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Message: 2700 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:48:46 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> OK, but CS ==> PB in all "reasonable" cases where the unison vectors >> are not "ridiculously large" relative to the step sizes -- right? >> (Clearly a definition of "ridiculously large" is needed.) >> Presumably, however you define it, PB requires convexity,I just said "any shape that tiles the plane". Certainly, I've also tended to impose convexity on top of the PB property whenever a JI, untempered scale is meant. Do any of your examples still apply?
Message: 2701 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:51:24 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> I just said "any shape that tiles the plane". Certainly, I've also > tended to impose convexity on top of the PB property whenever a JI, > untempered scale is meant.Does "shape" entail connectedness, or can it be scattered islands all over the place? I also wonder about my second example, for CS. Does it apply--you tell me!
Message: 2702 - Contents - Hide Contents Date: Tue, 25 Dec 2001 03:35:10 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> I have really no time to define all with precision on that forum. If you want more > > precision, you could find someone to translate that: the first definitions permitting > > to explicit my chord theorem which suggests the chordicity as next axiom,just > > before the closure giving then the abelian group.After struggling through this, I still don't know why you want to mess around with groupoids. Why not simply go to the abelian group, and stay there? You present some definitions, but they seem unmotivated, in other words.
Message: 2703 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:58:16 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> I just said "any shape that tiles the plane". Certainly, I've also >> tended to impose convexity on top of the PB property whenever a JI, >> untempered scale is meant. >> Does "shape" entail connectedness, or can it be scattered islands > all over the place?The latter. Especially as preimages of ETs, such constructs would be just fine.> I also wonder about my second example, for CS. Does it apply--you > tell me!I'll look at it!
Message: 2704 - Contents - Hide Contents Date: Tue, 25 Dec 2001 03:45:13 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> P. I don't go to the abelian group for closure is not possible (without temperament) > in finite JI group. How determine a pertinent region in an infinite JI group?This doesn't strike me as a very good reason--why not work within the group, and define whatever regions or limitations you need? Ordinarily you wouldnot limit yourself to a groupoid when there is a group available; in fact the opposite is more commonly seen--when a group is not immediately available, we construct it. The point is always to make things as easy and elegantas possible.
Message: 2705 - Contents - Hide Contents Date: Tue, 25 Dec 2001 09:00:22 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 1--9/8--5/4--4/3--1024/675--5/3--15/8 > > would be an example of a scale which is CS, but neither epimorphic > nor PB.OK you're probably right, but epimorphic does still look like PB, and all the examples that were ever made by Wilson, Grady, et. al., who introduced the CS terminology, were PBs. So it is them I am thinking about when I say CS.
Message: 2706 - Contents - Hide Contents Date: Tue, 25 Dec 2001 01:07:35 Subject: Re: For Pierre, from tuning From: Pierre Lamothe Gene wrote: This doesn't strike me as a very good reason--why not work within the group, and define whatever regions or limitations you need? Ordinarily you wouldnot limit yourself to a groupoid when there is a group available; in fact the opposite is more commonly seen--when a group is not immediately available, we construct it. The point is always to make things as easy and elegant as possible. Gene, It's not my style to define arbitrarily limitations I would need. Since centuries all searchers don't face the conflict between justness of chords and closure: they seek only the compromise of good temperaments. I choose another way. It may seem, at your viewpoint, less elegant to find the founding axioms of the paradigmatic operative structures in music. I don't think so. I would like to add I don't work with groupoid since groupoid has only one axiom which is precisely the closure axiom. I constructed the chordoid structure which have not the closure axiom, but all others axioms of the abelian group. Could you construct a finite JI group using independant primes? Pierre. [This message contained attachments]
Message: 2707 - Contents - Hide Contents Date: Tue, 25 Dec 2001 09:06:20 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Does "shape" entail connectedness, or can it be scattered islands >> all over the place? >> The latter. Especially as preimages of ETs, such constructs would be > just fine.Under that definition, PB <==> epimorphic. Are you sure it is the accepted one?
Message: 2708 - Contents - Hide Contents Date: Tue, 25 Dec 2001 06:23:49 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> Could you construct a finite JI group using independant primes?Every element of a finite group has a finite order. However, why would I want to look at a finite JI set as an algebraic object, unless I was going touse the morphisms of the corresponding category somehow? Is this what you do? I'm trying to find out how you think this point of view helps.
Message: 2709 - Contents - Hide Contents Date: Tue, 25 Dec 2001 09:12:55 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> Does "shape" entail connectedness, or can it be scattered islands >>> all over the place? >>>> The latter. Especially as preimages of ETs, such constructs would be >> just fine. >> Under that definition, PB <==> epimorphic. Are you sure it is the > accepted one?The only published articles on PBs are Fokker's. Inferring strict definitions from these articles would suggest that a parallelepiped (or N-dimensional equivalent) are the only accepted shape (thus I call these _Fokker_ periodicity blocks, or FPBs), and that if there is an even number of notes, one needs to produce two alternative versions so that symmetry about 1/1 is maintainted.
