Transcript of “Yes, But Could The Martians Understand Bach?” the syntax and epistemology of classical tonal...
- Slide 1
- Slide 2
- Yes, But Could The Martians Understand Bach? the syntax and
epistemology of classical tonal harmony Dmitri Tymoczko Princeton
University http://dmitri.tymoczko.com
- Slide 3
- Todays 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
- Slide 4
- 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-to- chord
successions Rameau, Riemann, Piston, McHose, Kostka & Payne
Largely an American enterprise, at least recently Melodic
templates, procedures, conventions Schenker
http://dmitri.tymoczko.com
- Slide 5
- 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
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- 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
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- 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
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- 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 R
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- 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 werejust that some
consonance underlies every dissonance. http://dmitri.tymoczko.com R
F R R F R R
- Slide 10
- 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! http://dmitri.tymoczko.com R F R R F R R
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- The Fundamental Problem How 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 R
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- 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 PT
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- 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 PT
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- 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 PT
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- RN analysis is hard (1) http://dmitri.tymoczko.com g: i V 6 i V
65 /III III F: I V 6 I vii 6 I 6 PT M2 b3 interpreted
differently!!!
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- 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 iiI progressions (excluding
cadential @) account for less than 2% of the destinations from ii.
http://dmitri.tymoczko.com
- Slide 17
- 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
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- 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 doesnt even consider putting two chords on beat 3.
Political note. http://dmitri.tymoczko.com
- Slide 19
- Other people who are wrong Yitzak Sadai Aldwell and Schachter
Martin Rohrmeier Craig Sapp, Helen Budge Insert politician here
http://dmitri.tymoczko.com
- Slide 20
- 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 taught).
http://dmitri.tymoczko.com
- Slide 21
- 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
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- 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? Doesnt that bias our tests?
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- 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
- Slide 24
- 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.
http://dmitri.tymoczko.com
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- 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% accurate Tymoczko~95% accurate
http://dmitri.tymoczko.com
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- First strategy: embrace the circle Different resolution of
nonharmonic tones might make a 5% difference, but not a 13%
difference. Indeed, it doesnt even boost K & P above my theory.
Rameau/Meeus ~78% accurate Riemann basic function theory~79%
accurate Kostka/Payne~92% accurate Tymoczko~95% accurate
http://dmitri.tymoczko.com
- Slide 27
- 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
- Slide 28
- 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
- Slide 29
- 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 dont want to
stipulate that iii is rare or ii doesnt 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
- Slide 30
- 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 *
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- The case against I 6 -ii-I: ii-I almost never happens, in any
chordal inversion, without the intervening vii 6. In particular, we
find ii-I 6, 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 PT
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- In ii-I 6, 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-V
6 (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%)
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- 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
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- 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-I 6 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
- Slide 35
- 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 well- grounded.
Maybe we can make analytic decisions in a theory-neutral way.
Perhaps the Martians could understand Bach after all!
http://dmitri.tymoczko.com
- Slide 36
- But where will we find our martians????
http://dmitri.tymoczko.com
- Slide 37
- martians.py http://dmitri.tymoczko.com
- Slide 38
- martians.py martians.py is a computer program for analyzing
Bach chorales. I wrote it in python using Michael Cuthberts 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, wed use
slick tools from machine learning. http://dmitri.tymoczko.com
- Slide 39
- 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
- Slide 40
- 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
- Slide 41
- 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 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).
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- 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
- Slide 43
- Kill the Circle stage 1 http://dmitri.tymoczko.com Correct key
81.1%, correct chord 90.5% This music is largely unambiguous!
- Slide 44
- 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.
- Slide 45
- V 7 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.)
- Slide 46
- 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 6
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- 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
- Slide 48
- Kill the Circle stage 3 http://dmitri.tymoczko.com 1: I IV 6 vi
V 6 2: I IV 6 IV 6 V 6 3: I vi vi V 6 4: I IV 6 IVmaj# V 6 0* 6 5 0
#1 is 0 because we dont 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.
- Slide 49
- Kill the Circle stage 3 http://dmitri.tymoczko.com 1: I iii 6 V
vi 2: I iii 6 iii 6 vi 3: I V V vi 4: I iii 6 iii# vi 00900090
- Slide 50
- 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 dont agree at that level; also, higher-level
complexities, etc. Were really close! This method deals with all
the problematic cases mentioned earlier.
http://dmitri.tymoczko.com
- Slide 51
- 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 10:143:1 PT
- Slide 52
- 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 10:143:1
PT
- Slide 53
- RN analysis is hard (reprise) http://dmitri.tymoczko.com g: i V
6 i V 6 /III III F: I V 6 I vii 6 I 6 7:1 (NB: no V# ) 4.3:1
- Slide 54
- Using Mechanical Analyses Rameau/Meeus ~73% accurate Riemann
basic function theory~74% accurate Kostka/Payne~84% accurate
Tymoczko~86% accurate Cf. Human: Rameau/Meeus ~78% accurate Riemann
basic function theory~79% accurate Kostka/Payne~92% accurate
Tymoczko~95% accurate http://dmitri.tymoczko.com
- Slide 55
- 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.com
- Slide 56
- 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
- Slide 57
- 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
- Slide 58
- 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
- Slide 59
- Slow development Rameau (1720) ~78% accurate Riemann (1880)~79%
accurate McHose (1945)~76% accurate Kostka/Payne (1970)~92%
accurate Tymoczko (2011) ~95% accurate A pretty odd progression!
http://dmitri.tymoczko.com
- Slide 60
- 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
- Slide 61
- 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
- Slide 62
- Thank you! http://dmitri.tymoczko.com for more information
- Slide 63
- OUT TAKES http://dmitri.tymoczko.com
- Slide 64
- 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
- Slide 65
- 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
- Slide 66
- 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
- Slide 67
- 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! http://dmitri.tymoczko.com