“Yes, But Could The Martians - PowerPoint Presentation

Slide1

“Yes, But Could The Martians Understand Bach?”the syntax and epistemology of classical tonal harmony

Dmitri TymoczkoPrinceton University

http://dmitri.tymoczko.comSlide2

Today’s storyi

s about the conflict between embodied musical knowledge …… and scientific methodology,

which has produced serious misunderstandings of classical music——a cautionary tale about the difficulties of mixing music and science …

But everything works out OK in the end!

http://dmitri.tymoczko.comSlide3

SyntaxClassical music has several broad features that might deserve the name “syntactical.”

Tonal and thematic patterns as embodied in sonata, rondo, (etc.) form.Kaplan,

Hepokoski & Darcy, etc.Harmonic principles governing chord-to-chord successionsRameau, Riemann, Piston,

McHose

,

Kostka

& PayneLargely an American enterprise, at least recently

Melodic templates, procedures, conventions

Schenker

http://dmitri.tymoczko.comSlide4

Our Topic: Local Harmonic Laws

In Chapter 7 of GOM, I propose a theory of chord progressions in tonal harmony.Tested against substantial corpora:371 Bach chorales

All the Mozart piano sonatas40% of the Beethoven piano sonatas (and counting)

http://dmitri.tymoczko.comSlide5

Our Topic: Local Harmonic LawsCurrently our best theory of tonal harmony?

Captures ~95% of chord progressions in the Bach choralesCaptures ~97% of the nonsequential

progressions in Mozart.Captures ~98% of the progressions in Beethoven.

http://dmitri.tymoczko.comSlide6

Why is this important?

TrueUnder repeated attack:CPE Bach

SchenkerSchoenbergQuinn

Crucial for understanding the development

of

tonality

Tonicity and entropy?Crucial for understanding contemporary

music

http://dmitri.tymoczko.comSlide7

The Fundamental Challenge

Traditional harmonic theory says that there are two kinds of chords in classical music.“Harmonic” (or “real”) chords“Contrapuntal” (or “fake”) chords produced by melodic motion

between harmonic chords.The rules apply only to “real” chords.

http://dmitri.tymoczko.com

R F R R F R RSlide8

The Fundamental ChallengeThe rules for producing “fake” chords were borrowed from the Renaissance:

In the Renaissance, it was not necessary to specify what the

“real” chords were–just that some consonance underlies every dissonance.

http://dmitri.tymoczko.com

R F R R F R RSlide9

The Fundamental ChallengeOnce real harmonies evolved a grammar, we crucially need to distinguish the “real” harmonies from the “fake” ones.

The inherited contrapuntal rules did not change to make this any easier!

http://dmitri.tymoczko.com

R F R R F R RSlide10

The Fundamental ProblemHow can we separate “real” chords from “fake” chords in a principled way?

“The Quinn challenge”Thanks to IQ and DH

http://dmitri.tymoczko.com

R F R R F R RSlide11

RN analysis is hard (1)What is the best (C major) analysis?

http://dmitri.tymoczko.com

C: IV I

6

PT

C: ii vii°

6

I

6

PT

C: I vii°

6

I

6

“the ii-vii°

6

idiom”

C: V V

2

I

6

PTSlide12

RN analysis is hard (1)Note that all analyses suppress a (fake) ii-I!

http://dmitri.tymoczko.com

C: IV ii

6

I

6

C: ii I

6

C: I vii°

6

ii

6

I

6

“the ii-vii°

6

idiom”

C: V ii

6

I

6

PTSlide13

RN analysis is hard (1)So

what is the force of “ii-I is rare”?

http://dmitri.tymoczko.com

C: IV ii

6

I

6

C: ii I

6

C: I vii°

6

ii

6

I

6

“the ii-vii°

6

idiom”

C: V ii

6

I

6

PTSlide14

RN analysis is hard (1)

http://dmitri.tymoczko.com

g:

i

V

6

i

V

6–5

/III III

F: I V6 I vii°6 I

6

PT

M2 b3 interpreted

d

ifferently!!!Slide15

RN analysis is hard (2)

Even the pros make mistakes:

As far as I can tell, this is off by more than an order of magnitude; in Bach, Haydn, Mozart, and Beethoven ii–I

progressions

(excluding cadential

@

) account for less than 2% of the destinations from ii.

http://dmitri.tymoczko.comSlide16

RN analysis is hard (2)Even

the pros make mistakes:

Huron probably includes I

@

as I, which is unwise,

because I

@ plays a very particular syntactical role.

