Transcript The Music of the Spherical Quotients

http://dmitri.tymoczko.com
“Yes, But Could The Martians
Understand Bach?”
the syntax and epistemology
of classical tonal harmony
Dmitri Tymoczko
Princeton University
http://dmitri.tymoczko.com
Today’s story
• is 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.com
Syntax
• Classical 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-tochord successions
• Rameau, Riemann, Piston, McHose, Kostka &
Payne
• Largely an American enterprise, at least recently
– Melodic templates, procedures, conventions
http://dmitri.tymoczko.com
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 sonatas
– 40% of the Beethoven piano sonatas (and
counting)
http://dmitri.tymoczko.com
Our Topic: Local Harmonic Laws
• Currently our best theory of tonal
harmony?
– Captures ~95% of chord progressions in the
Bach chorales
– Captures ~97% of the nonsequential
progressions in Mozart.
– Captures ~98% of the progressions in
Beethoven.
http://dmitri.tymoczko.com
Why is this important?
• True
• Under repeated attack:
–
–
–
–
CPE Bach
Schenker
Schoenberg
Quinn
• Crucial for understanding the development
of tonality
– Tonicity and entropy?
• Crucial for understanding contemporary
music
http://dmitri.tymoczko.com
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.
RF RRFR R
http://dmitri.tymoczko.com
The Fundamental Challenge
• The 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.
RF RRFR R
http://dmitri.tymoczko.com
The Fundamental Challenge
• Once 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!
RF RRFR R
http://dmitri.tymoczko.com
The Fundamental Problem
• How can we separate “real” chords from
“fake” chords in a principled way?
– “The Quinn challenge”
• Thanks to IQ and DH
RF RRFR R
http://dmitri.tymoczko.com
RN analysis is hard (1)
• What is the best (C major) analysis?
“the ii-vii°6 idiom”
PT
I6
C: IV
C: ii vii°6 I6
PT
PT
C: V
V2
I6
C: I
vii°6
I6
http://dmitri.tymoczko.com
RN analysis is hard (1)
• Note that all analyses suppress a (fake)
ii-I!
“the ii-vii°6 idiom”
PT
C: IV
C: V
ii6
I6
C: ii
I6
ii6
I6
C: I
vii°6 ii6 I6
http://dmitri.tymoczko.com
RN analysis is hard (1)
• So what is the force of “ii-I is rare”?
“the ii-vii°6 idiom”
PT
C: IV
C: V
ii6
I6
C: ii
I6
ii6
I6
C: I
vii°6 ii6 I6
http://dmitri.tymoczko.com
RN analysis is hard (1)
V6
g: i
i V6–5/III III
PT
F: I
V6
I
vii°6 I6
M2 b3 interpreted
differently!!!
http://dmitri.tymoczko.com
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.com
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.com
Huron and the ii-vii°6 idiom
– Huron 2007 explicitly considers the ii-vii°6
idiom occurring at the very beginning of a Dmajor passage.
– He doesn’t even consider putting two chords
on beat 3.
– Political note.
http://dmitri.tymoczko.com
Other people who are wrong
•
•
•
•
•
Yitzak Sadai
Aldwell and Schachter
Martin Rohrmeier
Craig Sapp, Helen Budge
Insert politician here 
http://dmitri.tymoczko.com
First Plateau
• Music 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
http://dmitri.tymoczko.com
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.com
Is the circle good or bad?
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?
http://dmitri.tymoczko.com
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.com
First strategy: embrace the
circle
• Perhaps 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.
basic criteria
of analytical goodness
basic criteria
of analytical goodness
THE ONE CORRECT ANALYSIS
A RANGE OF PLAUSIBLE ANALYSES
Theory A
Theory B
Analysis 1
Analysis 2
Analysis 3
Theory A
Theory B
Theory C
Theory C
http://dmitri.tymoczko.com
First strategy: embrace the
circle
• In 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%
accurate
– Riemann basic function theory
~79%
accurate
– Kostka/Payne
~92%
http://dmitri.tymoczko.com
First strategy: embrace the
circle
• Different 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%
accurate
– Riemann basic function theory
~79%
accurate
– Kostka/Payne
~92%
http://dmitri.tymoczko.com
The problem with giving up
• Makes 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.com
The problem with giving up
• Also, 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.com
Second plateau
• It 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.com
The ii-vii°6 idiom: a case
study
• What is the best interpretation? (and how
do we tell?)
• The case against I6-vii°6-I:
– The ii chord is often leapt to, and hence
harmonic.
– Premise: incomplete neighbors are rare.
R74 m. 1
*
PT
C: I6
vii°6
I
F: I6 ii vii°6 I
http://dmitri.tymoczko.com
The ii-vii°6 idiom: a case
study
6
• The case against I -ii-I:
– ii-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!
*
PT
C: I6
ii
I
http://dmitri.tymoczko.com
The ii-vii°6 idiom: a case
study
– In ii-I , the default voice leading should send
6
the fifth of the chord (^6) down (to ^5) rather
than up (to ^8).
– This is what happens in the analogous
diatonic progressions.
