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Message: 8450 - Contents - Hide Contents 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 - Contents - Hide Contents 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 * [with cont.] However, "bivector" apparently means grade-n multivector Bivector -- from MathWorld * [with cont.] 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 P... * [with cont.] (Wayb.) 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: 8452 - Contents - Hide Contents Date: Fri, 21 Nov 2003 16:08:24 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> > wrote: >>> But yes, I would also include *both* 108:125 and 1024:1215 as ratios >> (or *roles*, as Dave and I call them when a Sagittal symbol is used >> to represent multiple ratios) that would be covered by the interval >> function of an augmented 2nd in traditional diatonic harmony, just as >> I would include both 8:9 and 9:10 as roles in which the interval of a >> major 2nd functions. My only requirement is that a single ratio >> *not* be represented by multiple vectors (or in this case multiple >> scalars) in the scale construct. >> If you add the condition that *every* ratio (within the prime limit) > be represented by one 'scalar in the scale construct',Yes, I would have no problem with that.> then I believe > you *are* speaking of epimorphism -- and equivalently, periodicity, > as presented in this paper of mine: > > http://lumma.org/tuning/erlich/erlich-tFoT.pdf - Type Ok * [with cont.] (Wayb.) > > Please read it if you haven't yet -- it should be a breeze for you.Okay. I think I looked at it before but didn't get very far with it because there were too many other things competing for my attention. I'm sure that I'll get more out of it now, since I'll be looking for things that will shed light on our present discussion.>> I mentioned vectors in the previous paragraph, because in the case of >> the 11-limit hexatonic otonal scale that we've been using as our >> other example, the tones occur in a 4-dimensional structure. And >> there will be only one ratio (or role) for each tone, since the scale >> is JI. >> Just because a scale is in JI doesn't mean there's only one role for > each tone, in my opinion. But that may be a separate discussion > from 'functional disorientation' . . .Yes, I agree with your first statement. I should have said "there will be only one role for each *interval*, since there is only one ratio for each *tone* (since the scale is JI)." When I was referring to the ratio for a *tone*, I was actually thinking of the ratio for the *interval* that a tone makes with 1/1. I recall that Harry Partch went to great lengths to make the point that a tone can have multiple roles (or "identities") in JI. Even in a diatonic scale I think we would agree that *tones* have multiple roles, e.g., the 5th degree of the scale functions as both the root of the dominant triad and the fifth of the tonic triad; but I would say that the *interval* of a (perfect) 5th has only a single role (but, on second thought, perhaps "identity" is a better term) in a diatonic scale. --George
Message: 8453 - Contents - Hide Contents 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 - Contents - Hide Contents 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 * [with cont.] (Wayb.) -monz
Message: 8456 - Contents - Hide Contents Date: Fri, 21 Nov 2003 18:12:08 Subject: Using Scala to insert pitch-bends (Was t-m: "does not work in the 11-limit") From: George D. Secor I am replying on the main list and changing the subject line, since the topic was no longer appropriate. --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote (message #7898):> > 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.I'm sorry, but I guess that what I said was misleading. Forget about everything that I said previously and let me start fresh. What I originally had in mind was to create a midi file in Cakewalk without any pitch bends. All of the notes in each track would be on the same channel, with no simultaneous notes in any track being separated by an interval other than an octave (or multiple octave). Each track would have its own channel, and a change of channel would never occur on any track, so that there would be a one-to-one correspondence between tracks and channels. Once the file was created, I then wanted to import it into Scala, load a scale, and then have Scala insert the pitch-bends into my file (deleting any pitch-bends that might already be there), but without making any changes to the track or channel assignments in my source file. I would then save the result as a midi file and import it back into Cakewalk. If Scala doesn't allow this at present, I wouldn't think that it would be difficult to provide that option (hint, hint). However, let me ask something. In a Cakewalk work file I'm able to specify whether a note will display as a sharp or flat (e.g., G# or Ab), but if I save it as a midi file, would that distinction be lost (such that Scala would not tell them apart)? If so, I would then have to restrict myself to scales having no more than 12 pitches in the octave. Unless: if the insert-pitch-bend operation would allow the user to select which channel(s) would be affected in a pass, then a new scale could be loaded between passes to allow for additional pitches (supposing, for example, that my midi source file were to have two channels on each track (say channels 1 and 2 on track 1, channels 3 and 4 on track 2, chs. 5 and 6 on trk. 3, etc.) Does this make any sense?>> 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.I did see that feature in the documentation, but I've come to the conclusion that doing this a track at a time may be as time-consuming as inserting pitch-bends manually, if this involves importing and exporting each track to and from Scala as separate midi files -- unless it is possible to generate the pitch-bends in Scala one track at a time, then reset the channel exclusions and then do the next track, etc., as I suggested above.> 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.And how convenient is it to display and edit the notes in a seq file? Cakewalk has features that I would not want to lose. --George
Message: 8457 - Contents - Hide Contents 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 - Contents - Hide Contents Date: Fri, 21 Nov 2003 12:08:22 Subject: Re: "does not work in the 11-limit" From: Carl Lumma>> >f 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents
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 * [with cont.]