Huron may misread the ii-vii°

6

idiom.

http://dmitri.tymoczko.comSlide17

Huron and the ii-vii°6 idiom

Huron 2007 explicitly considers the

ii-vii°6 idiom occurring at the very beginning of a D-major passage. He doesn’t

even consider putting two chords on beat 3.

Political note.

http://dmitri.tymoczko.comSlide18

Other people who are wrongYitzak

Sadai

Aldwell and SchachterMartin

Rohrmeier

Craig Sapp, Helen Budge

Insert politician here



http://dmitri.tymoczko.comSlide19

First PlateauMusic is syntactically more ambiguous than language, since we are constantly confronted with passages that can be read in multiple ways.

In resolving these ambiguities human beings rely on intuitive models of what is most likely to occur

.This is circular, since the intuitive models themselves depend on theory-laden analyses (and what we have been taught).

http://dmitri.tymoczko.comSlide20

Is the circle good or bad?From a simple scientific (or crude

scientistic) perspective, it is bad, since a fundamental methodological principle is to separate evidence gathering from theory.

cf. Doyen et al. (2012) failing to replicate

Bargh

et al.

(1996)

http://dmitri.tymoczko.comSlide21

Is the circle good or bad?

http://dmitri.tymoczko.com

The problem is that each of these four different harmonic theories provides a

different model

of what is likely to happen

in music.So in creating a corpus to test these theories, what model should we use?Doesn’t that bias our tests?Slide22

Is the circle good or bad?In other contexts, we learn to live with

circles. “the hermeneutic circle

”cf. pragmatics, artificial intelligence, etc.

“I am going to the bank.”

http://dmitri.tymoczko.comSlide23

First strategy: embrace the circlePerhaps we can only resolve ambiguities in a theory-dependent way.

If so, each theory should get tested on its own preferred identification of “fake” chords.

http://dmitri.tymoczko.comSlide24

First strategy: embrace the circleIn practice, this is only necessary when comparing roughly equally good theories.

Of the 4-5 major theories of tonal harmony, two are much more accurate than the others:Rameau/Meeus

~78% accurateRiemann basic function theory ~79% accurateKostka/Payne ~92% accurate

Tymoczko ~95% accurate

http://dmitri.tymoczko.comSlide25

First strategy: embrace the circleDifferent resolution of nonharmonic tones might make a 5% difference, but not a 13% difference.

Indeed, it doesn’t even boost K & P above my theory.Rameau/Meeus

~78% accurateRiemann basic function theory ~79% accurateKostka/Payne ~92% accurate

Tymoczko ~95% accurate

http://dmitri.tymoczko.comSlide26

The problem with giving upMakes it unclear how one could learn to analyze the music just by studying it.

“Yes but could the Martians understand Bach?”Abandons some important intellectual projects:

Providing the deepest possible justification of our analytical practices.Striving

to be theory

-neutral

if we can.

http://dmitri.tymoczko.comSlide27

The problem with giving upAlso, there is a genuine question here:

To what extent can traditional harmonic theory be inferred from the music itself, and to what extent is it an interpretive grid

laid over the music?We all have our (potentially divergent) intuitions, but nobody has ever tried to provide a rigorous answer

to this question.

http://dmitri.tymoczko.comSlide28

Second plateauIt would be nice to be able to provide an effective

theory-neutral approach to distinguishing “harmonic” (real) from “contrapuntal” (fake) chords.

NB: this means we don’t want to stipulate that “iii is rare” or “ii doesn’t usually go to I.”Of course, we have to make some assumptions.

