Enumerating pitch class sets algebraically

I took all 4095 nonempty subsets of 12 notes, and for each of these I
formed a corresponding sum of 12-th roots of unity, mod 13^8, of the
form sum z^i, i in I, where "I" is the subset of notes, and z is
-288746898, which is a primitive 12th root of unity mod 13^8. In order
to make this independent of circular permutations, I then raised all
of these to the twelvth power. This resulted in 361 different pitch
class sets. Pitch class sets so counted are invariant under
transposition. It turns out we need only go as far as 13^3 = 2197,
with z = 418, in order to insure all of the hash values for the pitch
class sets are distinct.

Does anyone know if 361 is what other people have gotten for this?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Does anyone know if 361 is what other people have gotten for this?

It occurred to me that if this method was going to work, it should
distinguish all 4095 pitch class sets to start out with, and since the
primitive 12th root of unity is algebraic of degee four (with
polynomial x^4 - x^2 + 1) it was by no means obvious this is true; in
fact, it isn't.

Oh well.

Gene - pc-sets are usually enumerated using Polya's method. IIRC there are
351 in 12-tet. Robert Walker made some good posts illustrating this a
couple of years ago on the main list, I *think* the subject had the number
351 in it, if you want to search for the thread. I started to write a FAQ
entry about enumerating pitch-class sets (there's still an 11k email on
this in my postponed messages folder), but like so many other things, it
died (as did the whole FAQ project, I think). I've also generated all
prime-form set-classes (and their interval vectors) for ets up to 31, and
have them sitting zipped up on another computer. One day I will put them
up somewhere in a searchable database--at the moment I use grep and crazy
regular expressions to dig out chords with the properties I'm looking for.
It's funny - the limit of 31 works well because 31 is such a popular
temperament, but actually I reached that limit because of 32-bit
architecture and the bit-packing procedures I use in the C program.

Just recently a very good article appeared in "Music Theory Online", the
most recent issue, by Rick Cohn. He has come up with a very simple formula
for enumerating tetrachordal set-classes in even ETs. It would be great if
you could see how to generalise that formula to n-chords! It's based on a
semitonal voice-leading model that stacks all the 12-tet tetrachords in a
tetrahedral structure.

Jon

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> I've also generated all
> prime-form set-classes (and their interval vectors) for ets up to
31, and
> have them sitting zipped up on another computer. One day I will put
them
> up somewhere in a searchable database--at the moment I use grep and
crazy
> regular expressions to dig out chords with the properties I'm
looking for.
> It's funny - the limit of 31 works well because 31 is such a popular
> temperament, but actually I reached that limit because of 32-bit
> architecture and the bit-packing procedures I use in the C program.
> >
> Jon

I would love to see your listings up to 31 of set-classes and their
interval vectors. Have you also generated counts that reduce for
(Allen Forte's) Z-relation? For example there are 35 unique interval
vectors for hexachords in Z12 and also 35 unique interval vectors
for pentachords in Z12. Of course 351 is based on (1,6,19,43,66,
80,66,43,19,6,1,1) (Notice I have not included the empty set)Reducing
this for backwards chords gives (1,6,12,29,38,50,38,29,12,6,1,1)
Reducing this for the "Z-relation" this is when two chords of
different Tn/TnI class have the same interval vector - gives
(1,6,12,28,35,35,28,12,6,1,1). Hexachords have 15 Z-related sets
all of which are interlocking (They are z-related with their
complement).

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
> >
> > I've also generated all
> > prime-form set-classes (and their interval vectors) for ets up to
> 31, and
> > have them sitting zipped up on another computer. One day I will
put
> them
> > up somewhere in a searchable database--at the moment I use grep
and
> crazy
> > regular expressions to dig out chords with the properties I'm
> looking for.
> > It's funny - the limit of 31 works well because 31 is such a
popular
> > temperament, but actually I reached that limit because of 32-bit
> > architecture and the bit-packing procedures I use in the C
program.
> > >
> > Jon
>
> I would love to see your listings up to 31 of set-classes and their
> interval vectors. Have you also generated counts that reduce for
> (Allen Forte's) Z-relation? For example there are 35 unique
interval
> vectors for hexachords in Z12 and also 35 unique interval vectors
> for pentachords in Z12. Of course 351 is based on (1,6,19,43,66,
> 80,66,43,19,6,1,1) (Notice I have not included the empty set)
Reducing
> this for backwards chords gives (1,6,12,29,38,50,38,29,12,6,1,1)
> Reducing this for the "Z-relation" this is when two chords of
> different Tn/TnI class have the same interval vector - gives
> (1,6,12,28,35,35,35,28,12,6,1,1). Hexachords have 15 Z-related sets
> all of which are interlocking (They are z-related with their
> complement).

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> Gene - pc-sets are usually enumerated using Polya's method. IIRC
there are
> 351 in 12-tet.

Thanks, I'll research that. I don't see how there can be 351; 361
should be a minimum. Anyway, my method isn't certain of working
unless the et is prime, which 12 is a long way from being.

Robert Walker made some good posts illustrating this a
> couple of years ago on the main list, I *think* the subject had the
number
> 351 in it, if you want to search for the thread. I started to write
a FAQ
> entry about enumerating pitch-class sets (there's still an 11k
email on
> this in my postponed messages folder), but like so many other
things, it
> died (as did the whole FAQ project, I think).

I'd be interested in what you can dig out; if you are willing to
forward the email that would be nice.

I've also generated all
> prime-form set-classes (and their interval vectors) for ets up to
31, and
> have them sitting zipped up on another computer.

Is prime-form a reduced form?

> Just recently a very good article appeared in "Music Theory
Online", the
> most recent issue, by Rick Cohn. He has come up with a very simple
formula
> for enumerating tetrachordal set-classes in even ETs. It would be
great if
> you could see how to generalise that formula to n-chords!

I had in mind exploring how to do such things, but I didn't know what
had been done, since googling didn't turn anything up.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
> >
> > Gene - pc-sets are usually enumerated using Polya's method. IIRC
> there are
> > 351 in 12-tet.
>
> Thanks, I'll research that. I don't see how there can be 351; 361
> should be a minimum.

352 is the number I remember -- maybe that includes the empty set.
This got discussed on the music theory yahoogroup (from which I've
since unsubscribed due to anti-semitic and anti-german postings).

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> Gene - pc-sets are usually enumerated using Polya's method.

Remarkably, there seems to be an entire book on this: "From
Polychords to Polya", by Michael Keith. I suppose if I was a
combinatorialist, it would have been the first thing I tried, but I
still would like to do it using number theory.

Alas, not only is not to be found at San Jose State, it isn't
anywhere in the interlibrary loan system we have either. I've put it
on my Christmas wish list, but if I don't get it I may buy it.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> 352 is the number I remember -- maybe that includes the empty set.
> This got discussed on the music theory yahoogroup (from which I've
> since unsubscribed due to anti-semitic and anti-german postings).

Flavell acting up?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
> > >
> > > I've also generated all
> > > prime-form set-classes (and their interval vectors) for ets up
to
> > 31, and
> > > have them sitting zipped up on another computer. One day I will
> put
> > them
> > > up somewhere in a searchable database--at the moment I use grep
> and
> > crazy
> > > regular expressions to dig out chords with the properties I'm
> > looking for.
> > > It's funny - the limit of 31 works well because 31 is such a
> popular
> > > temperament, but actually I reached that limit because of 32-bit
> > > architecture and the bit-packing procedures I use in the C
> program.
> > > >
> > > Jon
> >
> > I would love to see your listings up to 31 of set-classes and
their
> > interval vectors. Have you also generated counts that reduce for
> > (Allen Forte's) Z-relation? For example there are 35 unique
> interval
> > vectors for hexachords in Z12 and also 35 unique interval vectors
> > for pentachords in Z12. Of course 351 is based on (1,6,19,43,66,
> > 80,66,43,19,6,1,1) (Notice I have not included the empty set)
> Reducing
> > this for backwards chords gives (1,6,12,29,38,50,38,29,12,6,1,1)
> > Reducing this for the "Z-relation" this is when two chords of
> > different Tn/TnI class have the same interval vector - gives
> > (1,6,12,28,35,35,35,28,12,6,1,1). Hexachords have 15 Z-related
sets
> > all of which are interlocking (They are z-related with their
> > complement).

