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Message: 8425 - Contents - Hide Contents Date: Thu, 20 Nov 2003 22:36:00 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Dave Keenan wrote: >>> "m" here is the grade of the object, i.e. the number of nested >> brakets. A more intuitive (for me) alternative to m(m+1)/2 is >> Ceiling(m/2). If the sum of the indices plus this quantity is even >> then you negate it when complementing. >> That's an interesting short cut. The way I've been doing it is: > > Join the old basis on the left and the new basis on the right to get a > list of all primitive bases (or whatever they are). If this is an odd > permuatation, negate the coefficient. > > This assumes all bases are being stored in numerical order of their > components (what's all this talk about alphabetical order?) > > You can test for an odd permutation by swapping adjacent pairs of > primitive bases until their in the right order, and it's an odd > permutation if you did an odd number of swaps. This follows from the > antisymmetry of the wedge product, which is the main thing you need to > know about it. > You can also compare every pair of numbers in the basis, > and it's an odd permutation iff an odd number of them are the wrong way > round. > > > GrahamYes, the odd vs. even permutation thing is what (I think) Gene originally stated in this thread, and seems the clearest and most general way to think about it. What does 'primitive basis' mean? Mathworld doesn't seem to have this usage . . .

Message: 8426 - Contents - Hide Contents Date: Thu, 20 Nov 2003 22:36:49 Subject: Re: Finding the complement From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Dave Keenan wrote: >>> It should be mentioned that taking the complement of the complement >> doesn't always give you back what you started with, sometimes it's the >> negative of what you started with. So in those cases it's analogous to >> multiplying by i (the square root of -1). This depends on the >> dimension and the grade. But taking the complement four-times always >> gives you back exactly what you started with. >> Are you sure? Do you have an example?Yes. Disturbing isn't it? It occurs only for all odd grades in all even dimensions (where the dimension is the index of the limiting prime). So the simplest case is for a 3-limit vector (2D grade 1). The complement of <a b] is [-b a>, and so the complement of [-b a> must be <-a -b] which is -<a b] ~~<a b] = ~[-b a> = <-a -b] = -<a b] The fact that it occurs for 2D makes it clear there is no trick of reordering indices that is going to get rid of it. The next ocurrence is for 7-limit vectors and trivectors (4D grades 1 and 3). If you look at the 13-limit examples Gene gave in this thread, these are 6D so you should find that the vector, trivector and 5-vector complements have the same property.

Message: 8427 - Contents - Hide Contents Date: Thu, 20 Nov 2003 22:39:09 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> yes, as you know i (and especially graham) like that idea very much -- >> BUT 88cET has no octaves! >> You can make 88cET equivalent with respect to some other interval -- > like 88 cents for example. > > > GrahamWell, that would mean that one hears every pitch of this tuning as the same pitch class -- pretty absurd, really. I think octave similarity may never go away, so that any two notes whose ratio is an approximations to a power of 2 in such tunings will be heard as somewhat 'similar'.

Message: 8428 - Contents - Hide Contents Date: Thu, 20 Nov 2003 22:42:58 Subject: Re: Finding the compliment From: Paul Erlich> This is what Browne calls the Euclidean compliment,I tried to compliment you before but maybe i need to find the right compliment . . . oh, you're talking about the compl*e*ment!

Message: 8429 - Contents - Hide Contents Date: Thu, 20 Nov 2003 22:45:45 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> Grassman himself apparently used a prefix vertical bar. John Browne >> uses a horizontal bar above the symbol (or above a whole expression) >> exactly as you describe for logical complements. But this is usually >> translated to a prefix tilde ~ in ASCII, and it has the advantage of >> looking similar to a minus sign - which you say is more analogous, >> but is different from a prefix minus sign which would have the more >> obvious interpretation of negating _all_ the coefficients (and not >> reversing their order or the brakets). >

Message: 8430 - Contents - Hide Contents Date: Thu, 20 Nov 2003 22:52:44 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>>> OK -- but it's interesting to note that the cross product > immediately>>> gives you the quantity of interest in 3D, regardless of indexing >>> conventions. >>>> Paul. You must have missed where I explained that the cross-product >> stays the same no matter what the indexing conventions, because the >> wedge-product and the complement change in "complementary" ways when >> you change the indexing and A(x)B = ~(A^B). >> I don't know why you think I missed that, because (even if I did) > it's perfectly clear to me, and I was never confused about that. It > was the remark that followed that one which was my main point.Sorry Paul. I must have been reading my own confusion into what you wrote. I assumed you were hoping for an indexing convention that would make the 3D complement involve no changes of sign or reversals of ordering.

