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Message: 8350 - Contents - Hide Contents Date: Tue, 18 Nov 2003 22:17:06 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" > <gdsecor@y...> >> wrote: >>>>>> exactly . . . the two champions would have to be the diatonic >>>> pentatonic and heptatonic scales . . . >>>>>>>>> If I'm using a pentatonic scale made from a 9-limit otonal >> chord:>>>>> 8 : 9 : 10 : 12 : 14 : 16 >>>>> then I have two intervals each of 2:3 (both > pentatonic "4ths") >>> and>>>>> 3:4 (both pentatonic "3rds"). >>>>>>>> personally, i'm not fond of this as a scale or melodic entity> at all ->>>> - when i improvise over a dominant ninth chord, simply using > its>>>> notes is about the worst way to come up with a melody . . . >>>>>> I understand, and I wouldn't have much to say about the harmonic >>> possibilities either. But I think that we've been spoiled by the >>> harmonic sophistication of the major-minor system to such an > extent>>> that it's difficult to appreciate the resources of a simple > scale.>>> We would have to immerse ourselves in gamelan music (particularly >>> slendro) to get in the proper frame of mind to be able to even > begin>>> to create something decent with such limited tonal resources. >>> (Again, we're off on another topic.) >>>> i don't know . . . i mentioned the diatonic pentatonic scale above. >> that's an equally simple scale, isn't it, and yet i could probably >> live a happy life with no other melodic resources. so it seems you >> missed my point entirely. >> No, I don't think I did. > I recognize that a scale that is > essentially a just dominant 9th chord makes it a little more > difficult to imply any changes in the harmonic element other than > alternation between 4:5:6 to 6:7:9 triads in an accompaniment. An > unaccompanied melody using a diatonic pentatonic scale, on the other > hand (such as _Auld Lang Syne_), can easily evoke a heptatonic > chordal accompaniment in our imaginations -- it takes only the > presence of the 5/3 in the scale to imply that there should be a 4/3 > (subdominant) somewhere in context.I didn't mean to imply any harmonic dimension whatsoever, or at least not any harmonic changes. My original comment, above, concerned improvising over *one* chord. So I think you did misundestand me.> My motive in suggesting "slendro therapy" was to experience that > there is a feeling of satisfaction that can be achieved with music > that has little or no harmonic motion.I'm the last person that needs to be convinced of this -- one of my microtonal examples on mp3.com used to be about 9 minutes over an unchanging "harmony" or drone! Also, one of my main musical activities, sometimes quite lucrative actually, is improvising dextrously on acoustic guitar with open strings tuned to a 1/1-3/2 drone.>> i realized, since i made my original post, that the "dominant >> pentatonic" is not CS in 12-equal. perhaps that's one source of my >> difficulty? >> I wouldn't think so.Well, I'm interested in investigating further . . .>>> 2) But if there are two intervals in a scale that are *not >>> functionally different* (such as the two 2:3s or 3:4s in our 11- > limit >>> hexatonic otonality), >>>> why aren't they functionally different? because we don't have a > well->> defined sense of hexatonic musical function, while we know all too >> much about the history and theory of the diatonic scale? i don't >> think that the "happen to" above can be defined in any precise or >> perceptually relevant sense -- though it would be nice . . . >> As I see it, interval function is independent of the number of tones > in the scale, but instead has to do with the (just) *ratio* that is > either directly expressed (in JI) or implied (in a temperament) by > that interval. So two tempered intervals that (in a given context) > are implying the same just interval are functionally the same, even > if they are not exactly the same size (such as in a well- > temperament). But two tempered intervals that (by context) imply > different just intervals are functionally different, even if they are > exactly the same size in a particular tuning.What just interval does the 12-equal augmented second imply? And how is this implication effected, exactly?> In the context of a diatonic scale the tones are all assumed to be in > a chain of fifths. If one member of that chain is taken to represent > 1/1, then each of the other members can be assigned at least one > (rational) ratio that is unique to that member. An augmented 4th and > diminished 5th (or a minor 3rd and augmented 2nd, etc.) will > therefore be considered to be serving different harmonic functions, > since they represent different ratios.I'd like to see this made more explicit.> In an 8:9:10:11:12:14:16 scale there is no question that the two 2:3s > (or the two 3:4s) are for all intents and purposes identical (since > this is JI),What if I tuned a harmonic minor scale in JI with a 6:5 augmented second?> so on a *harmonic* level they are functionally > equivalent. But since these pairs of intervals subtend different > steps in the scale, the potential for _functional scale > disorientation_ (if you don't like the term, then please suggest > something else) exists.I'm hoping we can make this precise. Right now it seems fuzzy, with meaning adapted differently to fit this fact and that. Please help me remove the ambiguity. Perhaps we are talking about epimorphic vs. non-epimorphic scales? If so, realizing this could be a breakthrough. At least we could have a precise (and very relevant to the material on this list) mathematical characterization of what makes a scale have or not have "functional scale disorientation" to you. That could be very helpful. Gene, would you chime in?