I realize I should clarify that 351 is the sum of
1 monad
6 diads
19 triads
43 tetrads
66 pentads
80 hexads
66 septads
43 octads
19 nonads
6 decads
1 modecads
1 dodecads
-----
351 chords in 12-et (Thanks to John Rahn)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > 352 is the number I remember -- maybe that includes the empty
set.
> > This got discussed on the music theory yahoogroup (from which
I've
> > since unsubscribed due to anti-semitic and anti-german postings).
>
> Flavell acting up?

Yeah, that was the name. Are you familiar with it from elsewhere?

Gene wrote:

> Remarkably, there seems to be an entire book on this: "From
> Polychords to Polya", by Michael Keith. I suppose if I was a
> combinatorialist, it would have been the first thing I tried, but I
> still would like to do it using number theory.
>
> Alas, not only is not to be found at San Jose State, it isn't
> anywhere in the interlibrary loan system we have either.

I checked the library system at Harvard too, and there's nothing between
"From Poltava to Prut" and "From Ponkapog to Pesth", though they both
sound like good reads ;-)

Did you manage to read Rick Cohn's article in Music Theory Online? The
permissions seem to be wrong at the moment; I couldn't get through. It
should be at

http://www.societymusictheory.org/mto/issues/mto.03.9.4/toc.9.4.html

Jon

You can find the thread starting on 12th December 2000, called "351
possible scales". Robert's relevant messages which explain well are 16468
and 16486. See also message 16494 where I give the generating function for
19-tet, as Maple expanded it. Yes, the figure of 351 for 12-tet does not
include the empty set, and yes, those are Tn types not Tn/TnI.

The files I generated are of Tn/TnI classes so mirror-inversions don't
appear separately. Yes Gene, "prime form" is a canonic representative of
each set-class orbit--I think Allen Forte invented this name. It's usually
defined as the most compact, then most left-packed member of the
equivalence class, but "least right-packed" is technically more accurate.

I'd be happy to share the files with Paul Hjelmstad and anyone else. The
problem is that they are *huge*. They aren't reduced by the Z-relation,
but it is trivial to do so from the format they're in now, because I ran a
job on the file that counts how many times that chord's interval vector
appears, and prepends that number to the line. For 12-tet that number
is always 1 or 2, but for 16-tet and higher you can find larger groups of
Z-related chords. With suitable regular expressions you can query the file
for (for example) all groups of exactly 7 Z-related nonachords in 28-tet.
There is one such group; you can get an idea of the file format from the
following:

% grep ^7 chords28 | grep __9
7 [2,4,2,3,2,4,2,3,2,4,2,3,2,1]__28__9 (0,1,2,6,8,10,12,15,18)__28__9
7 [2,4,2,3,2,4,2,3,2,4,2,3,2,1]__28__9 (0,1,3,7,9,11,13,14,19)__28__9
7 [2,4,2,3,2,4,2,3,2,4,2,3,2,1]__28__9 (0,1,6,8,10,12,15,16,18)__28__9
7 [2,4,2,3,2,4,2,3,2,4,2,3,2,1]__28__9 (0,1,7,9,11,13,14,17,19)__28__9
7 [2,4,2,3,2,4,2,3,2,4,2,3,2,1]__28__9 (0,2,3,4,6,10,12,17,22)__28__9
7 [2,4,2,3,2,4,2,3,2,4,2,3,2,1]__28__9 (0,2,6,7,8,10,12,18,21)__28__9
7 [2,4,2,3,2,4,2,3,2,4,2,3,2,1]__28__9 (0,3,5,9,11,13,14,15,21)__28__9

You can also search for chords whose interval vectors have certain entries
within certain ranges, which I've found useful. The bad news is that the
size of the file for 31-tet is 1528081389 bytes (they double in size,
roughly, for each temperament). I can zip it to about a tenth of that
(about 150Mb if I've counted digits right). But that's still impractical.

Best -Jon Wild

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

>Yes Gene, "prime form" is a canonic representative of
> each set-class orbit--I think Allen Forte invented this name. It's
usually
> defined as the most compact, then most left-packed member of the
> equivalence class, but "least right-packed" is technically more
accurate.

Thanks. I found a url by googling for this, which gives the complete
351 (or 352, for the null chord is listed as number zero.)

http://music.theory.home.att.net/pcsets.htm#Explanation%20of%20This%
20Table

I wonder what other mathematical tidbits are back there in the stuff
before I came on board?

> Thanks. I found a url by googling for this, which gives the complete
> 351 (or 352, for the null chord is listed as number zero.)
>
> http://music.theory.home.att.net/pcsets.htm#Explanation%20of%20This%
> 20Table

Interesting. I didn't know Forte's methodology could be challenged. After
reading the explanation on this page, I'm still not convinced it can be. I
don't think introducing that kind of redundancy into the prime form list is
going to do anything but create confusion. Noone said that different
inversional and transpositional forms of prime sets sound the same, thats
not the point. The point is reducibility. "Tonal" theory is a limiting case
of set theory, just like Newtonian physics is a limiting case of relativity.

Dante

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> > Thanks. I found a url by googling for this, which gives the
complete
> > 351 (or 352, for the null chord is listed as number zero.)
> >
> > http://music.theory.home.att.net/pcsets.htm#Explanation%20of%
20This%
> > 20Table
>
> Interesting. I didn't know Forte's methodology could be challenged.
After
> reading the explanation on this page, I'm still not convinced it
can be.

I'm in complete agreement with the author of the page.

> I
> don't think introducing that kind of redundancy into the prime form
list is
> going to do anything but create confusion. Noone said that different
> inversional and transpositional forms of prime sets sound the same,
thats
> not the point. The point is reducibility. "Tonal" theory is a
limiting case
> of set theory, just like Newtonian physics is a limiting case of
relativity.
>
> Dante

Hi Dante. I must be totally ignorant of how this 'limiting' happens,
but what you are saying seems impossible. If Forte's methodology
eliminates the distinction between mirror inverses, how can any
limiting case of it possible restore that distinction?

Hi Paul-

The distinction is not "restored", it simply doesn't exist from the
set-theoretic perspective. Now, you may then say that this perspective is
therefore useless to "explain" tonal music, which may very well be. But any
music (tonal or not) can very well be >described< from a set-theoretic
perspective. Functional harmony, as a cultural construct, will not
necessarily "show up" in this type of description. I find this kind of set
stuff more useful for precompositional material than analysis (see Carter's
"Harmony" book).

Dante

> -----Original Message-----
> From: Paul Erlich [mailto:perlich@aya.yale.edu]
> Sent: Tuesday, December 02, 2003 2:07 AM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Enumerating pitch class sets algebraically
>
>
> --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> > > Thanks. I found a url by googling for this, which gives the
> complete
> > > 351 (or 352, for the null chord is listed as number zero.)
> > >
> > > http://music.theory.home.att.net/pcsets.htm#Explanation%20of%
> 20This%
> > > 20Table
> >
> > Interesting. I didn't know Forte's methodology could be challenged.
> After
> > reading the explanation on this page, I'm still not convinced it
> can be.
>
> I'm in complete agreement with the author of the page.
>
> > I
> > don't think introducing that kind of redundancy into the prime form
> list is
> > going to do anything but create confusion. Noone said that different
> > inversional and transpositional forms of prime sets sound the same,
> thats
> > not the point. The point is reducibility. "Tonal" theory is a
> limiting case
> > of set theory, just like Newtonian physics is a limiting case of
> relativity.
> >
> > Dante
>
> Hi Dante. I must be totally ignorant of how this 'limiting' happens,
> but what you are saying seems impossible. If Forte's methodology
> eliminates the distinction between mirror inverses, how can any
> limiting case of it possible restore that distinction?
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

Since the distinction does exist in tonal theory, the analogy to
Newtonian and relativistic gravitation, or calling tonal theory
a 'limiting case' or 'special case' of Fortean set theory, seems
totally wrong. In what sense is it right?