Message: 8431 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:02:24 Subject: Re: Finding the wedge product? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> the coefficient is 5. The result's value for e1e2 is 1+5=5.1*5=5, you musta meant.

Message: 8432 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:02:49 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Yes, the odd vs. even permutation thing is what (I think) Gene > originally stated in this thread, and seems the clearest and most > general way to think about it.Yes it's the most general and fundamental, but not the most practical for efficient computation, whether by human or machine. Certainly for humans, counting odd numbers in a numerically-ordered compound index will be far quicker and less error-prone.> What does 'primitive basis' mean? Mathworld doesn't seem to have this > usage . . .I think I used "simple basis" for the same thing, meaning a basis whose elements are not compounded of other basis elements, and therefore have a single-digit index in the indexing scheme I'm using. i.e. the basis of the vector, not the bi-vector or any higher grade multivector.

Message: 8433 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:05:17 Subject: Re: Finding the complement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Dave Keenan wrote: >>> It should be mentioned that taking the complement of the complement >> doesn't always give you back what you started with, sometimes it's the >> negative of what you started with. So in those cases it's analogous to >> multiplying by i (the square root of -1). This depends on the >> dimension and the grade. But taking the complement four-times always >> gives you back exactly what you started with. >> Are you sure? Do you have an example? > > > Grahamthat's easy -- in 3-dimensional space, the dual of e1^e2^e3 is 1, while the dual of 1 is -e1^e2^e3.

Message: 8434 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:07:30 Subject: Re: Finding Generators to Primes etc From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Imagine a triangle representing > > 1. Generators to Primes > 2. Commas > 3. Temperaments (such as 12&19) > > I am solid in my understanding of the leg between 2 and 1.(Even > though I understand that going from 1 to 2 is more difficult because > of contorsion). I have some understanding of the leg between 2 and 3 > (by mapping Linear Temperaments as lines on the Zoom diagrams, these > also represent commas, even though I am not sure how to extract them)The leg between 2 and 3 is actually the easiest, it seems to me. Our recent discussion on wedge products, with the particular example of cross products, should be helpful to you here.

Message: 8435 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:10:34 Subject: Re: Finding the complement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:>>> dimension and the grade. But taking the complement four-times always >>> gives you back exactly what you started with. >>>> Are you sure? Do you have an example? >> Yes. Disturbing isn't it? It occurs only for all odd grades in all > even dimensionsit occurs in odd dimensions too.> (where the dimension is the index of the limiting > prime). So the simplest case is for a 3-limit vector (2D grade 1). The > complement of <a b] is [-b a>, and so the complement of [-b a> must be > <-a -b] which is -<a b] > > ~~<a b] > = ~[-b a> > = <-a -b] > = -<a b] > > The fact that it occurs for 2D makes it clear there is no trick of > reordering indices that is going to get rid of it. The next ocurrence > is for 7-limit vectors and trivectors (4D grades 1 and 3).you missed 5-limit scalars and pseudoscalars (3D grades 0 and 3).

Message: 8436 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:12:09 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Yes, the odd vs. even permutation thing is what (I think) Gene >> originally stated in this thread, and seems the clearest and most >> general way to think about it. >> Yes it's the most general and fundamental, but not the most practical > for efficient computation, whether by human or machine. Certainly for > humans, counting odd numbers in a numerically-ordered compound index > will be far quicker and less error-prone. >>> What does 'primitive basis' mean? Mathworld doesn't seem to have this >> usage . . . >> I think I used "simple basis" for the same thing, meaning a basis > whose elements are not compounded of other basis elements, and > therefore have a single-digit index in the indexing scheme I'm using. > i.e. the basis of the vector, not the bi-vector or any higher grade > multivector.so the vector or 1-vector basis, yes?

