Dmitri Tymoczko Princeton University

The Music of the Spherical Quotients

Yes, But Could The Martians Understand Bach?

the syntax and epistemology of classical tonal harmonyDmitri TymoczkoPrinceton University

http://dmitri.tymoczko.com1Todays storyis about the conflict between embodied musical knowledge and scientific methodology,which has produced serious misunderstandings of classical musica cautionary tale about the difficulties of mixing music and science But everything works out OK in the end!

http://dmitri.tymoczko.com2SyntaxClassical 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 recentlyMelodic templates, procedures, conventionsSchenker

http://dmitri.tymoczko.com3Our Topic: Local Harmonic LawsIn Chapter 7 of GOM, I propose a theory of chord progressions in tonal harmony.Tested against substantial corpora:371 Bach choralesAll the Mozart piano sonatas40% of the Beethoven piano sonatas (and counting)

http://dmitri.tymoczko.com

4Our 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.


5Why is this important?TrueUnder repeated attack:CPE BachSchenkerSchoenbergQuinnCrucial for understanding the development of tonalityTonicity and entropy?Crucial for understanding contemporary music


6The Fundamental ChallengeTraditional harmonic theory says that there are two kinds of chords in classical music.Harmonic (or real) chordsContrapuntal (or fake) chords produced by melodic motion between harmonic chords.The rules apply only to real chords.


R F R R F R R7The 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 werejust that some consonance underlies every dissonance.


R F R R F R R8The 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!


R F R R F R R9The Fundamental ProblemHow can we separate real chords from fake chords in a principled way?The Quinn challengeThanks to IQ and DH


R F R R F R R10RN analysis is hard (1)What is the best (C major) analysis?


C: IV I6 PT

C: ii vii6 I6

PTC: I vii6 I6 the ii-vii6 idiomC: V V2 I6

PT11RN analysis is hard (1)Note that all analyses suppress a (fake) ii-I!


C: IV ii6 I6

C: ii I6

C: I vii6 ii6 I6 the ii-vii6 idiomC: V ii6 I6

PT12RN analysis is hard (1)So what is the force of ii-I is rare?


C: IV ii6 I6

C: ii I6

C: I vii6 ii6 I6 the ii-vii6 idiomC: V ii6 I6

PT13RN analysis is hard (1)


g: i V6 i V65/III IIIF: I V6 I vii6 I6PT

M2 b3 interpreteddifferently!!!14RN 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.


15RN 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-vii6 idiom.


16Huron and the ii-vii6 idiomHuron 2007 explicitly considers the ii-vii6 idiom occurring at the very beginning of a D-major passage. He doesnt even consider putting two chords on beat 3.Political note.


17Other people who are wrongYitzak SadaiAldwell and SchachterMartin RohrmeierCraig Sapp, Helen BudgeInsert politician here

http://dmitri.tymoczko.com18First 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.com19Is 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.com20Is 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?

Doesnt that bias our tests?21Is the circle good or bad?In other contexts, we learn to live with circles. the hermeneutic circlecf. pragmatics, artificial intelligence, etc.I am going to the bank.

http://dmitri.tymoczko.com22First 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.


23First 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% accurateTymoczko~95% accurate

http://dmitri.tymoczko.com24First strategy: embrace the circleDifferent 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% accurateRiemann basic function theory~79% accurateKostka/Payne~92% accurateTymoczko~95% accurate

http://dmitri.tymoczko.com25The 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.com26The 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.com27Second 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 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.com28What is the best interpretation? (and how do we tell?)The case against I6-vii6-I:The ii chord is often leapt to, and hence harmonic.Premise: incomplete neighbors are rare.


