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Message: 8678 - Contents - Hide Contents Date: Thu, 04 Dec 2003 14:43:09 Subject: Re: enumerating pitch class sets algebraically From: Carl Lumma>> >an you give any examples of pre-serial atonal music? >>You cant do any better than Webern op 5-16. You can download mp3s of 0p 5 >and 6 (both landmark works) from here: > >Anton Webern * [with cont.] (Wayb.)Funny, sounds just like serial atonal music (to my novice ear). -Carl

Message: 8679 - Contents - Hide Contents Date: Thu, 04 Dec 2003 14:45:52 Subject: Re: enumerating pitch class sets algebraically From: Carl Lumma>> >nd while I'm on it, serial tonal music? >>Can't help you there,Is the final fugue of the WTC1 a serial piece? Why or why not? -Carl

Message: 8681 - Contents - Hide Contents Date: Thu, 04 Dec 2003 01:28:29 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:> Cool. This was what Jon Wild was talking about. I would like to see > how one works this into Polya's algorithm. I was introduced to this > One-Seven symmetry by John Rahn, its neat to see the group- theoretical > basis for this (D(4)XS(3)) ThanksI gave the polynomial for it in another posting. If we have a permutation group, then for each element we form a product of (x^d+1) for each d-cycle; the degree of the resulting polynomial is necesarily the degee of the permutation group. Dividing by the order of the group gives the polynomial. There is something similar I've long worked with, called the Molien series, by the way, where instead of (x^d+1) one takes 1/(1-x^d); expanding the resulting rational function in a power series gives the number of invariants of each degree m as the coefficient of x^m. That, I suppose, might have its uses also.

Message: 8682 - Contents - Hide Contents Date: Thu, 04 Dec 2003 01:31:34 Subject: Re: C(24,12) reduction for Z relations From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote:>>> 984+41676/2+138/3+12512/4+876/6+600/8+36/12=25220 which is >>> 2^2*5*13*97. >> As I was saying, the z-relations are based on all the divisors of > 24. (1,2,3,4,6,8,12). Now for the factorization of 25220. > One possible pattern is (3+1)(4+1)(12+1)(96+1). Anyone think I am > onto something here?Not everyone knows what a Z-relation is, I'm afraid. I recall seeing the term somewhere, but that's it. I'll have to analyze more of Jon Wild's sets> before I really know anything...C(24,6) reduces to 2635 which is > 5*17*31, kind of interestingThis is also confusing to me. C(24,6) normally means the number of combinations of 24 things taken 6 at a time.

Message: 8684 - Contents - Hide Contents Date: Thu, 04 Dec 2003 10:27:20 Subject: p-optimal and w-optimal linear temperament generators From: Paul Erlich These were posted on the tuning list in February; thought the other Paul, at least, might like to see them if he hasn't already: Yahoo groups: /tuning/files/perlich/pop.gif * [with cont.] Yahoo groups: /tuning/files/perlich/paj.gif * [with cont.] Yahoo groups: /tuning/files/perlich/woptimal.gif * [with cont.] Yahoo groups: /tuning/files/perlich/wopaj.gif * [with cont.]

