving the least number of fingers for
the least amount.
So a fundamental point here, is that any
collection of notes, so we talk about
chords, or motives, or sets, or scales,
can be associated, in principle, with two
different scales. One, is the external
scale, or the enclosing scale. We can say
chromatic, diatonic, or any other
collection of pitches that we define as a
scale. And the other one is what
we call the intrinsic scale, that is a
scale formed by the own notes of our
collection that we start from. So in a
chord setting, the notes of the chord are
what an intrisic scale is, for that chord.
Now, this can be represented graphically
if you want, with this kind of diagram,
where we move from chordal degrees, at the
top of these figures, and we map this onto
an intrisic scale, and then we map this
scale into an actual scale, an external
scale and we can actually map the external
scale eventually, on the full collection
of all the possible pitches. So in our
twelve tone tradition, this would be the
chromatic scale. Now, all the numbers
that you see on this picture
can be confusing; this is
a way of actually designing an algorithm
that allows you to operate on all these
different levels, on all these different
collections. And this algorithm is
implemented into the musicntwrk
library, and I'll talk briefly at the end
of this lecture, about the implementation.
Let's talk about transposition along
the collection. So, transposition along
the chord and transposition along the
scale combine, together, to form kind of
a parallel motion, doubly parallel motion
along different
nested collections.
So this, again, I understand
that these might be concepts a little
complex to explain in a short lecture like
this, but I invite you also to look into
supplementary materials for fundamental
textbooks where I took all of these ideas
from: "Tonality: An Owner's Manual",
by Dmitri Tymoczko, and also to follow
some of Dmitri presentations that are
linked on the website. So, let's kind of
formalize this double transposition idea
in terms of notations and operations. And
here, I introduce three different
operators: tx, which is a transposition
along the chord, by a set number of steps,
and I refer here to the lecture on harmony
where I explain briefly what it means to
do a transposition along the chord. Then,
we have a transposition along a given
scale so if, for instance, I have a
C Major chord (C, E, G) that lives on the
C Major scale (C, D, E, F, G, A, B), then
if I transpose along this scale, I use
this operator Tx, where x is again the
number of steps I use in
the transposition.
And then, finally, the highest level of
the hierarchy is Boldface Tx, which is
the transposition along the chromatic
collection. So I can transpose my scale
in the chromatic collection and then
transpose the chord along the scale and
create all this movement in the voice
leading of my piece. Now, the useful
and kind of enlightening thing to me to
represent these operations graphically is
the introduction
of what we call
the spiral diagrams. A spiral diagram is
a diagram that allows us to visualize all
these transpositions as a single image, in
a single context, where everything is
described graphically. So, in order to
construct the space, because this is
a representation of the space where
all these operations happen, we build
what we call "spirals". Spiral diagrams
come from the drawing of spirals actually,
that represent a way of combining together
chords and scales, so that then we can
use this visualization to generate
different protocol progressions.
The number of loops that you have
in a spiral corresponds to
the cardinality of the chord, that means
how many notes there are in a chord. If
you have two notes, then you design a
spiral with n=2, so n with two loops.
If you have a triad, three chords,
three-note chords, then your spiral will
have three loops, and so on.
How do we design this, or draw
this, is we start from 12 o'clock and move
inwards, clockwise, and then after I go
for n loops, so for n number
of notes in the chord loops,
we stop at 9 o'clock, and from there,
we join the starting point. And so here,
you see a representation again in the same
image here, you start from 12 o'clock,
you go inward clockwise and then you get
to 9 o'clock, and then you join another
loop in order to close the two loop
spiral. So this is the skeleton of
a spiral diagram, now we need to
populate the spiral diagram with notes
or chords. So here, we need to separate,
or distinguish between a number of notes
in the chords and the number of notes
in the scale. So let's talk now about the
notes in the scale. If I have k notes in
the scale, what I do, I divide the circle
that I generated in k slices, that
intercept the spiral at regular intervals.
