Transcript 1. Functional Harmony - Dmitri Tymoczko

http://dmitri.tymoczko.com
BIG DATA, LITTLE DATA,
MUSICAL DATA
Dmitri Tymoczko
Princeton University
[email protected]
http://dmitri.tymoczko.com
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
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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.
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MY CORPUS
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TODAY’S QUESTIONS
1. What particular practices characterize
classical harmony?
2. How did functionality develop?
3. What is the deep nature of harmonic
functionality?
4. What are “nonharmonic tones” and are
they merely decorative?
5. How can statistical information change
our understanding of specific pieces?
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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.
http://dmitri.tymoczko.com
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)
78% accurate
http://dmitri.tymoczko.com
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)
79% accurate
http://dmitri.tymoczko.com
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) 92% accurate
http://dmitri.tymoczko.com
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)
95-97% accurate
http://dmitri.tymoczko.com
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.
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1. Functional Harmony
• There are important and systematic
differences among tonal composers.
Secondary
triads or
secondary
dominants?
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1. Functional Harmony
• There are important and systematic
differences among tonal composers.
What chord
harmonizes
^2 in the
bass?
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1. Functional Harmony
• There are important and systematic
differences among tonal composers.
How does
the IV chord
resolve?
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1. Functional Harmony
• There are important and systematic
differences among tonal composers.
In the 16th-century ii, IV, and vi are all common precursors to V; ii emerges
Does
directly to V?
redominant
onlyviinprogress
the early 1700s.
vi–V(7)
Corelli
24%
Bach
14%
Mozart
0%
Beethoven
0%
Chopin
0%
Brahms
0%
The proportion of vi chords that proceed directly to V or V7 (nonsequential,
ting contexts only). The vi–V progression is still relatively common in Cor
mpletely disappears in the music of Mozart, Beethoven, Chopin, and Brahms
y no uncontroversial examples in the corpus. In this music, vi has become
nt, progressing to V only by way of IV or ii.
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1. Functional Harmony
CONCLUSION:
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1. Functional Harmony
Warning: what follows
might be upsetting …
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A lot of the information in
textbooks is wrong or
incomplete.
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2. Development of Tonality
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2. Development of Tonality
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2. Development of Tonality
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2. Development of Tonality
• The harmonic cycle is the default scheme:
I  [vi  (IV or ii)]  vii° or V  I
• The harmonic cycle gets built backward along the
circle of fifths from dominant to predominant to prepredominant.
– First, the dominant increasingly goes to I.
– Then ii chord emerges in the early (or mid-) 17th-century.
vii° or V  I
ii)]  vii° or V  I
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2. Development of Tonality
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2. Development of Tonality
• The harmonic cycle is the default scheme:
I  [vi  (IV or ii)]  vii° or V  I
• The harmonic cycle gets built backward along the
circle of fifths from dominant to predominant to prepredominant.
– 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.
vii° or V  I
ii)]  vii° or V  I
I  [vi  (IV or ii)]  vii° or V  I
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2. Development of Tonality
• The harmonic cycle is the default scheme:
I  [vi  (IV or ii)]  vii° or V  I
• 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.
In the 16th-century
and
vi aregoes
all common
to V; ii emerges as t
• Priorii,toIV,
that,
it often
directly toprecursors
V
predominant
only primary
in the early
– The
role1700s.
of the iii chord is not to exist!
vi–V
(7)
Corelli
24%
Bach
14%
Mozart
0%
Beethoven
0%
Chopin
0%
Brahms
0%
The proportion of vi chords that proceed directly to V or V7 (nonsequential,
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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?
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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)
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THE TENDENCY HISTOGRAM
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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?
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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.
RF RRFR R
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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?
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The ineliminable suspension
*
Analytical reduction fails to produce triads!
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Suspensions license parallel
fifths
A-G
D-C
Happens
once every
~344
chords
(about
once per
mass
movement)
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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])
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A paradox
• Eliminating nonharmonic tones is central
to music-theoretical understanding.
• This works only some of the time.
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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.
RF RRFR R
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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!
RF RRFR R
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The Fundamental Challenge
• How do we separate “real” chords from
“fake” chords in a principled way?
• i.e. are our handmade analytical corpora
reliable???
RF RRFR R
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RN analysis is hard (1)
• What is the best (C major) analysis?
“the ii-vii°6 idiom”
PT
I6
C: IV
C: ii vii°6 I6
PT
PT
C: V
V2
I6
C: I
vii°6
I6
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RN analysis is hard (1)
• This is how I have intuitively done it.
“the ii-vii°6 idiom”
PT
I6
C: IV
C: ii vii°6 I6
PT
PT
C: V
V2
I6
C: I
vii°6
I6
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RN analysis is hard (1)
• Each analysis suppresses a (fake) ii-I!
“the ii-vii°6 idiom”
PT
C: IV
C: V
ii6
I6
C: ii
I6
ii6
I6
C: I
vii°6 ii6 I6
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RN analysis is hard (1)
• So what is the force of “ii-I is rare”?
“the ii-vii°6 idiom”
PT
C: IV
C: V
ii6
I6
C: ii
I6
ii6
I6
C: I
vii°6 ii6 I6
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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.
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RN analysis is hard (1)
V6
g: i
i V6–5/III III
PT
F: I
V6
I
vii°6 I6
M2 b3 interpreted
differently!!!
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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.
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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.
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Justifying Analysis – stage 1
• Correct key 81.1%, correct chord 90.5%
• This music is largely unambiguous!
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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!
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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
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.
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Justifying Analysis – 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
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RN analysis is hard (reprise)
• What is the best (C major) analysis?
5.8:1
7.8:1
PT
I6
C: IV
43:1
C: ii vii°6 I6
10:1
PT
PT
C: V
V2
I6
C: I
vii°6
I6
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RN analysis is hard (reprise)
• Ratio of my preferred analysis to the best alternative.
5.8:1
7.8:1
PT
C: IV
C: V
ii6
43:1
I6
ii6
I6
I6
C: ii
10:1
C: I
vii°6 ii6 I6
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RN analysis is hard (reprise)
V6
g: i
F: I
V6
i V6/III III
I
vii°6 I6
7:1 (NB: no V# )
4.3:1
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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.
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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)
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5. A source for Schubert?
I
V6
IV6
“I@”
“ii6”
“I6”
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Thank you!
For further info:
http://web.mit.edu/music21/
http://mtosmt.org/issues/mto.13.19.3/mto.13.19.3.tymoczko.html