We want to kill the circle instead of embracing it!

http://dmitri.tymoczko.comSlide29

What is the best interpretation? (and how do we tell?)The case against I

6-vii°6-I:

The ii chord is often leapt to, and hence harmonic.Premise: incomplete neighbors are rare.

http://dmitri.tymoczko.com

C: I

6

vii°

6

I

PT

The ii-vii°

6

idiom: a case study

R74 m. 1

F

: I

6

ii vii°

6

I

*Slide30

The case against I6-ii-I:i

i-I almost never happens, in any chordal inversion, without the intervening vii°6.

In particular, we find ii-I6, with ^6 going to

^5

,

much less often than we should!

http://dmitri.tymoczko.com

The ii-vii°

6

idiom: a case study

*

C: I

6

ii I

PTSlide31

In ii-I6, the default voice leading

should send the fifth of the chord (^6) down (to ^

5) rather than up (to ^8).This is what happens in the analogous diatonic progressions.

I-vii°

6

(83%)

vi-V6 (75%)

http://dmitri.tymoczko.com

The ii-vii°

6

idiom: a case study

*

C: I

6

ii I

PT

•

V

-IV

6

(47%)*

• “ii-I

6

”

(17%

)Slide32

Instead, with ii-(vii°6)-I, the fifth (^6

) most commonly moves up through ^7 to

^1. Furthermore, in the ii-vii°

6

progression, the fifth is doubled less than half as often as one would expect (~ 3% vs. 6%).

http://dmitri.tymoczko.com

The ii-vii°

6

idiom: a case study

C: I

6

ii I

PT

C: I

6

ii vii°

6

I Slide33

In other words, Bach clearly goes out of his way to create a leading tone and vii°6 harmony.

The best evidence for the harmonic status of vii°6 is holistic, focusing in part on what does

not happen, namely frequent ii-I or ii-I6 progressions.We can justify this only if we have reliable, extensive corpus data (explicit or implicit)!

http://dmitri.tymoczko.com

The ii-vii°

6

idiom: a case study

*

C: I

6

ii I

PT

C: I

6

ii vii°

6

I Slide34

GeneralizingClearly, this sort of detailed case-by-case reasoning will take us only so far.

We need a way of formalizing

and generalizing this thought process.If we could do this, then we could show that traditional harmonic analysis is well-grounded.

Maybe we

can

make analytic decisions in a theory-neutral way.

Perhaps the Martians could

understand Bach after all!

http://

dmitri.tymoczko.comSlide35

But …

… where will we find our martians????

http://dmitri.tymoczko.comSlide36

martians.py

http://dmitri.tymoczko.comSlide37

martians.py

“martians.py” is a computer program for analyzing Bach chorales.I wrote it in python using Michael Cuthbert’s music21 module.

The goal is not to achieve the best possible results, but rather to probe the justificatory structure of our music-analytical practices.An exercise in computational epistemology.

Using as few postulates as possible.

If we cared only about results, we’d use slick tools from machine learning.

http://dmitri.tymoczko.comSlide38

martians.pyDoes pretty well:

Correct key 87.6% (on a per eighth-note basis).

Correct chord (given correct key) 92.7%Average correctness (per chorale)

81.75%

Wrong key 4860 Wrong chord 2491

Correct chord 31758 (eighth notes)

Compare

Aarden

:

Correct key

62.2% Correct chord (given correct key)

35.1%

Average correctness (per chorale)

23.5%

http://dmitri.tymoczko.comSlide39

martians.pyDoes pretty well (best in the world?):

Correct key 87.6% (on a per eighth-note basis).

Correct chord (given correct key) 92.7%Average correctness (per chorale)

81.75%

Wrong key 4860 Wrong chord 2491

Correct chord 31758 (eighth notes)

Compare

Aarden

:

Correct key

62.2% Correct chord (given correct key)

35.1%

Average correctness (per chorale)

23.5%

http://dmitri.tymoczko.comSlide40

Second strategy: kill the circleBasic plan

Stage 1: create a raw analysis of the chorales, considering every triad and seventh chord to be a harmony.