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> Hi Paul-
>
> The distinction is not "restored", it simply doesn't exist from the
> set-theoretic perspective. Now, you may then say that this
perspective is
> therefore useless to "explain" tonal music, which may very well be.
But any
> music (tonal or not) can very well be >described< from a set-
theoretic
> perspective. Functional harmony, as a cultural construct, will not
> necessarily "show up" in this type of description. I find this kind
of set
> stuff more useful for precompositional material than analysis (see
Carter's
> "Harmony" book).
>
> Dante
>
> > -----Original Message-----
> > From: Paul Erlich [mailto:perlich@a...]
> > Sent: Tuesday, December 02, 2003 2:07 AM
> > To: tuning-math@yahoogroups.com
> > Subject: [tuning-math] Re: Enumerating pitch class sets
algebraically
> >
> >
> > --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
wrote:
> > > > Thanks. I found a url by googling for this, which gives the
> > complete
> > > > 351 (or 352, for the null chord is listed as number zero.)
> > > >
> > > > http://music.theory.home.att.net/pcsets.htm#Explanation%20of%
> > 20This%
> > > > 20Table
> > >
> > > Interesting. I didn't know Forte's methodology could be
challenged.
> > After
> > > reading the explanation on this page, I'm still not convinced it
> > can be.
> >
> > I'm in complete agreement with the author of the page.
> >
> > > I
> > > don't think introducing that kind of redundancy into the prime
form
> > list is
> > > going to do anything but create confusion. Noone said that
different
> > > inversional and transpositional forms of prime sets sound the
same,
> > thats
> > > not the point. The point is reducibility. "Tonal" theory is a
> > limiting case
> > > of set theory, just like Newtonian physics is a limiting case of
> > relativity.
> > >
> > > Dante
> >
> > Hi Dante. I must be totally ignorant of how this 'limiting'
happens,
> > but what you are saying seems impossible. If Forte's methodology
> > eliminates the distinction between mirror inverses, how can any
> > limiting case of it possible restore that distinction?
> >
> >
> >
> > To unsubscribe from this group, send an email to:
> > tuning-math-unsubscribe@yahoogroups.com
> >
> >
> >
> > Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
> >
> >

All I meant was in set theory [0,3,7] is just another trichord with no
priveleged status. Maybe a better analogy is how Euclidean geometry is "just
another geometry" within generalized geometry theory?

Dante

> -----Original Message-----
> From: Paul Erlich [mailto:perlich@aya.yale.edu]
> Sent: Tuesday, December 02, 2003 3:36 AM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Enumerating pitch class sets algebraically
>
>
> Since the distinction does exist in tonal theory, the analogy to
> Newtonian and relativistic gravitation, or calling tonal theory
> a 'limiting case' or 'special case' of Fortean set theory, seems
> totally wrong. In what sense is it right?
>
> --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> > Hi Paul-
> >
> > The distinction is not "restored", it simply doesn't exist from the
> > set-theoretic perspective. Now, you may then say that this
> perspective is
> > therefore useless to "explain" tonal music, which may very well be.
> But any
> > music (tonal or not) can very well be >described< from a set-
> theoretic
> > perspective. Functional harmony, as a cultural construct, will not
> > necessarily "show up" in this type of description. I find this kind
> of set
> > stuff more useful for precompositional material than analysis (see
> Carter's
> > "Harmony" book).
> >
> > Dante
> >
> > > -----Original Message-----
> > > From: Paul Erlich [mailto:perlich@a...]
> > > Sent: Tuesday, December 02, 2003 2:07 AM
> > > To: tuning-math@yahoogroups.com
> > > Subject: [tuning-math] Re: Enumerating pitch class sets
> algebraically
> > >
> > >
> > > --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
> wrote:
> > > > > Thanks. I found a url by googling for this, which gives the
> > > complete
> > > > > 351 (or 352, for the null chord is listed as number zero.)
> > > > >
> > > > > http://music.theory.home.att.net/pcsets.htm#Explanation%20of%
> > > 20This%
> > > > > 20Table
> > > >
> > > > Interesting. I didn't know Forte's methodology could be
> challenged.
> > > After
> > > > reading the explanation on this page, I'm still not convinced it
> > > can be.
> > >
> > > I'm in complete agreement with the author of the page.
> > >
> > > > I
> > > > don't think introducing that kind of redundancy into the prime
> form
> > > list is
> > > > going to do anything but create confusion. Noone said that
> different
> > > > inversional and transpositional forms of prime sets sound the
> same,
> > > thats
> > > > not the point. The point is reducibility. "Tonal" theory is a
> > > limiting case
> > > > of set theory, just like Newtonian physics is a limiting case of
> > > relativity.
> > > >
> > > > Dante
> > >
> > > Hi Dante. I must be totally ignorant of how this 'limiting'
> happens,
> > > but what you are saying seems impossible. If Forte's methodology
> > > eliminates the distinction between mirror inverses, how can any
> > > limiting case of it possible restore that distinction?
> > >
> > >
> > >
> > > To unsubscribe from this group, send an email to:
> > > tuning-math-unsubscribe@yahoogroups.com
> > >
> > >
> > >
> > > Your use of Yahoo! Groups is subject to
> http://docs.yahoo.com/info/terms/
> > >
> > >
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

Dante,

That would be fine if tonal theory did nothing more than say [0,3,7]
was a priveledged trichord, etc. But it does more than that -- it
distinguishes two instances of the trichord, one the mirror inverse
of the other, as well as, for example, two instances of the
tetrachord [0,2,6,9], one the mirror inverse of the other, which have
very different functions!

Not only that, but it distinguishes, functionally, enharmonically
equivalent sonorities, that not only cannot be distinguished in
Fortean set theory, but can't be distinguished *physically* in 12-
equal without looking at the surrounding context.

However, Newtonian theory cannot make any distinctions that cannot be
made in Relativity theory (except, perhaps, for physically
meaningless, useless vestiges of Newton's philosophy, such as
Absolute Space -- but don't tell me dominant seventh vs. half-
diminished seventh is physically meaningless!), nor can Euclidean
geometry make any distinctions that cannot also be made in
generalized geometry theory.

-Paul

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> All I meant was in set theory [0,3,7] is just another trichord with
no
> priveleged status. Maybe a better analogy is how Euclidean geometry
is "just
> another geometry" within generalized geometry theory?
>
> Dante
>
> > -----Original Message-----
> > From: Paul Erlich [mailto:perlich@a...]
> > Sent: Tuesday, December 02, 2003 3:36 AM
> > To: tuning-math@yahoogroups.com
> > Subject: [tuning-math] Re: Enumerating pitch class sets
algebraically
> >
> >
> > Since the distinction does exist in tonal theory, the analogy to
> > Newtonian and relativistic gravitation, or calling tonal theory
> > a 'limiting case' or 'special case' of Fortean set theory, seems
> > totally wrong. In what sense is it right?
> >
> > --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
wrote:
> > > Hi Paul-
> > >
> > > The distinction is not "restored", it simply doesn't exist from
the
> > > set-theoretic perspective. Now, you may then say that this
> > perspective is
> > > therefore useless to "explain" tonal music, which may very well
be.
> > But any
> > > music (tonal or not) can very well be >described< from a set-
> > theoretic
> > > perspective. Functional harmony, as a cultural construct, will
not
> > > necessarily "show up" in this type of description. I find this
kind
> > of set
> > > stuff more useful for precompositional material than analysis
(see
> > Carter's
> > > "Harmony" book).
> > >
> > > Dante
> > >
> > > > -----Original Message-----
> > > > From: Paul Erlich [mailto:perlich@a...]
> > > > Sent: Tuesday, December 02, 2003 2:07 AM
> > > > To: tuning-math@yahoogroups.com
> > > > Subject: [tuning-math] Re: Enumerating pitch class sets
> > algebraically
> > > >
> > > >
> > > > --- In tuning-math@yahoogroups.com, "Dante Rosati"
<dante@i...>
> > wrote:
> > > > > > Thanks. I found a url by googling for this, which gives
the
> > > > complete
> > > > > > 351 (or 352, for the null chord is listed as number zero.)
> > > > > >
> > > > > > http://music.theory.home.att.net/pcsets.htm#Explanation%
20of%
> > > > 20This%
> > > > > > 20Table
> > > > >
> > > > > Interesting. I didn't know Forte's methodology could be
> > challenged.
> > > > After
> > > > > reading the explanation on this page, I'm still not
convinced it
> > > > can be.
> > > >
> > > > I'm in complete agreement with the author of the page.
> > > >
> > > > > I
> > > > > don't think introducing that kind of redundancy into the
prime
> > form
> > > > list is
> > > > > going to do anything but create confusion. Noone said that
> > different
> > > > > inversional and transpositional forms of prime sets sound
the
> > same,
> > > > thats
> > > > > not the point. The point is reducibility. "Tonal" theory is
a
> > > > limiting case
> > > > > of set theory, just like Newtonian physics is a limiting
case of
> > > > relativity.
> > > > >
> > > > > Dante
> > > >
> > > > Hi Dante. I must be totally ignorant of how this 'limiting'
> > happens,
> > > > but what you are saying seems impossible. If Forte's
methodology
> > > > eliminates the distinction between mirror inverses, how can
any
> > > > limiting case of it possible restore that distinction?
> > > >
> > > >
> > > >
> > > > To unsubscribe from this group, send an email to:
> > > > tuning-math-unsubscribe@yahoogroups.com
> > > >
> > > >
> > > >
> > > > Your use of Yahoo! Groups is subject to
> > http://docs.yahoo.com/info/terms/
> > > >
> > > >
> >
> >
> >
> > To unsubscribe from this group, send an email to:
> > tuning-math-unsubscribe@yahoogroups.com
> >
> >
> >
> > Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
> >
> >

