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Message: 8450 Date: Fri, 21 Nov 2003 16:55:55 Subject: Re: "does not work in the 11-limit" From: Manuel Op de Coul >I already do MTS -- if that means "Most Things Slowly." ;-) Scala seq files are eminently suitable for that :-) >But seriously, I don't know what you're referring to by "MTS". >Please enlighten me. The MIDI Tuning Standard. Some softsynths support it. It has better resolution than pitch bends, and no inherent channel limitation. See the MMM archive for more info. Manuel
Message: 8451 Date: Fri, 21 Nov 2003 03:27:26 Subject: Re: Definition of val etc. From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > hi Dave, > > > thanks *very* much for these suggestions. > just a couple of things before i actually do > incorporate them ... > > > in general, the shortest and most compact terminology > is the one that gets the most use anyway, and the one > i prefer to promote. It isn't that simple. Yes the shortest one gets the most use, but is least meaningful to a newbie. There is no need to "promote" the shortest term, it's shortness is all the promotion it needs. It's nice to have a path of gradual tradeoff between descriptiveness and shortness. Often the shortest term will have some ambiguity. For example it may be used to refer to two or more slightly different kinds of thing, but in any particular discussion only one will be relevant. At the start of the discussion you will use the most explicit term and as you get into it you shorten it, and we say the meaning is made clear from the context. And sometimes other possible meanings for the same short term only become apparent later (as you describe happening with "vector" below). So I think it is best to put the specific definition with the term that is most specific, or carries the most "context" with it. > thus, i think that the nice new definition you gave > below should be that for "map", with all the other > terms pointing to *it*. I guess I'm not too worried about this, so long as links take you to the definition from all the equivalent terms. > and likewise for "monzo". > > ... of course i also have my own reasons for promoting > that particular term. ;-) I've been meaning to talk to you about that. :-) Don't you thing you're famous enough already? The tuning dictionary alone should be sufficient. And you've got the monzisma (although I can never remember even roughly how small that is, or what prime factors it has other besides 2 and 3 ;-). Really, the term "monzo" has exactly the same problem as "val" to the uninitiated, complete and utter meaninglessness. The only reason I haven't objected to this until now is that (a) I didn't want to offend you (I still don't), and (b) I didn't have a good suggestion for something to replace it, as the ultimate shortening of "prime exponent vector", except "vector", which, as you point out, is a bit too general. Even a map (val) can be called a vector. But somehow we managed to get by without any other terms for at least the last 5 years! How long have you been using them Monz? So I have to ask why do we need one now? I did think of "expo". The only problem is that it starts with a vowel, so adding the latin number prefixes is awkward. "bi-expo", "tri-expo". > also, i believe that "vector" should retain its general > defintion. there's another term "interval vector" which > i haven't yet put in the Dictionary but which is common > currency in atonal music-theory. Fair enough. But as it stands, your definition of "vector" is exactly what we're now calling a "monzo", except for the second sentence. By the way, everybody, It seems we should not be using the terms "n-vector", or "4-vector", "5-vector" etc, to refer to grade-n multivectors in the Grassman algebra. According to mathworld this already means n-dimensional vector. n-Vector -- from MathWorld * However, "bivector" apparently means grade-n multivector Bivector -- from MathWorld * Could it be that I've caught Gene out on a matter of mathematical rigour here? :-) And so when specifying the grade I suggest we use the latin prefixes all the way up. See Numerical Adjectives, Greek and Latin Number Prefixes * How about (uni)vector bivector trivector quadrivector quintivector sexivector septivector octivector nonivector decivector If we ever need something beyond that, I think we should just write "grade-11-multivector". If you wrote simply "11-vector", a newbie will most likely assume you mean a vector with 11 components, an 11-dimensional vector. Same goes for the maps. I'm pleased to find there's no conflicting definition for bimaps etc., or even n-maps, in Mathworld. Now all we have to settle is whether a 6D-bivector is one with 6 components (7-limit) or one with 6C2 = 15 components (13-limit)?
