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Dmitri Tymoczko Princeton University

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. i s about the conflict between embodied musical knowledge …

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Dmitri Tymoczko Princeton University

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  1. http://dmitri.tymoczko.com “Yes, But Could The Martians Understand Bach?”the syntax and epistemology of classical tonal harmony Dmitri Tymoczko Princeton University

  2. 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!

  3. 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-to-chord successions • Rameau, Riemann, Piston, McHose, Kostka & Payne • Largely an American enterprise, at least recently • Melodic templates, procedures, conventions • Schenker

  4. 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)

  5. 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 nonsequentialprogressions in Mozart. • Captures ~98% of the progressions in Beethoven.

  6. 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

  7. 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 motionbetween harmonic chords. • The rules apply only to “real” chords. R F R R F R R

  8. 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. R F R R F R R

  9. 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! R F R R F R R

  10. 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 R F R R F R R

  11. http://dmitri.tymoczko.com RN analysis is hard (1) • What is the best (C major) analysis? “the ii-vii°6 idiom” PT C: IV I6 C: ii vii°6 I6 PT PT C: V V2 I6 C: I vii°6 I6

  12. 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 ii6 I6 C: ii I6 C: V ii6 I6 C: I vii°6 ii6 I6

  13. 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 ii6 I6 C: ii I6 C: V ii6 I6 C: I vii°6 ii6 I6

  14. http://dmitri.tymoczko.com RN analysis is hard (1) M2 b3 interpreted differently!!! g: i V6 i V6–5/III III PT F: I V6 I vii°6 I6

  15. 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.

  16. 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.

  17. 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 D-major passage. • He doesn’t even consider putting two chords on beat 3. • Political note.

  18. http://dmitri.tymoczko.com Other people who are wrong • YitzakSadai • Aldwell and Schachter • Martin Rohrmeier • Craig Sapp, Helen Budge • Insert politician here 

  19. 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 taught).

  20. 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 Barghet al. (1996)

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

  22. 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.”

  23. 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.

  24. 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% accurate • Tymoczko ~95% accurate

  25. 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% accurate • Tymoczko ~95% accurate

  26. 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.

  27. 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 answerto this question.

  28. http://dmitri.tymoczko.com Second plateau • It would be nice to be able to provide an effectivetheory-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!

  29. 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

  30. http://dmitri.tymoczko.com The ii-vii°6 idiom: a case study • The case against I6-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

  31. http://dmitri.tymoczko.com The ii-vii°6 idiom: a case study • 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%) • • V-IV6 (47%)* • “ii-I6”(17%) * PT C: I6 ii I

  32. http://dmitri.tymoczko.com The ii-vii°6 idiom: a case study • 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%). PT C: I6 ii I C: I6 ii vii°6 I

  33. 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

  34. 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 formalizingand 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!

  35. http://dmitri.tymoczko.com But … … where will we find our martians????

  36. http://dmitri.tymoczko.com martians.py

  37. 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.

  38. 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%

  39. 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%

  40. 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 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. – Stage 1b. Improve key finding with various rules (e.g. dorian scale regions).

  41. http://dmitri.tymoczko.com Kill the Circle – stage 1 • Go through the chorale to find maximal regionsbelonging 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 everytertianverticality (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.

  42. http://dmitri.tymoczko.com Kill the Circle – stage 1 • Correct key 81.1%, correct chord 90.5% • This music is largely unambiguous!

  43. 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.

  44. 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.)

  45. 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) I V6 vii° i V iivii°6

  46. 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!

  47. 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.

  48. 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

  49. 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.

  50. http://dmitri.tymoczko.com RN analysis is hard (reprise) • What is the best (C major) analysis? 5.8:1 7.8:1 PT C: IV I6 C: ii vii°6 I6 43:1 10:1 PT PT C: V V2 I6 C: I vii°6 I6

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