1. BIG DATA, LITTLE DATA, MUSICAL DATA
BIG DATA, LITTLE DATA, MUSICAL DATA
Dmitri Tymoczko
Princeton University 1

2. A NEW WORLD The internet has made it easy to share musical scores.
A NEW WORLD
The internet has made it easy to share musical scores.
The development of cross-platform methods of representing musical notation
MIDI files
MusicXML
Millions of volunteers who are encoding music.
Various computer tools for reading and manipulating musical scores.
music21

3. BUT … All this data is difficult to interpret.
BUT …
All this data is difficult to interpret.
Chords have different meanings in different keys.
Some things that look like chords are not chords, but rather the byproduct of nonharmonic tones.
For some questions we do not need chords or keys.
For some questions we can guess the chord and key.
But, for a detailed and truly useful perspective on musical structure we need to put this information in by hand.
Following procedures in linguistics.

4.
MY CORPUS

5.
TODAY’S QUESTIONS
What particular practices characterize classical harmony?
How did functionality develop?
What is the deep nature of harmonic functionality?
What are “nonharmonic tones” and are they merely decorative?
How can statistical information change our understanding of specific pieces?

6. 1. Functional Harmony Chords follow a syntax.
1. Functional Harmony
Chords follow a syntax.
Bad news: roughly first-order Markov (boring)
Good news: masked by radical ambiguity (exciting)
Studied for three centuries.
We can, for the first time, test these theories against substantial and accurate corpora.
~50,000 chords in Bach, Mozart, Beethoven, etc.
A BORING SYNTAX!
Yosef Goldenberg helped
----- Meeting Notes (2/11/13 13:01) -----
ALLOWABLE AND not ALLOWABLE

7. 1. Functional Harmony Chords follow a syntax.
1. Functional Harmony
Chords follow a syntax.
Bad news: roughly first-order Markov (boring)
Good news: masked by radical ambiguity (exciting)
Studied for three centuries.
We can, for the first time, test these theories against substantial and accurate corpora.
~50,000 chords in Bach, Mozart, Beethoven, etc.
Rameau (1722) % accurate

8. 1. Functional Harmony Chords follow a syntax.
1. Functional Harmony
Chords follow a syntax.
Bad news: roughly first-order Markov (boring)
Good news: masked by radical ambiguity (exciting)
Studied for three centuries.
We can, for the first time, test these theories against substantial and accurate corpora.
~50,000 chords in Bach, Mozart, Beethoven, etc.
Riemann (~1880) % accurate

9. 1. Functional Harmony Chords follow a syntax.
1. Functional Harmony
Chords follow a syntax.
Bad news: roughly first-order Markov (boring)
Good news: masked by radical ambiguity (exciting)
Studied for three centuries.
We can, for the first time, test these theories against substantial and accurate corpora.
~50,000 chords in Bach, Mozart, Beethoven, etc.
Kostka/Payne (1970) % accurate

10. 1. Functional Harmony Chords follow a syntax.
1. Functional Harmony
Chords follow a syntax.
Bad news: roughly first-order Markov (boring)
Good news: masked by radical ambiguity (exciting)
Studied for three centuries.
We can, for the first time, test these theories against substantial and accurate corpora.
~50,000 chords in Bach, Mozart, Beethoven, etc.
Tymoczko (2011) % accurate

11.
1. Functional Harmony
Music may be less recursive, and less hierarchical, than many of us think.
The nature of long-range structure is an open question.

12.
1. Functional Harmony
There are important and systematic differences among tonal composers.
Secondary triads or secondary dominants?

13.
1. Functional Harmony
There are important and systematic differences among tonal composers.
What chord harmonizes ^2 in the bass?

14.
1. Functional Harmony
There are important and systematic differences among tonal composers.
How does the IV chord resolve?

15.
1. Functional Harmony
There are important and systematic differences among tonal composers.
Does vi progress directly to V?

16.
1. Functional Harmony
CONCLUSION:

17. Warning: what follows might be upsetting …
1. Functional Harmony
Warning: what follows might be upsetting …

18. A lot of the information in textbooks is wrong or incomplete.
A lot of the information in textbooks is wrong or incomplete.

19. 2. Development of Tonality
2. Development of Tonality

20. 2. Development of Tonality
2. Development of Tonality

21. 2. Development of Tonality
2. Development of Tonality

22. 2. Development of Tonality
2. Development of Tonality
The harmonic cycle is the default scheme:
The harmonic cycle gets built backward along the circle of fifths from dominant to predominant to pre-predominant.
First, the dominant increasingly goes to I.
Then ii chord emerges in the early (or mid-) 17th-century.
I  [vi  (IV or ii)]  vii° or V  I
vii° or V  I
ii)]  vii° or V  I

23. 2. Development of Tonality
2. Development of Tonality

24. 2. Development of Tonality
2. Development of Tonality
The harmonic cycle is the default scheme:
The harmonic cycle gets built backward along the circle of fifths from dominant to predominant to pre-predominant.
First, the dominant increasingly goes to I.
Then ii chord emerges in the early 17th-century.
The vi chord emerges as a pre-predominant only in the 18th-century.
I  [vi  (IV or ii)]  vii° or V  I
vii° or V  I
ii)]  vii° or V  I
I  [vi  (IV or ii)]  vii° or V  I

25. 2. Development of Tonality
2. Development of Tonality
The harmonic cycle is the default scheme:
The harmonic cycle gets built backward from dominant to predominant to pre-predominant.
First, the dominant increasingly goes to I.
Then ii chord emerges in the early 17th-century.
The vi chord emerges as a pre-predominant only in the 18th-century.
Prior to that, it often goes directly to V
The primary role of the iii chord is not to exist!
I  [vi  (IV or ii)]  vii° or V  I

26.
CONCLUSION
Tonality developed long before its theoretical principles were codified in writing.
It was an oral tradition, a series of conventions learned by ear and known in practice.
Similar to jazz and rock
Very different from the artificial musical languages of the twentieth-century
Maybe there’s a lesson here for how we think about musical language?