Message: 8687 - Contents - Hide Contents Date: Fri, 05 Dec 2003 18:38:49 Subject: Re: Digest Number 862 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:>> Funny, sounds just like serial atonal music (to my novice ear). >> Yeah funnily enough even stochastic music with the right parameters will > sound like serial atonal music with just a casual listen.But completely atonal stochastic music with triadic harmony will not sound at all like what people think atonal music ought to sound like.>> It sounds ugly either way, but shouldn't that be "class-set", >> not "set-class"? And what is a "set-class correspondence"? >> A pitch is, for example, {16}. It belongs to the pitch-class {4}. > > A pitch-set is, e.g. {3,6,7}. It belongs to the set-class [014]. [014] > is an equivalence class of pitch sets, so its a pitch-set equivalence > class, or set-class for short.This is getting totally out of hand. A pitch denoted by an integer. A pitch-class is denoted by an element of Z/12Z, as an integer reduced to the range 0-11 mod 12. A set-class, short for set-pitch-class, is a element of an equivalence class of sets of pitch-classes under the dihedreal group acting as a permutation group on Z/12Z. What next? In general, one might for "pitch" want an element of a fintely generated free abelian group with a specififed mapping from the positive rationals, to the reals, or both, determining what pitch it is. In this case however we want a rank one group with a mapping T to the reals, which sends 0 to the base pitch B, such that n-->T(n)/B is a homomorphism. Then the number N such that T(N)/B = 2 is the number of octave divisions. For instance in the above case, we can map n to 261.2*2^(n/12) Hz, defining a specific pitch for it. Is "pitch" really the right word for this concept, given that it is only actually a pitch after we've mapped it? Next we can reduce this group modulo octaves, which means reducing a group isomorphic to Z modulo N to obtain Z/NZ. If N is 12, the residues mod 12 are {0,1,...11}; it seems to me that instead of the equivalence classes of numbers mod 12 you really are working with the residues in terms of nomenclature, so you could call them "residues" and not "pitch classes". That is, clearly you don't really mean "the pitch-class {4}" but the mod 12 residue 4, so why not call it that? The pitch-class containing 4 is not {4}, but {... -20, -8, 4, 16, 30, ...}, an infinite set, so you really do seem to be trying for this anyway. Now you can take a permutation group G on the residues, and a set s of residues, and define Pfred(s, G) as I did before--we associate a number Ba(s) to s by taking the sum 2^i for i in s. Then Pfred(G,s) is the least Ba(t) among all the sets t in the G-orbit of s. Whatever name one gave this, it certainly shouldn't be something as confusing as pitch-class-set!

Message: 8688 - Contents - Hide Contents Date: Fri, 05 Dec 2003 12:13:07 Subject: Re: Digest Number 862 From: Carl Lumma>> >s the final fugue of the WTC1 a serial piece? Why >> or why not? >>I feel like this is a trick question, but: No, because it's not written >with a tone-row. Can you derive the countersubject from the subject via >serial procedures?No trick questions from me, unless I'm tricking myself! So does the subject fail to be a tone row because some notes are used more than once before all of them are used? As for transforming it into the countersubject, can you give me two subjects of the same length that cannot be transformed into one another with serial procedures? I'll believe you if you say yes. What are the allowed serial procedures? transpostion? contrary motion? mirror inversion? retrograde? augmentation? diminution? others? -Carl

Message: 8691 - Contents - Hide Contents Date: Fri, 05 Dec 2003 13:33:48 Subject: Re: (unknown) From: Carl Lumma>but what else do you want to call it that makes sense to musicians?I don't think the Magical Floating Head will permit this argument on behalf of serialists, PC Set theorists, or whatever this lot call themselves. -Carl

Message: 8692 - Contents - Hide Contents Date: Fri, 05 Dec 2003 22:14:39 Subject: Re: Digest Number 862 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote: >>>> Funny, sounds just like serial atonal music (to my novice ear). >>>> Yeah funnily enough even stochastic music with the right parameters > will>> sound like serial atonal music with just a casual listen. >> But completely atonal stochastic music with triadic harmony will not > sound at all like what people think atonal music ought to sound like. >>>> It sounds ugly either way, but shouldn't that be "class-set", >>> not "set-class"? And what is a "set-class correspondence"? >>>> A pitch is, for example, {16}. It belongs to the pitch-class {4}. >> >> A pitch-set is, e.g. {3,6,7}. It belongs to the set-class [014]. > [014]>> is an equivalence class of pitch sets, so its a pitch-set > equivalence>> class, or set-class for short. >> This is getting totally out of hand. A pitch denoted by an integer. A > pitch-class is denoted by an element of Z/12Z, as an integer reduced > to the range 0-11 mod 12. A set-class, short for set-pitch-class, is > a element of an equivalence class of sets of pitch-classes under the > dihedreal group acting as a permutation group on Z/12Z. What next? > > In general, one might for "pitch" want an element of a fintely > generated free abelian group with a specififed mapping from the > positive rationals, to the reals, or both, determining what pitch it > is. In this case however we want a rank one group with a mapping T to > the reals, which sends 0 to the base pitch B, such that n-->T(n)/B is > a homomorphism. Then the number N such that T(N)/B = 2 is the number > of octave divisions. For instance in the above case, we can map n to > 261.2*2^(n/12) Hz, defining a specific pitch for it. Is "pitch" > really the right word for this concept, given that it is only > actually a pitch after we've mapped it? > > Next we can reduce this group modulo octaves, which means reducing a > group isomorphic to Z modulo N to obtain Z/NZ. If N is 12, the > residues mod 12 are {0,1,...11}; it seems to me that instead of the > equivalence classes of numbers mod 12 you really are working with the > residues in terms of nomenclature, so you could call them "residues" > and not "pitch classes". That is, clearly you don't really mean "the > pitch-class {4}" but the mod 12 residue 4, so why not call it that? > The pitch-class containing 4 is not {4}, but {... -20, -8, 4, 16, > 30, ...}, an infinite set, so you really do seem to be trying for > this anyway. > > Now you can take a permutation group G on the residues, and a set s > of residues, and define Pfred(s, G) as I did before--we associate a > number Ba(s) to s by taking the sum 2^i for i in s. Then Pfred(G,s) is > the least Ba(t) among all the sets t in the G-orbit of s. Whatever > name one gave this, it certainly shouldn't be something as confusing > as pitch-class-set!Somehow, pitch-class-set is immediately clear to me, while the above is practically indecipharable. And I'm no fan of PC set theory!