Stage 2: gather statistics on the Stage 1 analyses

Stage 3: use these statistics to “prune” the Stage 1 analysis, removing fake or “merely contrapuntal” chords.

http://dmitri.tymoczko.com

– Stage 1b. Improve key finding with various

rules (e.g.

dorian

scale regions). Slide41

Kill the Circle – stage 1

Go through the chorale to find maximal regions belonging to the three standard tonal scales.

Stay in each region as long as possible.For each region, locate its earliest possible starting pointWhen a strong beat has no triadic sonority, attempt to resolve suspensions and accented passing tones.

Within each region label every

tertian

verticality (triad and seventh chord).

By convention, weak-eighth sevenths (with the same root as a strong-eighth triad) are only labeled if they are V.

http://dmitri.tymoczko.comSlide42

Kill the Circle – stage 1

http://dmitri.tymoczko.com

Correct key 81.1%, correct chord 90.5%

This music is largely unambiguous!Slide43

Kill the Circle – stage 1

http://dmitri.tymoczko.com

The lack of ambiguity provides statistical purchase, allowing us to build a relatively theory-free model.Slide44

V7 is special

http://dmitri.tymoczko.com

Numbers are percentages of all chords in the raw data, counting every triad and seventh.

Suggests V

7

is by far the most common seventh chord, and syntactically unusual. (NB: iii is suppressed.)Slide45

Kill the Circle – stage 2

Using only the 4/4 chorales, gather rhythmicized data on the harmonic progressions.

For each quarter, gather a 4-tuple:(prev. harmony, strong eighth harmony, weak eighth harmony, next harmony)

http://dmitri.tymoczko.com

I V

6

vii°

i

V

i

i

vii°

6Slide46

Kill the Circle – stage 3

When we find a quarter note containing a pair of eighth-note harmonies, ask:Could the first be the product of nonharmonic tones?

Could the second?Could they represent a motion from a triad to an incomplete seventh chord on the same root?Using the preliminary statistics choose the most likely of the available readings.

Penalize accented passing and neighboring tones.

These are rare in the raw data!

http://dmitri.tymoczko.comSlide47

Kill the Circle – stage 3

http://dmitri.tymoczko.com

1: I – IV

6

– vi – V

6

2:

I – IV

6

– IV

6 – V63: I – vi – vi – V

6

4:

I – IV

6

–

IVmaj

#

– V

6

0*

6

5

0

#1 is 0 because we don’t count the progression itself (and because we gather our initial stats using 4/4 chorales);

since #3 requires an accented neighbor, it is penalized; #4 is 0 by convention.Slide48

Kill the Circle – stage 3

http://dmitri.tymoczko.com

1: I – iii

6

– V – vi

2:

I –

iii

6

– iii

6 – vi3: I – V– V – vi4: I – iii

6

–

iii

#

–

vi

0

0

9

0Slide49

Kill the Circle – stage 3This brings the within-key accuracy from 90.5% to ~92.5%, fixing ~21% of the errors.

In practice, 95% is probably about as close as we can get to perfection, since expert humans don’t agree at that level; also, higher-level complexities, etc.We’re really close!

This method deals with all the problematic cases mentioned earlier.

http://dmitri.tymoczko.comSlide50

RN analysis is hard (reprise)What is the best (C major) analysis?

http://dmitri.tymoczko.com

C: IV I

6

PT

C: ii vii°

6

I

6

PT

C: I vii°

6

I

6

C: V V

2

I

6

5.8:1

7.8:1

10:1

43:1

PTSlide51

RN analysis is hard (reprise)Ratio of my preferred analysis to the

best alternative.

http://dmitri.tymoczko.com

C: IV I

6

PT

C: ii vii°

6

I

6

PT

C: I vii°

6

I

6

C: V V

2

I

6

5.8:1

7.8:1

10:1

43:1

PTSlide52

RN analysis is hard (reprise)

http://dmitri.tymoczko.com

g:

i

V

6

i

V

6

/III III

F: I V6 I vii°6 I

6

7:1 (NB: no V

#

)

4.3:1Slide53

Using Mechanical AnalysesRameau/

Meeus ~73% accurateRiemann basic function theory ~74% accurate

Kostka/Payne ~84% accurateTymoczko ~86% accurate

Cf. Human:

Rameau/

Meeus

~78% accurateRiemann basic function theory ~79% accurateKostka

/Payne ~92% accurate

Tymoczko ~95% accurate

http://dmitri.tymoczko.comSlide54

Conclusion

We have solved the Fundamental Problem! Provided a theory-free justification for our complex, seemingly inconsistent analytical practices.Large stretches of classical harmony are basically unambiguous (90%)

These ambiguities give us good statistical grounds for making our analytical decisions.The rarity of the iii chord, or of the ii-I progression, is not

simply an artifact of our analytical methods.

