Tuning-Math Digests messages 8651 - 8675

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Message: 8651

Date: Wed, 03 Dec 2003 22:18:29

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

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


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Message: 8652

Date: Wed, 03 Dec 2003 01:07:24

Subject: Re: Enumerating pitch class sets algebraically

From: Carl Lumma

> 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


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Message: 8653

Date: Wed, 03 Dec 2003 22:21:01

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

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


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Message: 8656

Date: Wed, 03 Dec 2003 22:28:33

Subject: Re: 301 "set theories"

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "hstraub64" <hstraub64@t...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> wrote:
> > There are 301 transitive permutation groups of degree 12, any one 
of 
> > which one could use as a the basis of a 12-et "set theory". 
Unless 
> > they contain a 12-cycle they will not equate things under 
> > transposition, but even those cases might be interesting, since 
they 
> > include groups of low order. In fact, there are five different 
> > transitive permutation groups of degree and order 12; one of 
these, 
> > of course, is the cyclic group of order 12. The others are
> > 
> > E(4) x C(3)
> > 
> > The 2-elementary group of order 4 (Klein 4 group) times the 
cyclic 
> > group of order 3.
> > 
> > Generators
> > 
> > (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> > (0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
> > (0, 3)(1, 10)(2, 5)(4, 7)(6, 9)(8, 11)
> > 
> > 
> > D6(6) x 2
> > 
> > Dihedral group of order 6, times cyclic group of order 2
> > 
> > Generators
> > 
> > (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> > (0, 1)(2, 3)(4, 5)(6, 7)(8, 9)(10, 11)
> > (0, 10)(1, 11)(2, 8)(3, 9)(4, 6)(5, 7)
> > 
> > 
> > A4(12)
> > 
> > The regular representation of the alternating group of degree 
four.
> > 
> > Generators
> > 
> > (0, 4, 8)(1, 11, 6)(2, 9, 7)(3, 10, 5)
> > (0, 11, 10)(1, 9, 5)(2, 4, 3)(6, 8, 7)
> > 
> > 
> > (1/2)[3:2]4
> > 
> > Generators: a, h, Z
> > 
> > (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> > (0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
> > (0, 3, 6, 9)(1, 8, 7, 2)(4, 11, 10, 5)
> > 
> > The names and generators are those found in "On Transitive 
> > Permutation Groups", Conway, Hulpke and McKay.
> 
> I think there is another one: the Mathieu group M12.

Gene didn't list all 301, so there is much more than just one other 
one! He's mentioned the Mathieu group quite a lot before!


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Message: 8658

Date: Wed, 03 Dec 2003 22:30:09

Subject: Re: Transitive groups of degree 12 and low order containing a 12-cycle

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> <paul.hjelmstad@u...> wrote:
> 
> >What application do the other polynomials have 
> > to music theory?
> 
> They enumerate distinct 12-equal chord forms under other 
permutation 
> groups. For instance, we might be interested in enumnerating 
> disctinct chords under D(4) x S(3); this group is generated by the 
> cyclic permutation giving transpositions, inversion, and the 
> operation of converting the circle of semitones to a circle of 
> fifths, by sending the ith note to 7i mod 12.

I think you've touched on something that was just being discussed on 
this list! Though I wasn't following very closely . . .


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Message: 8665

Date: Wed, 03 Dec 2003 00:59:32

Subject: yet another reason to buy a tablet PC

From: Carl Lumma

xThink *

xThink *

-Carl


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Message: 8669

Date: Wed, 03 Dec 2003 19:51:57

Subject: Re: Enumerating pitch class sets algebraically

From: Carl Lumma

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


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Message: 8673

Date: Thu, 04 Dec 2003 11:07:52

Subject: Re: enumerating pitch class sets algebraically

From: Carl Lumma

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


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