Re: enumerating pitch class sets algebraically

Responding to several posts at once here:

Carl asked, re pc-set analysis:

> () Does it generalize the serial technique, or is it different?

It can be used usefully for pre-serial atonal music, when it's
appropriate.

> () Was it started/coined by Babbitt?

Allen Forte once said in a keynote SMT speech: "I didn't invent the
unordered pitch-class set, that was due to a higher power... and I don't
mean Milton Babbitt!"

Forte's first listing of set-classes was flawed as it reduced for the
Z-relation. It was David Lewin who pointed out that there were no
consistent transformations that got you from one Z-related pair to
another, so from a group-theorteical perspective it's more useful to treat
them separately. The "canonic" list is in Forte's "The Structure of Atonal
Music" from 1972.

> () Does it claim to be / is it a prescriptive (ie algo comp) process,
> a descriptive process, or both?

It's really just a labelling scheme and an assertion that it's meaningful
to talk about set-classes, their abstract relationships, etc. The former
is what Dante says is unimpeachable, and the latter two are what Paul E
objects to in the context of music that the scheme wasn't designed to
address in the first place.

> () What's the best piece for a beginner to start with, and what
> should he listen for?

The thing to be wary of is the segmentation process - what's a set, and
what isn't. It's easy for people to go "cherry-picking", and take the
notes they want, with no particular musical justification, to get the sets
they want. This is a very valid objection. Forte tries to argue that
Schoenberg consciously uses any member of a set-class to "represent" his
"signature" set, EsCHBEG, or Eb-C-H-B-E-G (derived from his name). But if
you're looking for members of this set-class you'll find them of course,
and ultimately his argument (that Schoenberg was using unordered
set-classes because we can find them in the music, and we're allowed to
look for them in the music because Schoenberg was using them) is circular.
There's a good published rebuttal of the argument that is exactly what I
had wanted to write ever since first coming across this statement. I can't
remember who wrote it though!

Gene wrote:

> Not everyone knows what a Z-relation is, I'm afraid. I recall seeing
> the term somewhere, but that's it.

When two pitch-class sets have the same interval vector but are not
related by the operations in the group (usually transposition and
inversion) then Forte calls them Z-related. The generalised hexachordal
theorem shows that in universes of cardinality 2n, an n-chord is either a
transformation of its complement, or the two are Z-related. There are
several proofs of this.

> This is also confusing to me. C(24,6) normally means the number of
> combinations of 24 things taken 6 at a time.

Yes I gather Paul Hj. means you start with the set choose(24,6), then
reduce it by eliminating anything that's not in prime-form.

--Jon

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> Responding to several posts at once here:
>
> Forte's first listing of set-classes was flawed as it reduced for
the
> Z-relation. It was David Lewin who pointed out that there were no
> consistent transformations that got you from one Z-related pair to
> another, so from a group-theorteical perspective it's more useful
to treat them separately.

There are obviously different flavors of Z-relations. I have
discovered that (in 24-et at least, for every subset 4 thru 9 and
for 12, that I have tested) the Z-relations are based on the divisors
of 24 (except for 24) (1,2,3,4,6,8,12) So there is some kind
of "clumping" going on. In 19-et though, there are pairs and triples
of z-relations and 3 is obviously not a divisor of 19. (I should point
out that I am counting Z-related sets AFTER reducing for transposition
and inversion). The final set counts are then based on unique
interval vectors. If I am wasting my time looking for patterns of Z-
relations, and of set counts based on unique interval vectors, you
guys would tell me, right?

>
> Gene wrote:
>
> > Not everyone knows what a Z-relation is, I'm afraid. I recall
seeing
> > the term somewhere, but that's it.
>
> When two pitch-class sets have the same interval vector but are not
> related by the operations in the group (usually transposition and
> inversion) then Forte calls them Z-related. The generalised
hexachordal
> theorem shows that in universes of cardinality 2n, an n-chord is
either a
> transformation of its complement, or the two are Z-related. There
are
> several proofs of this.

Thanks Jon. You said it better than I did. Can you point me to these
theorems regarding Z-relations?