Message: 8693 - Contents - Hide Contents Date: Fri, 05 Dec 2003 23:30:34 Subject: Re: Digest Number 862 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:> Unfortunately, these are the names that have stuck; I'm afraid > that they are the accepted system of nomenclature for musical > set theory, no matter how wrong they are.I'm not saying they are wrong, but they are confusing in a rocco way. Moreover calling it "set theory" is a little bizarre. Remember this> bastardization of mathematical set theory for supposed musical > purposes originally had nothing to do with tunings outside of 12 > equal.It isn't a bastardization of set theory. Set theory as an area of study does not concern itself with finite sets; if you look at the cardinality of such sets you get numbers, and therefore number theory; and you can construct all the other animals of finite math, such as permutation groups, designs, finite geometries, matroids and so forth using finte sets. This is not set theory to a mathematician. Algebra, combinatorics and number theory are the relevant areas, not set theory. If a set isn't so big we aren't even sure if it can exist without self-contradiction set theorists think of it as tiny.> I agree that the nomenclature is mostly misleading and silly. But > then I also feel that terms like "unison vector" are silly.Point taken, but don't blame me for that one. :)> It may also be a good idea to keep in mind that it does become > tiresome to have to wade through a bunch of re-definitions of > conventional terminology just to get at any meaning out of > someone's writing.True, but failing to use standard math terminology is also a matter of redefinition.