Paul-

But set theory is mathematically precise whereas tonal theory is not. In
painting, for example, different colors are scientifically describeable by
freq or wavelength of light, but when painters discuss how colors interact
on a canvas, even if they come up with "rules" it is subjective and
cultural. Western common-practice functional harmony is really an art, a
cultural construct, not a science. PC set theory is a science. So, set
theory can say that 7-35 (major scale) is likely to be musically interesting
because it has unique interval vector entries, but it cannot say how
functional harmony came out of this simple fact. Even if one traces the
dominant-tonic relationship to the harmonic series, its not inevitable that
this aspect (out of many) of the harmonic series must be made foundational.

So I think my analogies fall down because tonal theory and set theory are
apples and oranges: one is an arbitrary cultural construct and the other is
a abstract mathematical descriptive contraption that maps onto notes, if one
wishes.

Dante

> -----Original Message-----
> From: Paul Erlich [mailto:perlich@aya.yale.edu]
> Sent: Tuesday, December 02, 2003 10:32 AM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Enumerating pitch class sets algebraically
>
>
> Dante,
>
> That would be fine if tonal theory did nothing more than say [0,3,7]
> was a priveledged trichord, etc. But it does more than that -- it
> distinguishes two instances of the trichord, one the mirror inverse
> of the other, as well as, for example, two instances of the
> tetrachord [0,2,6,9], one the mirror inverse of the other, which have
> very different functions!
>
> Not only that, but it distinguishes, functionally, enharmonically
> equivalent sonorities, that not only cannot be distinguished in
> Fortean set theory, but can't be distinguished *physically* in 12-
> equal without looking at the surrounding context.
>
> However, Newtonian theory cannot make any distinctions that cannot be
> made in Relativity theory (except, perhaps, for physically
> meaningless, useless vestiges of Newton's philosophy, such as
> Absolute Space -- but don't tell me dominant seventh vs. half-
> diminished seventh is physically meaningless!), nor can Euclidean
> geometry make any distinctions that cannot also be made in
> generalized geometry theory.
>
> -Paul
>
>
> --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> > All I meant was in set theory [0,3,7] is just another trichord with
> no
> > priveleged status. Maybe a better analogy is how Euclidean geometry
> is "just
> > another geometry" within generalized geometry theory?
> >
> > Dante
> >
> > > -----Original Message-----
> > > From: Paul Erlich [mailto:perlich@a...]
> > > Sent: Tuesday, December 02, 2003 3:36 AM
> > > To: tuning-math@yahoogroups.com
> > > Subject: [tuning-math] Re: Enumerating pitch class sets
> algebraically
> > >
> > >
> > > Since the distinction does exist in tonal theory, the analogy to
> > > Newtonian and relativistic gravitation, or calling tonal theory
> > > a 'limiting case' or 'special case' of Fortean set theory, seems
> > > totally wrong. In what sense is it right?
> > >
> > > --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
> wrote:
> > > > Hi Paul-
> > > >
> > > > The distinction is not "restored", it simply doesn't exist from
> the
> > > > set-theoretic perspective. Now, you may then say that this
> > > perspective is
> > > > therefore useless to "explain" tonal music, which may very well
> be.
> > > But any
> > > > music (tonal or not) can very well be >described< from a set-
> > > theoretic
> > > > perspective. Functional harmony, as a cultural construct, will
> not
> > > > necessarily "show up" in this type of description. I find this
> kind
> > > of set
> > > > stuff more useful for precompositional material than analysis
> (see
> > > Carter's
> > > > "Harmony" book).
> > > >
> > > > Dante
> > > >
> > > > > -----Original Message-----
> > > > > From: Paul Erlich [mailto:perlich@a...]
> > > > > Sent: Tuesday, December 02, 2003 2:07 AM
> > > > > To: tuning-math@yahoogroups.com
> > > > > Subject: [tuning-math] Re: Enumerating pitch class sets
> > > algebraically
> > > > >
> > > > >
> > > > > --- In tuning-math@yahoogroups.com, "Dante Rosati"
> <dante@i...>
> > > wrote:
> > > > > > > Thanks. I found a url by googling for this, which gives
> the
> > > > > complete
> > > > > > > 351 (or 352, for the null chord is listed as number zero.)
> > > > > > >
> > > > > > > http://music.theory.home.att.net/pcsets.htm#Explanation%
> 20of%
> > > > > 20This%
> > > > > > > 20Table
> > > > > >
> > > > > > Interesting. I didn't know Forte's methodology could be
> > > challenged.
> > > > > After
> > > > > > reading the explanation on this page, I'm still not
> convinced it
> > > > > can be.
> > > > >
> > > > > I'm in complete agreement with the author of the page.
> > > > >
> > > > > > I
> > > > > > don't think introducing that kind of redundancy into the
> prime
> > > form
> > > > > list is
> > > > > > going to do anything but create confusion. Noone said that
> > > different
> > > > > > inversional and transpositional forms of prime sets sound
> the
> > > same,
> > > > > thats
> > > > > > not the point. The point is reducibility. "Tonal" theory is
> a
> > > > > limiting case
> > > > > > of set theory, just like Newtonian physics is a limiting
> case of
> > > > > relativity.
> > > > > >
> > > > > > Dante
> > > > >
> > > > > Hi Dante. I must be totally ignorant of how this 'limiting'
> > > happens,
> > > > > but what you are saying seems impossible. If Forte's
> methodology
> > > > > eliminates the distinction between mirror inverses, how can
> any
> > > > > limiting case of it possible restore that distinction?
> > > > >
> > > > >
> > > > >
> > > > > To unsubscribe from this group, send an email to:
> > > > > tuning-math-unsubscribe@yahoogroups.com
> > > > >
> > > > >
> > > > >
> > > > > Your use of Yahoo! Groups is subject to
> > > http://docs.yahoo.com/info/terms/
> > > > >
> > > > >
> > >
> > >
> > >
> > > To unsubscribe from this group, send an email to:
> > > tuning-math-unsubscribe@yahoogroups.com
> > >
> > >
> > >
> > > Your use of Yahoo! Groups is subject to
> http://docs.yahoo.com/info/terms/
> > >
> > >
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> Paul-
>
> But set theory is mathematically precise whereas tonal theory is
>not.

Eytan Agmon, for one, might disagree with you there.