Message: 8437 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:15:04 Subject: Re: Finding the compliment From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> This is what Browne calls the Euclidean compliment, >> I tried to compliment you before but maybe i need to find the right > compliment . . . oh, you're talking about the compl*e*ment!Come to think of it, what would a Euclidean compliment be? Perhaps something like, "My, your triangles are looking very congruent this morning Mrs Aristotle". :-)

Message: 8438 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:33:50 Subject: Re: Finding the complement From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> Dave Keenan wrote: >>>>> It should be mentioned that taking the complement of the > complement>>> doesn't always give you back what you started with, sometimes > it's the>>> negative of what you started with. So in those cases it's > analogous to>>> multiplying by i (the square root of -1). This depends on the >>> dimension and the grade. But taking the complement four-times > always>>> gives you back exactly what you started with. >>>> Are you sure? Do you have an example? >> that's easy -- in 3-dimensional space, the dual of e1^e2^e3 is 1, > while the dual of 1 is -e1^e2^e3.No, that second one is not correct. In other words you're saying that in 3D the complement of 1 is <<<-1]]]. By the rules posted recently, the coefficient is negated if and only if the sum of the indices plus Ceiling(g/2) is odd, where g is the grade. The grade of a scalar is zero and the sum of the indices is zero (since there aren't any). So you do not negate.

Message: 8439 - Contents - Hide Contents Date: Thu, 20 Nov 2003 23:46:38 Subject: Re: Finding the complement From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> you missed 5-limit scalars and pseudoscalars (3D grades 0 and 3).I don't think so. See page 10 of http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/bookpdf/TheComplement.pdf - Type Ok * [with cont.] (Wayb.)

Message: 8440 - Contents - Hide Contents Date: Fri, 21 Nov 2003 01:50:07 Subject: Re: Definition of val etc. From: Dave Keenan Gene, It looks like you're mostly still objecting to my lack of mathematical rigour, even when this is clearly being done in favour of educational efficiency. I really hoped we had got beyond that. There's room for both of us (both kinds of definition), really there is. --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > {{A "prime exponent mapping", sometimes shortened to "prime mapping", > "exponent mapping", "mapping" or simply "map", is a list of numbers > (integers) enclosed in < ... ] that tell you how a particular > temperament maps each prime number (up to some limit) to numbers of a > particular "generator" in that temperament.}} > > This assumes that all such mappings are (equal, and you need to say > that) temperaments, which is not true.How does it assume that? In the case of linear or higher-D temperaments we have more than one generator. The mapping from primes to a single one of those generators is still a val isn't it?> Also, simply calling it > a "map" won't work as a specific shorthand, since that already has a > well-established meaning, as another word for "function" which is > more often used in some contexts ("homomorphic map" being one > example.) Huh?Isn't this your definition of "val"?: Definitions of tuning terms: val, (c) 2001 by ... * [with cont.] (Wayb.) It starts: "A map ... ". How many other kinds of map do we use in this application of Grassman algebra, or in tuning theory in general?> I find "prime exponent mapping" too clumsy, too confusing, > and too verbose, and have no plans to use the term.Sure it's clumsy and verbose, but it's _meaningful_. I won't use it most of the time either (I'll use "map" or "mapping"), but what could be more confusing for a newbie than a term that carries absolutely no meaning for them whatsoever, except maybe as a person's name. (You reading, Monz?) I should think it would at least be clear from "prime exponent mapping" that whatever it is, it maps prime exponents to something (and from "map" that it maps something to something).> {{The prime numbers here represent frequency ratios.}} > > This is at best confusing; the prime numbers are prime numbers and > don't represent anything else. Tuning is another matter.I completely fail to understand how you could imagine that tuning is "another matter" in a tuning dictionary. What else could the primes represent, in a tuning dictionary.> {{When an interval is represented in the complementary form...}} > > "Complimentary form" is not a good phrase to use here.I agree, which is why I wrote "compl_e_mentary form". But assuming you don't like that either, please tell me why? You might suggest alternatives.> {{...as a prime-exponent-vector, we can find the number of generators > corresponding to it in some temperament by multiplying each number in > the temperament's map by the corresponding number in the vector, and > adding up the results.}} > > This is assuming the mapping in question defines an equal temperament > (and again leaves out the word equal), which is hardly always the > case.As I said, It does not assume equal temperaments at all. It applies equally well to finding the number of fourth generators for meantone (or the number of octave "generators"). Perhaps a more valid criticism of this definition is that it excludes prime mapping _matrices_, since these give you the counts of _all_ the generators at once. But these can be seen as a stack of prime-mapping pseudo-vectors (vals) one above the other, right? So we should extend the definition of prime-exponent-mapping and all its abbreviations (not including "val"), so that it includes these matrices. It will usually be clear from the context, and from the notation, whether one is talking about a (pseudo-)matrix or a (pseudo-)vector (val). If necessary, one can distinguish them by using the words "matrix" and "vector". And then my proposed preamble to the definition of val will need to say "When applied to tuning it usually represents a prime exponent mapping for a single generator of a temperament". With the appropriate links.> {{In mathematical terms this is called the dot-product, scalar- > product or inner-product of the map with the vector.}} > > This is unfortunately the case--unfortunate, in that these in most > contexts mean a product defined on a single vector space, not on a > vector space with its dual. As if that were not enough, "interior > product" in the context of exterior algebra has a specialized meaning > that we probably don't want to mess with. What about simply calling > it the bracket and leaving it at that?Gene, have you ever heard of the Principle of Parsimony, otherwise known as Ockham's Razor? "Entia non sunt multiplicanda praeter necessitatum" "Do not multiply entities beyond necessity" I agree it is necessary to distinguish this operation from a "true" dot-product in pure math and maybe in other applications, but since it is the only way we're using it in tuning, there is no need to confuse people with distinctions irrelevant to their application. It's a tuning dictionary. The actual manipulations of the numbers (the button presses on the calculator or the formulae in the spreadsheet) are the same as for a dot product, which some readers may at least have a vague memory of from high school, or be able to look up in an old textbook. If the reader's education proceeds in this area, they will eventually come to understand such distinctions, but nothing is gained by trying to include them all from the start. This is the difference between something that aims to educate or introduce people to something new, as opposed to a repository of precise definitions for reference by existing practitioners. I note that mathworld.com is pretty much one of the latter, which is why most of its definitions are next-to incomprehensible to a non-mathematician. I would hope that Monz's tuning dictionary would not become like that. In education we start by introducing simplified versions of things. Often they are _so_ simplified that an experienced practitioner could be forgiven for being horrified at the _lies_ being told. But that was one of the geniuses of Richard Feynman as a teacher of one of the most difficult and heavily mathematical subjects, quantum mechanics. He knew exactly what lies to tell (simplifications to make), and when, and how to appeal to the reader's/listener's existing knowledge and intuitions. I believe I've managed to avoid telling any actual lies in my proposed definitions in this thread, although I have of course left many things unsaid.

