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Making the Old New:
non-Euclidean perspectives
on the [broadly] classical style
Dmitri Tymoczko
Princeton University
http://dmitri.tymoczko.com
A simplistic view of the classical
tradition
• 18th and early 19th-century music involves
“functional tonality,” which is well-understood
– Consists in ii-V-I norms as described by
Rameau/Riemann/Piston/McHose
• Late 19th-century music involves efficient
chromatic voice leading
– Alternative structures described by
Hauptmann/Riemann and recent music theory
• Two “systems,” distinguished chronologically.
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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 19th-century music extends these
contrapuntal procedures to the chromatic
domain.
• The two systems are already fully present in
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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.”
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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 wellunderstood.
– There are specific mathematical relations between harmony
and counterpoint that have been overlooked.
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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!
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Two “rules of the octave”
traditional (incl. partimenti,
etc.):
(note TDT
harmony)
fauxbourdon:
(note nonfunctional harmony with
I-V-IV in the first chords!)
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Fauxbourdon ROTO variants
Not: V/V
V
or V2!
Source of a large number of idioms!
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Two “rules of the octave”
70 Bach
chorales
Mozart
sonatas
0%
20%
100%
80%
+ variants
+ variants
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Two “rules of the octave”
70 Bach
chorales
Mozart
sonatas
0%
23%
100%
77%
+ variants
+ variants
http://dmitri.tymoczko.com
Two “rules of the octave”
70 Bach
chorales
Mozart
sonatas
38%
7%
62%
93%
+ variants
+ variants
http://dmitri.tymoczko.com
Two “rules of the octave”
70 Bach
chorales
Mozart
sonatas
0%
37%
100%
63%
+ variants
+ variants
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!
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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?
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Two “rules of the octave”
traditional (incl. partimenti,
etc.):
(note TDT harmony)
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A few examples
This fauxbourdon pattern is not just a curious or
decorative feature of keyboard music; it’s a
legitimate harmonic component of the style itself!
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The fauxbourdon rule of the octave
provides the main material of the
Quartett-Satz
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The fauxbourdon rule of the octave
provides the main material of the
Quartett-Satz
I
V6
IV6
[email protected]
“ii6”
“I6”
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Geometrically, the
fauxbourdon
ROTO moves
stepwise down
the lattice at the
center of diatonic
chord space
(GOM, chs. 3 and
7)
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The common
variants involve
efficient voiceleading-based
substitutions for the
stepwise
descending,
fauxbourdon
background
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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.
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It is NOT just
“generic voiceleading goo.”
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Chromatic sequences in the QuartettSatz
similar to the fauxbourdon passages!
RP = –1, 3, 7 or –1 mod 4 (GOM, §8.4)
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The gorgeous secondtheme extension
implicitly
cyclical
q q
7
Af
7
Ef
3 7 7
bf[Df]
Af Ef
Ef
bf
Af
Q1: what is the familiar root progression?
RP = –1, 3, 7 or –1 mod 4 (GOM,
§8.4)
Q2:
is something missing?
Q3: does this remind you of anything?
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The gorgeous secondtheme extension
implicitly
cyclical
q q
Functional tonality or chromaticism?
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Chromatic sequences
in the Quartett-Satz
RP
= –1, transfers
2, 5, 8 or in
–1Schubert
mod 4 (GOM,
Octave
often §8.5–6)
occur where the underlying voice
leading leads to changes in inversion.
They restore sequential structure.
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Summary
• The stepwise descending passages we’ve looked at
account for a substantial majority of the Quartett-Satz’s
exposition. A structural motive?
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In the development, the music
ascends!
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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 space
– Chromatic tetrachordal space
• Crucially, 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
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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
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Piece 2: A Bach Chorale
i v6 iv6 III6
possible: D:
ii°6 i6 vii°6 i
(no root!)
I6
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.
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Piece 2: A Bach Chorale
i v6 iv6 III6
ii°6 i6 vii°6 i
(no root!)
• 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!
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Piece 2: A Bach Chorale
i v6 iv6 III6
ii°6 i6 vii°6 i
(no root!)
• 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!
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The chorale’s second phrase
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
phrase
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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.
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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.
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Piece 3: Chopin A minor prelude
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Chopin A minor prelude
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.
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Chopin A minor prelude
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Chopin A minor prelude
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One continuous process!
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A minor prelude, “fauxbourdon” “background”
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A minor prelude, “fauxbourdon” “background”
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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!
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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 …
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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 17th-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.
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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.
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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 cryptofunctionality 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.
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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 higherlevel 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.”
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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!
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Schenkerian Themes
• Or the beautiful Ef neighboring passage in the
Quartett-Satz.
• Presumably, the compositional intention is to decorate
the Ef 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.)
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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
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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!
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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.
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for more information …
Thank you!

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