> So, set
> theory can say that 7-35 (major scale) is likely to be musically
interesting
> because it has unique interval vector entries, but it cannot say how
> functional harmony came out of this simple fact.

It didn't (in my opinion), and the unique interval vector entries
only occur when the diatonic scale is assumed to be in 12-equal, a
reversal of historical facts.

> Even if one traces the
> dominant-tonic relationship to the harmonic series, its not
inevitable that
> this aspect (out of many) of the harmonic series must be made
foundational.

I have a different view of the dominant-tonic relationship, as my
paper shows.

> So I think my analogies fall down because tonal theory and set
theory are
> apples and oranges: one is an arbitrary cultural construct and the
other is
> a abstract mathematical descriptive contraption that maps onto
notes, if one
> wishes.

I'm still in complete agreement with the keeper of that music theory
page that Fortean set theory is severely deficient even as an
abstract mathematical contraption that maps onto notes, because the
classing together of a pitch set and its mirror inverse is aurally
indefensible. Let's have mathematical precision, and let's
distinguish what needs to be distinguished at the same time. There's
no reason we can't have both.

But where do you draw the line then? If inversions are distinct, why not
transpositions? Why not distinguish pitches in different octaves, since
these too are aurally distinguishable? I think if you're going to go the
reductionist route (Forte) then go all the way, and at least have that to
play with. The ways in which 0,3,7 and 0,4,7 are the same is real, not
imaginary, AND they are different as well, on another level. All I'm saying
is that the level that they are different on is not the one that set theory
is talking about.

As far as tonal theory being a science, you only have to look at or try to
analyze some Brahms passages, or Wagner et al to see that it is far from
being so (IMO).

Dante

> -----Original Message-----
> From: Paul Erlich [mailto:perlich@aya.yale.edu]
> Sent: Tuesday, December 02, 2003 12:24 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Enumerating pitch class sets algebraically
>
>
> --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> > Paul-
> >
> > But set theory is mathematically precise whereas tonal theory is
> >not.
>
> Eytan Agmon, for one, might disagree with you there.
>
> > So, set
> > theory can say that 7-35 (major scale) is likely to be musically
> interesting
> > because it has unique interval vector entries, but it cannot say how
> > functional harmony came out of this simple fact.
>
> It didn't (in my opinion), and the unique interval vector entries
> only occur when the diatonic scale is assumed to be in 12-equal, a
> reversal of historical facts.
>
> > Even if one traces the
> > dominant-tonic relationship to the harmonic series, its not
> inevitable that
> > this aspect (out of many) of the harmonic series must be made
> foundational.
>
> I have a different view of the dominant-tonic relationship, as my
> paper shows.
>
> > So I think my analogies fall down because tonal theory and set
> theory are
> > apples and oranges: one is an arbitrary cultural construct and the
> other is
> > a abstract mathematical descriptive contraption that maps onto
> notes, if one
> > wishes.
>
> I'm still in complete agreement with the keeper of that music theory
> page that Fortean set theory is severely deficient even as an
> abstract mathematical contraption that maps onto notes, because the
> classing together of a pitch set and its mirror inverse is aurally
> indefensible. Let's have mathematical precision, and let's
> distinguish what needs to be distinguished at the same time. There's
> no reason we can't have both.
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

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

> Yes Gene, "prime form" is a canonic representative of
> each set-class orbit--I think Allen Forte invented this name. It's
usually
> defined as the most compact, then most left-packed member of the
> equivalence class, but "least right-packed" is technically more
accurate.

It seems to me the simplest definition would be to say it is the
smallest number among the orbit of sets if we take the sets to be
numbers base 2--that is, the sum 2^i for i in I.

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

> Interesting. I didn't know Forte's methodology could be challenged.
After
> reading the explanation on this page, I'm still not convinced it
can be.

I'm not sure what you mean, but certainly we can take permutation
groups of various kinds, and do enumerations for any of them. In this
case we are doing the cyclic group of order 12, C12, but if inversion
is included we would get the dihedral group D12.

I
> don't think introducing that kind of redundancy into the prime form
list is
> going to do anything but create confusion.

For some purposes, obviously, we want to regard a major triad as
different from a minor triad.

Noone said that different
> inversional and transpositional forms of prime sets sound the same,
thats
> not the point. The point is reducibility. "Tonal" theory is a
limiting case
> of set theory, just like Newtonian physics is a limiting case of
relativity.

I know little of "set theory" beyond a wish people wouldn't call it
that, since "set theory" has an establshed meaning in mathematics,
but this analogy is at best puzzling.

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> Hi Paul-
>
> The distinction is not "restored", it simply doesn't exist from the
> set-theoretic perspective. Now, you may then say that this
perspective is
> therefore useless to "explain" tonal music, which may very well be.
But any
> music (tonal or not) can very well be >described< from a set-
theoretic
> perspective.

This can't possibly be true, since drums can make music.

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

> All I meant was in set theory [0,3,7] is just another trichord with
no
> priveleged status.

And 12 is just another equal division of the octave with no
priivledge status, and equal divisions are without a priveledged
status either.

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

> But where do you draw the line then? If inversions are distinct,

Not inversions, just inverses.

> why not
> transpositions?

I can "play" any familiar piece of music in my head, but as realistic
as it sounds to me, it often turns out to be in the wrong key.
Transposition seems to make little aural difference.

> Why not distinguish pitches in different octaves, since
> these too are aurally distinguishable?

One can construct a theory that does this, but I think octave-
similarity (and if you believe Agmon, true *octave-equivalence* in
harmonic *function*) allows one to make great simplifications in the
space of possibilities to consider without throwing out too-coarse
distinctions.

I think if you're going to >go the
> reductionist route (Forte) then go all the way, and at least have
>that to
> play with.

I think most differently from you here. Besides, one could go "even
further" and, say, not distinguish a pitch set from its complement,
or what have you . . . I have a pretty firm sense of where a
reasonable place is to draw the line, and in that I seem to be in
close agreement with the author of the webpage in question.

> The ways in which 0,3,7 and 0,4,7 are the same is real, not
> imaginary,

Not really -- the patterns of coinciding partials, of combinational
tones, of just about everything that distinguishes a *physical*
realization of these chords from their Fortean set theoretic
abstractions, are markedly different in character for these two
chords. The Fortean set theoretic abstraction would apply very well
to a system of objects for which, by viewing them at a different
angle, we would see the order of the intervals reversed. For example,
Fortean set theory would desribe excellently the arrangement of
tokens on an unmarked clock face, given that we are allowed to both
rotate the clock face by an arbitrary angle as well as being able to
flip it around and view it from behind (without ever knowing which
side is the "front" and which is the "back")

> AND they are different as well, on another level. All I'm saying
> is that the level that they are different on is not the one that
set theory
> is talking about.

Does that latter level have any perceptual or musical relevance? I
would argue, "not a whole lot".

Question authority -- think for yourself! (and I went to the same
school Forte was prof at . . .)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
wrote:
>
> > All I meant was in set theory [0,3,7] is just another trichord
with
> no
> > priveleged status.
>
> And 12 is just another equal division of the octave with no
> priivledge status, and equal divisions are without a priveledged
> status either.

Take that, Forteans!

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

> But set theory is mathematically precise whereas tonal theory is
not.

Can you clue the ignorant here? I see the mathemaical precision in
looking at permutation groups acting on sets, but it seems to me that
this by itself doesn't take us very far. Moreover, one can classify
chords and their relationships in contexts other than equal divisions
of the octave under the assumption of octave-equivalence.

In
> painting, for example, different colors are scientifically
describeable by
> freq or wavelength of light, but when painters discuss how colors
interact
> on a canvas, even if they come up with "rules" it is subjective and
> cultural.

Far more is involved with color than this, but let's not go astray.

> So I think my analogies fall down because tonal theory and set
theory are
> apples and oranges: one is an arbitrary cultural construct and the
other is
> a abstract mathematical descriptive contraption that maps onto
notes, if one
> wishes.

Eh? I think you've got it backwards. Tonal music relates to how your
ears hear, whereas using 12 notes to the octave without reference to
he fact that 12 provides good approximations seems like an arbitrary
cultural construct.