Message: 8694 - Contents - Hide Contents Date: Fri, 05 Dec 2003 23:54:24 Subject: (unknown) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:>> This is getting totally out of hand. A pitch denoted by an integer. A >> pitch-class is denoted by an element of Z/12Z, as an integer reduced >> to the range 0-11 mod 12. A set-class, short for set-pitch-class, is >> a element of an equivalence class of sets of pitch-classes under the >> dihedreal group acting as a permutation group on Z/12Z. What next?> I'll tell you what's next: Klumpenhouwer networks, which describe > complexes of set-class relationships abstracted from any particular > set-class.I've seen the buzzword when googling; is there something online which gives the definition?> "Set-class" isn't short for "set-pitch-class". It's a class of sets of > pitches==>a class of pitch sets==>a "pitch-set class", if anything.This is why I'm complaining. If "pitch" is an integer, a "pitch- class" a mod 12 equivalence class of integerss, then a set of such classes is a set of pitch classes, and so should be a set-pitch-class. Isn't that what you are calling by the hideous name "set-class"?> Alternatively you could think of it as a class of sets of pcs, i.e. a > class of pc-sets==>"pc-set class" or "pitch-class-set class".Now you've got a set of sets of equivalence classes of sets, right? Is that a so-called "set-class"? Sure it> sounds ridiculous if you use full names like that, but in general it's > pretty easy to say something like "such-and-such a sc has 3 pcs from one > wholetone collection and 2 pcs from the other"It would be even better with less confusing name behind the acronyms, but a good beginning would be clear definitions, which I'm not sure we have here. We seem to be conflating pitches with inegers representing them, and equivalence classes mod 12 with mod 12 residues, sets of such classes with sets of residues, and then we try to sort out what, exactly, we are defining when we look at orbits of such sets of classes under a G-action, and take a representive of the orbit. This has gotten to the point where you need to get mathematical about it, and give definitions which you really mean.> I'm just reporting on what general accepted usage is. I don't much like > the name "pitch" either, really, but what else do you want to call it that > makes sense to musicians? Yes of course it's a mod 12 residue, but it's > easier to say "pc 4" (you're right, I shouldn't have written "pc {4}") > than "the mod 12 residue {... -20, -8, 4, 16, 30...}".{... -20, -8, 4, 16, 30...} isn't a residue mod 12, it is an equivalence class mod 12; it can also be called an element of the group Z/12Z, whose 12 elements are precisely these classes. The corresponding residue mod 12 would be 4. It's a useful> musical concept--it's nice to have a simple name to refer to any or all > E's and Fb's in any or every octave--so it gets a simple name for the > benefit of musicians.This is not simple, this is a crazy-quilt of needless complexity. Most musicians don't care that transposition and> inversion form a dihedral group acting as a permutation group on Z/12Z, > though presumably tuning-math readers do.Most musicians probably don't give much of a hoot for any of this, but what do music theorists call it, if not the usual? Tuning-math writers, then, can> use the terminology that everyone who's *not* a tuning-math writer uses, > or they can invent their own.Are you sure music theorists don't know we are talking about the dihedral group of degree 12? There's a certain amount of group awareness percolating through academia--e.g., Lewin.> Calling them "pitches" doesn't imply anything about the tuning. In this > sense of "pitches", a score contains the same pitches whether or not the > piece is performed at A-440 or A-445 or A-415.Is the definition of "pitch" an integer associated to standard musical notation in such a way that C corresponds to 0, sharp to the addition of 1, and flat to the subtraction of 1? If so, at some point it ought to be said; if not, whatever else it means ought to be given. I presume someone actually has done this?

Message: 8696 - Contents - Hide Contents Date: Fri, 05 Dec 2003 09:12:24 Subject: Re: enumerating pitch class sets algebraicallyy From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dante Rosati" <dante@i...> wrote:>> Can you give any examples of pre-serial atonal music? >> You cant do any better than Webern op 5-16. You can download mp3sof 0p 5 Do you know of a good source for midi versions of Webern?

Message: 8697 - Contents - Hide Contents Date: Fri, 05 Dec 2003 09:15:54 Subject: (unknown) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:>> How did we manage to get from sets to some kind of compositional >> process? >> Sorry, I can see how you thought that's what I meant, but I was talking > about the analytic process, since that's what is being "attacked". It's > easy for an analyst who wants to defend set-class analysis to find > set-class correspondances between sets in the music being analysed-- it's > harder to show that you're doing this in a meaningful way and not just > "cherry-picking" to get the sets you want.It sounds ugly either way, but shouldn't that be "class-set", not "set-class"? And what is a "set-class correspondence"?> In my opinion, the thing you can reproach set-class theory for, is that it > makes it easy to do bogus analyses that look like they're meaningful, > because readers will look at all the sets you've found that enjoy some > sort of set-class relationship, nod their heads and say "ah yes it's all > so logical", but when you look at the music you wonder "why did they pick > *those* sets and not others?"Same question re "set-class relationhip". How are these defined? Is this vertical, horizontal, or both?

Message: 8698 - Contents - Hide Contents Date: Fri, 05 Dec 2003 09:18:45 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:>>>> C(4) x C(3) = C(12) >>>> Order 12 >> * What do C and S stand for? (I've tried Eric Weisstein's site...)C(n) is the cyclic group of degree and order n, and S(n) is the symmetric group of degree n.>>>> 1/2[3:2]cD(4) = D(12) >> * What does the above stand for?D(12) is the dihedral group of degree 12 (and order 24)--the group of the dodecagon.

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