Message: 8453 Date: Fri, 21 Nov 2003 05:46:56 Subject: Re: Finding the compliment From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: > --- 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". :-) that is just *too* Monty Python-esque. well, at least it's *one* thing that i've understood in the recent tuning-math discussions! otherwise, i have not a clue what you guys are going on about. :( -monz
Message: 8455 Date: Fri, 21 Nov 2003 05:53:39 Subject: Re: Definition of val etc. From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: > 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" just splitting hairs ... i've seen it as: "Pluralitas non est ponenda sine neccesitate." the english translation is the same. see: http://phyun5.ucr.edu/~wudka/Physics7/Notes_www/node10.html * -monz
Message: 8457 Date: Fri, 21 Nov 2003 05:58:45 Subject: Re: Definition of val etc. From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- 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. Of course we can all agree with this. It's just that what seems confusing or incorrect to you, isn't necessarily so to everyone else on this list. What may be confusing to you, may be clear to others on this list because all the other possible meanings that you, as a mathematician, can ascribe to it, simply do not occur to them. What may be incorrect to you, may be a sufficiently good approximation for others on this list. > > > {{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. Please give a non-trivial example of the use of a val in regard to tuning, where it doesn't 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. Yes I understand that you don't want a stack of vals to be called a val, and that's fine. But I don't see any harm in calling a stack of prime-mapping vectors a prime-mapping matrix, and thereby calling them all maps. It's not a distinction that I've ever felt the need to make before. > You must also mean generator > in a broad sense, including, for instance, octaves. Yes. But that's something that belongs to the definition of "generator", not here. There's no need to spell this out in the definition of "prime mapping". > > 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. Sure, but that doesn't say anything about other tuning applications. > Tuning defines a map. OK. I should have said "How many other things that we actually _call_ "maps", do we use in tuning. > Multivals define maps on > corresponding multimonzos, which is a specifically Grassman algebra > fact for you. OK. But I don't see any harm in calling these "multimaps". > 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. Yes. But we've never had any urge to refer to any of these by the term "map". The terms "indexing" or "function" serve us just fine for these. So there are no name conficts with "map" there that I can see. > Why do we need to keep arguing this stuff? Because you have mathematical knowledge that I don't have, and I have some insights into how to explain things to non-mathematicians, that you apparently don't have. And it is apparently difficult for either of us to sort out what are valid objections to my terminology and expositions, and what are mere nit-picks. So we engage in arguments. It can be tedious, but the end result can sometimes be very satisfying for both of us, as well as the onlookers. I should hope that has already been the case in the thread explaining how to compute complements. > > > 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? No question there for most tuners. We don't usually try to compute things with infinite numbers of coefficients. :-) > 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? Yes. That's true. They may wonder if it's a mapping _from_ prime exponents, or _to_ prime exponents, and what's on the other side. But this still seems to be getting us a lot closer to the intended meaning than a randomly chosen girl's name would. :-) > > 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. I'm afraid I've never been able to _hear_ numbers. Unless of course they get interpreted as corresponding to some quantity in the physics of vibrating matter, although I suppose you could count singing the names of the numbers in some spoken language. :-) Otherwise, I'm afraid you're off-topic for this list. ;-) > > > {{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. I actually thought of it as the _same_ meaning - the form that their Grassman complement would take. But you're right. That's only true of the _direction_ of the brackets, not their number. > > 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. Agreed. "in prime-exponent-vector form". > > > {{...