Our background knowledge can justify our treating superficially similar passages in different ways.

http://dmitri.tymoczko.comSlide55

ConclusionWe have built a

martian, and it is us! … or at least, it understands Bach like we do …

The approach is (very) loosely inspired by Bayes, using the raw analysis to build a set of prior probabilities.It

is simply impossible to do good harmonic analysis without good priors.

Harmonic analysis is minimally a two-stage process.

http://dmitri.tymoczko.comSlide56

ConclusionOur results have a practical consequences for music analysis.

We should not be afraid to use our intuitions of likelihood when doing RN analysis.

Artificial corpus data can provide a useful check on

these intuitions

.

In the chorales, the importance of harmonic rhythm is easily overstated.

You get better results if you try to maximize harmonic likelihood, rather than insisting on one chord per quarter note.

http://dmitri.tymoczko.comSlide57

ConclusionThis issue has

bedeviled many corpus studies (Rohrmeier, Huron?)

.The frame of mind of the corpus builder is scientific, objective, and seemingly reluctant to engage in the kind of intuitive judgment that is necessary for harmonic analysis.

Is this why Huron ends up more than an order of magnitude wrong about the ii chord?

Is this why it took so long to develop theories of harmonic progression?

http://dmitri.tymoczko.comSlide58

Slow developmentRameau (1720) ~78% accurate

Riemann (1880) ~79% accurateMcHose (1945) ~“76% accurate”

Kostka/Payne (1970) ~92% accurateTymoczko (2011)

~95% accurate

A pretty odd progression!

http://dmitri.tymoczko.comSlide59

The role of perceptionTwo approaches:

A good analyst has internalized, by ear, the conventions of the style.

The practice of RN analysis represents a genuine and embodied knowledge.– My “experiment” with R243.

Who cares?

http://dmitri.tymoczko.comSlide60

Conclusion (even more general)

If we want to study the psychology or neuroscience of music, it helps to understand the internal, syntactical structure of music really well.We still have a long way to go here ...Formalization of voice leading

Connection between voice leading and modulationThe foundations of harmonic theoryTraditional theory is a mess, and we are only just starting to clean up.

http://dmitri.tymoczko.comSlide61

Thank you!

http://dmitri.tymoczko.com

for more information …Slide62

OUT TAKES

http://dmitri.tymoczko.comSlide63

Where Does It Fail?

A few small but noteworthy failures:Specific contrapuntal idiomscadential ii6-ii6/5

Occasionally wipes out dominant chordsCan improve the accuracy by ~.1% by telling it not to (cheating)Perhaps I want to preserve anything over a certain likelihood?

“Obligatory passing chords”

http://dmitri.tymoczko.comSlide64

Computer Analysis, summaryAnalyzing the chorales involves five basic steps:

Key findingChord identification

Key consolidationChord pruningLarge-scale pattern matching

http://dmitri.tymoczko.comSlide65

Some interesting chorale detailsDiscursive modulation (115 m3, 96 m7, 95 m9, 103 m3)

EQE nonharmonic tones (275 m3, 120 cadence, 226)

Sources of “Modality”Minor vDiscursive modulation

Tonal plan

Some genuinely modal chorales

V2-I progressions

V-IV-I

IV-I containing quasi-V chords

http://dmitri.tymoczko.comSlide66

Political NoteIf we want to sell traditional music theorists on quantitative, corpus-based methods, we our basic musical skills need to be beyond reproach.

Issues like this create will really annoy people!

http://dmitri.tymoczko.com