2. POP QUIZ: QUESTION 1 Below you will find the 10 most-common major-mode chord progressions in the music of composer X, along with the percentage of all progressions belonging to that category. In making this table, I have grouped together root position and first- inversion triads and all sevenths, so V 6 –I, V 7 –I, V–I 6 (etc.) all count as “V–I progressions.” However, 6/4 chords are their own category. http://dmitri.tymoczko.com Today's first quiz question is: who is the composer? Don’t dither, just answer off the top of your head.

3. POP QUIZ: QUESTION 1I In the Bach chorales, the most common destination for V 2 is I 6. What is the second most common destination? http://dmitri.tymoczko.com

4. Tonal Functionalities (within-key harmonic patterns) Dmitri Tymoczko Princeton University http://dmitri.tymoczko.com

5. Answers to the Quiz http://dmitri.tymoczko.com JosquinPalestrina

6. Syntax Classical music has at least two features that might deserve the name “syntactical.” –Harmonic principles governing chord-to- chord successions Rameau, Riemann, Piston, McHose, Kostka & Payne Largely an American enterprise, at least recently –Contrapuntal syntax governing “harmonic” and “nonharmonic” tones Neighbor tones, passing tones, suspensions, etc. Schenker http://dmitri.tymoczko.com

7. Exploring these Syntaxes Constructing large databases, with two parts: –Roman Numeral analyses of music –The score as an XML or KRN file –Analysis using python and Myke Cuthbert’s music21. Provides unprecedented access to syntactical detail: –What note gets doubled in a first-inversion triad? –Does the tritone typically resolve in vii° 6 –I? –Do nonharmonic tones ever license parallel fifths? –How did tonal harmony develop? http://dmitri.tymoczko.com

8. Examples http://dmitri.tymoczko.com Analysis file XML Labeled chorale Nonharmonic tones removed Some simple calculations all V 2 –I progressions in the chorales

9. Example http://dmitri.tymoczko.com

10. The Corpus: a snapshot http://dmitri.tymoczko.com To do: frottola, Chopin, Brahms, ? 107,902 chords

11. Methodological Question What does it mean to use Roman Numerals with modal music? –“V6” and “V” are uninterpreted labels, with no necessary relation between them. “V6” means “the 6/3 chord on the leading tone” –Label almost all consonances, except where harmonic rhythm suggests otherwise. –Incomplete root-third is a root position triad, except when it is contained within the previous sonority (ex. C-E followed by E-G). –Can reanalyze the data from other perspectives. http://dmitri.tymoczko.com

12. Sample topics How accurate is standard harmonic theory? How did tonal harmony develop? Are nonharmonic tones “merely decorative” or do they serve a structural function? Is Roman Numeral Analysis justified? More general show and tell http://dmitri.tymoczko.com

13. Topic I: Accuracy Four grammars http://dmitri.tymoczko.com

14. Topic I: Accuracy http://dmitri.tymoczko.com major

15. Topic I: Accuracy http://dmitri.tymoczko.com minor

16. Topic I: Accuracy http://dmitri.tymoczko.com both

17. Three Grades of Modal Involvement http://dmitri.tymoczko.com True modality (~65%)

18. Three Grades of Modal Involvement http://dmitri.tymoczko.com “Tonally tinged modality” (~75%)

19. Three Grades of Modal Involvement http://dmitri.tymoczko.com “Modally tinged tonality” (~85%)

20. Three Grades of Modal Involvement http://dmitri.tymoczko.com “True tonality” (> 90%)

21. Topic II: Development http://dmitri.tymoczko.com

22. Topic II: Development http://dmitri.tymoczko.com

23. Topic II: Development The rule of the octave http://dmitri.tymoczko.com ( )

24. Topic II: Development The rule of the octave: degrees 1, 5, 7. –Strong preference evident in Josquin. –Grows even stronger over the 16 th century. http://dmitri.tymoczko.com 5353 5353 6363 ( )

25. Topic II: Development The rule of the octave: degree 3. –Mild preference for I 6 in Palestrina, grows strong in Morley, and even stronger after that. http://dmitri.tymoczko.com 5353 5353 6363 6363 ( ) Amazing fact!

