Tuning-Math Digests messages 8605 - 8629

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Message: 8605

Date: Tue, 02 Dec 2003 15:28:06

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

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@xxxxxxxxxxx.xxxx "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@xxxxxxxxxxx.xxx
> > 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@xxxxxxxxxxx.xxxx "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@xxxxxxxxxxx.xxx
> > > > Subject: [tuning-math] Re: Enumerating pitch class sets
> > algebraically
> > > >
> > > >
> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "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.)
> > > > > >
> > > > > > Table of Pitch Class Sets (Set Classes) *
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@xxxxxxxxxxx.xxx
> > > >
> > > >
> > > >
> > > > Your use of Yahoo! Groups is subject to
> > Yahoo! Terms of Service *
> > > >
> > > >
> >
> >
> >
> > To unsubscribe from this group, send an email to:
> > tuning-math-unsubscribe@xxxxxxxxxxx.xxx
> >
> >
> >
> > Your use of Yahoo! Groups is subject to 
Yahoo! Terms of Service *
> >
> >


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Message: 8607

Date: Tue, 02 Dec 2003 15:32:00

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

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@xxxxxxxxxxx.xxxx "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@xxxxxxxxxxx.xxx
> > 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@xxxxxxxxxxx.xxxx "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@xxxxxxxxxxx.xxx
> > > > Subject: [tuning-math] Re: Enumerating pitch class sets
> > algebraically
> > > >
> > > >
> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "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.)
> > > > > >
> > > > > > Table of Pitch Class Sets (Set Classes) *
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@xxxxxxxxxxx.xxx
> > > >
> > > >
> > > >
> > > > Your use of Yahoo! Groups is subject to
> > Yahoo! Terms of Service *
> > > >
> > > >
> >
> >
> >
> > To unsubscribe from this group, send an email to:
> > tuning-math-unsubscribe@xxxxxxxxxxx.xxx
> >
> >
> >
> > Your use of Yahoo! Groups is subject to 
Yahoo! Terms of Service *
> >
> >


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Message: 8611

Date: Tue, 02 Dec 2003 17:23:36

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "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.


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Message: 8624

Date: Tue, 02 Dec 2003 18:26:34

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "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 . . .)


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