So, I start again from 12 o'clock, move
clockwise and place a point every n
intersection. So, what I do is the
following. Let's look at this particular
image here: on the left, what I want to do
is I want to construct the spiral
diagram of major triads,
so I have a three-note chord,
for a seven-note diatonic scale.
Again, the example of the C Major
chord (C, E, G), three notes, on the
C Major scale set of notes. So, what
I do, is I start from 12 o'clock, I draw
my slices, seven of them, and then
once I've drawn them, I populate them
with the different chords that I
identified with the base notes. So
C Major would be C, D, E, F, G. The
minor chord in the diatonic scale
of C Major would be D, and so on.
And what I do, I start form the innermost
12 o'clock point, and then I start placing
the note of the scale on each of the point
in descending order, and these points are
separated from one another by the number
of notes in the chord. So every three
slices, I put an entry, I put a note.
So, let's go again to this drawing here,
on the left, you see I start from C, then
I start from the innermost loop, I skip
the first two intersections of the
sections, at the third intersection I put
my B. And then again, I skip two and I put
my A, I skip two, and so on. I keep going
until I populate the whole diagram.
Now, the important thing in this kind of
representation is that each point
represents a complete chord, so even if
I write on the A, or B, or C, in this
particular diagram, each of these
letters correspond to their representing
chord. C would be C, E, G, the C Major
chord, and so on. And I can do this for
any arbitrary number of notes in a
chord, any arbitrary scale, so on the
right, for instance, we have a collection
of all the possible diatonic scales of
seven pitches embedded into the
structure of the chromatic scales,
so now we have seven loops and
twelve sections.
Now, the usefulness of these kinds of
diagrams, is that they allow us to
calculate all the possible voice
leading that exists within this collection
of chords, onto that scale. And this,
again, can be done geometrically, so
for instance, we can calculate a path
going from the C triad to the E triad,
here, on a chromatic scale, and this is
represented by a trajectory into this
spiral. So I go, as you see here, in
bold arrow, that shows the particular
trajectory that I am following. And on the
right, I have the result of this
trajectory, that indeed you can verify
even just by looking at it, corresponds
to the most efficient voice leading, so
the most efficient way of going from that
chord to the next, so form C Major to
E Major in this case.
Spiral diagrams can be used to break down
chord progressions, for instance. In this
particular example, I am showing here a
motion between a C Major chord to a
B Major chord, and that's the bold line
on the right, but I can break this down
following trajectories within this
spiral diagram. So I can, for instance,
instead of going from C to B to B-flat,
I can actually go from C to E-flat
to B-flat and use this other trajectory
that produces a different voice leading.
And I can do this in many different ways.
So, this a graphical representation of
how this combination of double
transpositions along the chord and
along the scale gives rise to particular
chord progressions.
This is possible because the
transpositions, both along the chord
and along the scale, satisfy both the
commutative property and the associative
property, mathematically speaking, so
you can actually invert the order of
the operations and you get the same
results. And using the spiral diagrams,
you can build visually any progression
in the history of Western music within
the tonal tradition.
So this is again another representation
of this spiral diagram from triads, where
I would actually like to mention here
that using the geometry of these diagrams,
we can define a distance between different
chords, so chords that are closer together
and further apart, and the betweenness
idea, which is what I just explained,
that you can factor into smaller
moves and larger voice leadings.
Now, all of this is completey generic,
it's not limited to any particular choice
of chord, so you can start with your
triads like major chords, you can use
minor chords,
diminished chords, cluster chords,
whatever, and this is geometry, this
operation is still valid. So, you can
construct very complex harmonic spaces
just by operating graphically, if you want
on these particular diagrams.
So these operations on spiral diagrams are
actually implemented into the musicntwrk
library through a class, which is
called "MIDIset", that is demonstrated at
the end of the notebook on harmony,
although I am not talking about that at
all, so I leave you this exercise, if you
wish to try to play with all these kinds
of transpositions and spiral diagrams
using the material in that notebook.
Here, you have a summary of both the
attributes and the methods that are
implemented into the MIDIset class.
But I definitely think that this is a bit
more advanced than what this course
is supposed to be, so I leave this just to
the people that are more interested and
computationally minded.