Tuning-Math Archive Section 9: 8651

Message: 8651 - Contents - Hide Contents 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.

Message: 8652 - Contents - Hide Contents 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

Message: 8653 - Contents - Hide Contents 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!

Message: 8654 - Contents - Hide Contents Date: Wed, 03 Dec 2003 22:21:29 Subject: Re: Transitive groups of degree 12 and low order containing a 12-cycle From: Gene Ward Smith --- 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.

Message: 8655 - Contents - Hide Contents Date: Wed, 03 Dec 2003 03:21:52 Subject: Re: Transitive groups of degree 12 and low order containing a 12-cycle From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: Here are the polynomials for these, giving the number chords of each size

> C(4) x C(3) = C(12) > Order 12 x^12+x^11+6*x^10+19*x^9+43*x^8+66*x^7+80*x^6+66*x^5+43*x^4+19*x^3+6*x^2+x+1 > S(3) x C(4) > Order 24 x^12+x^11+5*x^10+12*x^9+28*x^8+38*x^7+48*x^6+38*x^5+28*x^4+12*x^3+5*x^2+x+1 > 1/2[3:2]cD(4) = D(12) > Order 24 x^12+x^11+6*x^10+12*x^9+29*x^8+38*x^7+50*x^6+38*x^5+29*x^4+12*x^3+6*x^2+x+1 > D(4) x C(3) > Order 24 x^12+x^11+5*x^10+13*x^9+28*x^8+40*x^7+50*x^6+40*x^5+28*x^4+13*x^3+5*x^2+x+1 > [3^2]4 > Order 36 x^12+x^11+5*x^10+9*x^9+22*x^8+26*x^7+36*x^6+26*x^5+22*x^4+9*x^3+5*x^2+x+1

The generators of this were given incorrectly, it should have been: e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] kk := [[12, 4, 8], [2, 10, 10]]

> [3^2]4' > Order 36 x^12+x^11+4*x^10+9*x^9+18*x^8+26*x^7+32*x^6+26*x^5+18*x^4+9*x^3+4*x^2+x+1

As distinct from:

> e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] > r := [[12, 4, 8], [2, 6, 10]] > D(4) x S(3) > Order 48 x^12+x^11+5*x^10+9*x^9+21*x^8+25*x^7+34*x^6+25*x^5+21*x^4+9*x^3+5*x^2+x+1

Message: 8656 - Contents - Hide Contents 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!

Message: 8657 - Contents - Hide Contents Date: Wed, 03 Dec 2003 04:36:50 Subject: Diatonic PCs From: Gene Ward Smith Partly because it is a smaller number and partly because it is a prime number, there are far few transitive permutation groups of degree 7 than there are of degree 12--in fact, only seven of them. These can be thought of in terms of 7-equal, or in terms of 7-note scales--the diatonic scale in particular. Here's the scoop: C(7) Cyclic group of order 7 (1,2,3,4,5,6,7) x^7+x^6+3*x^5+5*x^4+5*x^3+3*x^2+x+1 D(7) Dihedral group of order 14 (1,2,3,4,5,6,7) and (1,6)(2,5)(3,4) x^7+x^6+3*x^5+4*x^4+4*x^3+3*x^2+x+1 F21(7) Frobenius group of order 21 (1,2,3,4,5,6,7) and (1,2,4)(3,6,5) x^7+x^6+x^5+3*x^4+3*x^3+x^2+x+1 F42(7) Frobenius group of order 42 (affine line group) (1,2,3,4,5,6,7) and (1,3,2,6,4,5) x^7+x^6+x^5+2*x^4+2*x^3+x^2+x+1 L(3,2) Group of Fano plane of order 168 (1,2,3,4,5,6,7) and (1,2)(3,6) x^7+x^6+x^5+2*x^4+2*x^3+x^2+x+1 A7 Alternating group, order 7!/2 = 2520 x^7+x^6+x^5+x^4+x^3+x^2+x+1 S7 Symmetric group, order 7! = 5040 x^7+x^6+x^5+x^4+x^3+x^2+x+1 We have five cyclic triad forms, four dihedreal triad forms, three F21 triad forms, two F42 triad forms and two L(3,2) triad forms; for A7 and S7 all chords with the same number of notes are the same.

Message: 8658 - Contents - Hide Contents 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 . . .

Message: 8664 - Contents - Hide Contents Date: Wed, 03 Dec 2003 05:50:36 Subject: Re: Transitive groups of degree 12 and low order containing a 12-cycle From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> I recognize the first and third polynomials as being counts for 12- et > Tn-types and TnI-types respectively (for 12 down to 1). Is this just > a coincidence? Do these polynomials (also) have another function?

Not at all. What you are calling Tn types and TnI types should probably be called C(12) types and D(12) types or something of that sort, since this explicitly names the permutation group in question. What is Tn short for?

Message: 8665 - Contents - Hide Contents Date: Wed, 03 Dec 2003 00:59:32 Subject: yet another reason to buy a tablet PC From: Carl Lumma xThink * [with cont.] (Wayb.) xThink * [with cont.] (Wayb.) -Carl

Message: 8669 - Contents - Hide Contents 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

Message: 8672 - Contents - Hide Contents Date: Thu, 04 Dec 2003 18:35:30 Subject: Re: enumerating pitch class sets algebraically From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:

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

> It's really just a labelling scheme and an assertion that it's meaningful > to talk about set-classes, their abstract relationships, etc.

Isn't this almost the same as saying it is worth looking at all possible 12-et chords (or scales), under the assumption of octave equivalence? We then can go further and look at all chords assuming transpostion, or transposition and inversion, etc. Incidentally, I started thinking about this because it occurs to me that PC theory may be a good starting point for adaptive tuning. There is a small enough set of chords that one can work with it pretty easily in 12-et, especially after group reductions. Of course, we also want the context the chord is in, and this now begins to look like theory.

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

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

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

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

Message: 8673 - Contents - Hide Contents 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