httpdmitritymoczkocom While were waiting please provide a Roman numeral analysis of the following Making the Old New nonEuclidean perspectives on the broadly classical style Dmitri Tymoczko ID: 531179
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Slide1
Pop Quiz
http://dmitri.tymoczko.com
While we’re waiting, please provide a Roman numeral analysis of the following:Slide2
Making the Old New:
non-Euclidean perspectives on the [broadly] classical style
Dmitri TymoczkoPrinceton University
http://dmitri.tymoczko.comSlide3
A simplistic view of the classical tradition
18th and early 19
th-century music involves “functional tonality,” which is well-understoodConsists in ii-V-I norms as described by Rameau/Riemann/Piston/McHose
Late 19th-century music involves efficient chromatic voice leadingAlternative structures described by Hauptmann/Riemann and recent music
theory
Two “systems,” distinguished chronologically.
http://dmitri.tymoczko.comSlide4
Another (simplistic?) view
Elementary diatonic
harmony involves relatively few genuinely harmonic (ii-V-I) progressions.And a much larger number of embellishing/prolongational
/contrapuntal progressions, which come in an almost uncatalogueable
variety.
Late
19
th
-century music
extends these contrapuntal procedures to the chromatic domain.
T
he two systems are already fully present in elementary diatonic music.
http://dmitri.tymoczko.comSlide5
Unsettling Our Complacency
Standard T-S-D-T tonal functionality is not as well understood as we think (
A Geometry of Music, ch. 7).There’s no consensus on what the “TSDT” harmonic norms actually are.
The very same ideas that help us understand “chromatic voice leading” are also useful for understanding classical tonality.Recent accounts of functional harmony exploit and depend on features of diatonic voice-leading space, closely analogous to the properties underwriting chromaticism
, e.g. the circle of thirds.
Also: fantasias, development sections, etc. There’s no moment at which the classical tradition became “chromatic.
”
http://dmitri.tymoczko.comSlide6
Unsettling Our Complacency
There are deep and as-yet unanswered methodological questions surrounding basic Schenkerian
concepts like prolongation, or the claim that “counterpoint produces harmony.”In particular, though there are nonfunctional contrapuntal passages in diatonic classical music, they are not ubiquitous, but rather confined to a small number of idioms.
Also, voice-leading itself isn’t perfectly well-understood.There are specific mathematical relations between harmony and counterpoint that have been overlooked.
http://dmitri.tymoczko.comSlide7
Unsettling Our Complacency
Neither of our two pictures is fully sustainable when you look at the details.Functional
harmony is not a unified phenomenon, but rather contains of a variety of oft-neglected subsystems including fauxbourdon, sequences, and other specific idioms.Today I’m going to illustrate this by considering very familiar
pieces:focusing mainly on Schubert’s Quartett-Satz
but also taking a look at a Bach chorale, a Mozart sonata movement, and a Chopin prelude
Not radical or avant-garde works!
http://dmitri.tymoczko.comSlide8
Two
“rules of the octave”
traditional (incl. partimenti, etc.):
fauxbourdon:
http://dmitri.tymoczko.com
(note TDT harmony)
(note nonfunctional harmony with
I-V-IV in the first chords!)Slide9
Fauxbourdon ROTO variants
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Source of a large number of idioms!
Not:
V/V V or V
2
!Slide10
Two
“rules of the octave”
http://dmitri.tymoczko.com
70 Bach Mozart
chorales sonatas
20%
80%
0%
100%
+ variants
+ variantsSlide11
http://dmitri.tymoczko.com
23%
77%
0%
100%
+ variants
Two
“
rules of the octave
”
70 Bach Mozart
chorales sonatas
+ variantsSlide12
http://dmitri.tymoczko.com
7%
93%
38%
62%
+ variants
Two
“
rules of the octave
”
70 Bach Mozart
chorales sonatas
+ variantsSlide13
http://dmitri.tymoczko.com
37%
63%
0%
100%
+ variants
Two
“
rules of the octave
”
70 Bach Mozart
chorales sonatas
+ variantsSlide14
http://dmitri.tymoczko.com
Two
“
rules of the octave”
Note: no analogous difficulties with the ascending “rule of the octave.” There, the
pedagogical formula corresponds to actual compositional practice.
Tonal harmony uses functional harmony for stepwise ascending bass lines; while stepwise descending bass lines are often nonfunctional!Slide15
http://dmitri.tymoczko.com
Two
“
rules of the octave”
Note: no analogous difficulties with the ascending “rule of the octave.” There, the pedagogical formula corresponds to actual compositional practice.