26. Topic II: Development Given these basic preferences, many tonal routines appear naturally. –3 up to 5, 5 up to 8, 8 down to 5. http://dmitri.tymoczko.com 5353 5353 6363 6363 ( )

27. Topic II: Development Others still remain. –1 up to 3, 3 down to 1, and 5 down to 3 are seventeenth-century developments. http://dmitri.tymoczko.com 5353 5353 6363 6363 ( )

28. Topic II: Development What about the idea that 4–1 should be harmonized with root position triads, rather than as ii 6 –I? –In Josquin and Palestrina, fourth and fifth motion in the bass almost always involves root position. http://dmitri.tymoczko.com

29. Mutiple Tonalities “Riemannian Tonality” –I, IV, and V are primary triads, with I-IV-I counterbalancing I-V-I –Developed first –Present in Palestrina (and Encina, the Carter Family, Ramones, etc.) “Schenkerian Tonality” –I and V are primary triads –ii displaces IV, and serves primarily as a connector between I and V. –Classical tonality. –Jazz? http://dmitri.tymoczko.com

30. Topic II: Development http://dmitri.tymoczko.com

31. Topic II: Development http://dmitri.tymoczko.com

32. Topic II: Development http://dmitri.tymoczko.com

33. Topic III: Nonharmonic tones Do nonharmonic tones “embellish” an underlying, structural “harmonic skeleton”? First answer (16 th century): –For passing tones, neighboring tones, anticipations, and incomplete neighbors: yes. –For suspensions: no. Two reasons: –Suspensions are not removable. –Suspensions license parallels, unlike the others. http://dmitri.tymoczko.com

34. Topic III: Nonharmonic tones http://dmitri.tymoczko.com

35. Topic III: Nonharmonic tones http://dmitri.tymoczko.com A-G D-C

36. Topic III: Nonharmonic tones Second answer (Bach): –Again, only suspensions license parallels. They occur at almost exactly the same rate in Bach and Palestrina. Bach almost never suspends the fifth! They almost never involve the bass. –In Bach, a number of progressions seem to require passing tones: IV 6 –I V 2 –I –Contrapuntal schemas http://dmitri.tymoczko.com

37. So … Suspensions do NOT “represent” their underlying consonances. But … –It sometimes happens that a chord changes before a suspension resolves. –In Bach, this occurs most often over a changing bass, like i–i 6. –In Palestrina, this occurs most often over a fixed bass, like I-vi 6. http://dmitri.tymoczko.com

38. Topic IV: Justification The harmonic syntax applies only to “real” chords. How can we separate “truly harmonic” chords from “merely contrapuntal” chords in a principled way? http://dmitri.tymoczko.com R F R R F R R

39. Topic IV: Justification 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

40. Topic IV: Justification 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

41. Topic IV: Justification 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

42. Topic IV: Justification 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!!!

43. Topic IV: Justification Even the pros make mistakes: –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

44. RN analysis as generalization Write a computer program to: –Stage 1: create a raw analysis of the chorales, identifying keys with scales, and 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).

45. RN analysis as generalization http://dmitri.tymoczko.com Correct key 81.1%, correct chord 90.5% This music is largely unambiguous!

46. RN analysis as generalization 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. RN analysis as generalization When we find a quarter note containing a pair of eighth-note harmonies, ask: –Could the first be the product of nonharmonic tones (according to standard contrapuntal theory). –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. RN analysis as generalization 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. RN analysis as generalization 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. RN analysis as generalization 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. 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 RN analysis as generalization

52. Ratio of my preferred analysis to the best alternative. http://dmitri.tymoczko.com C: IV ii 6 I 6 C: ii I 6 C: I vii° 6 ii 6 I 6 C: V ii 6 I 6 PT RN analysis as generalization 5.8:1 7.8:1 10:143:1

53. RN analysis as generalization 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. Conclusions Combining analyses and scores gives us a lot of power Chords really do follow a local grammar Tonality evolved very slowly, in many different genres simultaneously 16 th -century harmony resembles rock harmony in some interesting ways RN analysis exploits intuitive knowledge of statistical probabilities I’m just getting started thinking of questions to ask –Maybe you can help? http://dmitri.tymoczko.com

55. Thank you! http://dmitri.tymoczko.com for more stuff like this …

56. Thank you! http://dmitri.tymoczko.com for more stuff like this …