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The Music of the Spherical Quotients

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http://dmitri.tymoczko.comWhile were waiting, please provide a Roman numeral analysis of the following:1Making the Old New: non-Euclidean perspectives on the [broadly] classical styleDmitri TymoczkoPrinceton University

http://dmitri.tymoczko.com2A simplistic view of the classical tradition18th and early 19th-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 theoryTwo systems, distinguished chronologically.

http://dmitri.tymoczko.com3Another (simplistic?) viewElementary 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 elementary diatonic music.

http://dmitri.tymoczko.com4Unsettling Our ComplacencyStandard T-S-D-T tonal functionality is not as well understood as we think (A Geometry of Music, ch. 7).Theres 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. Theres no moment at which the classical tradition became chromatic.

http://dmitri.tymoczko.com5Unsettling Our ComplacencyThere 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 isnt perfectly well-understood.There are specific mathematical relations between harmony and counterpoint that have been overlooked.

http://dmitri.tymoczko.com6Unsettling Our ComplacencyNeither 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 Im going to illustrate this by considering very familiar pieces:focusing mainly on Schuberts Quartett-Satzbut also taking a look at a Bach chorale, a Mozart sonata movement, and a Chopin preludeNot radical or avant-garde works!

http://dmitri.tymoczko.com7Two rules of the octave

traditional (incl. partimenti, etc.):fauxbourdon:http://dmitri.tymoczko.com(note TDT harmony)(note nonfunctional harmony withI-V-IV in the first chords!)

8Fauxbourdon ROTO variants

http://dmitri.tymoczko.comSource of a large number of idioms!

Not: V/V V or V2!9Two rules of the octave

http://dmitri.tymoczko.com70 Bach Mozartchorales sonatas20%

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+ variantsTwo rules of the octave70 Bach Mozartchorales sonatas+ variants11


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http://dmitri.tymoczko.comTwo rules of the octaveNote: 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!14

http://dmitri.tymoczko.comTwo rules of the octaveNote: 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?15Two rules of the octave

traditional (incl. partimenti, etc.):http://dmitri.tymoczko.com(note TDT harmony)16A few examples

http://dmitri.tymoczko.com

This fauxbourdon pattern is not just a curious or decorative feature of keyboard music; its a legitimate harmonic component of the style itself!17The fauxbourdon rule of the octave provides the main material of the Quartett-Satz


18The fauxbourdon rule of the octave provides the main material of the Quartett-Satz


I V6 IV6 I@ ii6 I619Geometrically, the fauxbourdon ROTO moves stepwise down the lattice at the center of diatonic chord space (GOM, chs. 3 and 7)


20The common variants involve efficient voice-leading-based substitutions for the stepwise descending, fauxbourdon background


21This is part of the first practiceof functional harmony but it is not clearly T-S-D-T tonal functionality. It is an important idiom or second subsystem.


22It is NOT just generic voice-leading goo.


23Chromatic sequences in the Quartett-Satz


RP = 1, 3, 7 or 1 mod 4 (GOM, 8.4)

similar to the fauxbourdon passages!24The gorgeous second-theme extension


Q1: what is the familiar root progression? RP = 1, 3, 7 or 1 mod 4 (GOM, 8.4)

Af Ef bf Af EfqqAf Ef bf[Df] Af EfQ2: is something missing?Q3: does this remind you of anything?7 7 3 7 7implicitlycyclical

25The gorgeous second-theme extension


Functional tonality or chromaticism?implicitlycyclical

qq

26Chromatic sequences in the Quartett-Satz


RP = 1, 2, 5, 8 or 1 mod 4 (GOM, 8.56)

Octave transfers in Schubert often occur where the underlying voice leading leads to changes in inversion. They restore sequential structure.

27SummaryThe stepwise descending passages weve looked at account for a substantial majority of the Quartett-Satzs exposition. A structural motive?


28In the development, the music ascends!


29SummaryWeve 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 spaceChromatic 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.com30Next up Ill now turn to two other passages that we can understand by way of Schuberts.The opening of Bachs Chorale #16Chopins A minor prelude

http://dmitri.tymoczko.com31Piece 2: A Bach Chorale

http://dmitri.tymoczko.comi v6 iv6 III6 ii6 i6 vii6 i (no root!)possible: D: I6 ii vii6 INB: 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). Its somewhat harder to get the alto to B3, however.32Piece 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 v6 iv6 III6 ii6 i6 vii6 i (no root!)33Piece 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 v6 iv6 III6 ii6 i6 vii6 i (no root!)34The chorales 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

35MoralsRN 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 doesnt 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.com36MoralsWe can probably learn to algorithmatize RN analysis, but only after we have a substantial body of human-constructed analyses.Its 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.com37Piece 3: Chopin A minor prelude


38Chopin A minor prelude


B2 is the only note missing in the descending pattern. Its 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.39Chopin A minor prelude


40Chopin A minor prelude


41One continuous process!


42A minor prelude, fauxbourdon background


43A minor prelude, fauxbourdon background


44Bach, Schubert, ChopinBach 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 youre 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.com45Schenkerian ThemesA 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.Weve seen, for example, that this depends on the background scale that youre using.We can actually produce mathematical formulas relating the two!Theres sometimes a bit of hand-waving in Schenkerian analysis What Im doing is a kind of local or non-prolongational Schenkerianism, made precise by voice-leading geometry.

http://dmitri.tymoczko.com46Schenkerian ThemesIn 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.

http://dmitri.tymoczko.com47Schenkerian ThemesSchenkerians sometimes suggest that the existence of idioms poses problems for traditional harmonic theorists.E.g. Salzer on the Pachelbel progression in Bachs 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.com48Schenkerian ThemesThough 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 Chopins A minor prelude offers a fauxbourdon background. If theres any deep structure here, its not the crypto-functionality of the Ursatz, but rather fauxbourdon-esque parallelism.Why not allow this as a kind of tonal background?If youre not committed to the unity of the language, it makes perfect sense.

http://dmitri.tymoczko.com49Schenkerian ThemesSchenker felt that there was a conflict between his voice-leading-first views and traditional harmonic theoryTheres no conflict. The Roman-numeral principles provide grammatical constraints within which higher-level compositional intentions operate.Intuitively: Chopin mightve said something like oh, Ill base my piece on descending voice leading, but I will continue to obey the harmonic norms.Compare: Ill dribble toward the basket while obeying the rules of basketball. Or: Ill write a poem in which E is the only vowel, but Ill continue to obey the norms of English grammar. Or: Ill control the center of the board by moving my knight to Q3. The principles operate at completely different psychological levels.

http://dmitri.tymoczko.com50


51Schenkerian ThemesConsider 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!


52Schenkerian ThemesOr 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, IVI and viiI progressions, etc.)


53Final word about methodIn this talk, my goal has been to use our new knowledge about how voice-leading worksour understanding of the underlying contrapuntal geometryin order to do some detailed analytical work on very canonical pieces, and in order to think more generally about the style.You might think thered 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.com54Final word about methodNote 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 doesnt necessarily accomplish anything analytically useful!

http://dmitri.tymoczko.com55Final word about methodOf 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 theyre published or because computers will be able to generate them on the fly.

http://dmitri.tymoczko.com56Thank you!


for more information 57

Chopin: Prelude #2 In A Minor, Op. 28/2Vladimir AshkenazyChopin: The Piano Works [Disc 1]1997iTunes 10.4.1, QuickTime 7.6.6

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