Traditional pedagogues ignored the more popular, nonfunctional ROTO, in favor of the less popular
version. Why?Slide16
Two
“rules of the octave”
traditional (incl. partimenti, etc.):
http://dmitri.tymoczko.com
(note TDT harmony)Slide17
A few examples
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This fauxbourdon pattern is not just a curious or decorative feature of keyboard music; it’s a legitimate harmonic component of the style itself!Slide18
The fauxbourdon rule of the octave provides the main material of the
Quartett-Satz
http://dmitri.tymoczko.comSlide19
The fauxbourdon rule of the octave provides the main material of the
Quartett-Satz
http://dmitri.tymoczko.com
I V
6
IV
6
“I
@
” “ii
6
” “I
6
”Slide20
Geometrically, the fauxbourdon ROTO moves stepwise down the lattice at the center of diatonic chord space
(GOM, chs. 3 and 7)
http://dmitri.tymoczko.comSlide21
The common variants involve efficient voice-leading-based substitutions for the stepwise descending, fauxbourdon background
http://dmitri.tymoczko.comSlide22
This is part of the
“first practice”of functional harmony but it is not clearly
T-S-D-T tonal functionality. It is an important idiom or second subsystem.
http://dmitri.tymoczko.comSlide23
It is NOT just “generic voice-leading goo.”
http://dmitri.tymoczko.comSlide24
Chromatic sequences in the
Quartett-Satz
http://dmitri.tymoczko.com
RP = –1, 3, 7 or –1 mod 4 (GOM, §8.4)
similar to the fauxbourdon passages!Slide25
The gorgeous second-theme extension
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Q1: what is the familiar root progression?
RP = –1, 3, 7 or –1 mod 4 (GOM, §8.4)
A
f
E
f
b
f
A
f
E
f
q
q
A
f
E
f
b
f
[D
f
] A
f
E
f
Q2: is something missing?
Q3: does this remind you of anything?
7 7 3 7 7
implicitly
cyclicalSlide26
The gorgeous second-theme extension
http://dmitri.tymoczko.com
Functional tonality or chromaticism?
implicitly
cyclical
q
qSlide27
Chromatic sequences in the
Quartett-Satz
http://dmitri.tymoczko.com
RP = –1, 2, 5, 8 or –1 mod 4 (GOM, §8.5–6)
Octave transfers in Schubert often occur where the underlying voice leading leads to changes in inversion. They restore sequential structure.Slide28
Summary
The stepwise descending passages we’ve looked at account for a substantial majority of the Quartett-Satz’
s exposition. A structural motive?
http://dmitri.tymoczko.comSlide29
In the development, the music ascends!
http://dmitri.tymoczko.comSlide30
Summary
We’ve encountered a number of passages that exemplify the same basic procedure, descending stepwise voice leading between familiar harmonic objects, in three different musico-geometrical environments:Diatonic triadic space
Chromatic triadic spaceChromatic tetrachordal spaceCrucially, the examples straddle the line between familiar, diatonic “
first-practice” routines, clearly part of the shared tonal inheritance, and more radical, chromatic “
second practice
”
styles.
Having recognized the structural similarities between these passages, we are now better positioned to understand both the connections and discontinuities between these two tonal languages.
http://dmitri.tymoczko.comSlide31
Next up …
I’ll now turn to two other passages that we can understand by way of Schubert’s.The opening of Bach’s Chorale #16
Chopin’s A minor prelude
http://dmitri.tymoczko.comSlide32
Piece 2: A Bach Chorale
http://dmitri.tymoczko.com
i
v
6
iv
6
III
6
ii°
6
i
6
vii°
6
i
(no root!)
possible:
D: I
6
ii vii°
6
I
NB: if there were a D major chord in m. 2, Bach could have raised the alto to A4. (See Chorale 1, m. 8 or Chorale 26, m. 1). It’s somewhat harder to get the alto to B3, however.Slide33
Piece 2: A Bach Chorale
http://dmitri.tymoczko.com
• in general, we prefer unaccented passing tones to accented passing tones
• in general we prefer missing fifths to missing roots
• by overriding both defaults we find the standard fauxbourdon ROTO progression!
i
v
6
iv
6
III
6
ii°
6
i
6
vii°
6
i
(no root!)Slide34
Piece 2: A Bach Chorale
http://dmitri.tymoczko.com
• I find it very plausible to think that Bach noticed that the tune could be harmonized by a descending fauxbourdon sequence, and was amused by this fact.