27. Important to distinguish zeroth order from first order function:
3. What is function?
Important to distinguish zeroth order from first order function:
If a music uses a lot of V chords and a lot of I chords, this will typically mean it has a lot of V–I progressions.
It does not mean that V has an unusual tendency to move to I.
To measure that we need P(I|V) – P(I)

28. THE TENDENCY HISTOGRAM
THE TENDENCY HISTOGRAM

29.
3. What is function?
At the deepest level, tonic and dominant are oriented oppositely in time.
Dominants can be approached freely, but move only in specific ways (e.g. to I or vi)
Tonic chords can progress freely, but are approached only in specific ways (e.g. from V or vii°).
These opposite orientations interact with phrase structure:
Tonics begin phrases, dominants end phrases.
Seeds of an abstract notion of function?
Arguably in rock harmony IV has something of the function of a dominant.
Useful information from a compositional perspective?

30.
4. Nonharmonic tones
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.
Harmonic syntax applies only to “real” chords.
Nonharmonic tones are inessential or decorative, and can be removed without changing musical structure.
----- Meeting Notes (2/11/13 12:25) -----
To understand the syntax we need to answer the fundamental challenge
R F R R F R R

31. 4. Nonharmonic tones This leads to two questions:
4. Nonharmonic tones
This leads to two questions:
Can we eliminate nonharmonic tones? Are they “merely decorative”?
If we can eliminate nonharmonic tones, is there a principled way to do it?
----- Meeting Notes (2/11/13 12:25) -----
To understand the syntax we need to answer the fundamental challenge

32. The ineliminable suspension
The ineliminable suspension *
Fascinating schenkerian questions about reduction!
Analytical reduction fails to produce triads! 32

33. Suspensions license parallel fifths
Suspensions license parallel fifths
A-G
Happens once every ~344 chords
(about once per mass movement)
D-C 33

34.
Bach Chorales
Suspensions continue to license parallel fifths (not octaves).
These occur at roughly the same rate in Bach and Palestrina (~once every 286 chords [6 chorales])

35.
A paradox
Eliminating nonharmonic tones is central to music-theoretical understanding.
This works only some of the time.

36. The Fundamental Challenge
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

37. The Fundamental Challenge
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

38. The Fundamental Challenge
The Fundamental Challenge
How do we separate “real” chords from “fake” chords in a principled way?
i.e. are our handmade analytical corpora reliable???
Thanks to IQ and DH.
R F R R F R R

39. RN analysis is hard (1) What is the best (C major) analysis?
RN analysis is hard (1)
What is the best (C major) analysis?
“the ii-vii°6 idiom” PT
----- Meeting Notes (2/10/13 14:40) -----
Intuitions of LIKELIHOOD!
C: IV I6
C: ii vii°6 I6 PT PT
C: V V2 I6
C: I vii°6 I6

40. RN analysis is hard (1) This is how I have intuitively done it.
RN analysis is hard (1)
This is how I have intuitively done it.
“the ii-vii°6 idiom” PT
----- Meeting Notes (2/10/13 14:40) -----
Intuitions of LIKELIHOOD!
C: IV I6
C: ii vii°6 I6 PT PT
C: V V2 I6
C: I vii°6 I6

41. RN analysis is hard (1) Each analysis suppresses a (fake) ii-I!
RN analysis is hard (1)
Each analysis suppresses 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

42. RN analysis is hard (1) So what is the force of “ii-I is rare”?
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

43.
RN analysis is hard (1)
CLAIM: when looking at harmonic syntax, we should resolve ambiguities so as to produce the most plausible (or statistically likely) reading.
The problem is determining what this is.

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

45. RN analysis is hard (2) Even the pros make mistakes:
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 account for less than 2% of the destinations from ii.
Sweet Anticipation

46. Harmonic and nonharmonic tones
Harmonic and nonharmonic tones
Basic plan
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.

47. Justifying Analysis – stage 1
Justifying Analysis – stage 1
Correct key 81.1%, correct chord 90.5%
This music is largely unambiguous!

48. Justifying Analysis – stage 3
Justifying Analysis – 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 (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!

49. Justifying Analysis – stage 3
Justifying Analysis – 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
#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.

50. Justifying Analysis – stage 3
Justifying Analysis – stage 3
1: I – iii6 – V – vi
2: I – iii6 – iii6 – vi
3: I – V– V – vi
4: I – iii6 – iii# – vi 9

51. RN analysis is hard (reprise)
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

52. RN analysis is hard (reprise)
RN analysis is hard (reprise)
Ratio of my preferred analysis to the best alternative.
5.8:1
7.8:1 PT
C: IV ii6 I6
C: ii I6
43:1
10:1
C: V ii6 I6
C: I vii°6 ii6 I6

53. RN analysis is hard (reprise)
RN analysis is hard (reprise)
g: i V i V6/III III
7:1 (NB: no V# )
F: I V I vii°6 I6
4.3:1

54.
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 theories of harmonic syntax continue to be controversial?
Traditional music analysis looks unscientific, even though it isn’t.

55. 5. WHAT DOES IT MEAN FOR ANALYSIS?
5. WHAT DOES IT MEAN FOR ANALYSIS?
Don’t be afraid to draw on tonal concepts when looking at 16th-century music!
Palestrina, Tu Es Petrus (5v)

56. 5. A source for Schubert? http://dmitri.tymoczko.com
I V IV “ii6” “I6”

57. Thank you! http://dmitri.tymoczko.com For further info: