Tuning-Math Archive Section 9: 8604

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: