WEBVTT

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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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MICHAEL SCOTT ASATO CUTHBERT:
Hello, computational music

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theorists.

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Our topic for the
coming week is going

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to be on intervals
and equivalents

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and working with what equal
and the same and so on mean.

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I am very much trying
to treat everybody

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in this class and
everybody I meet as equals,

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But that doesn't
mean that I think

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that everybody is the same.

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In fact, that would be a very
unequal way to treat people.

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So we can have this notion
that equality does not

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mean the same thing, and that
equal does not mean identical.

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This is something I hope
we practice in our lives,

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but it's also very important
in thinking about how

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we compare musical structures.

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I'm asking you over the weekend
to read a very interesting

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chapter from Dmitri Tymoczko's
A Geometry of Music in which he

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talks about five kinds of
equivalents, a few more,

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but mostly about five
kinds that he calls optic,

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octave, permutation,
transposition, inversion,

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and cardinality.

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These ways of thinking about
equality of notes or melodies

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or chords or other musical
structures even whole pieces

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is going to be
very important when

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we want to find particular
musical gestures

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or we want to identify
particular things using

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a computer.

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I'm sitting at the
piano because I'm

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going to demonstrate
a few of the things

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that we might think mean
that something is equal

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or something is the same but
maybe they're not identical.

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But first, I want to play
something that doesn't

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work very well on the piano.

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So listen to these two pieces.

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[MUSIC PLAYING]

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FRED ASTAIRE: (SINGING)
The way you wear your hat

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The way you sip your tea

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The memory of all that

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No, no, they can't
take that away from me

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The way your smile just beams

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The way you sing off key

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[MUSIC PLAYING]

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ELLA FITZGERALD: (SINGING)
The way you wear your hat

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The way you sip your tea

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The memory of all that

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No, no, they can't
take that away from me

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The way your smile just beams

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The way you sing off key

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MICHAEL SCOTT ASATO
CUTHBERT: I asked

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you to listen to two pieces.

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Actually I just played excerpts.

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But did I play two pieces, or
did I play one piece twice?

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Were those two pieces
the same piece?

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Were they the same recording?

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I think we'll think
that, well, they

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were the same piece but
different recordings,

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different performers,
different styles,

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different interpretations,
but there was something

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the same about them.

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So we can think of an
equivalence class or a kind

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of equivalence between pieces.

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That doesn't necessarily
mean that they are identical.

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Now let's turn to some
musical structures

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that we can demonstrate
at the piano.

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So I want to talk
concretely about some ways

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that we might think
about equivalence

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and what do we mean
by equivalence when

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we're talking about notes or
chords or melodies or pieces.

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And I want to think
about some of the things

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that we might be doing.

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So, for instance, can we
think of a context where

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notes as a group are the
same regardless of the order

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that we talk about them?

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Let's start with just a
C major chord C, E, G.

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[PLAYING PIANO]

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And if I say C3, E3,
G3, that's the chord.

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Or if I say that it's G3, E3,
C3, it's still the same chord.

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Or if I say that the
chord is E3, C3, G3,

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it really doesn't matter.

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So this could be, when
we're talking about chords,

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a place where there is a
permutation equivalence.

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Doesn't matter what
order they're in.

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On the other hand,
if we're talking

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about a melody or something,
some sort of a piece that

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has [PLAYING PIANO]

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Then it matters if we go
[PLAYING PIANO]-- sorry--

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[PLAYING PIANO]

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Those don't seem to be
equivalent in some context

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that we might think
of as equivalents.

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So there could be
a context where P,

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permutation equivalence, is
active and a context where it

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isn't.

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We can also think of
cardinality, especially

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on the piano.

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It doesn't matter if I play
C-E-G [PLAYING PIANO] as a chord

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or if I play C, C,
E, G [PLAYING PIANO].

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You can't even tell.

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Maybe I can put two fingers on
the C or something like that.