C: I6 vii6 I PTThe ii-vii6 idiom: a case study

R74 m. 1F: I6 ii vii6 I *29The case against I6-ii-I:ii-I almost never happens, in any chordal inversion, without the intervening vii6.In particular, we find ii-I6, with ^6 going to ^5, much less often than we should!

http://dmitri.tymoczko.comThe ii-vii6 idiom: a case study*

C: I6 ii I PT

30In 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-vii6 (83%)vi-V6 (75%)


C: I6 ii I PT

V-IV6 (47%)*

ii-I6(17%)31Instead, with ii-(vii6)-I, the fifth (^6) most commonly moves up through ^7 to ^1. Furthermore, in the ii-vii6 progression, the fifth is doubled less than half as often as one would expect (~ 3% vs. 6%).

http://dmitri.tymoczko.comThe ii-vii6 idiom: a case study

C: I6 ii I PT

C: I6 ii vii6 I 32In other words, Bach clearly goes out of his way to create a leading tone and vii6 harmony.The best evidence for the harmonic status of vii6 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)!


C: I6 ii I PT

C: I6 ii vii6 I 33GeneralizingClearly, 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.com34But where will we find our martians????

http://dmitri.tymoczko.com35martians.py


36martians.pymartians.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.com37martians.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.com38martians.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.com39Second strategy: kill the circleBasic planStage 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). 40Kill the Circle stage 1Go 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.com41Kill the Circle stage 1


Correct key 81.1%, correct chord 90.5%This music is largely unambiguous!

42Kill the Circle stage 1


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

43V7 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.)44Kill the Circle stage 2Using 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)


I V6 vii i V i i vii645Kill the Circle stage 3When 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.com46Kill the Circle stage 3


1: I IV6 vi V62: I IV6 IV6 V63: I vi vi V64: I IV6 IVmaj# V6

0*650#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.47Kill the Circle stage 3


1: I iii6 V vi2: I iii6 iii6 vi3: I V V vi4: I iii6 iii# vi

009048Kill 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 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.com49RN analysis is hard (reprise)What is the best (C major) analysis?


C: IV I6 PT

C: ii vii6 I6

PTC: I vii6 I6 C: V V2 I6 5.8:17.8:110:143:1

PT50RN analysis is hard (reprise)Ratio of my preferred analysis to the best alternative.


C: IV I6 PT

C: ii vii6 I6

PTC: I vii6 I6 C: V V2 I6 5.8:17.8:110:143:1

PT51RN analysis is hard (reprise)


g: i V6 i V6/III IIIF: I V6 I vii6 I67:1 (NB: no V# )4.3:152Using Mechanical AnalysesRameau/Meeus ~73% accurateRiemann basic function theory~74% accurateKostka/Payne~84% accurateTymoczko~86% accurateCf. Human:Rameau/Meeus ~78% accurateRiemann basic function theory~79% accurateKostka/Payne~92% accurateTymoczko~95% accurate

http://dmitri.tymoczko.com53ConclusionWe 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.com54ConclusionWe 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.com55ConclusionOur 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.com56ConclusionThis 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.com57Slow developmentRameau (1720) ~78% accurateRiemann (1880)~79% accurateMcHose (1945)~76% accurateKostka/Payne (1970)~92% accurateTymoczko (2011) ~95% accurate

A pretty odd progression!

http://dmitri.tymoczko.com58The 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.com59Conclusion (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 leadingConnection 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.com60Thank you!


for more information 61OUT TAKES

http://dmitri.tymoczko.com62Where Does It Fail?A few small but noteworthy failures:Specific contrapuntal idiomscadential ii6-ii6/5Occasionally 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.com63Computer Analysis, summaryAnalyzing the chorales involves five basic steps:Key findingChord identificationKey consolidationChord pruningLarge-scale pattern matching

http://dmitri.tymoczko.com64Some interesting chorale detailsDiscursive modulation (115 m3, 96 m7, 95 m9, 103 m3)EQE nonharmonic tones (275 m3, 120 cadence, 226)Sources of ModalityMinor vDiscursive modulationTonal planSome genuinely modal choralesV2-I progressionsV-IV-IIV-I containing quasi-V chords

http://dmitri.tymoczko.com65Political 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!


6623902.12CHORDTRIADSSEVENTHS

I302

V1511

IV93

ii44

vi62

vii42

iii32

Dmitri Tymoczko Princeton University

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