Message: 8351 - Contents - Hide Contents Date: Tue, 18 Nov 2003 22:38:58 Subject: contravariant vs. covariant vectors From: Paul Erlich 403 Forbidden * [with cont.] (Wayb.)
Message: 8352 - Contents - Hide Contents Date: Tue, 18 Nov 2003 22:57:34 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> E.g. With Pauls's example of the syntonic comma and diaschisma and > 12-ET, wedging the two comma monzos gives > > [-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo > > whereas their cross product gives > > [-4 4 -1> (x) [-11 4 2> = <12 19 28] a map > > and one is the complement of the other > > ~[[28 19 12>> = <12 19 28] > > So why no problems with minus signs in 3D?maybe this is why: Tensor -- from MathWorld * [with cont.] "While the distinction between covariant and contravariant indices must be made for general tensors, the two are equivalent for tensors in three-dimensional Euclidean space, and such tensors are known as Cartesian tensors."
Message: 8353 - Contents - Hide Contents Date: Tue, 18 Nov 2003 00:31:13 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Dave Keenan wrote: >>> And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a >> 7-limit trimonzo will have coefficients ordered [[[e357 e572 e723 e235>>>. >> >> Is this how your software does it too Graham? >> The wedgies are stored in a dictionary, indexed by the bases. So the > order only becomes important for some display functions. I order them > by increasing index. And everything uses increasing numbers left to > right. So it'd be [[[e235 e237 e257 e357>>>.OK. That's the "alphabetical" ordering that John Browne uses. I suppose it's a path-of-least-resistance when writing software using a "dictionary", but it's definitely not the most useful ordering for human consumption, and nor is the one I gave.>> But how do you order the coefficents of a 7-limit bimonzo or bimap >> (bival) so it's its own complement??? >> Gene does it so you reverse the order to do the complement.Aha! Since posting my previous message on this, I had figured out that was the best way to do it too.> But he's > never given the general case, and I haven't worked it out. If I could, > I might be able to go on to write an efficient implementation in C.OK. Well I think we have to work this out, and standardise on it, since it seems to work so well with the new notation. For example, from the Pascal's triangle of types I posted earlier it seems that in the 3-limit (2 dimensions) if you want to know what comma vanishes in an ET you just should just flip the ET's map left for right, brackets and all. The 3-limit map for 12-tET is <12 19], which we read as saying there are 12 generators (steps, in this case) per octave and 19 per tritave. The comma that vanishes is of course the Pythagorean comma whose monzo (prime-exponent-vector) is [-19 12>, which we read as the ratio 2^-19 * 3^12. So where did the minus sign come from, on the 19? In 41-ET the map is <41 65], and the comma is [65 -41>. The minus sign's on the other side here?
Message: 8354 - Contents - Hide Contents Date: Tue, 18 Nov 2003 00:42:09 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:>> Don't we always fix the prime limit anyway? Why might this be a > problem? >> sometimes you want to use a set of nonconsecutive primes, as you've > mentioned yourself, dave. Good point.There must be a convenient way of dealing with these. Does it actually matter if you use non-consecutive primes, as long as you do it consistently throughout the calculations. Isn't it really just the _dimension_ of the multi-vectors that must be fixed for any given set of calculations?
Message: 8355 - Contents - Hide Contents Date: Tue, 18 Nov 2003 01:10:46 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Dave Keenan wrote: >>> And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a >> 7-limit trimonzo will have coefficients ordered [[[e357 e572 e723 e235>>>. >> >> Is this how your software does it too Graham? >> The wedgies are stored in a dictionary, indexed by the bases. So the > order only becomes important for some display functions. I order them > by increasing index. And everything uses increasing numbers left to > right. So it'd be [[[e235 e237 e257 e357>>>. >>> But how do you order the coefficents of a 7-limit bimonzo or bimap >> (bival) so it's its own complement??? >> Gene does it so you reverse the order to do the complement. But he's > never given the general case, and I haven't worked it out. If I could, > I might be able to go on to write an efficient implementation in C.And I might be able to write an Excel Add-in, inefficiently in VBA :-) (Visual Basic for Applications). So the 3D wedge product is not quite the same as the cross-product. The cross product is actually the wedge-product followed by a complementation. What should we use for the complement operator? Tilde? E.g. With Pauls's example of the syntonic comma and diaschisma and 12-ET, wedging the two comma monzos gives [-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo whereas their cross product gives [-4 4 -1> (x) [-11 4 2> = <12 19 28] a map and one is the complement of the other ~[[28 19 12>> = <12 19 28] So why no problems with minus signs in 3D?