A little tidbit for everyone: In 12-et, counting hexachords, after
reducing for transposition and mirror images, you get 50 sets. 20 are
sets which have a normal complement (6 symmetrical sets, 14
asymmetrical sets) The 6 symmetrical sets are all-combinatorial.
Their complements are the same set. 3 have self-symmetry. 1
asymmetrical set has a complement which is the same set:
0,1,3,4,5,8). The remaining 13 asymmetrical sets (that have a normal
complement) have a complement that is the mirror-image of the set.
(One of these sets also has self-symmetry (0,1,3,6,7,9)) Now, the
last 30 sets are 15-pairs of z-related sets. They are all z-related
to their complement. (I've also played with pentachords and
tetrachords which have some interesting properties)

Jon, regarding your files: unfortunately I don't have the ability to
scp files. And my email here only accepts attachments up to 10 mg.
Any ideas? Could you burn your zipfile to a CD? Or you could ftp it
to my website (uptownjazz.com). Whenever...

Paul

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:

> > () Does it claim to be / is it a prescriptive (ie algo comp)
process,
> > a descriptive process, or both?
>
> It's really just a labelling scheme and an assertion that it's
meaningful
> to talk about set-classes, their abstract relationships, etc.

Isn't this almost the same as saying it is worth looking at all
possible 12-et chords (or scales), under the assumption of octave
equivalence? We then can go further and look at all chords assuming
transpostion, or transposition and inversion, etc.

Incidentally, I started thinking about this because it occurs to me
that PC theory may be a good starting point for adaptive tuning.
There is a small enough set of chords that one can work with it
pretty easily in 12-et, especially after group reductions. Of course,
we also want the context the chord is in, and this now begins to look
like theory.

> > () What's the best piece for a beginner to start with, and what
> > should he listen for?
>
> The thing to be wary of is the segmentation process - what's a set,
and
> what isn't. It's easy for people to go "cherry-picking", and take
the
> notes they want, with no particular musical justification, to get
the sets
> they want.

How did we manage to get from sets to some kind of compositional
process?

> Yes I gather Paul Hj. means you start with the set choose(24,6),
then
> reduce it by eliminating anything that's not in prime-form.

It seems to me that the set of prime-form reductions of n things
taken m at a time modulo some permutation group of degree G could be
given a name--"(n,m) reduced G" or "G{n,m}" or something--and then
we'd really be cooking. The prime form in question could be defined
as the least base-2 number in the orbit. You might also want a name
for the function which takes a chord or PC or whatever you wish to
call it to its G-reduction. Something like "Pfred(s, G)" where s is
the PC set and G is the permuation group.

Hi Jon!

>Carl asked, re pc-set analysis:
>
>> () Does it generalize the serial technique, or is it different?
>
>It can be used usefully for pre-serial atonal music, when it's
>appropriate.

Can you give any examples of pre-serial atonal music?

And while I'm on it, serial tonal music?

>> () Does it claim to be / is it a prescriptive (ie algo comp) process,
>> a descriptive process, or both?
>
>It's really just a labelling scheme and an assertion that it's meaningful
>to talk about set-classes, their abstract relationships, etc. The former
>is what Dante says is unimpeachable, and the latter two are what Paul E
>objects to in the context of music that the scheme wasn't designed to
>address in the first place.

And it's the latter that my hunch says is simply wrong.

>> () What's the best piece for a beginner to start with, and what
>> should he listen for?
>
>The thing to be wary of is the segmentation process - what's a set, and
>what isn't. It's easy for people to go "cherry-picking", and take the
>notes they want, with no particular musical justification, to get the sets
>they want. This is a very valid objection. Forte tries to argue that
>Schoenberg consciously uses any member of a set-class to "represent" his
>"signature" set, EsCHBEG, or Eb-C-H-B-E-G (derived from his name). But if
>you're looking for members of this set-class you'll find them of course,
>and ultimately his argument (that Schoenberg was using unordered
>set-classes because we can find them in the music, and we're allowed to
>look for them in the music because Schoenberg was using them) is circular.
>There's a good published rebuttal of the argument that is exactly what I
>had wanted to write ever since first coming across this statement. I can't
>remember who wrote it though!

Well if you find it, pass the ref. along.