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But on-- in a chordal
context on the piano,

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we would normally say, well,
that cardinality doesn't really

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matter in terms of the sound.

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If you're going to be notating
it, then, yeah, it matters.

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We also can think of
octave equivalence.

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[PLAYING PIANO]

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That all those
notes are C. When we

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say something like all
of these things are,

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we're saying that they're
equivalent in some way.

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So we're saying that
in this case when

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I say all these
notes are C, we're

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talking about an octave
equivalence relationship.

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But I can't say that all
of those notes are C3--

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[PLAYING PIANO]

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Only that note is C3,
[PLAYING PIANO] C2,

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[PLAYING PIANO] C4.

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If you were, a
instrumental teacher

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and you ask somebody on
an instrument like flute

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or the clarinet or
the violin to play

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a scale in three
octaves and they

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just played the same octave--

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[PLAYING PIANO]--

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You'd really say no, no, no.

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I asked for three octaves,
so we can think of a world

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where there isn't an
octave equivalence.

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And we can put these
two things together

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that we can think C, E, G, C--

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[PLAYING PIANO]

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Which have a kind of
world where we say, well,

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that's the same as C, E,
G in the sense of play

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the tonic triad in C major.

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It doesn't matter if
I play [PLAYING PIANO]

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Those all fulfill this
particular equivalence class.

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And there are other ones.

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One of the ones that Tymoczko
assumes throughout is

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and isn't even a part
of his optic, o-p-t-i-c,

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world is spelling, so
it doesn't matter or--

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C major's a little
bit hard-- but it

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doesn't matter if I
think of a single note,

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[PLAYING PIANO] play E-flat,
[PLAYING PIANO] play D-sharp,

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it's the same thing.

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So we can think of a
spelling equivalence.

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Now if you were to be
in theory one class

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and you're asked to
write a C minor triad,

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and you write C
[PLAYING PIANO], D-sharp, G,

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you're probably going
to get marked wrong

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because that's a context
where there is not

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spelling equivalence.

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We can also think of a world
where there's a transposition

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equivalence.

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So, for instance,
[PLAYING PIANO]

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And [PLAYING PIANO]

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We can say, oh well, those
are both the same piece.

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Those are both "Mary
Had a Little Lamb."

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So that there's a transposition
equivalence that makes that

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piece still "Mary
Had a Little Lamb."

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And we definitely
use that when we're

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talking about chord types--

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so we talk about
[PLAYING PIANO],

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that's a major triad,
[PLAYING PIANO] that's a major

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triad, [PLAYING PIANO]
that's a major triad--

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so that we can have--

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talk about a concept
like major triad

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that takes into account
that there's a transposition

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equivalence.

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Probably the one that
people like the least

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and think that it's not
really a context that we want

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to be working with is
the inversion context,

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and when we play something
in a tonal context like a C

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major triad
[PLAYING PIANO] and then we

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invert it so let's see
from [PLAYING PIANO] E to G

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is a minor third, so we'll go
from G, B-flat is a minor third

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and E to C is a major third.

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So B-flat to D is
a is a major third.

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So we can say [PLAYING PIANO] in
an inversion and transposition

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contexts that those--

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well, I guess those are not
even a transposition context--

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that those two are
equivalent and that

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might go, whoa, whoa, whoa,
what are we talking about.

00:10:58.640 --> 00:11:02.980 align:middle line:84%
But there are some places
in post-tonal theory

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where that's important, but we
can even think about it even

00:11:06.250 --> 00:11:09.200 align:middle line:84%
in the tonal or in this
case a pre-tonal context.

00:11:09.200 --> 00:11:13.150 align:middle line:84%
Here I'm going to
play four pieces.

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One goes-- well, I'll do
the one that we just played.

00:11:16.030 --> 00:11:19.040 align:middle line:90%
[PLAYING PIANO]

00:11:19.040 --> 00:11:22.090 align:middle line:90%


00:11:22.090 --> 00:11:23.780 align:middle line:90%
Here's another piece.