Message: 8356 - Contents - Hide Contents Date: Tue, 18 Nov 2003 02:19:19 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:>> Don't we always fix the prime limit anyway? Why might this be a > problem? >> sometimes you want to use a set of nonconsecutive primes, as you've > mentioned yourself, dave.I suspect if you just put in "don't cares" for some of the coefficients, they will propagate sensibly. e.g. Use NaNs (Not-a-number) in IEEE Floating point, and display them as "X"s.
Message: 8357 - Contents - Hide Contents Date: Tue, 18 Nov 2003 04:25:25 Subject: Re: Vals? From: Dave Keenan Further to my question of whether the cross-product is the 3D wedge product followed by complementation, is it the case that the dot-product is complementation of the second argument followed by the wedge-product followed by complementation? i.e. for a map M and a monzo E (for exponents), of the same prime-limit p (dimension d), M.E = ~(M ^ ~E) i.e. <m2 m3 m5 ... mp] . [e2 e3 e5 ... ep> = ~( <m2 m3 m5 ... mp] ^ ~[e2 e3 e5 ... ep> ) = ~( <m2 m3 m5 ... mp] ^ <d-1< ep ... e5 e3 e2 ]d-1] ) (with some minus sign on some of the e's?) The notation <g< ... ]g] is meant to indicate g nested brackets, a g-vector, where g is the grade. = ~<d< a2*b2 + a3*b3 + a5*b5 + .... ap*mp ]d] = a2*b2 + a3*b3 + a5*b5 + .... ap*mp Is that correct? And what _is_ the general complement operation in terms of scalar multiply, add, and negate operations? For that matter, what is the general wedge-product in terms of scalar multiply, add, and negate operations? Is there a convenient recursive definition?
Message: 8358 - Contents - Hide Contents Date: Tue, 18 Nov 2003 04:30:44 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: Oops I changed my variables in the first part and forgot to fix them up in the second part. That should have been:> i.e. for a map M and a monzo E (for exponents), of the same > prime-limit p (dimension d), > > M.E = ~(M ^ ~E) > > i.e. <m2 m3 m5 ... mp] . [e2 e3 e5 ... ep> > > = ~( <m2 m3 m5 ... mp] ^ ~[e2 e3 e5 ... ep> ) > > = ~( <m2 m3 m5 ... mp] ^ <d-1< ep ... e5 e3 e2 ]d-1] ) > > (with some minus sign on some of the e's?) > > The notation <g< ... ]g] is meant to indicate g nested brackets, a > g-vector, where g is the grade. > > = ~<d< m2*e2 + m3*e3 + m5*e5 + .... mp*ep ]d] > > = m2*e2 + m3*e3 + m5*e5 + .... mp*ep > > Is that correct? And what _is_ the general complement operation in > terms of scalar multiply, add, and negate operations? > > For that matter, what is the general wedge-product in terms of scalar > multiply, add, and negate operations? Is there a convenient recursive > definition?I guess it would be better to use the index of the prime as the subscript, rather than the prime itself, in describing the general complement and wedge-product operations in terms of scalar operations.
Message: 8359 - Contents - Hide Contents Date: Wed, 19 Nov 2003 05:06:57 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> GABLE gives 28*e2^e3 + 12*e3^e5 + 19*e5^e2, where "e" is the unit > vector.Sounds great. What's GABLE? Now you know why I tried to sweep all of this under the rug.
