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2 Pop Quiz While we’re waiting, please provide a Roman numeral analysis of the following:

3 Making the Old New: non-Euclidean perspectives on the [broadly] classical style Dmitri Tymoczko Princeton University

4 A simplistic view of the classical tradition 18 th and early 19 th -century music involves “functional tonality,” which is well-understood –Consists in ii-V-I norms as described by Rameau/Riemann/Piston/McHose Late 19 th -century music involves efficient chromatic voice leading –Alternative structures described by Hauptmann/Riemann and recent music theory Two “systems,” distinguished chronologically.

5 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. The two systems are already fully present in elementary diatonic music.

6 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.”

7 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.

8 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!

9 Two “rules of the octave” traditional (incl. partimenti, etc.): fauxbourdon: (note TDT harmony) (note nonfunctional harmony with I-V-IV in the first chords!)

10 Fauxbourdon ROTO variants Source of a large number of idioms! Not: V/V V or V 2 !

11 Two “rules of the octave” 70 Bach Mozart chorales sonatas 20% 80% 0% 100% + variants

12 23% 77% 0% 100% + variants Two “rules of the octave” 70 Bach Mozart chorales sonatas + variants

13 7% 93% 38% 62% + variants Two “rules of the octave” 70 Bach Mozart chorales sonatas + variants

14 37% 63% 0% 100% + variants Two “rules of the octave” 70 Bach Mozart chorales sonatas + variants

15 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!

16 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?

17 Two “rules of the octave” traditional (incl. partimenti, etc.): (note TDT harmony)

18 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!

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

20 The fauxbourdon rule of the octave provides the main material of the Quartett-Satz I V 6 IV 6 “I@” “ii 6 ” “I 6 ”

21 Geometrically, the fauxbourdon ROTO moves stepwise down the lattice at the center of diatonic chord space (GOM, chs. 3 and 7)

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

23 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.

24 It is NOT just “generic voice- leading goo.”

25 Chromatic sequences in the Quartett- Satz RP = –1, 3, 7 or –1 mod 4 (GOM, §8.4) similar to the fauxbourdon passages!

26 The 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 Ef q q Af Ef bf[Df] Af Ef Q2: is something missing? Q3: does this remind you of anything? 7 7 3 7 7 implicitly cyclical

27 The gorgeous second- theme extension Functional tonality or chromaticism? implicitly cyclical q q

28 Chromatic sequences in the Quartett-Satz 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.

29 Summary The stepwise descending passages we’ve looked at account for a substantial majority of the Quartett-Satz’s exposition. A structural motive?

30 In the development, the music ascends!

31 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 languages.

32 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

33 Piece 2: A Bach Chorale 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.

34 Piece 2: A Bach Chorale 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!)

35 Piece 2: A Bach Chorale 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!)

36 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

37 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.

38 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.

39 Piece 3: Chopin A minor prelude

40 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.

41 Chopin A minor prelude

42 Chopin A minor prelude

43 One continuous process!

44 A minor prelude, “fauxbourdon” “background”

45 A minor prelude, “fauxbourdon” “background”

46 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!

47 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.

48 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.

49 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.

50 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.

51 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.


53 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!

54 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.)

55 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.

56 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!

57 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.

58 Thank you! for more information …

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