-Carl

>Can you give any examples of pre-serial atonal music?

You cant do any better than Webern op 5-16. You can download mp3s of 0p 5
and 6 (both landmark works) from here:

http://www.antonwebern.com/

Dante

>>Can you give any examples of pre-serial atonal music?
>
>You cant do any better than Webern op 5-16. You can download mp3s of 0p 5
>and 6 (both landmark works) from here:
>
>http://www.antonwebern.com/

Funny, sounds just like serial atonal music (to my novice ear).

-Carl

>>And while I'm on it, serial tonal music?
>
>Can't help you there,

Is the final fugue of the WTC1 a serial piece? Why
or why not?

-Carl

If you compare op5 with any of his classic serial works like the symphony op
21 you will easily hear the difference. However, the difference may not be
so much the pitch content as the handling of the material. The symphony is
written to explicate the structures, whereas op 5 or 6 creates moods, colors
with the textures and unearthly scoring. The symphony does sound like, as
Stravinsky put it, a "perfect diamond" with its crystalline angles and
symmetries. The earlier atonal works sound like alien orchids.

In high school I had an LP of the Quartetto Italiano playing Webern, I wore
out the side with op 5 and op 9 bagatelles, but couldn't get into the op 28
quartet. I can appreciate it more now, but I'd still rather hear the earlier
works- they still blow me away.

Dante

> -----Original Message-----
> From: Carl Lumma [mailto:ekin@lumma.org]
> Sent: Thursday, December 04, 2003 5:43 PM
> To: tuning-math@yahoogroups.com
> Subject: RE: [tuning-math] Re: enumerating pitch class sets
> algebraically
>
>
> >>Can you give any examples of pre-serial atonal music?
> >
> >You cant do any better than Webern op 5-16. You can download mp3s of 0p 5
> >and 6 (both landmark works) from here:
> >
> >http://www.antonwebern.com/
>
> Funny, sounds just like serial atonal music (to my novice ear).
>
> -Carl
>
>
>
> To unsubscribe from this group, send an email to:
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> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> >Can you give any examples of pre-serial atonal music?
>
> You cant do any better than Webern op 5-16. You can download mp3s
of 0p 5
> and 6 (both landmark works) from here:
>
> http://www.antonwebern.com/

Do you know of a good source for midi versions of Webern?

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:

> > Can you give any examples of pre-serial atonal music?
>
> You cant do any better than Webern op 5-16. You can
> download mp3s of 0p 5 and 6 (both landmark works) from here:
>
> http://www.antonwebern.com/
>
> Dante

i've made mp3's of some samples from my MIDI-files,
of course with my usual exaggerated rubato (for which
i will never apologize) :

Schoenberg's 1st Quartet and Webern's "String Quartet (1905)"
are the two earliest examples (AFAIK) of atonality in music.

Webern - climax from "String Quartet (1905)"
http://sonic-arts.org/monzo/webern/qt1905-a.mp3

Schoenberg - opening of 1st Quartet
http://sonic-arts.org/monzo/schoenberg/1stqt/schbg-qt1-beg.mp3

and since Dante mentioned an mp3 of op. 6, here's mine:

Webern - 1st piece from _6 Orchesterstücke, op. 6_
http://sonic-arts.org/monzo/webern/op6-1.mp3

and this is perhaps the most famous example of non-serial
atonal orchestral music:

Schoenberg - opening of 1st piece from _5 Orchesterstücke, op. 16_
http://sonic-arts.org/monzo/schoenberg/5pcs/5pcs1.mp3

there's lots more in my "A Century of New Music in Vienna",
between 1905 and 1920.

http://sonic-arts.org/monzo/schoenberg/Vienna1905.htm

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
wrote:
> > >Can you give any examples of pre-serial atonal music?
> >
> > You cant do any better than Webern op 5-16. You can download mp3s
> of 0p 5
> > and 6 (both landmark works) from here:
> >
> > http://www.antonwebern.com/
>
> Do you know of a good source for midi versions of Webern?

i have made several, but i believe that they're all still
under copyright and thus i'm not supposed to share them.

(but i could be wrong about the pieces from 1905-1909.)

-monz