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. I already said at the start of the def that it related to a particular generator. To spell this out again here would just complicate the language and risk losing the reader. There is no harm, (and in fact there may be some actual benefit to the educational process), if they don't pick this up on a first pass. They can then go away and play with ETs (or octave equivalent LTs) until they are familiar. They may never even need to progress beyond those, but if they do, the more general reading is there waiting. I've told no lies. It's really quite interesting that you're forcing me to elucidate this thinking on what I regard as good explanatory writing. > This would imply the mapping you had in mind must be a > matrix. No. As explained above. > > 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. Sigh. From my point of view, and probably most on this list, any other random word involving two consonants with a vowel between them would have done just as well. Which is not well enough, in my view. What's wrong with calling them mapping matrices, or val matrices for that matter? I agreed to assume you know what you're doing in the pure math department, but you shouldn't be surprised if I don't want to adopt these obscure new terms when applying it to tuning, at least not without a serious struggle. :-) > 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?? It just occurred to me. If vals are pseudo-vectors then when you stack them up to make a matrix, surely it must actually be a pseudo-matrix. If not, why not? I'm trying to talk pure-math here. I wouldn't put stuff like this in the tuning dictionary. > If necessary, one can distinguish them by using > > the words "matrix" and "vector". > > Not for anyone who assumes "matrix" includes "vector". So are you telling me that an nx1 icon can't also be considered as a val? Why would it be a problem if an nx1 mapping matrix can also be considered as a mapping 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. But readers of the tuning dictionary are unlikely to go on to do undergraduate pure math. Sure a few will, but I expect they will cope somehow. I really don't think they will be irreparably damaged by a lack of rigour in the tuning dictionary. And the other 99.9% of dictionary readers will be spared some unnecessary complication. Please give a URL for your web pages on this stuff. A Google search did not reveal. Although it revealed a very funny (but true) quote of yours. Poor spelling does not prove poor knowledge, but is fatal to the argument by intimidation. -Gene Ward Smith :-)
Message: 8458 Date: Fri, 21 Nov 2003 12:08:22 Subject: Re: "does not work in the 11-limit" From: Carl Lumma >> If you can do MTS, I don't know what it does to the track->chan >> mapping. Maybe Manuel can chime in. >> >> -Carl > >I 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. Scala supports at least two general retuning methods. One is pitch bend, which as you point out won't work for you. Another is the Midi Tuning Standard, which is just a block of data at the beginning which cakewalk will ask you if you want included or not (something like "send sysex messages?"). It won't mess up your channels. So Gene's suggestion, I believe, is to type your score as a .seq file in a text editor, then render it to MIDI with Scala. Or something. -Carl
Message: 8459 Date: Fri, 21 Nov 2003 05:59:52 Subject: Re: Definition of val etc. From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: > 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. well, in fact, in a few months the Tuning Dictionary will no longer exist in its present form. it will morph into the full-fledged Encyclopedia of Tuning. as such, i plan for it to be simple enough to act as a primer for total newbies, but also comprehensive enough to be a repository of all the accumulated knowledge of the experts. as Chris (my business partner) is taking over most of the business and programming stuff that concerns my software, i'll be focusing on the Encyclopedia and tutorial. the Encyclopedia will be bundled with the software, and eventually the two will be completely interactive. -monz
Message: 8460 Date: Fri, 21 Nov 2003 12:09:57 Subject: Re: "does not work in the 11-limit" From: Carl Lumma >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 :-) Really? Why does it do this? -Carl
Message: 8461 Date: Fri, 21 Nov 2003 07:07:09 Subject: Re: Definition of val etc. From: Dave Keenan So here's another run at the fence. I suggest that the definition of val stay as Gene had it, as a definition of a pure math term. But that we add something like the following text at the start of it. ---------------------------------------------------------------------- "val" is a term coined by Gene Ward Smith for the mathematical object described below. When vals are applied to tuning theory they are usually interpreted as prime exponent mappings (or maps) for a single generator of a temperament. ---------------------------------------------------------------------- with links for both "prime exponent mapping" (and/or "map") and "generator" and "temperament". ---------------------------------------------------------------------- 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. The prime numbers here represent frequency ratios. The simplest case is an equal temperament where the generator is the step interval. For example, the 5-limit map for 12-equal is <12 19 28] which means it takes 12 steps to make an octave (1:2), 19 steps to make a twelfth (1:3), and 28 steps to make a 1:5 interval. When an interval is represented as a prime exponent vector, we can find out how many of some generator correspond to it in some temperament by multiplying each number in the map (for that generator) by the corresponding number in the exponent vector, and adding up the results. In mathematical terms this is called the dot-product, scalar-product or inner-product of the map with the exponent vector. For example the interval 3:5 (a major sixth), has the 5-limit exponent vector [0 -1 1>. To find how many steps of 12-equal it maps to, we write <12 19 28].