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: <i>tx</i>, 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 <i>Tx</i>, where x is again the

number of steps I use in

the transposition.

And then, finally, the highest level of

the hierarchy is <i>Boldface Tx</i>, 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 <i>k</i> notes in

the scale, what I do, I divide the circle

that I generated in <i>k</i> slices, that

intercept the spiral at regular intervals.

So, I start again from 12 o'clock, move

clockwise and place a point every <i>n</i>

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 <i>betweenness</i>

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.