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

3. 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

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

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

6. 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

7. 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

8. 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

9. 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. http://dmitri.tymoczko.com R F R R F R R

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

11. 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

12. 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

13. 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

14. 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

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

16. 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

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

18. 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.com

19. Other people who are wrong Yitzak Sadai Aldwell and Schachter Martin Rohrmeier Craig Sapp, Helen Budge Insert politician here http://dmitri.tymoczko.com

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

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

22. 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?

23. 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

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

25. 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

26. 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 http://dmitri.tymoczko.com

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

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

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 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

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 *

31. 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

32. –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%)

33. –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

34. –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

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

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

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

38. 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

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

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

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).

42. 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

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

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.

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.)

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

47. 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

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 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.

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

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 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

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

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

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

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

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

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

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

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

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

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

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

62. Thank you! http://dmitri.tymoczko.com for more information …

63. OUT TAKES http://dmitri.tymoczko.com

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

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

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

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