00:11:23.780 --> 00:11:26.732 align:middle line:90%
[PLAYING PIANO]

00:11:26.732 --> 00:11:33.490 align:middle line:90%


00:11:33.490 --> 00:11:36.164 align:middle line:90%
And here's another piece.

00:11:36.164 --> 00:11:38.980 align:middle line:90%
[PLAYING PIANO]

00:11:38.980 --> 00:11:43.420 align:middle line:90%


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Play that one more time
so you get your ears.

00:11:46.126 --> 00:11:48.400 align:middle line:90%
[PLAYING PIANO]

00:11:48.400 --> 00:11:50.710 align:middle line:90%


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And then we can have
a piece like this.

00:11:53.350 --> 00:11:55.360 align:middle line:90%
[PLAYING PIANO]

00:11:55.360 --> 00:12:01.255 align:middle line:90%


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I played in some sense four
different melodies or something,

00:12:05.930 --> 00:12:08.440 align:middle line:84%
but maybe I only played
three different pieces

00:12:08.440 --> 00:12:11.230 align:middle line:84%
because we can recognize,
oh, those last two

00:12:11.230 --> 00:12:15.920 align:middle line:84%
are the same piece but one is
upside down from the other.

00:12:15.920 --> 00:12:19.900 align:middle line:84%
This is actually a piece that
is really famous in the 15th

00:12:19.900 --> 00:12:22.520 align:middle line:84%
century called "L'homme
Armé" or "The Armed Man,"

00:12:22.520 --> 00:12:25.880 align:middle line:84%
and composers at that time
liked to play around with it.

00:12:25.880 --> 00:12:27.320 align:middle line:90%
They transposed it.

00:12:27.320 --> 00:12:32.590 align:middle line:84%
They did diatonic transpositions
so that you play in--

00:12:32.590 --> 00:12:34.490 align:middle line:90%
well, different modes.

00:12:34.490 --> 00:12:39.560 align:middle line:84%
So instead of [PLAYING PIANO],
here you play some--

00:12:39.560 --> 00:12:42.946 align:middle line:84%
maybe you play something
like [PLAYING PIANO]

00:12:42.946 --> 00:12:45.340 align:middle line:90%


00:12:45.340 --> 00:12:48.355 align:middle line:84%
And it's a slightly different
piece but still the same piece,

00:12:48.355 --> 00:12:50.730 align:middle line:84%
so you can think of a diatonic
transposition equivalence.

00:12:50.730 --> 00:12:52.910 align:middle line:84%
But they also definitely
played with what

00:12:52.910 --> 00:12:58.410 align:middle line:84%
I was showing the inversion
equivalence context so that

00:12:58.410 --> 00:13:05.990 align:middle line:84%
[PLAYING PIANO] at
[PLAYING PIANO] are in some ways

00:13:05.990 --> 00:13:11.450 align:middle line:90%
the beginning of the same piece.

00:13:11.450 --> 00:13:13.880 align:middle line:84%
We're going to be asking
you in the problem set

00:13:13.880 --> 00:13:17.720 align:middle line:84%
to see if you can come
up with other contexts

00:13:17.720 --> 00:13:20.960 align:middle line:84%
where we talk about, well,
that's the same thing like is

00:13:20.960 --> 00:13:22.255 align:middle line:90%
this a C major triad.

00:13:22.255 --> 00:13:25.310 align:middle line:90%
[PLAYING PIANO] Good.

00:13:25.310 --> 00:13:26.840 align:middle line:84%
Well, this is a
fancy piano, so I'm

00:13:26.840 --> 00:13:29.520 align:middle line:84%
going to change the
temperament to meantone.

00:13:29.520 --> 00:13:30.990 align:middle line:90%
And now we go.

00:13:30.990 --> 00:13:34.460 align:middle line:90%
[PLAYING PIANO]

00:13:34.460 --> 00:13:35.640 align:middle line:90%
Maybe I'll resume.