Message: 8441 - Contents - Hide Contents Date: Fri, 21 Nov 2003 15:16:17 Subject: Re: "does not work in the 11-limit" From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > George wrote:>> Anyway, I haven't had much incentive to figure it out, because I >> don't use midi channels (in Cakewalk) the same way Scala does -- I >> prefer to keep a single melodic line in a single track and channel >> You might be wrong about that, because the tracks in .seq files don't > correspond with midi channels. > So if you want to get rid of bothering with pitch-bend events, it > should be quite easy. You can transform the Cakewalk midi file to a > .seq file, where each midi channel will be mapped to a different > track. Then add the tuning to the .seq file, and transform it back to > a midi file. If the note numbers don't correspond exactly with the > scale degrees, you can use a keyboard mapping as well. > > ManuelI need to clarify what I said. I wanted to see if I could create midi files (consisting of only a single track) from scratch in Scala (which would save me the trouble of calculating and manually inserting pitch-bends), which I could then import into Cakewalk (one track at a time). Your Scala documentation indicates that pitch-bend events are minimized, so that you are constantly *changing channels* from one note to the next (rather than inserting *pitch-bend events* for a single channel). Such a midi file would be of little use to me, since I choose to have a separate channel for each track in Cakewalk. This way, if I need to manually edit the pitches in my Cakewalk file, I can easily tell which pitch-bend events go with which notes. While this tends to *maximize* the pitch-bend events, it makes it much easier for me to keep track of what pitch-bend is currently set for each channel at any point in a file. If you minimize the pitch-bend events and constantly switch channels, then most tunings with more than 15 pitches in the octave are going to require occasional pitch-bend events on one or more channels. The amount pitch-bend in effect on a given channel at a given point in the file might have been set 10 or 20 measures previous to that point, and searching backward in the file to locate that event (and thereby verify what I believe to be the current value) would be very time-consuming and a constant source of frustration that I want to avoid at all costs. --George

Message: 8442 - Contents - Hide Contents Date: Fri, 21 Nov 2003 23:41:43 Subject: Re: Finding the complement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Then the dual must not be the same thing as the Euclidean > complement. >> What dual are we talking about?I was getting my information from the GABLE program and from this tutorial: http://carol.science.uva.nl/~leo/GABLE/tutorial.pdf - Type Ok * [with cont.] (Wayb.) see page 18.

