POP QUIZ: QUESTION 1 - PowerPoint Presentation

Slide1

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 V6–I, V7–I, V–I6 (etc.) all count as “V–I progressions.”  However, 6/4 chords are their own category.

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Today's first quiz

question is:

who

is the composer?  

Don’t dither, just answer off the top of your head.Slide2

POP QUIZ: QUESTION 1I

In the Bach chorales, the most common destination for V2 is I6. What is the second most common destination?

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Tonal Functionalities(within-key harmonic patterns)

Dmitri TymoczkoPrinceton University

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Answers to the Quiz

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Josquin

PalestrinaSlide5

Syntax

Classical music has at least two features that might deserve the name “syntactical.”Harmonic principles governing chord-to-chord successionsRameau, Riemann, Piston, McHose, Kostka & PayneLargely an American enterprise, at least recently

Contrapuntal syntax governing “harmonic” and “nonharmonic” tonesNeighbor tones, passing tones, suspensions, etc.Schenker

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Exploring these Syntaxes

Constructing large databases, with two parts:Roman Numeral analyses of musicThe score as an XML or KRN fileAnalysis 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?

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Examples

http://dmitri.tymoczko.comAnalysis file

XML

Labeled choraleNonharmonic tones removedSome simple calculations

all V

2–I progressions in the choralesSlide8

Example

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The Corpus: a snapshot

http://dmitri.tymoczko.comTo do: frottola

, Chopin, Brahms, ?

107,902 chordsSlide10

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.

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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

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Topic I

: AccuracyFour grammars

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Topic

I: Accuracy

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majorSlide14

Topic

I: Accuracy

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minorSlide15

Topic

I: Accuracy

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bothSlide16

Three Grades of Modal Involvement

http://

dmitri.tymoczko.comTrue modality (~65%)Slide17

Three Grades of Modal Involvement

http://

dmitri.tymoczko.com“Tonally tinged modality” (~75%)Slide18

Three Grades of Modal Involvement

http://

dmitri.tymoczko.com“Modally tinged tonality” (~85%)Slide19

Three Grades of Modal Involvement

http://

dmitri.tymoczko.com“True tonality” (> 90%)Slide20

Topic

II: Development

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Topic

II: Development

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Topic II

: DevelopmentThe rule of the octave

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( )Slide23

Topic II

: DevelopmentThe rule of the octave: degrees 1, 5, 7.Strong preference evident in Josquin.Grows even stronger over the 16

th century.

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5

3

5

363

( )Slide24

Topic II

: DevelopmentThe rule of the octave: degree 3.Mild preference for I6 in Palestrina, grows strong in Morley, and even stronger after that.

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5

3

5

3

6363

( )

Amazing fact!Slide25

Topic II

: DevelopmentGiven these basic preferences, many tonal routines appear naturally.3 up to 5, 5 up to 8, 8 down to 5.

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5

3

53

63

63

( )Slide26

Topic II

: DevelopmentOthers still remain.1 up to 3, 3 down to 1, and 5 down to 3 are seventeenth-century developments.

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5

3

53

63

63

( )Slide27

Topic II

: DevelopmentWhat about the idea that 4–1 should be harmonized with root position triads, rather than as ii6–I?In Josquin and Palestrina, fourth and fifth motion in the bass almost

always involves root position.

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Mutiple Tonalities

“Riemannian Tonality”I, IV, and V are primary triads, with I-IV-I counterbalancing I-V-IDeveloped firstPresent in Palestrina (and Encina, the Carter Family, Ramones, etc.)

“Schenkerian Tonality”I and V are primary triadsii displaces IV, and serves primarily as a connector between I and V.

Classical tonality.Jazz?

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Topic

II: Development

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Topic

II: Development

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Topic

II: Development

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Topic III

: Nonharmonic tonesDo nonharmonic tones “embellish” an underlying, structural “harmonic skeleton”?First answer (16th 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.

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Topic III

: Nonharmonic tones

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Topic III

: Nonharmonic tones

http://dmitri.tymoczko.comA-G

D-CSlide35

Topic III

: Nonharmonic tonesSecond 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:

IV6–IV

2–I

Contrapuntal schemas

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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-vi6.

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Topic IV:

JustificationThe harmonic syntax applies only to “real” chords.How can we separate “truly harmonic” chords from “merely contrapuntal” chords in a principled way?

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R F R R F R RSlide38

Topic IV: Justification

What is the best (C major) analysis?

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C: IV I6

PT

C: ii vii°

6 I6

PT

C: I vii°

6

I

6

“the ii-vii°

6

idiom”

C: V V

2

I

6

PTSlide39

Topic IV:

JustificationNote that all analyses suppress a (fake) ii-I!

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C: IV ii6 I6

C: ii I

6

C: I vii°6 ii6

I

6

“the ii-vii°

6

idiom”

C: V ii

6

I

6

PTSlide40

Topic IV:

JustificationSo what is the force of “ii-I is rare”?

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C: IV ii6 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

PTSlide41

Topic IV:

Justificationhttp://dmitri.tymoczko.com

g: i V

6 i V6–5/III III

F: I V

6 I vii°6 I6

PT

M2 b3 interpreteddifferently!!!Slide42

Topic IV:

JustificationEven 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.

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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.

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– Stage 1b. Improve key finding with various

rules (e.g. dorian scale regions). Slide44

RN analysis as generalization

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Correct key 81.1%, correct chord 90.5%

This music is largely unambiguous!Slide45

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)

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I V

6 vii° i

V i i vii°6Slide46

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!

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RN analysis as generalization

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1: I – IV

6 – vi – V62: I – IV6 – IV6 – V63: I –

vi – vi

– V64: I – IV6 – IVmaj# – V

6

0*650#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.Slide48

RN analysis as generalization

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1: I – iii

6 – V – vi2: I – iii6 – iii6 – vi3:

I – V– V –

vi4: I – iii6 – iii# –

vi

0090Slide49

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.

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What is the best (C major) analysis?

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C: IV I6

PT

C: ii vii°

6 I6

PT

C: I vii°6 I

6

C: V V

2

I

6

5.8:1

7.8:1

10:1

43:1

PT

RN analysis as generalizationSlide51

Ratio of my preferred analysis to the

best alternative.http://dmitri.tymoczko.com

C: IV ii6

I6

C: ii I

6

C: I vii°

6 ii6 I6

C: V ii

6

I

6

PT

RN analysis as generalization

5.8:1

7.8:1

10:1

43:1Slide52

RN analysis as generalization

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g: i V6 i

V6/III IIIF: I V

6 I vii°6

I67:1 (NB: no V

# )

4.3:1Slide53

Conclusions

Combining analyses and scores gives us a lot of powerChords really do follow a local grammarTonality evolved very slowly, in many different genres simultaneously16th-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 askMaybe you can help?

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Thank you!

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for more

stuff like this …Slide55

Thank you!

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for more

stuff like this …