[0 -1 1> = 12*0 + 19*-1 + 28*1 = 28 - 19 = 9 The term "prime exponent mapping" and its abbreviations may also be used to refer to the matrix formed by stacking, one above the other, the mappings for _all_ the generators of some temperament. For example the two generators for meantone may be taken as the octave and the fourth, in which case the complete 5-limit mapping may be given as <1 2 4] <0 -1 -4] The first row relates the primes to the octave generator, the second row relates them to the perfect fourth generator. And we'll use the prime exponent vector for the 3:5 major sixth again. [0 -1 1> We can calculate the individual dot-products, for each row in turn, or we can use software that has matrix operations (e.g. Microsoft Excel) and simply find the matrix-product of the mapping matrix with the transpose of the exponent vector. <1 2 4] [ 0 <2 <0 -1 -4] -1 = -3> 1> The result is a column vector <2 -3> which tells us that the 3:5 minor sixth is approximated in meantone by an interval 2 octaves up and 3 fourth-generators down. The mathematical definition of a mapping or map is far more general than those used here. See Map -- from MathWorld * ---------------------------------------------------------------------- Gene, Further to your objection to calling that operation the dot product. It seems there's a precedent here Tensor -- from MathWorld * It looks to me like the dot product is defined for tensors as a covariant by a contravariant. Am I interpreting this correctly? -- Dave Keenan
Message: 8462 Date: Fri, 21 Nov 2003 12:58:55 Subject: Re: Finding Generators to Primes etc From: Carl Lumma >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... The other Paul demonstrated this recently -- you take the cross product of two vals. So < 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? -Carl
Message: 8465 Date: Fri, 21 Nov 2003 02:31:55 Subject: Re: Definition of val etc. From: Carl Lumma I wish you guys wouldn't argue over the inclusion of the term "val". Dave, it isn't this that causes a problem. It's the complete lack, until now, of material like... >When an interval is represented as a >prime exponent vector, we can find out how many of some generator >correspond to it in some temperament by multiplying each number in >the map (for that generator) by the corresponding number in the >exponent vector, and >adding up the results. In mathematical terms this is called the >dot-product, scalar-product or inner-product of the map with the >exponent vector. > >For example the interval 3:5 (a major sixth), has the 5-limit exponent >vector [0 -1 1>. To find how many steps of 12-equal it maps to, we >write > ><12 19 28].[0 -1 1> >= 12*0 + 19*-1 + 28*1 >= 28 - 19 >= 9 ...which is pure gold. I don't care what you call the stuff, if you say how to use it! >[0 -1 1> > >We can calculate the individual dot-products, for each row in turn, or >we can use software that has matrix operations (e.g. Microsoft Excel) >and simply find the matrix-product of the mapping matrix with the >transpose of the exponent vector. Perfect example of what not to do. Introduce the word "transpose" without saying what the hell it is. It doesn't matter what word you use if you don't explain it. ><1 2 4] [ 0 <2 ><0 -1 -4] -1 = -3> > 1> > >The result is a column vector <2 -3> And how did you get that result? This stuff clearly isn't that hard unless you make it hard. Mainly by *leaving out* all-important definitions and examples. -Carl
Message: 8466 Date: Fri, 21 Nov 2003 23:40:06 Subject: Re: "does not work in the 11-limit" From: Manuel Op de Coul Carl wrote: >Really? Why does it do this? You need note numbers for the note on and note off messages. But there is no relation between pitches and note numbers anymore. So instead of a channel limitation, there's a note number limitation which means there can be at most 128 simultaneous pitches, but any pitch the MTS range allows. Manuel
Message: 8468 Date: Fri, 21 Nov 2003 22:55:04 Subject: Re: Finding the complement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: > --- 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. You're probably doing something wrong, then. This is correct according to both the GABLE tutorial and the program itself.
Message: 8470 Date: Fri, 21 Nov 2003 22:58:51 Subject: Re: Finding the complement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: > --- 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 > Index of /homes/browne/grassmannalgebra/book/bookpdf * TheComplement.pdf Then the dual must not be the same thing as the Euclidean complement.
Message: 8473 Date: Fri, 21 Nov 2003 23:07:49 Subject: Re: Definition of val etc. 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: > > > 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. That's exactly what Dave himself said! > 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. Again, exactly what Dave said. And nothing in his definition contradicted this. > Meantone uses two vals, and you are talking about it as if it used > only one. No, Dave's talking didn't look that way at all to me.
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