Message: 8443 - Contents - Hide Contents Date: Fri, 21 Nov 2003 02:15:15 Subject: Re: Definition of val etc. From: Dave Keenan Gene, Feel free to give us a pure-math definition of "map" to include along with the tuning-related stuff in the tuning dictionary entry. Or how about we just include this link? Map -- from MathWorld * [with cont.]

Message: 8444 - Contents - Hide Contents Date: Fri, 21 Nov 2003 15:19:14 Subject: Re: "does not work in the 11-limit" From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Are you entering notes from a keyboard? >>> >>> -C. >>>> No, I have to mouse back and forth around the screen, clicking on a >> note duration in one place (if it needs to be changed from what was >> set for the previous note) and then clicking on the staff in the >> appropriate place to draw the note. >> That's the only composition method I've ever used with a computer. >>> This part would go much faster >> if Cakewalk allowed me to use the keyboard to change the note >> durations (with the left hand; only about a half-dozen different keys >> would be needed) while I inserted the notes with the mouse >> When you say "keyboard"... many notation packages support computer > keyboard in such a fashion. And even Cakewalk supports a MIDI key- > board for choosing the notes.No help to me since I don't have a midi keyboard. Besides, a midi keyboard wouldn't be of much use to me in entering the notes anyway, since this is something that I can't do in real time. If this were ordinary diatonic music, then I would be able to compose on the fly and just play it right into the computer off the top of my head (fixing any mistakes afterwards). But in working with unconventional harmonic materials my method is much different -- much slower going.>> Then, if the note isn't something that occurs in the key signature >> (or if I chose not to have anything in the key signature), then I >> have to right-click on the note and set a chromatic alteration with >> the mouse. >> Again, MIDI keyboard to the rescue and/or Finale and Sibelius > have computer keyboard shortcuts for this.The minimal microtonal support available in Finale and Sibelius (and very low probability of any improvements along that line in the foreseeable future) would not justify the cost for me. I can put up with Cakewalk until Monz's software is available.> ... > I too like to keep one voice per MIDI channel. However, Scala's > seq format provides a higher-level music-description language. It > has "tracks". Right now, you have to code seq files by hand (if > someone were to come up with a 'Cakewalk' for seq files...). > > If your synth only supports pitch-bend retuning, Scala will > scramble things over MIDI channels in the end, and you won't have > very much flexibility with mixing patches no matter what you do.I don't have a synth (except for a Scalatron, but that's pre-midi). I'm doing everything on a computer.> If you can do MTS, I don't know what it does to the track->chan > mapping. Maybe Manuel can chime in. > > -CarlI already do MTS -- if that means "Most Things Slowly." ;-) But seriously, I don't know what you're referring to by "MTS". Please enlighten me. --George

Message: 8445 - Contents - Hide Contents Date: Fri, 21 Nov 2003 23:45:45 Subject: Re: Finding Generators to Primes etc From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >>> >>>> Imagine a triangle representing >>>> >>>> 1. Generators to Primes >>>> 2. Commas >>>> 3. Temperaments (such as 12&19) >>>> >>>> I am solid in my understanding of the leg between 2 and 1. (Even >>>> though I understand that going from 1 to 2 is more difficult >>> because>>>> of contorsion). I have some understanding of the leg between 2 >> and 3>>>> (by mapping Linear Temperaments as lines on the Zoom diagrams, >> these>>>> also represent commas, even though I am not sure how to extract >>> them) >>>>>> The leg between 2 and 3 is actually the easiest, it seems to me. >> Our>>> recent discussion on wedge products, with the particular example > of>>> cross products, should be helpful to you here. >>>> It would be cool if you or someone could give an example of the >> number crunching used to, say, get 81/80 from 12&19 Temperaments. >> Can this be done using matrices? I know the wedge product of the >> comma is equal to the wedge product of the val.. but still don't > see>> how you get from 12&19 TO 81/80... >> write down the representations of the primes {2,3,5} in 12: > > |12 19 28> > > and in 19: > > |19 30 44> > > now take the usual cross-product between these two and you get: > > <-4 4 -1| > > these is the "monzo" or prime-exponent-vector for 81/80, as you can > see by computing > > 2^(-4) * 3^4 * 5^(-1).sorry, i had the left-pointing and right-pointing notation backwards.

