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<TITLE>Re: [Smt-talk] Sequences</TITLE>
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<FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>Rather than make neo-Pistonian lists of root motions, I suggest Matthew Brown’s book, Explaining Tonality, chapter 3 (indeed, hello) as a good place to start for trying to understand tonal sequences. Steve Laitz’s textbook is also a place to find a useful compendium of such phenomena.<BR>
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The break in the descending 5th sequence below is, of course, where the tritone occurs. Check out Chopin’s Etude op. 25, no. 1 m. 15 for ways to smooth things with chromaticisms and “7ths” in combination with triads. See also Brown’s discussion of how chromaticisms are introduced into sequences.<BR>
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The orbifold approach will work for complete chords, but rather than the permutational equivalence that orbifolds are based on, something like 000 040 004 047 (CCC CEC CCE CEG) equivalence more closely matches tonal practice, to avoid constant careening from the middle to the edges and back to pick up incomplete chords (see Headlam and Brown reply to Tymoczko in Science). We also have the Chopin situation of combined triads and seventh chords that require coordination between orbifolds of different cardinalities.<BR>
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(For those interested, in a special double issue of Theory and Practice soon to appear, dedicated to the life and music of George Perle, I have a discussion of the relationship between the structure of orbifolds and Perle’s triadic arrays)<BR>
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My favorite sequence is (of course!) in Berg’s op. 2 songs (see the beginning of song II for the clearest presentation of this descending 5th adaptation), in one manifestation in III, we move from Ab to Fb = E A D of the tritone related D minor, in a recall of song I, a juxtaposition of the real world and the world of sleep ( = death, but also a return to one’s homeland) that pervades the first 3 songs, and a “tonal conundrum” of the direct tritone relation in the return to Ab — from this crossroads, Berg moves to the non-tonal path in the next piece, op. 2/IV. In opus 2/IV, Berg’s “descending 5th” sequence (from Ravel’s Gaspard no. 2, Le Gibet) alternates the two all-interval-class tetrachords, Bb-Ab-D-G to Eb-G-C#-Gb (+5 bass, -1 [016]s upper voices, in a pattern found in all of the opus 2 songs) for a locally “strong” non-tonal progression, but of course, one found in jazz, if not by then (1909-ish), at least soon after. See also Redlich’s Berg book (1957) and of course Perle’s writings for some interesting derivations of these and other Berg sequences and their expansion into the worlds of symmetry and cycles.<BR>
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Quote from earlier post: <BR>
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One piece of evidence for this is that there's a marked asymmetry between triadic and seventh chord routines: triads are more likely to move chromatically by major third and seventh chords are more likely to move by minor third or tritone. This reflects the underlying voice-leading relations: C major is "closer" (in voice leading terms) to E major than Eb major, but C7 is closer to Eb7 than E7. You can actually see this asymmetry in the statistics -- just count up all the chromatic progressions in Mozart or Schubert.<BR>
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I haven’t counted them all yet, but check out Coltrane’s “Giant Steps” for 7th chords in M3 cycles (albeit mediated by V-Is, ii-V-Is) and Messiaen’s Quartet for the end of Time V for triads in minor 3rd cycles.<BR>
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Dave Headlam<BR>
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On 3/5/09 5:20 PM, "Dmitri Tymoczko" <dmitri@Princeton.EDU> wrote:<BR>
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</SPAN></FONT><BLOCKQUOTE><BLOCKQUOTE><FONT FACE="Verdana, Helvetica, Arial"><FONT SIZE="4"><SPAN STYLE='font-size:18.0px'>Another example of a sequence where the typical part-writing is opposite the smoothest idealized voice leading is the ubiquitous descending fifth sequence. We usually associate descending part writing with the descending fifths sequence, but it has idealized ascending voice leading. <BR>
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This is an important point.<BR>
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In "strong" progressions, the root moves (diatonically) by descending third, descending fifth, or ascending step. When you're using plain triads, the most efficient voice leading for a strong progression is ascending. This creates a conflict between two tonal imperatives: 1) using efficient voice leading, and 2) creating phrases that use descending melodic contours. (Hello, Heinrich, nice to see you again!) Typically, then, a harmonic cycle such as I-IV-V-I or I-ii-V-I involves at least one non-minimal voice leading. The descending fifth sequence is a case in point -- usually one of the two voice leadings is non-minimal, as in (C, E, G)->(C, F, A)->(B, D, F)->(B, E, G), where the nonminimal voice leading is (C, F, A)->(B, D, F).<BR>
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With seventh chords, the situation is reversed -- now the most efficient voice leadings for strong progressions descend. This means that you can create sequences, or harmonic cycles, using maximally efficient descending voice leadings. You start to see this more and more often in nineteenth-century music, and it eventually becomes central to jazz voice-leading practice.<BR>
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DT<BR>
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</SPAN><FONT SIZE="2"><SPAN STYLE='font-size:12.0px'>Dmitri Tymoczko<BR>
Associate Professor of Music<BR>
310 Woolworth Center<BR>
Princeton, NJ 08544-1007<BR>
(609) 258-4255 (ph), (609) 258-6793 (fax)<BR>
<a href="http://music.princeton.edu/~dmitri">http://music.princeton.edu/~dmitri</a><BR>
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Dave Headlam<BR>
Professor of Music Theory<BR>
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Eastman School of Music<BR>
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