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Message: 8625 - Contents - Hide Contents Date: Tue, 02 Dec 2003 03:14:17 Subject: Re: polya counting method (fwd) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:> If you expand this polynomial (I advise the use of a software package > like Maple to do this), you get > > x^12 + x^11 + 6x^10 + 12x^9 + 29x^8 + 38x^7 + 50x^6 + 38x^5 + 29x^4 + > 12x^3 + 6x^2 + x + 1 > > and the coefficient of x^n in this expression is the number of n- chords.Thanks; I was just looking at this polynomial and trying to reverse engineer the process. This isn't hard at all, is it?

Message: 8626 - Contents - Hide Contents Date: Tue, 02 Dec 2003 18:28:32 Subject: Re: Enumerating pitch class sets algebraically From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- 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. >> 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!

Message: 8627 - Contents - Hide Contents Date: Tue, 02 Dec 2003 05:05:50 Subject: Re: Enumerating pitch class sets algebraically From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx 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.) Table of Pitch Class Sets (Set Classes) * [with cont.] (Wayb.) 20Table I wonder what other mathematical tidbits are back there in the stuff before I came on board?

Message: 8628 - Contents - Hide Contents Date: Tue, 02 Dec 2003 18:37:21 Subject: Re: Enumerating pitch class sets algebraically From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "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 ontonotes, 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.

Message: 8630 - Contents - Hide Contents Date: Tue, 02 Dec 2003 18:52:30 Subject: Re: Enumerating pitch class sets algebraically From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "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?

Message: 8631 - Contents - Hide Contents Date: Tue, 02 Dec 2003 07:06:46 Subject: Re: Enumerating pitch class sets algebraically From: Paul Erlich --- 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) * [with cont.] (Wayb.) 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. > > DanteHi 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?

Message: 8634 - Contents - Hide Contents Date: Tue, 02 Dec 2003 11:23:54 Subject: Re: Enumerating pitch class sets algebraically From: Carl Lumma>PC set theory is a science.What does PC stand for? -C.

Message: 8635 - Contents - Hide Contents Date: Tue, 02 Dec 2003 08:36:23 Subject: Re: Enumerating pitch class sets algebraically From: Paul Erlich 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) * [with cont.] (Wayb.) >> 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 * [with cont.] (Wayb.) >> >>

Message: 8636 - Contents - Hide Contents Date: Tue, 02 Dec 2003 19:27:08 Subject: Re: Enumerating pitch class sets algebraically From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> PC set theory is a science.hrmm . . .

Message: 8637 - Contents - Hide Contents Date: Tue, 02 Dec 2003 11:35:25 Subject: Re: Enumerating pitch class sets algebraically From: Carl Lumma>>> >C set theory is a science. >>hrmm . . .For the record, I didn't write that bit.

Message: 8640 - Contents - Hide Contents Date: Tue, 02 Dec 2003 21:03:47 Subject: 301 "set theories" From: Gene Ward Smith 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.

Message: 8646 - Contents - Hide Contents Date: Wed, 03 Dec 2003 12:35:03 Subject: Re: Enumerating pitch class sets algebraically From: Carl Lumma [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

Message: 8647 - Contents - Hide Contents Date: Wed, 03 Dec 2003 22:13:05 Subject: Re: Enumerating pitch class sets algebraically From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "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!

Message: 8648 - Contents - Hide Contents Date: Wed, 03 Dec 2003 22:14:31 Subject: Re: Enumerating pitch class sets algebraically From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "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 - Type Ok * [with cont.] (Wayb.) *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 . . .

Message: 8649 - Contents - Hide Contents Date: Wed, 03 Dec 2003 22:15:30 Subject: Re: 301 "set theories" From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "hstraub64" <hstraub64@t...> wrote:> I think there is another one: the Mathieu group M12. Or is that thelast one on the list?> M12 is quite interesting: not just transitive, but 5-transitive.I did say there were 301. M12 (order 95040) is number 295, M11(12) (order 7920) number 272, and M10(12) (order 720) is number 181. Many, many more, of course, including L(2,11) (order 660) at number 179, PGL(2,9) (order 720) at number 182, PGL(2,11) (order 1320) at number 218, and even A5(12) (order 60) at number 33.

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