Message: 2710 - Contents - Hide Contents Date: Tue, 25 Dec 2001 06:29:07 Subject: Re: For Pierre, from tuning From: paulerlich --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> Gene wrote: > This doesn't strike me as a very good reason--why not work within the group, > and define whatever regions or limitations you need? Ordinarilyyou would not> limit yourself to a groupoid when there is a group available; in fact the > opposite is more commonly seen--when a group is not immediately available, we > construct it. The point is always to make things as easy and elegant as > possible. > Gene, > > It's not my style to define arbitrarily limitations I would need. > > Since centuries all searchers don't face the conflict betweenjustness of chords> and closure: they seek only the compromise of good temperaments. I choose > another way.Sounds like a motivation for periodicity blocks; I wish to understand how it motivates you in yet another direction.> It may seem, at your viewpoint, less elegant to find the founding > axioms of the paradigmatic operative structures in music.I seriously doubt Gene would say that. In fact, I bet he could lay out an axiomatic system for the researches we engage in most of the time on this list lately.> > Could you construct a finite JI group using independant primes?This looks _exactly_ like what we and especially Gene have been doing here on this list. Using a set of unison vectors, you define a periodicity block. Now, by treating each unison vector as an equivalence relation (choice of either chromatic or commatic equivalence), you get a finite group, constructed using independent primes. If there are no commatic equivalences you wish to temper out, you're done -- you have a JI block (whose precise ratios can be chosen in a variety of ways, corresponding to different choices for the shape (normally convex) and the position of the PB shape, subject to the constraint that the shape tile the plane with the right symmetry group). We've been doing all these sorts of things all along, and yet you claim to understand nothing. So what are you adding to our understanding? So far, all I can see in your work is lots of pretty numbers. How odd! A very merry christmas to you, my friend, Pierre! -Paul
Message: 2711 - Contents - Hide Contents Date: Tue, 25 Dec 2001 06:31:26 Subject: Re: For Pierre, from tuning From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote: >>> Could you construct a finite JI group using independant primes? >> Every element of a finite group has a finite order. However, why >would I want to look at a finite JI set as an algebraic object, >unless I was going to use the morphisms of the corresponding >category somehow? Is this what you do? I'm trying to find out how >you think this point of view helps.Gene, if you think you're speaking Pierre's language, is there any way you might be able to try and explain what we're (you're) doing, in _his_ language? He said it looks like nothing but numbers to him.
Message: 2712 - Contents - Hide Contents Date: Tue, 25 Dec 2001 07:01:43 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Gene, if you think you're speaking Pierre's language, is there any > way you might be able to try and explain what we're (you're) doing, > in _his_ language? He said it looks like nothing but numbers to him. For Pierre:The p-limit positive rationals 2^e1 ... p^ek form a free group of rank k under multiplication. An equal temperament can be viewed as an epimorphism ofthis group to Z. This can be defined in two ways--by giving the map (the equal temperament view) or giving the kernel in terms of a set of generatorsfor the kernel. Then we have the quesiton of the tuning of this system, which involves an injective mapping into the reals. Normally, we tune by making octaves pure. We've been looking at temperaments, which generalize the above to maps fromthe group of rank k to a free group of rank 1<m<k; here we traditionally refer to the dimension as m-1, so that rank 2 is a linear temperament, rank 3 planar, and so forth. Just as in the equal temperament case, we have two ways of defining the temperament, by means of a mapping or by defining the kernel, and we also have the question of the tuning injection of the temperament, and the optimal choices for the precise values involved; this is an optimization problem, which suggests using least squares or linear programming for a solution. The question of precisely how to define the goodness ofa temperament has much occupied us of late, and invovles Diophantine approximation theory, just as it does in the case of equal temperaments. In order to survey temperaments, we need to have a standard invariant form to reduce them to. One possibility is to use lattice basis reduction on thekernel, and another is to construct a standard form of the mapping; however I've discovered that invoking multilinear algebra in the form of the wedge product allows one to define an invariant, starting either from mappings or kernels, which uniquely defines the resulting temperament. We've been using this to survey temperaments. We've also devoted quite a lot of time to JI scales, mostly of those (Fokker blocks and the like) which are preimages of an equal temperament mapping.
Message: 2713 - Contents - Hide Contents Date: Tue, 25 Dec 2001 07:35:14 Subject: Re: For Pierre, from tuning From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> For Pierre: > > The p-limit positive rationals 2^e1 ... p^ek form a free group of >rank k under multiplication. An equal temperament can be viewed as >an epimorphism of this group to Z. This can be defined in two ways-- >by giving the map (the equal temperament view) or giving the kernel >in terms of a set of generators for the kernel.Which we call unison vectors.> We've also devoted quite a lot of time to JI scales, mostly of >those (Fokker blocks and the like) which are preimages of an equal >temperament mapping.Or of a mapping to a partially tempered system, where the unison vectors that are not tempered out are called "chromatic unison vectors" -- the 81:80 tempered, and 25:24 (or 135:128) chromatic, case, correctly accounts for 99% of Western pitch usage since 1480 -- including Zarlino!
