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Message: 8678 Date: Thu, 04 Dec 2003 14:43:09 Subject: Re: enumerating pitch class sets algebraically From: Carl Lumma >>Can 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 * Funny, sounds just like serial atonal music (to my novice ear). -Carl
Message: 8679 Date: Thu, 04 Dec 2003 14:45:52 Subject: Re: enumerating pitch class sets algebraically From: Carl Lumma >>And 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: 8684 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 * Yahoo groups: /tuning/files/perlich/paj.gif * Yahoo groups: /tuning/files/perlich/woptimal.gif * Yahoo groups: /tuning/files/perlich/wopaj.gif *
Message: 8688 Date: Fri, 05 Dec 2003 12:13:07 Subject: Re: Digest Number 862 From: Carl Lumma >> Is 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 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 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!
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