----------------------------------------------------------------------
Gene,
Further to your objection to calling that operation the dot product.
It seems there's a precedent here
Tensor -- from MathWorld * [with cont.]
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 - Contents - Hide Contents 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: 8463 - Contents - Hide Contents Date: Fri, 21 Nov 2003 10:19:30 Subject: Re: Definition of val etc. From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> If you wrote simply "11-vector", a newbie will most likely assume you > mean a vector with 11 components, an 11-dimensional vector.I don't say n-vector, I say n-val or n-monzo, which makes it pretty clear the grade is intended. Since, despite what you suggest, things like 2-form etc actually are very standard, this is *not* a problem with standard terminology.
Message: 8465 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 8467 - Contents - Hide Contents Date: Fri, 21 Nov 2003 10:50:33 Subject: Re: Definition of val etc. From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> 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.This I already did--the exponent for some particular prime (the p-adic valuation for prime p) provides an example. Another would be the vals which turn up in connection with notation systems such as you have been working on.> 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"."Generator" should only come in as an example, not as a part of the definition. Otherwise, the defintion isn't correct.>> Tuning defines a map. >> OK. I should have said "How many other things that we actually _call_ > "maps", do we use in tuning.I meant tuning maps--that is, for example, maps from temperaments to real numbers, determined by giving a specific value to the generators, which define a tuning. Even more concretely, maps to Hertz.>>> 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".I do. It sounds as if it isn't a map, perhaps because it is a multi- valued function (which isn't, strictly speaking, a function at all.)> 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 you insist on rewriting standard mathematical terminology? That is asking for confusion.>> 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.It doesn't answer my question. Why do you seem hell-bent on tossing out standard mathematical terminology?>> 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. :-)So what is it we magically determine the domain to be--some prime numbers? That isn't the correct answer!>> 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. :-)"Val" comes from "valuation", and that *is* getting us nearer to where we want to be.>>> 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. :-)What you hear are sounds, which if you are lucky are more or less periodic and have a frequency expressible in Hertz. There are various mappings involved here--p-limit rational numbers to abstract temperaments, temperaments (via a tuning map) to real numbers, real numbers representing intervals to Hertz--and you are trying to gum them all together into one ugly, confusing mess with this idea that prime numbers are a ratio of frequencies.> 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.Who told you vals were pseudo-vectors?
Message: 8468 - Contents - Hide Contents 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: 8469 - Contents - Hide Contents Date: Fri, 21 Nov 2003 11:11:31 Subject: Re: Definition of val etc. From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> "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.Kill the "or maps". If you want an or, try "exponent maps" or some such thing.> A "prime exponent mapping", sometimes shortened to "prime mapping", > "exponent mapping", "mapping" or simply "map"Kill "mapping" and "map" for certain, and I would suggest ditching "prime mapping" as well, and sticking with "exponent mapping". , is a list of numbers> (integers)"List of integers" is both clearer and shorter. 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.Kill "temperament" and "generator" except as examples, and remove the claim that prime numbers represent frequency rations, which is incorrect anyway and screws up the math conceptually.> 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.More correctly, it sends 2 to 12 steps, 3 to 19 steps, and 5 to 28 steps--and in so doing, refutes your claim that 2, 3, and 5 were frequency ratios. The actual ratios turn out to be 2, 3^(19/12) and 5^(7/3).> 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>This should be <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]Now you've stuck yourself with making the monzos column vectors, which isn't the notation we had for them. Why not use standard math notation instead; then we can use column vectors corresponding to monzos, without trying to figure out what the above unfamiliar notation is supposed to mean.> <1 2 4] [ 0 <2 > <0 -1 -4] -1 = -3> > 1>This looks ghastly after it gets munged by Yahoo, and I think you'd be better off just using a regular column vector with a usual sort of matrix.> Gene, > > Further to your objection to calling that operation the dot product. > It seems there's a precedent here > Tensor -- from MathWorld * [with cont.] > It looks to me like the dot product is defined for tensors as a > covariant by a contravariant. Am I interpreting this correctly?This is what I said--the terminology is used all the time, but that doesn't mean it isn't confusing.
Message: 8470 - Contents - Hide Contents 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/b... * [with cont.] (Wayb.) TheComplement.pdfThen the dual must not be the same thing as the Euclidean complement.
Message: 8471 - Contents - Hide Contents Date: Fri, 21 Nov 2003 11:15:15 Subject: Re: Definition of val etc. From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> More correctly, it sends 2 to 12 steps, 3 to 19 steps, and 5 to 28 > steps--and in so doing, refutes your claim that 2, 3, and 5 were > frequency ratios. The actual ratios turn out to be 2, 3^(19/12) and > 5^(7/3).I'm falling asleep--this is 2^(19/12) and 2^(7/3), of course.
Message: 8472 - Contents - Hide Contents Date: Fri, 21 Nov 2003 23:11:01 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- 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>
Message: 8473 - Contents - Hide Contents 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.
Message: 8474 - Contents - Hide Contents Date: Fri, 21 Nov 2003 23:13:01 Subject: Re: Finding the complement From: Gene Ward Smith --- 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?
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