Message: 8446 - Contents - Hide Contents Date: Fri, 21 Nov 2003 02:26:30 Subject: Re: Definition of val etc. From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> It looks like you're mostly still objecting to my lack of mathematical > rigour, even when this is clearly being done in favour of educational > efficiency. I really hoped we had got beyond that.I don't think this is the case, and I reject the idea that confusing or incorrect definitions will serve.>> {{A "prime exponent mapping", sometimes shortened to "prime mapping", >> "exponent mapping", "mapping" or simply "map", is a list of numbers >> (integers) enclosed in < ... ] that tell you how a particular >> temperament maps each prime number (up to some limit) to numbers of a >> particular "generator" in that temperament.}} >> >> This assumes that all such mappings are (equal, and you need to say >> that) temperaments, which is not true. >> How does it assume that? In the case of linear or higher-D > temperaments we have more than one generator. The mapping from primes > to a single one of those generators is still a val isn't it?Whether you call them vals or anything else, it simply isn't the case that these necessarily have anything to do with any temperament. Moreover, by looking at higher-D temperaments you seem to be conflating with matricies. The individual rows or columns (depending on how you set things up) may be vals, but the whole thing isn't. The mathematical nit picks are that the mapping isn't from primes, but from the whole p-limit group, nor is it to generators, but to integers, which count generator steps. You must also mean generator in a broad sense, including, for instance, octaves.>> Also, simply calling it >> a "map" won't work as a specific shorthand, since that already has a >> well-established meaning, as another word for "function" which is >> more often used in some contexts ("homomorphic map" being one >> example.) > > Huh?> Isn't this your definition of "val"?: > Definitions of tuning terms: val, (c) 2001 by ... * [with cont.] (Wayb.) > > It starts: "A map ... ".It could have started "A function ..." but in this context you normally call that function a "map". "Map" is often used in a variety of mathematical contexts.> How many other kinds of map do we use in this application of Grassman > algebra, or in tuning theory in general?Any matrix defines a map on various sorts of vectors or matricies, just for starters. Tuning defines a map. Multivals define maps on corresponding multimonzos, which is a specifically Grassman algebra fact for you. The ordered steps of a scale define a map; in fact anything indexed is indexed by an indexing map from the index set to whatever is being indexed. The various sorts of goodness, badness, error, complexity and what not functions are maps because they are functions. Why do we need to keep arguing this stuff?>> I find "prime exponent mapping" too clumsy, too confusing, >> and too verbose, and have no plans to use the term. >> Sure it's clumsy and verbose, but it's _meaningful_.It's damned confusing. Is the domain the prime numbers, or some prime numbers? Is it the rational numbers, and does the map give prime expondents (which would mean they are p-adic valuations?) Is the mapping *from* prime exponents, and if so, how and to what? I won't use it> most of the time either (I'll use "map" or "mapping"), but what could > be more confusing for a newbie than a term that carries absolutely no > meaning for them whatsoever, except maybe as a person's name. (You > reading, Monz?) I should think it would at least be clear from "prime > exponent mapping" that whatever it is, it maps prime exponents to > something (and from "map" that it maps something to something).It most certainly is not clear.>> {{The prime numbers here represent frequency ratios.}} >> >> This is at best confusing; the prime numbers are prime numbers and >> don't represent anything else. Tuning is another matter. >> I completely fail to understand how you could imagine that tuning is > "another matter" in a tuning dictionary. What else could the primes > represent, in a tuning dictionary. Numbers.>> {{When an interval is represented in the complementary form...}} >> >> "Complimentary form" is not a good phrase to use here. >> I agree, which is why I wrote "compl_e_mentary form".Still no good, given that we have another meaning of complement and this really doesn't say anything.> But assuming you don't like that either, please tell me why?It conveys exactly nothing. You might> suggest alternatives.You could try "in monzo form" or "in prime-exponent-vector form" or "in ket vector form", for instance.>> {{...as a prime-exponent-vector, we can find the number of generators >> corresponding to it in some temperament by multiplying each number in >> the temperament's map by the corresponding number in the vector, and >> adding up the results.}} >> >> This is assuming the mapping in question defines an equal temperament >> (and again leaves out the word equal), which is hardly always the >> case. >> As I said, It does not assume equal temperaments at all. It applies > equally well to finding the number of fourth generators for meantone > (or the number of octave "generators").Meantone uses two vals, and you are talking about it as if it used only one. This would imply the mapping you had in mind must be a matrix.> Perhaps a more valid criticism of this definition is that it excludes > prime mapping _matrices_, since these give you the counts of _all_ the > generators at once. But these can be seen as a stack of prime- mapping > pseudo-vectors (vals) one above the other, right?Taken in some kind of order--but you aren't saying this, and this is intended to be a definition.> So we should extend the definition of prime-exponent-mapping and all > its abbreviations (not including "val"), so that it includes these > matrices.That gives you what I called an icon on my web page, BTW. It will usually be clear from the context, and from the> notation, whether one is talking about a (pseudo-)matrix or a > (pseudo-)vector (val).What in the world is a pseudo-matrix?? If necessary, one can distinguish them by using> the words "matrix" and "vector".Not for anyone who assumes "matrix" includes "vector".> I agree it is necessary to distinguish this operation from a "true" > dot-product in pure math and maybe in other applications, but since it > is the only way we're using it in tuning, there is no need to confuse > people with distinctions irrelevant to their application.If so, you'll need to make it clear by making it explicit. In particular, it is easy for people to think the product of a dual vector (bra) with a vector (ket), if it is called a "dot product", is really the same as the dot product of one vector with another. This is a classic undergraduate trap and the source of much confusion.