00:13:35.640 --> 00:13:38.100 align:middle line:84%
You can't hear the
difference as much,

00:13:38.100 --> 00:13:41.570 align:middle line:84%
but we can go to a
slightly different one

00:13:41.570 --> 00:13:48.666 align:middle line:84%
even [PLAYING PIANO]
or [PLAYING PIANO]

00:13:48.666 --> 00:13:49.550 align:middle line:90%


00:13:49.550 --> 00:13:53.460 align:middle line:84%
And those are all
the notes C. E,

00:13:53.460 --> 00:13:57.650 align:middle line:84%
G. They're not exactly the
same, but we think of them

00:13:57.650 --> 00:13:58.410 align:middle line:90%
as the same.

00:13:58.410 --> 00:14:02.330 align:middle line:84%
So we can think of a temperament
context or a tuning context

00:14:02.330 --> 00:14:05.120 align:middle line:84%
where we allow things
to be equivalent.

00:14:05.120 --> 00:14:06.740 align:middle line:84%
So I think this
is the background

00:14:06.740 --> 00:14:12.200 align:middle line:84%
that you need in order to go
through the reading of Dmitri

00:14:12.200 --> 00:14:15.770 align:middle line:84%
Tymoczko on optics
and equivalents.

00:14:15.770 --> 00:14:19.700 align:middle line:84%
As usual, you're going to be
implementing your code in Python

00:14:19.700 --> 00:14:22.520 align:middle line:84%
using musical objects
either that you create

00:14:22.520 --> 00:14:25.880 align:middle line:84%
or that you have
unlocked in music21.

00:14:25.880 --> 00:14:30.380 align:middle line:84%
And when you're
doing this, you think

00:14:30.380 --> 00:14:34.140 align:middle line:90%
about what two objects are.

00:14:34.140 --> 00:14:37.430 align:middle line:84%
What are two notes that
Python defines as equal

00:14:37.430 --> 00:14:41.040 align:middle line:84%
or that I have defined in
music21 defines as equal?

00:14:41.040 --> 00:14:43.200 align:middle line:90%
What do we mean by that?

00:14:43.200 --> 00:14:47.610 align:middle line:84%
And you'll see that each object
in the computational language

00:14:47.610 --> 00:14:52.980 align:middle line:84%
generally has only one
built-in equivalence

00:14:52.980 --> 00:14:56.850 align:middle line:84%
set that defines what
the equal equal operator

00:14:56.850 --> 00:14:59.260 align:middle line:90%
or whatever it is means.

00:14:59.260 --> 00:15:01.320 align:middle line:84%
And, well, there's
actually a second one

00:15:01.320 --> 00:15:04.600 align:middle line:84%
that is operator that
defines something else.

00:15:04.600 --> 00:15:09.840 align:middle line:84%
What do we mean by these
two objects are the same--

00:15:09.840 --> 00:15:11.950 align:middle line:84%
is the same item is
what I wanted to say.

00:15:11.950 --> 00:15:17.430 align:middle line:84%
But-- and what choice was
made, and how could we

00:15:17.430 --> 00:15:18.730 align:middle line:90%
have made a better choice?

00:15:18.730 --> 00:15:21.610 align:middle line:84%
And who's going to hate
me for the choice I made,

00:15:21.610 --> 00:15:25.110 align:middle line:84%
and who would have hated me if I
made another choice on defining

00:15:25.110 --> 00:15:29.310 align:middle line:84%
equality in a note
or in a court?

00:15:29.310 --> 00:15:31.300 align:middle line:84%
Thanks for listening
to this video.

00:15:31.300 --> 00:15:35.250 align:middle line:84%
And now let's get started with
reading what Dmitri Tymoczko has

00:15:35.250 --> 00:15:37.370 align:middle line:90%
to say on this subject.

00:15:37.370 --> 00:15:44.000 align:middle line:90%