Message: 2714 - Contents - Hide Contents Date: Tue, 25 Dec 2001 07:42:20 Subject: The epimorphic property From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> We've also devoted quite a lot of time to JI scales, mostly of those (Fokker blocks and the like) which are preimages of an equal temperament mapping.It occurs to me that this property, which is very important, hasn't been singled out or named so far as I know. I propose to call it the "epimorphic property". For instance, let's see if Margo's pelog-pentatonic is epimorphic. The scale is in 2^i 3^j 7^k, so we can leave 5 out of the map. If we denoteit by h, and if h(2)=a, h(3)=b and h(7)=c, we want h(28/27)=1, h(4/3)=2, h(3/2)=3. Solving the resulting linear equations gives a=5, b=8, c=15, and so h(14/9)=4. The scale therefore is epimorphic, or has the epimorphic property.
Message: 2715 - Contents - Hide Contents Date: Tue, 25 Dec 2001 07:44:26 Subject: Merry Christmas! From: genewardsmith Just felt like saying it. I hope no one objects.
Message: 2716 - Contents - Hide Contents Date: Tue, 25 Dec 2001 07:54:49 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> We've also devoted quite a lot of time to JI scales, mostly of >> those (Fokker blocks and the like) which are preimages of an equal >> temperament mapping. >> It occurs to me that this property, which is very important, hasn't > been singled out or named so far as I know.You're kidding? Isn't this equivalent to the PB property or the CS property for JI scales?> I propose to call it the "epimorphic property". For instance, let's >see if Margo's > pelog-pentatonic is epimorphic. > > The scale is in 2^i 3^j 7^k, so we can leave 5 out of the map. If >we denote it by h, and if h(2)=a, h(3)=b and h(7)=c, we want > h(28/27)=1, h(4/3)=2, h(3/2)=3. Solving the resulting linear >equations gives a=5, b=8, c=15, and so h(14/9)=4. The scale >therefore is epimorphic, or has the epimorphic property.It's a PB. It's CS. What's new?
Message: 2717 - Contents - Hide Contents Date: Tue, 25 Dec 2001 07:56:09 Subject: Re: Merry Christmas! From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Just felt like saying it. I hope no one objects.Merry Christmas to you too! Let mathematics be our path toward approaching the "idea of god".
Message: 2718 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:13:49 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> You're kidding? Isn't this equivalent to the PB property or the CS > property for JI scales?PB ==> epimorphic ==> CS but not conversely, if I've got the definitions right.
Message: 2719 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:17:11 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> You're kidding? Isn't this equivalent to the PB property or the CS >> property for JI scales? >> PB ==> epimorphic ==> CS but not conversely, if I've got the > definitions right.OK, but CS ==> PB in all "reasonable" cases where the unison vectors are not "ridiculously large" relative to the step sizes -- right? (Clearly a definition of "ridiculously large" is needed.)
Message: 2720 - Contents - Hide Contents Date: Tue, 25 Dec 2001 00:21:19 Subject: Re: a different example From: monz> From: dkeenanuqnetau <d.keenan@xx.xxx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 4:47 PM > Subject: [tuning-math] Re: a different example > > Here is 31-tET mapped onto the surface of a toroid as a 5-limit > lattice. If you print out the lattice below (in a monospaced font), > cut out the rectangle (cutting a half character width or height inside > the lines), loop and tape it first side to side and then top to > bottom, and you'll have it.Thanks, Dave! Actually, I seem to recall that you posted something like this once before. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2721 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:21:02 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> OK, but CS ==> PB in all "reasonable" cases where the unison vectors > are not "ridiculously large" relative to the step sizes -- right? > (Clearly a definition of "ridiculously large" is needed.)How does CS allow you to conclude you even have unison vectors? Having enough unison vectors to define the map is equivalent to being epimorphic, by the way.
Message: 2722 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:23:56 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> How does CS allow you to conclude you even have unison vectors? Having enough unison vectors to define the map is equivalent to being epimorphic, bythe way.Plus, the map has to correctly order the scale, so we do need a little more.
Message: 2723 - Contents - Hide Contents Date: Tue, 25 Dec 2001 00:37:40 Subject: Re: For Pierre, from tuning From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, December 24, 2001 10:29 PM > Subject: [tuning-math] Re: For Pierre, from tuning > > > --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote: >>> It's not my style to define arbitrarily limitations I would need. >> >> Since centuries all searchers don't face the conflict between >> justness of chords and closure: they seek only the compromise >> of good temperaments. I choose another way. >> Sounds like a motivation for periodicity blocks; I wish to understand > how it motivates you in yet another direction.That was exactly my first thought when I read this. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2724 - Contents - Hide Contents Date: Tue, 25 Dec 2001 00:43:47 Subject: Re: For Pierre, from tuning From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, December 24, 2001 11:01 PM > Subject: [tuning-math] Re: For Pierre, from tuning > > > We've also devoted quite a lot of time to JI scales, > mostly of those (Fokker blocks and the like) which are > preimages of an equal temperament mapping.Hmmm... this description sounds very much like what I'm trying to portray with my "acoustical rational implications of meantones" lattices. The JI periodicity-blocks I derive could be called "preimages of a meantone mapping", which in turn in many cases equate to an equal-temperament mapping. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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