[Smt-talk] Stravinsky, sonorities, and nomenclature

Stephen Guy Soderberg ssod at loc.gov
Tue Feb 17 09:49:44 PST 2009


One method I've found particularly useful was inspired by an idea of
David Lewin that has taken hold somewhat in the neo-Riemannian world,
and that I generalized to describe virtually any equal-tempered
sonority.  David's idea was to label, say, a C-major triad as (C,+) —
the C indicating the root and "+" indicating "major"; (F#,-) would be an
F#-minor triad, etc.  Surprisingly, this simple change of nomenclature
to "(pitch, sign)" produced some non-trivial ideas, but it could only go
so far.  The obvious generalization it suggested, though, allows
expansion to the entire universe of sonorities and rhythmic structures
(and even into sampling and mashing).  None of this is particularly
revolutionary.  Programmers and mathematicians worked out all the math
for strings a long time ago.  But as far as I know it hasn't caught on
yet in music applications and analyses.  Here's a mini-primer on
interval-string notation in 12 easy lessons:

- Lewin's (pitch, sign) becomes (pitch, interval string) or (p,s).  An
interval string in the usual chromatic (and without additional bells &
whistles) is nothing more than a (circular) ordered representation of
the intervals measured in semitones between pitches in a chord that add
up to 12.  So "a major triad in root position" — without identifying
which one — is simply <4,3,5>.
- In extended Lewin notation, (C,<4,3,5>) is the chord {C, C+4, C+4+5},
or {C,E,G}, or (C,+) or "(any) close root-position C-major triad."  [I
generally use numbers for pitches, but I'll stick to letter-names
- Rotate the string and you get chord inversions: (C,<3,5,4>) =
{C,Eb,Ab}, (any) Ab-major triad in first inversion.
- Read the string backwards and you get chromatic inversion or "mode
flip": (C,<4,5,3>) = {C,E,A} = (any) A-minor triad in first inversion.
- Specify octave of the pitch (getting outside pitch-classes) and you
nail down the chord: (C4,<4,3,5>) is THE close root-position major triad
on pitch C4.
- Change the rule that the string must sum to 12 to a rule that the
string must sum to some multiple of 12, and you can place the notes
anywhere: (C4,<7,9,8>) ={C4, G4, E5} is the open root-position major
triad on C4.
- Allow zeroes to get doublings: (C4,<4,3,0,5>) = {C4,E4,G4,G4}, ...
- ... and multiples of 12 can yield octaves: (C3,<16,8,7,5>) = (C3, E4,
C5, G5).
- You can consistently label any "traditional" chord outside maj/min
triads: (C,<3,3,6>) is a C-dim triad; (C,<4,3,3,2>) is a dom-7th on C,
- ... and, unlike other notation systems, you can — consistently — 
take it as far out of traditional as you want: (0,<1,2,2,2,2,3>) =
{013579} is  the Scriabin "Mystic Chord" or Forte's 6-34;
(0,<1,2,2,1,3,3>) = {013569} is Forte 6-Z28.
- For the exotic-minded, you can represent microtonal sonorities by
leaving behind the insistence on 12 as a 
"base modulus": in quartertone (mod 24): (0,<7,7,10>) = {0,7,14} is a
triad that splits the fifth equally.
- Finally: all the above can be used for rhythmic notation equally
well.  (p,s) becomes (t,s) or (time point, duration string).  E.g.,
Beethoven 7th generating rhythm: (t,<3,1,2>), dotted-eighth, sixteenth,
eighth — the entire first movement can be considered transformations
of this duration string.

For analytical problems, there are often interval- and duration-strings
(either or both) that stand out.  Once identified, I usually label these
stand-outs with a short-hand.  E.g., if I discover the string <21211212>
I might think of it as a bastard octatonic source string created when
(0,<2127>) met up with (6,<12126>) and label it "k"; so the various
transpositions of k would be labeled (p,k) with p=0,1,2,etc.  But I
always keep a "translation scratch pad" handy so I don't come across
<21211221> and assume it's a k & the analysis starts to go haywire.  

None of this is much help with works solidly within the common
practice/usual diatonic period since (p,s) notation tends to reduce to
the more familiar comfort-zone (i.e., traditional) notations for triads
and 7th chords — ground pretty well covered that doesn't really need
another notation system layered on top of all the others.  But get to
the edge of that period and beyond into the present and it's a different
story as strings "warp" and finally break into nontonal formations. 
There are certainly dozens of "histories" to these various new
sonorities & how they arise in the actual music, but the point here is
that, however different they are in context, they can be abstractly
represented and compared with strings in a consistent way that I often
find more revealing than using pc set theory alone.

This may be heresy but, for me, while "pitches" and the more abstract
concept "pitch classes" are certainly a sine qua non for most music to
happen, they are only a means to establish a complex of
multi-dimensional INTERVALLIC relationships (which, today at any rate,
is my definition of music).  Re Stravinsky: remember his cryptic answer
to an age-old question: "I compose with intervals."

For more on this, see "Transformational Etudes" chapter of _Music
Theory and Mathematics: Chords, Collections and Transformations_ 



Stephen Soderberg
Music Division
Library of Congress

>>> Rebecca Hyams <rebecca.hyams at gmail.com> 2/16/2009 2:12 PM >>>
    Currently, I'm in the process of working on my MA thesis, where I'm

looking at Stravinsky's alteration of his sources in Pulcinella. As I'm

working, my biggest challenge is dealing with harmonies and what to
them. I wanted to pose my conundrum to the theory community, and though

I realize that no single solution is perfect, I want to see what other

ideas are out there (or if perhaps there's a way to reconcile a method

I'm already familiar with with the realities of the music).
    My first instinct was to call them by set class, but that has its 
limitations as well as connotations that are not necessarily applicable

to this musical context. I know there's also an approach that attempts

to place non-triadic sonorities into an altered triadic context. While
agree that there's some instances of altered triads throughout the work

(after all, the source materials are clearly common practice) there's 
sections where the majority of material is added by Stravinsky. Some of

those sonorities, while they clearly have some sort of root, cannot be

explained by identifying them as some sort of triad, in part because of

the functional implications triads have from tonal music. Of course
while set theory can provide a name for the sonority and a method of 
relating it to other similar sonorities, it doesn't easily lend itself

to the centric-nature of the sonorities in question. I know there must

be some sort of middle ground or other approach that I have yet to be 
exposed to.
    (I have a specific section in the music that I've been milling over

that started a whole conversation between myself and my thesis advisor.

I would be happy to share that except of the score with anyone willing

to take a look at it.)

Thank you,
Rebecca Hyams
MA student in music theory
Queens College- CUNY
rebecca.hyams at gmail.com 
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