Message: 8447 - Contents - Hide Contents Date: Fri, 21 Nov 2003 16:52:34 Subject: Re: "does not work in the 11-limit" From: Manuel Op de Coul George wrote:>I wanted to see if I could create >midi files (consisting of only a single track) from scratch in Scala >(which would save me the trouble of calculating and manually >inserting pitch-bends), which I could then import into Cakewalk (one >track at a time).Ah, I assumed you were using Cakewalk to enter the notes more quickly, but you want to use Scala to enter the notes, and use Cakewalk to adjust the tuning at places afterwards. Well, this is a use case I hadn't envisioned, since with Scala you can change the tuning quickly, but typing note commands is very slowly.>Your Scala documentation indicates that pitch-bend >events are minimized, so that you are constantly *changing channels* >from one note to the next (rather than inserting *pitch-bend events* >for a single channel).This is not entirely true anymore, I forgot to update the documentation for that. There are also possibly program change and parameter change events involved in channel switching. So minimising pitch bend events doesn't make sense if it causes many more other messages. There may be a way to do what you want but I've never tried it. You can exclude midi channels from being used. So if you exclude all channels except the first for the first track, then generate the midi file for that track and for the next track exclude all channels except the second one, generate that, etc. I don't see why that wouldn't work. Generating MTS from .seq files isn't a good solution because it keeps the channels, but switches the note numbers on a round-robin basis. That will be even more confusing to look at in Cakewalk :-) But perhaps still the most efficient solution would be to discard Cakewalk from the process, and do all changes in one seq file, not looking at the midi file. Manuel

Message: 8448 - Contents - Hide Contents Date: Fri, 21 Nov 2003 23:44:48 Subject: Re: Finding Generators to Primes etc From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> At 03:11 PM 11/21/2003, you wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >>>>> < 12 19 28 | >>> >>> is h12 and >>> >>> < 19 30 44 | >>> >>> is h19. Except there's something about using the transpose of >>> one of them to get it into a form where the cross product will >>> give you a monzo. Which in this case is >>> >>> | -4 4 -1 > = 81/80 >>> >>> Do I have that right, guys? >>>> ~(<12 19 28| ^ <19 30 44|) = |-4 4 -1> >> ^ is the wedge product. yes. > ~ is ? Complement? yes. > So the wedging with > a complement is the same as crossing?no, look at the parentheses. the complement of the wedge product is the cross product (when you're dealing with a 3 dimensional problem).> Please answer each question, > I'm just guessing. > > I still don't know a simple procedure to calculate a wedge product.Graham just explained that.

Message: 8449 - Contents - Hide Contents Date: Fri, 21 Nov 2003 02:27:51 Subject: Re: Definition of val etc. From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Gene, > > Feel free to give us a pure-math definition of "map" to include along > with the tuning-related stuff in the tuning dictionary entry. Or how > about we just include this link? > > Map -- from MathWorld * [with cont.]That looks OK. It certainly isn't heavy-duty.

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