•
• vi-V6 (75%)
I-vii°6 (83%)
• “ii-I6”(17%)
*
PT
C: I6
ii
• V-IV6 (47%)*
I
http://dmitri.tymoczko.com
The ii-vii°6 idiom: a case
study
6
– Instead, with ii-(vii° )-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%).
PT
C: I6
ii
I
C: I6
ii vii°6 I
http://dmitri.tymoczko.com
The ii-vii°6 idiom: a case
study
– 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)!
*
PT
C: I6
ii
I
C: I6
ii vii°6 I
http://dmitri.tymoczko.com
Generalizing
• Clearly, 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 wellgrounded.
• Maybe we can make analytic decisions in a
theory-neutral way.
• Perhaps the Martians could understand Bach
after all!
http://dmitri.tymoczko.com
But …
… where will we find our
martians????
http://dmitri.tymoczko.com
martians.py
http://dmitri.tymoczko.com
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.com
martians.py
• Does 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.com
martians.py
• Does 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.com
Second strategy: kill the circle
• Basic plan
– Stage 1: create a raw analysis of the
chorales, considering every triad and
seventh chord to be a harmony.
– Stage 1b. Improve key finding with various
rules (e.g. dorian scale regions).
– 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
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
point
• When 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.com
Kill the Circle – stage 1
• Correct key 81.1%, correct chord 90.5%
• This music is largely unambiguous!
http://dmitri.tymoczko.com
Kill the Circle – stage 1
• The lack of ambiguity provides
statistical purchase, allowing us to
build a relatively theory-free model.
http://dmitri.tymoczko.com
V7 is special
Numbers are percentages
of all chords in the raw
data, counting every triad
and seventh.
Suggests V7 is by far the
most common seventh
chord, and syntactically
unusual. (NB: iii is
suppressed.)
http://dmitri.tymoczko.com
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)
V i i vii°6 I V6 vii° i
http://dmitri.tymoczko.com
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.com
Kill the Circle – stage 3
1: I – IV6 – vi – V6
2: I – IV6 – IV6 – V6
3: I – vi – vi – V6
4: I – IV6 – IVmaj# – V6
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.
http://dmitri.tymoczko.com
Kill the Circle – stage 3
1: I – iii6 – V – vi
2: I – iii6 – iii6 – vi
3: I – V– V – vi
4: I – iii6 – iii# – vi
0
0
9
0
http://dmitri.tymoczko.com
Kill the Circle – stage 3
• This 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.com
RN analysis is hard (reprise)
• What is the best (C major) analysis?
5.8:1
7.8:1
PT
I6
C: IV
43:1
C: ii vii°6 I6
10:1
PT
PT
C: V
V2
I6
C: I
vii°6
I6
http://dmitri.tymoczko.com
RN analysis is hard (reprise)
• Ratio of my preferred analysis to the best
alternative.
5.8:1
7.8:1
PT
I6
C: IV
43:1
C: ii vii°6 I6
10:1
PT
PT
C: V
V2
I6
C: I
vii°6
I6
http://dmitri.tymoczko.com
RN analysis is hard (reprise)
V6
g: i
F: I
V6
i V6/III III
I
vii°6 I6
7:1 (NB: no V# )
4.3:1
http://dmitri.tymoczko.com
Using Mechanical Analyses
– Rameau/Meeus
~73%
accurate
– Riemann basic function theory
~74%
accurate
– Kostka/Payne
~84%
accurate
– Tymoczko
~86%
accurate
• Cf. Human:
– Rameau/Meeus
accurate
~78%
http://dmitri.tymoczko.com
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
http://dmitri.tymoczko.com
Conclusion
• We 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.com
Conclusion
• Our 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.com
Conclusion
• This 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.com
Slow development
•
•
•
•
•
Rameau (1720)
Riemann (1880)
McHose (1945)
Kostka/Payne (1970)
Tymoczko (2011)
~78% accurate
~79% accurate
~“76% accurate”
~92% accurate
~95% accurate
A pretty odd progression!
http://dmitri.tymoczko.com
The role of perception
• Two approaches:
1. 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.
2. Who cares?
http://dmitri.tymoczko.com
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
modulation
– The foundations of harmonic theory
• Traditional theory is a mess, and we are
only just starting to clean up.
http://dmitri.tymoczko.com
for more information …
Thank you!
http://dmitri.tymoczko.com
OUT TAKES
http://dmitri.tymoczko.com
Where Does It Fail?
• A few small but noteworthy failures:
– Specific contrapuntal idioms
• cadential ii6-ii6/5
– Occasionally wipes out dominant chords
• Can 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.com
Computer Analysis, summary
• Analyzing the chorales involves five basic
steps:
– Key finding
– Chord identification
– Key consolidation
– Chord pruning
– Large-scale pattern matching
http://dmitri.tymoczko.com
Some interesting chorale details
• Discursive modulation (115 m3, 96 m7, 95 m9, 103
m3)
• EQE nonharmonic tones (275 m3, 120 cadence, 226)
• Sources of “Modality”
– Minor v
– Discursive modulation
– Tonal plan
– Some genuinely modal chorales
• V2-I progressions
• V-IV-I
• IV-I containing quasi-V chords
http://dmitri.tymoczko.com
Political Note
– If 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!