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

> But where do you draw the line then? If inversions are distinct,
why not
> transpositions? Why not distinguish pitches in different octaves,
since
> these too are aurally distinguishable? I think if you're going to
go the
> reductionist route (Forte) then go all the way, and at least have
that to
> play with.

Assuming the permutation group in question is D12 rather than C12 is
hardly going all the way. I suppose S12, the symmetric group, would
be that, which ends up saying two chords modulo octaves are the same
if and only if they have the same number of notes. But 12 is well-
supplied with other groups--the sporadic groups M12 and M11, the
special linear group Sl(2,11) which equates the 12 notes with the
projective line mod 11, and so forth. What *musical* meaning these
would have I don't know, but unless you are willing to relate your
classification to psychoacoustics rather than dismissing that as a
cultural contruct, you are not in a good position to dismiss such
classiifications.

The ways in which 0,3,7 and 0,4,7 are the same is real, not
> imaginary, AND they are different as well, on another level. All
I'm saying
> is that the level that they are different on is not the one that
set theory
> is talking about.

It is if your group is C12, and it isn't if your group is D12. What's
the big deal?

> As far as tonal theory being a science, you only have to look at or
try to
> analyze some Brahms passages, or Wagner et al to see that it is far
from
> being so (IMO).

It won't be any more scientific simply to look at sets of
equivalences class when analyzing Brahms, will it? Or are you saying
Brahms wrote unscientific music?

>PC set theory is a science.

What does PC stand for?

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >PC set theory is a science.

hrmm . . .

> What does PC stand for?
>
> -C.

pitch-class.

>> >PC set theory is a science.
>
>hrmm . . .

For the record, I didn't write that bit.

>> What does PC stand for?
>>
>> -C.
>
>pitch-class.

Thanks!

-Carl

I dont get it- who said 12 is anything but the system that is most used? Of
course you can generalize these methods to any edo you want. The fact
remains that reams of music has been written by composers thinking in this
way, like it or not. The fact remains that there are certain properties of
pc sets that remain invariant under certain transformations, and that some
sets are reducable to a single prime form. Is it a coincidence that the
major and minor triad are two versions of the same prime form? No. Is this
approach the One True System? Of course not- but it is elegant and useful
within a certain domain, that domain being music written in this way. I'm
not so sure about using it to analyze music that was NOT written this way,
like the Rite for example. Can a composer write a piece in which 037 and 047
are treated as transforms of each other. Of course. So whats the beef? If
you want to use tonal theory, fine. If you want to use atonal "set theory"
fine. Cant they all just get along? But I dont see the point of trying to
get one to be the other or visa versa.

Dante

> -----Original Message-----
> From: Paul Erlich [mailto:perlich@aya.yale.edu]
> Sent: Tuesday, December 02, 2003 1:29 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Enumerating pitch class sets algebraically
>
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
> wrote:
> >
> > > All I meant was in set theory [0,3,7] is just another trichord
> with
> > no
> > > priveleged status.
> >
> > And 12 is just another equal division of the octave with no
> > priivledge status, and equal divisions are without a priveledged
> > status either.
>
> Take that, Forteans!
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

> Eh? I think you've got it backwards. Tonal music relates to how your
> ears hear, whereas using 12 notes to the octave without reference to
> he fact that 12 provides good approximations seems like an arbitrary
> cultural construct.

All music is cultural construct. Its just a matter of what you're familiar
with and what floats your boat. 12 edo may have arisen as approximations of
ji but once its here, you can play with it any damn way you want. Some
people like to play with it as some kind of mod12 clockface and project it
into sound. If composer A writes a piece this way and listener B digs it,
then thats all the "justification" necessary (if you're into
justifications).

Dante

> > As far as tonal theory being a science, you only have to look at or
> try to
> > analyze some Brahms passages, or Wagner et al to see that it is far
> from
> > being so (IMO).
>
> It won't be any more scientific simply to look at sets of
> equivalences class when analyzing Brahms, will it? Or are you saying
> Brahms wrote unscientific music?

No, I'm saying Brahms wrote music that, at times, exhibits ambiguity when
subjected to traditional harmonic analysis. And no, I'm not saying Fortean
analysis will tell you anything here. My point was that the ambiguity
demonstrates that harmonic analysis is more of an art than a science.

Dante

> > AND they are different as well, on another level. All I'm saying
> > is that the level that they are different on is not the one that
> set theory
> > is talking about.
>
> Does that latter level have any perceptual or musical relevance? I
> would argue, "not a whole lot".

I repeat- if someone writes a piece using this equivilence, and someone else
likes how it sounds, then it is relevant to the music in question.

> Question authority -- think for yourself!

Thanks Paul, I never would have thot of that. ;-)

Dante

> I repeat- if someone writes a piece using this equivilence,
> and someone else likes how it sounds, then it is relevant to
> the music in question.

Actually, listener enjoyment by itself isn't justification for
anything.

-Carl

I'm going to go shoot myself now.

> -----Original Message-----
> From: Carl Lumma [mailto:ekin@lumma.org]
> Sent: Tuesday, December 02, 2003 8:07 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Enumerating pitch class sets algebraically
>
>
> > I repeat- if someone writes a piece using this equivilence,
> > and someone else likes how it sounds, then it is relevant to
> > the music in question.
>
> Actually, listener enjoyment by itself isn't justification for
> anything.
>
> -Carl
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

[Dante]
>> > I repeat- if someone writes a piece using this equivilence,
>> > and someone else likes how it sounds, then it is relevant to
>> > the music in question.
>>
>> [Carl]
>> Actually, listener enjoyment by itself isn't justification for
>> anything.
>
> [Dante]
>I'm going to go shoot myself now.

People enjoy all sorts of things. For an algorithmic comp. method
to be justified it should at least produce results that are distinct
from other methods. That means listeners should be able to identify
it. Now PC Set Theory may meet this condition, although it probably
demands some training. I certainly have nothing against PC Set Theory
or training (fugues certainly take some training to fully appreciate).
In fact, I'd like to learn more about PC Set Theory...

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

() Was it started/coined by Babbitt?

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

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

-Carl

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

> I dont get it- who said 12 is anything but the system that is most
>used? Of
> course you can generalize these methods to any edo you want.

But you're still restricted to equal divisions!

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
>
> > Eh? I think you've got it backwards. Tonal music relates to how
your
> > ears hear, whereas using 12 notes to the octave without reference
to
> > he fact that 12 provides good approximations seems like an
arbitrary
> > cultural construct.
>
> All music is cultural construct. Its just a matter of what you're
familiar
> with and what floats your boat.

I recommend reading this paper:

http://homepage.mac.com/cariani/CarianiWebsite/CarianiNP99.pdf

*harmony*, to a certain degree, is innate.

> If composer A writes a piece this way and listener B digs it,
> then thats all the "justification" necessary (if you're into
> justifications).

Agreed. Music speaks louder than words, theory, etc . . .

--- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...> wrote:
> > > As far as tonal theory being a science, you only have to look
at or
> > try to
> > > analyze some Brahms passages, or Wagner et al to see that it is
far
> > from
> > > being so (IMO).
> >
> > It won't be any more scientific simply to look at sets of
> > equivalences class when analyzing Brahms, will it? Or are you
saying
> > Brahms wrote unscientific music?
>
> No, I'm saying Brahms wrote music that, at times, exhibits
ambiguity when
> subjected to traditional harmonic analysis. And no, I'm not saying
Fortean
> analysis will tell you anything here. My point was that the
ambiguity
> demonstrates that harmonic analysis is more of an art than a
science.
>
> Dante

Or it might just demonstrate that Bramhs's music exhibits ambiguity --
maybe because he wanted it to! Anyway, I don't think any of these
modalities of musical analysis are anywhere near a "science", but
certainly ambiguity is something that can be understood, described,
and predicted in a scientific way. For example, the pitch of an
inharmonic spectrum, as I've been discussing with Kurt on the tuning
list lately.

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

> > Question authority -- think for yourself!
>
> Thanks Paul, I never would have thot of that. ;-)

I thought the person who created that webpage we were looking at made
a very valid case for the challenge he(?) was making to Forte, and I
expressed my support for it. You said you didn't think Forte was
possible to challenge, or something like that. That got me going!

