[Smt-talk] Sequences

[Dave Headlam]

Rather than make neo-Pistonian lists of root motions, I suggest Matthew
Brown¹s book, Explaining Tonality, chapter 3 (indeed, hello) as a good place
to start for trying to understand tonal sequences.  Steve Laitz¹s textbook
is also a place to find a useful compendium of such phenomena.

The break in the descending 5th sequence below is, of course, where the
tritone occurs.  Check out Chopin¹s Etude op. 25, no. 1 m. 15 for ways to
smooth things with chromaticisms and ³7ths² in combination with triads.  See
also Brown¹s discussion of how chromaticisms are introduced into sequences.

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.

(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
Perle¹s triadic arrays)

My favorite sequence is (of course!)  in Berg¹s 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 one¹s
homeland) that pervades the first 3 songs,  and a ³tonal conundrum² 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,
Berg¹s ³descending 5th² sequence (from Ravel¹s 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 ³strong² non-tonal progression, but of course, one
found in jazz, if not by then (1909-ish), at least soon after.  See also
Redlich¹s Berg book (1957) and of course Perle¹s writings for some
interesting derivations of these and other Berg sequences and their
expansion into the worlds of symmetry and cycles.

---
Quote from earlier post:

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.
---

I haven¹t counted them all yet, but check out Coltrane¹s ³Giant Steps² for
7th chords in M3 cycles (albeit mediated by V-Is,  ii-V-Is) and Messiaen¹s
Quartet for the end of Time V for triads in minor 3rd cycles.



Dave Headlam


On 3/5/09 5:20 PM, "Dmitri Tymoczko" <dmitri at Princeton.EDU> wrote:

>> 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. 
> 
> This is an important point.
> 
> 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).
> 
> 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.
> 
> DT
> 
>  
> Dmitri Tymoczko
> Associate Professor of Music
> 310 Woolworth Center
> Princeton, NJ 08544-1007
> (609) 258-4255 (ph), (609) 258-6793 (fax)
> http://music.princeton.edu/~dmitri
> 
> 
> 
> 
>  
> 
> 
> 
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> rg


-----------------------

Dave Headlam
Professor of Music Theory

Eastman School of Music
26 Gibbs St
Rochester, NY 14604
(585) 274-1568 office
dheadlam at esm.rochester.edu
http://theory.esm.rochester.edu/dave_headlam


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