• It seems like a (pretty good) inside joke!
i
v
6
iv
6
III
6
ii°
6
i
6
vii°
6
i
(no root!)Slide35
The chorale’s second phrase
http://dmitri.tymoczko.com
every bass note is part of an ascending group, and every harmony is part of an ascending root
progression …
elegantly balancing the persistent descent of the first phraseSlide36
Morals
RN analysis is a process of reducing complex musical surfaces to a small set of familiar patterns or schemas.
While we can articulate preference rules for this reduction (e.g. “prefer a missing fifth to a missing root”), these are at best heuristics that are extremely hard to make algorithmic.RN analysis requires training, extensive musical experience, and a hefty does of good musical judgment.The fauxbourdon ROTO is a great example of an important idiom that needs to be internalized.
We have a good theoretical account of why this is so hard.(General) Enjoyment doesn’t require (detailed) comprehension (GOM, §1.4).(Technical) Classical dissonance treatment is largely preserved from the modal tradition, in which there is no need to determine whether the underlying sequence of harmonies conforms to a grammar or not.
http://dmitri.tymoczko.comSlide37
Morals
We can probably learn to algorithmatize RN analysis, but only after we have a substantial body of human-constructed analyses.
It’s probably a fairly complex endeavor, analogous to computational natural-language processing.We need to bootstrap, beginning with data whose reliability itself may be open to question.
http://dmitri.tymoczko.comSlide38
Piece 3: Chopin A minor prelude
http://dmitri.tymoczko.comSlide39
Chopin A minor prelude
http://dmitri.tymoczko.com
B2 is the only note missing in the descending pattern. It’s in the melody as B3 instead. (The melody gives D4-B3 instead of D3-B2 in the accompaniment.
) The figuration changes at this point, making it unclear what the third
accompanimental
voice is.Slide40
Chopin A minor prelude
http://dmitri.tymoczko.comSlide41
Chopin A minor prelude
http://dmitri.tymoczko.comSlide42
One
continuous process!
http://dmitri.tymoczko.comSlide43
A minor prelude, “fauxbourdon” “background”
http://dmitri.tymoczko.comSlide44
A minor prelude, “fauxbourdon” “background”
http://dmitri.tymoczko.comSlide45
Bach, Schubert, Chopin
Bach and Schubert both use the same particular idiom, the descending fauxbourdon ROTO pattern.You might not immediately see it, particularly if you associate “functional harmony” with TSDT progressions.
Once you’re sensitized it jumps right out at you.Schubert and Chopin make more general and flexible use of stepwise descending voice leading, which appears in both diatonic and chromatic versions, with both triads and seventh chords.
In these pieces, we see very tight connection between root progressions and voice leading, driven by the underlying scalar geometry. (Different progressions for sevenths and triads.)One can construct a “fauxbourdon” background for the Chopin!
http://dmitri.tymoczko.comSlide46
Schenkerian Themes
A key Schenkerian idea is that “voice leading produces harmony.” That is, apparently “harmonic” progressions are being created by subterranean and more fundamental voice-leading forces.
But Schenker lacked a principled understanding of how particular kinds of voice leading (e.g. stepwise descending voice leading) produces different kinds of harmonic results.We’ve seen, for example, that this depends on the background scale that you’re using.
We can actually produce mathematical formulas relating the two!There’s sometimes a bit of hand-waving in Schenkerian analysis …
What I’m doing is a kind of local or non-prolongational Schenkerianism, made precise by voice-leading geometry.
http://dmitri.tymoczko.comSlide47
Schenkerian Themes
In a way that is somewhat reminiscent of Schenker, I emphasize that descending stepwise bases are often harmonized
nonfunctionally, via the fauxbourdon ROTO.The textbook ROTO (including 17
th-century pedagogy) shoehorns this practice into TSDT functionality.Unlike Schenker
, I see this as a particularly important
specific idiom
.
Schenker
saw it as just one manifestation of a very general tendency (voice leading) which could appear in many different forms.
I agree that voice leading can appear in many different forms, but think that “basic tonal harmony” consists of a very small number of idiomatic moves, along with
a purely harmonic syntax.
http://dmitri.tymoczko.comSlide48
Schenkerian Themes
Schenkerians sometimes suggest that the existence of idioms poses problems for traditional harmonic theorists.