Hi Carl:

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

"Serial technique" sounds like compostional technique, pc set theory is more
analytical.

> () Was it started/coined by Babbitt?

Forte attributes the term and concept of pc sets to Babbitt. I'm not sure
how much else he came up with prior to Forte's systematization in "The
Structure of Atonal Music". There's also Lewin's "Musical Intervals and
Transformations" which is over my head mathematically but the parts that I
have looked at where he develops a vector concept applicable to intervals
(in both atonal and tonal contexts, btw) and time intervals is interesting.

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

Not intended to be perscriptive, but this way of looking at pc sets is used
by Carter and expecially Babbitt. See Carter's "Harmony Book" which is
basically lists and tables of sets and chords that he uses for his
compositions.

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

Not sure what you mean here. It starts with Schoenberg and Webern, developed
by Boulez and Babbitt Carter etc, and then carried on by multitudes of
others in various approaches. I love Boulez's radical "total serialization"
works from the 50s- Piano Sonata 2 and the two books of Structures for two
pianos, Marteau Sans Maitre, etc. He's still going strong- recent works
include Sur Incise & Explosante fixe, though I'm not sure how serial or pc
set his process is these days. Babbitt and Carter I can do without for the
most part, although I'm open to learning to hear what they're doing some
day. I suppose Babbitt is the ultimate example of this type of thinking
applied to composition.

I believe a piece of music must work as sound first, and not rely on
familiarity with techniques to appreciate it. If a piece is only
interesting if you know how it was put together, then it doesn't cut it for
me. Certain symmetries and transformations are audible in, e.g, Webern's
Symphonie, STring Quartet and Piano Variations due to the simplicity of
presentation. Mostly you can't really hear this stuff, especially in dense
presentations, but thats neither here nor there- either it sounds good to
you or not- thats the bottom line. It does take alot of listening to give it
a fair chance, though.

Dante

> > > Question authority -- think for yourself!
> >
> > Thanks Paul, I never would have thot of that. ;-)
>
> I thought the person who created that webpage we were looking at made
> a very valid case for the challenge he(?) was making to Forte, and I
> expressed my support for it. You said you didn't think Forte was
> possible to challenge, or something like that. That got me going!

I just think that in what Forte tried to do, limited as that may be, its
pretty much cut and dried. Either you're going to accept the methodology of
normal orger, best normal order and prime form, or not. If you do, then you
get his tables. Is the extended enumeration of forms found on that webpage
derivable from a set of simple rules like forte's best normal order, etc?

BTW, can anyone point to a published critique of Forte's methodolgy?

Dante

Dante,

Thanks for the crash-course. I've got Carter's string quartets,
which I'm not fond of but which Norman Henry thinks are the bomb,
and there are few whose opinion on music I respect more than his.
He once played me a piano piece by Carter that I thought was quite
good, but I forget which one it was.

I have a disc with Babbitt's Elizabethan Sextette and some others
which I am listening to for the 2nd time as I write this. I find
I like the contrapuntal nature of it, but not the harmonic nature
of it. The two aren't mutually exclusive, so it seems like just
throwing away an opportunity for harmony and tonal matter.

I've never heard Boulez, except some excerpts of one of his piano
sonatas on Amazon. As for Schoenberg, his early string quartets I
think are some of the best music I've ever heard, but I don't
think they're serial. I remember liking a piano piece from 32
short films about Glenn Gould, which I later obtained a recording
of, which I think *was* serial...

...looks like it was either the Gigue from the Op. 25 Suite for
Piano, or Little Pieces for Piano (Op. 19)...

Ok, it was the Gigue. Gould says, "I can think of no composition
for solo piano from the first 1/4 of this century which can stand
as its equal. Nor is my affection for it influenced by S.'s
total reliance on 12-tone procedures. ... From out of an arbitrary
rationale of elementary mathematics and debatable historical
perception came a rare joie de vivre, a blessed enthusiasm for the
making of music."

Sounds like ol' Glenn was on to something there.

It's long been a hunch of mine that I (or you) could take the rules
of Forte et al and change them arbitrarily and as often as not use
the result to carry out just as effective an analysis, or compose
just as listenable a composition, as with the real rules. If my
hunch were wrong, that's what it takes to justify such rules. It's
the kind of hunch that's very wrong for most of the common practice
theory of Brahms' day, and of the stuff we do here on tuning-math.

-Carl

hi Dante and paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
wrote:
> > > >
> > > > As far as tonal theory being a science, you only
> > > > have to look at or try to analyze some Brahms passages,
> > > > or Wagner et al to see that it is far from
> > > > being so (IMO).
> > >
> > > It won't be any more scientific simply to look at sets
> > > of equivalences class when analyzing Brahms, will it?
> > > Or are you saying Brahms wrote unscientific music?
> >
> > No, I'm saying Brahms wrote music that, at times,
> > exhibits ambiguity when subjected to traditional harmonic
> > analysis. And no, I'm not saying Fortean analysis will
> > tell you anything here. My point was that the ambiguity
> > demonstrates that harmonic analysis is more of an art
> > than a science.
> >
> > Dante
>
> Or it might just demonstrate that Bramhs's music exhibits
> ambiguity -- maybe because he wanted it to! Anyway, I don't
> think any of these modalities of musical analysis are
> anywhere near a "science", but certainly ambiguity is
> something that can be understood, described, and predicted
> in a scientific way. For example, the pitch of an inharmonic
> spectrum, as I've been discussing with Kurt on the tuning
> list lately.

i think the main reason harmonic analysis would be
characterised as an "art" is precisely *because* of
the ambiguity available to a composer like Brahms,
whether his intended tuning is 12edo or a meantone
(the only two likely possibilities for Brahms IMO).

my point: that *temperament* allows composers to play
the kinds of games ("punning") that aren't possible
in JI. and of course JI is the tuning which offers
the straightforward "scientific" approach to harmonic
analysis.

in reality, JI harmonic analysis is merely a lot simpler
than those cases in which temperament must considered.
but harmonic analysis of tempered music can indeed be
put on a similarly scientific basis, which i think is
mostly what's going on on this list every day.

... and yes, for the JI enthusiasts out there: it is
certainly possible to compose harmonically ambiguous
music in JI as well. as with temperaments, the business
of periodicity-blocks and unison-vectors tells the story.
but in contrast to temperaments, JI music like this *exposes*
the tiny discrepancies rather than eliminates them.

-monz

hi Carl and Dante,

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

> Hi Carl:
>
> > () Was it [the term "pitch class"] started/coined by Babbitt?
>
> Forte attributes the term and concept of pc sets to Babbitt.
> I'm not sure how much else he came up with prior to Forte's
> systematization in "The Structure of Atonal Music". There's
> also Lewin's "Musical Intervals and Transformations"

the term "pitch-class" was coined by Babbitt in his 1960
paper "Twelve-Tone Invariants as Compositional Determinants"
(The Musical Quarterly, vol 46, p 247).

thanks for asking about it, Carl ... it prompted me to
add a citation to Babbitt in my definition:

http://sonic-arts.org/dict/pc.htm

> > () What's the best piece for a beginner to start with, and what
> > should he listen for?
>
> Not sure what you mean here. It starts with Schoenberg
> and Webern, developed by Boulez and Babbitt Carter etc,
> and then carried on by multitudes of others in various
> approaches. I love Boulez's radical "total serialization"
> works from the 50s- Piano Sonata 2 and the two books of
> Structures for two pianos, Marteau Sans Maitre, etc. He's
> still going strong- recent works include Sur Incise &
> Explosante fixe, though I'm not sure how serial or pc
> set his process is these days. Babbitt and Carter I can
> do without for the most part, although I'm open to learning
> to hear what they're doing some day. I suppose Babbitt is
> the ultimate example of this type of thinking applied to
> composition.

