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Message: 8604 - Contents - Hide Contents Date: Tue, 02 Dec 2003 23:17:52 Subject: Transitive groups of degree 12 and low order containing a 12-cycle From: Gene Ward Smith Here are generators for some of these; since they contain a 12-cycle the chords of these systems are transposible, and they might even make sense in musical terms. Groups containing 12-cycle C(4) x C(3) = C(12) Order 12 a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] S(3) x C(4) Order 24 a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] b := [[1, 5], [2, 10], [4, 8], [7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] 1/2[3:2]cD(4) = D(12) Order 24 a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] w := [[1, 11], [2, 10], [3, 9], [4, 8], [5, 7]] D(4) x C(3) Order 24 a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] k := [[1, 7], [3, 9], [5, 11]] [3^2]4 Order 36 e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] kk := [[12, 4, 8], [2, 6, 10]] [3^2]4' Order 36 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 a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] b := [[1, 5], [2, 10], [4, 8], [7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] k := [[1, 7], [3, 9], [5, 11]]

Message: 8605 - Contents - Hide Contents 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) * [with cont.] (Wayb.) 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 * [with cont.] (Wayb.) >>>> >>>> >> >> >>>> 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: 8607 - Contents - Hide Contents 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) * [with cont.] (Wayb.) 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 * [with cont.] (Wayb.) >>>> >>>> >> >> >>>> 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: 8611 - Contents - Hide Contents 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 ontonotes, 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.

Message: 8617 - Contents - Hide Contents Date: Tue, 02 Dec 2003 18:13:09 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.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.

Message: 8618 - Contents - Hide Contents Date: Tue, 02 Dec 2003 18:18:53 Subject: Re: Enumerating pitch class sets algebraically From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "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.

Message: 8619 - Contents - Hide Contents Date: Tue, 02 Dec 2003 18:20:38 Subject: Re: Enumerating pitch class sets algebraically From: Gene Ward Smith --- 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.This can't possibly be true, since drums can make music.

Message: 8621 - Contents - Hide Contents Date: Tue, 02 Dec 2003 18:22:58 Subject: Re: Enumerating pitch class sets algebraically From: Gene Ward Smith --- 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.

Message: 8624 - Contents - Hide Contents 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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