E.g. Salzer on the “Pachelbel progression” in Bach’s WTC. (Itself part of the fauxbourdon ROTO package!)Sophisticated versions of traditional harmonic theory acknowledge that tonal harmony is not unified.
Beyond the standard harmonic norms, there are sequences and idioms such as the fauxbourdon ROTO.It is a nontrivial fact that most of the exceptional progressions in the tonal literature (i.e. those seeming to violate the basic harmonic grammar) belong to a small number of idioms. These exceptions really do prove the rule!
Schenkerians
falsely drew a general conclusion (
“little or no
harmonic grammar
”)
from the existence of a few exceptions.
http://dmitri.tymoczko.comSlide49
Schenkerian Themes
Though he detested Riemann and Rameau,
Schenker remained fundamentally rooted in the TSDT paradigm.The only acceptable “backgrounds” are interpretable as I-V-I progressions.
Committed to the unity of the classical harmonic language.Or rather: I-V-I patterns plus
a LOT of “
voice leading
goo.
”
My reading of Chopin’s A minor prelude offers a “fauxbourdon background.”
If there’s any “deep structure” here, it’s not the crypto-functionality of the
Ursatz
, but rather fauxbourdon-
esque
parallelism.
Why not allow this as a kind of tonal background?
If you’re not committed to the unity of the language, it makes perfect sense.
http://dmitri.tymoczko.comSlide50
Schenkerian Themes
Schenker felt that there was a conflict between his “voice-leading-first” views and traditional harmonic theory
There’s no conflict. The Roman-numeral principles provide grammatical constraints
within which higher-level compositional intentions operate.Intuitively: Chopin might’ve said something like “oh, I’ll base my piece on descending voice leading, but I will continue to obey the harmonic norms.”
Compare: “I’ll
dribble toward the basket while obeying the rules of
basketball.” Or: “I’ll write a poem in which E is the only vowel, but I’ll continue to obey the norms of English grammar.” Or: “I’ll control the center of the board by moving my knight to Q3.”
The principles operate at completely different psychological levels.
http://dmitri.tymoczko.comSlide51
http://dmitri.tymoczko.comSlide52
Schenkerian Themes
Consider this response with respect to the opening of the Chopin prelude:
The art here lies precisely in the fact that the descending stepwise voice leading
also
forms syntactical (but slightly weird) harmonic progressions!
http://dmitri.tymoczko.comSlide53
Schenkerian Themes
Or the beautiful Ef
neighboring passage in the Quartett-Satz.
Presumably, the compositional intention is to decorate the
E
f
major triad with a series of melodic neighboring motions.
This intention is carried out within a harmonic grammar that licenses only a small number of familiar moves (common-tone diminished sevenths, IV–I and vii°–I progressions, etc.)
http://dmitri.tymoczko.comSlide54
Final word about method
In this talk, my goal has been to use our new knowledge about how voice-leading works—our understanding of the underlying contrapuntal geometry—in order to do some detailed analytical work on very canonical pieces, and in order to think more generally about the style.
You might think there’d be nothing left to say about these topics, but
they actually seem like unexplored territory to me!One fundamental set of tools are lattices describing voice-leading possibilities for chords of any size in any scale.
These are the subject of §3.11 of my book.
If you can internalize the discrete lattices described there, you have a powerful set of tools for understanding the inner workings of a wide range of musical
procedures.
http://dmitri.tymoczko.comSlide55
Final word about method
Note that the point of the geometry is not simply to allow us to draw pretty pictures of music.Instead, it is to sensitize us to a range of common voice-leading possibilities, and to allow us to see that those
are the only possibilities.To teach us the
inherent logic of voice leading.Having
understood this
, we can “throw away the ladder” if we want. You just need to know what the patterns are, and be able to find them in pieces.
Merely making a picture doesn’t necessarily accomplish anything analytically useful!
http://dmitri.tymoczko.comSlide56
Final word about method
Of course, you also need to study a lot of music carefully, so that you identify
basic patterns and idioms!Here analytical corpora are very
useful tools.In working on this talk, I began with the various voice-leading patterns in the Schubert piece.At a certain point I searched through my databases of RN analyses to look for the fauxbourdon ROTO progression.
This turned up both the chorale and the Mozart movement.
Looking again at the chorale I came to the analysis proposed in the talk (the analysis in my corpus was wrong!)
Someday extensive analytical corpora will be available to everyone, either because they’re published or because computers will be able to generate them on the fly.
http://dmitri.tymoczko.comSlide57
Thank you!
http://dmitri.tymoczko.com
for more information …