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Message: 5701

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!


top of page bottom of page up down Message: 5703 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.
top of page bottom of page up down Message: 5704 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]
top of page bottom of page up down Message: 5707 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.
top of page bottom of page up down Message: 5708 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? Ordinarily you 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 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. > 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
top of page bottom of page up down Message: 5709 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.
top of page bottom of page up down Message: 5711 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!
top of page bottom of page up down Message: 5714 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?
top of page bottom of page up down Message: 5715 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".
top of page bottom of page up down Message: 5717 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.)
top of page bottom of page up down Message: 5718 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 Setup *
top of page bottom of page up down Message: 5721 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 Setup *
top of page bottom of page up down Message: 5722 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 Setup *
top of page bottom of page up down Message: 5723 Date: Wed, 26 Dec 2001 11:43:43 Subject: Re: a different example From: monz Hi Gene and Paul, I'm finally getting around to replying to the "different example" periodicity-blocks we created a couple of days ago. Gene, you must have been sleepy, because there are three errors in your post; I'll correct them in the quote. > From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 3:31 PM > Subject: [tuning-math] a different example (was: coordinates from unison-vectors) > > > ... So anyway, I put in the [(3,5) unison-vector] matrix: > > ( 6 -14) > (-4 1) > > and could see that the resulting periodicity-block had a strong > correlation (in the sense of my meantone-JI implied lattices) > with the neighborhood of 2/7- to 3/11-comma meantone. > > The determinant is 50, so this agrees with my observation. > From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 8:04 PM > Subject: [tuning-math] Re: a different example > > > ... First I put the 2 back into the above commas, and get > > q1 = 2^23 3^6 5^-15 and q2 = 80/81. Er... there's a typo in that last exponent: it should be q1 = 2^23 3^6 5^-14. > ... I have one et, h50, such that h50(q1) = h50(q2) = 0. > Now I search for something where h(q1)=0 and h(q2)=1, > obtaining h19, h69, -h31 and -h81. The simplest of these > is h19, and I choose it. > > Next, I look for something such that h(q1)=1 and h(q2)=0, > and I get h16, -h34, -h84; I choose h16. > > Now I form a 3x3 matrix from these, and invert it: > > [ 50 19 16] > [ 79 30 25]^(-1) = > [116 44 37] > > [-10 -1 5] > [ 23 6 -14] > [ 4 -4 1] > > The rows of the inverted matrix correspond to the commas > q0 = 2^-10 3^-1 5 = 3125/3072 (small diesis), Typo: that should be q0 = 2^-10 3^-1 5^5 > q1 = 2^23 3^6 5^14, and q2=80/80. Typo: according to the signs on the integers in the bottom row of the last matrix, q2 = 80/81. If that's not right, then it should be q2 = [-4 4 -1] = 81/80. > > Now I calculate the scale; the nth step is > > scale[n] = q0^n * q1^round(19n/50) * q2^round(16n/50), > > where "round" rounds to the nearest integer. It doesn't matter > which paticular hn I selected when I do this, or where I start > and end; though my definition of "nearest integer" does matter. > > I got in this way: > > 1, 3125/3072, 128/125, 25/24, 82944/78125, 16/15, 625/576, > 3456/3125, 10/9, 15625/13824, 144/125, 125/108, 18432/15625, > 6/5, 625/512, 768/625, 5/4, 15625/12288, 32/25, 125/96, > 20736/15625, 4/3, 3125/2304, 864/625, 25/18, 110592/78125, > 36/25, 625/432, 4608/3125, 3/2, 15625/10368, 192/125, 25/16, > 24576/15625, 8/5, 625/384, 1024/625, 5/3, 15625/9216, 216/125, > 125/72, 27648/15625, 9/5, 3125/1728, 1152/625, 15/8, 78125/41472, > 48/25, 125/64, 6144/3125 > > How does this compare with your results? > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 8:33 PM > Subject: [tuning-math] Re: a different example (was: coordinates from unison-vectors) > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > With 50 notes, some arbitrary decision has to be made -- no note can > be exactly in the center, since 50 is an even number. But you should > be getting the following block or its reflection through the origin: > > > p5's M3's > ---- ----- > > 3 -7 > 4 -7 > 1 -6 > 2 -6 > 3 -6 > 4 -6 > 1 -5 > 2 -5 > 3 -5 > 0 -4 > 1 -4 > 2 -4 > 3 -4 > 0 -3 > 1 -3 > 2 -3 > 3 -3 > 0 -2 > 1 -2 > 2 -2 > -1 -1 > 0 -1 > 1 -1 > 2 -1 > -1 0 > 0 0 > 1 0 > -2 1 > -1 1 > 0 1 > 1 1 > -2 2 > -1 2 > 0 2 > -3 3 > -2 3 > -1 3 > 0 3 > -3 4 > -2 4 > -1 4 > 0 4 > -3 5 > -2 5 > -1 5 > -4 6 > -3 6 > -2 6 > -1 6 > -4 7 Sorry about the long quote, but I wanted both of these sets of data to be together here, because after going thru all the trouble of prime-factoring Gene's set, I see that it's identical to Paul's. Now, how does this compare with my results? Well... the shape and size of the periodicity-block is exactly the same; the only difference is that my block is not quasi-centered on 1/1, as Gene/Paul's is and as I wanted mine to be. I'd appreciate it if you guys could fix my pseudo-code so that I can use my own spreadsheet method and still get the correct results. Again, my spreadsheet is at Yahoo groups: /tuning-math/files/monz/matrix math.xls * Thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 5724 Date: Wed, 26 Dec 2001 01:37:58 Subject: Re: My top 5--for Paul From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > Yes. It is a fine example of the musical irrelevance of a flat > badness > > measure. I think musicians would rate it somewhere between 5 and > > infinity times as bad as the other four you listed. 50 notes for > one > > triad? The problem, as usual is that an error of 0.5 c is > > imperceptible and so an error of 0.0002 c is no better, and does > not > > compensate for a huge number of generators. Sorry if I'm sounding > like > > a stuck record. > > Let's not make decisions for musicians. Many theorists have delved > into systems such as 118, 171, and 612. We would be doing no harm to > have something to say about this range, even if we don't personally > feel that it would be musically useful. But Paul! You _are_ making decisions for musicians! You can't help but do so. Unless you plan to publish an infinite list of temperaments, the fact that you rate cases like this highly means that you will include fewer cases having more moderate numbers of gens per consonance. Shouldn't the question be rather whether you are making a _good_ decision for musicians? There's nothing terribly personal about the fact that an error of 0.5 c is imperceptible by humans. Theorists have delved into systems such as 118, 171, 612-tET, but has anything musical ever come of it? And if it has or does, surely we would be looking at subsets, not the entire 118 notes per octave etc. i.e. we'd be looking at temperaments within these ETs where consonant intervals are produced by considerably fewer than 50 notes in a chain (or chains) of generators.
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