Dante is right, Carl ... start at the beginning, with
Schoenberg's _Suite for Piano, op. 25_ (1921-23),
the first dodecaphonic serial piece.

http://www.wwnorton.com/classical/covers/52664.htm

the introduction of serialism also ushered in Schoenberg's
neoclassical style, which i never really cared for.
the pieces he composed from then until his "temporary death"
in 1946 are the ones of his which i like least. but then,
*after* he was revived ...

i'd say that by far the best introductions to
Schoenberg's serial work are his two late compositions
_String Trio_ (1946) and _A Survivor From Warsaw_ (1947).
both of those should knock your socks off. (i've raved
about both of them many times on these lists.)

here's a good "list of works" with links to recordings,
of Schoenberg's work:

http://www.usc.edu/isd/archives/schoenberg/as_disco/works/worklist.htm

then follow the development thru Berg and Webern.

Berg's _Lyric Suite_ for string quartet (1925-26) was the
first piece in which he used serialism (only in half of the
six movements), and of course his opera _Lulu_ (1929-35,
left unfinished at his death) and _Violin Concerto_ (1935)
are masterpieces.

(and of course there's also the non-serial opera _Wozzeck_,
which you shouldn't miss if you're going to delve into
this repertoire.)

*see* the operas if possible.

you can't get a proper understanding of Webern's life-work
if you don't give enough attention to his vocal works, which
make up more than half of his total output, and which
unfortunately is exactly what has happened in the literature.

but that said, the volume of literature with an analytical
focus on Webern's _Symphonie, op. 21_ (1927-28) and
_Piano Variations, op. 27_ (1935-36) does IMO do justice
to the quality of those two pieces. in addition, his
_String Quartet, op. 28_ (1936-38)is a marvel.

Boulez's _Le Marteau Sans Maitre_ (19654-55) is a stunning
virtuosic display of total serialism, and a piece which i
absolutely love, and see live every time i find a performance
of it.

Babbitt's work is similarly virtuosic in compositional
technique, and at the same time i find it very listenable.

in particular, i like his _Three Compositions for Piano_
(1947) and _2nd String Quartet_ (1954). Babbitt is a
huge fan of Broadway shows, and i think that somehow that
populist lyricism breaks thru his convoluted theoretical
apparatus.

a lot more has happened since 1955, but those are pretty
much the big "classics" of serialism.

a good book offering an alternative theoretical summary
to Forte's is George Perle's _Serial Composition and
Atonality_, published originally in 1962.

http://www.amazon.com/exec/obidos/tg/detail/-
/0520074300/ref=pd_sxp_f/104-5664280-6894348?v=glance&s=books

OR

http://tinyurl.com/ygec

> I believe a piece of music must work as sound first, and
> not rely on familiarity with techniques to appreciate it.
> If a piece is only interesting if you know how it was put
> together, then it doesn't cut it for me. Certain symmetries
> and transformations are audible in, e.g, Webern's Symphonie,
> STring Quartet and Piano Variations due to the simplicity of
> presentation. Mostly you can't really hear this stuff,
> especially in dense presentations, but thats neither here
> nor there- either it sounds good to you or not- thats the
> bottom line. It does take alot of listening to give it
> a fair chance, though.
>
> Dante

i find that as the years pass and i listen to and study
Webern and his music, it appeals to me more and more.

this is something that also continues to be true of Mahler
... altho i certainly loved his music a lot right from
the very beginning.

but what i find interesting is that very parallel, because
Webern admired Mahler with a fanaticism just like mine.

i hope that means that people will continue to admire *my*
compositions more and more as they study them! ;-)

-monz

>i'd say that by far the best introductions to
>Schoenberg's serial work are his two late compositions
>_String Trio_ (1946) and _A Survivor From Warsaw_ (1947).
>both of those should knock your socks off. (i've raved
>about both of them many times on these lists.)
//
>Berg's _Lyric Suite_ for string quartet (1925-26) was the
>first piece in which he used serialism (only in half of the
>six movements), and of course his opera _Lulu_ (1929-35,
>left unfinished at his death) and _Violin Concerto_ (1935)
>are masterpieces.
//
>but that said, the volume of literature with an analytical
>focus on Webern's _Symphonie, op. 21_ (1927-28) and
>_Piano Variations, op. 27_ (1935-36) does IMO do justice
>to the quality of those two pieces. in addition, his
>_String Quartet, op. 28_ (1936-38)is a marvel.
//
>Boulez's _Le Marteau Sans Maitre_ (19654-55) is a stunning
>virtuosic display of total serialism, and a piece which i
>absolutely love, and see live every time i find a performance
>of it.
>
>Babbitt's // _Three Compositions for Piano_ (1947) and
>_2nd String Quartet_ (1954). Babbitt is a huge fan of
>Broadway shows, and i think that somehow that populist
>lyricism breaks thru his convoluted theoretical apparatus.

Thanks monz!!!

-Carl

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i think the main reason harmonic analysis would be
> characterised as an "art" is precisely *because* of
> the ambiguity available to a composer like Brahms,
> whether his intended tuning is 12edo or a meantone
> (the only two likely possibilities for Brahms IMO).

Speaking as a veteran retuner, meantone is fine for Francois
Couperin, Meade Lux Lewis or Buddy Holly, but don't try it with
Brahms. It won't work. Speaking of which, I've recently finished a
grail version of his Piano Concerto #2, which does work, but I've
corrupted the piano into a jazz piano in this version. It makes me
think it wasn't just Beethoven who was one step away from total
boogie.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Dante and paul,
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, "Dante Rosati" <dante@i...>
> wrote:
> > > > >
> > > > > As far as tonal theory being a science, you only
> > > > > have to look at or try to analyze some Brahms passages,
> > > > > or Wagner et al to see that it is far from
> > > > > being so (IMO).
> > > >
> > > > It won't be any more scientific simply to look at sets
> > > > of equivalences class when analyzing Brahms, will it?
> > > > Or are you saying Brahms wrote unscientific music?
> > >
> > > No, I'm saying Brahms wrote music that, at times,
> > > exhibits ambiguity when subjected to traditional harmonic
> > > analysis. And no, I'm not saying Fortean analysis will
> > > tell you anything here. My point was that the ambiguity
> > > demonstrates that harmonic analysis is more of an art
> > > than a science.
> > >
> > > Dante
> >
> > Or it might just demonstrate that Bramhs's music exhibits
> > ambiguity -- maybe because he wanted it to! Anyway, I don't
> > think any of these modalities of musical analysis are
> > anywhere near a "science", but certainly ambiguity is
> > something that can be understood, described, and predicted
> > in a scientific way. For example, the pitch of an inharmonic
> > spectrum, as I've been discussing with Kurt on the tuning
> > list lately.
>
>
>
> i think the main reason harmonic analysis would be
> characterised as an "art" is precisely *because* of
> the ambiguity available to a composer like Brahms,
> whether his intended tuning is 12edo or a meantone
> (the only two likely possibilities for Brahms IMO).
>
> my point: that *temperament* allows composers to play
> the kinds of games ("punning") that aren't possible
> in JI.

Irrelevant -- Dante and I were talking about 'conventional' tonal
harmonic analysis, which never distinguishes any 81:80s anyway.

> and of course JI is the tuning which offers
> the straightforward "scientific" approach to harmonic
> analysis.

BS.

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> > i think the main reason harmonic analysis would be
> > characterised as an "art" is precisely *because* of
> > the ambiguity available to a composer like Brahms,
> > whether his intended tuning is 12edo or a meantone
> > (the only two likely possibilities for Brahms IMO).
> >
> > my point: that *temperament* allows composers to play
> > the kinds of games ("punning") that aren't possible
> > in JI.
>
> Irrelevant -- Dante and I were talking about 'conventional' tonal
> harmonic analysis, which never distinguishes any 81:80s anyway.

OK, my bad. i didn't follow the thread from the beginning
and probably should have just stayed out of it.

... in fact, my eyes have glazed over with nearly every
post i've seen here over the last week, except for this one.

> > and of course JI is the tuning which offers
> > the straightforward "scientific" approach to harmonic
> > analysis.
>
> BS.

i guess i didn't express myself clearly enough. you and
i both already know each other's viewpoints on this.

anyway, since i did miss so much here in the last week,
it's probably not worth it for me to try to clarify now ...

-monz