Message: 8360 - Contents - Hide Contents Date: Wed, 19 Nov 2003 17:15:02 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:>>> According to John Browne, the above bimonzo is correct if its basis >> is >>>>>> lg(2)^lg(3) lg(5)^lg(2) lg(3)^lg(5) (indices 12 31 23) >>> >>> where ^ is the wedge-product operator, not exponentiation. >>> >>> = [[ (-4*4)-(4*-11) (-1*-11)-(-4*2) (4*2)-(-1*4) >> >>> = [[28 19 12>> >>> >>> But if the basis is instead >>> >>> lg(2)^lg(3) lg(2)^lg(5) lg(3)^lg(5) (indices 12 13 23) >>> >>> (just swapped the order of lg(2) and lg(5) in the middle one) >>> then the bimonzo is >>> [[28 -19 12>> >>>> right, but if you keep the (directed) angle between the two vectors >> in each basis bivector the same, you don't get this behavior in 3D >> (since you use e5^e2 and not e2^e5 in your basis) -- but you *do* get >> it in 2D. >>>>> And it starts to look like the general complement (for any grade and >>> dimension) should not only reverse the order of coefficients, but >>> negate every second one. >>>> what could be special about every second one? think about this purely >> geometrically, so the order of the primes loses itssignificance . . .> > True, but we have to agree on _some_ standard ordering of the > coefficients in a multivector of any grade and dimension, and in > addition to that, we have to agree on the ordering of the grade-1 > basis components making up higher-grade basis components. > > It should be something we can easily remember for any grade and > dimension. > > Lexicographic ("alphabetical") ordering (in both of the above cases), > is something that's easy to remember. It's what Browne uses in his > Mathematica package. And it seems like it might give rise to a uniform > complementation rule of "negate every second one and reverse the order".It sure doesn't seem that way to me, for the reasons I tried to convey to you.
Message: 8361 - Contents - Hide Contents Date: Wed, 19 Nov 2003 23:31:40 Subject: Re: "does not work in the 11-limit" From: Manuel Op de Coul 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 channelYou 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. Manuel
Message: 8362 - Contents - Hide Contents Date: Wed, 19 Nov 2003 05:08:44 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> GABLE gives 28*e2^e3 + 12*e3^e5 + 19*e5^e2, where "e" is the unit > vector.Sounds great. What's GABLE? Now you know why I tried to sweep all of this under the rug.
Message: 8363 - Contents - Hide Contents Date: Wed, 19 Nov 2003 14:33:50 Subject: Re: "does not work in the 11-limit" From: Carl Lumma>> >re 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 mouseWhen 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.>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. But again, back in the day, I used Encore and had to apply accidentals with a tool. "respell" was a drop-down menu option. -Carl
Message: 8364 - Contents - Hide Contents Date: Wed, 19 Nov 2003 17:17:06 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> if linear temperaments are 2-dimesional as you always stress, why >> would these be 0-dimensional and not 1-dimensional? >> Don't blame me--you are the one who insisted linear temperaments were > to be called linear, not planar. If they are linear--ie 1D, then what > are really 1D temperaments (ets) now have to be called 0D. > > for example,>> 88cET has a single generator of 88 cents . . . seems 1 dimensional > to >> me! >> Of course, but I was mugged for saying this sort of thing in the > first place. If you have octave equivalence, you can reduce mod > octaves, and get cyclic groups, which is about as 0D a thing as this > business will afford you.yes, as you know i (and especially graham) like that idea very much -- BUT 88cET has no octaves!
Message: 8365 - Contents - Hide Contents Date: Wed, 19 Nov 2003 05:11:25 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Lexicographic ("alphabetical") ordering (in both of the above cases), > is something that's easy to remember. It's what Browne uses in his > Mathematica package. And it seems like it might give rise to a uniform > complementation rule of "negate every second one and reverse the order".It doesn't. It is, however, what we've been using ever since Graham got me to switch by basis choice to conform to his.
Message: 8366 - Contents - Hide Contents Date: Wed, 19 Nov 2003 14:49:07 Subject: Re: "does not work in the 11-limit" From: Carl Lumma>> >here's always the possibility of simply creating a Scala seq file >> directly. >>I haven't figured out how to do that yet. (Yes, I did see the recent >postings about that on the main list.) > >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 so >I can copy and paste something from one track to another, including >the pitch-bend events. I then change the channel for each note in >the new track to another number (which goes fairly quickly in >Cakewalk, once I figured out how to do it). Another reason for >assigning channels this way is that I experienced that having two >different patches assigned to the same channel results tends to >corrupt the quality of the sound. George,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. If you can do MTS, I don't know what it does to the track->chan mapping. Maybe Manuel can chime in. -Carl
Message: 8367 - Contents - Hide Contents Date: Wed, 19 Nov 2003 17:24:09 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> OK. With lexicographic ordering of the indices, it isn't as simple as > negating every second coefficient. There's sometimes a hiccup in the > middle. It's explained in Section 5.4 of > > Index of /homes/browne/grassmannalgebra/book/b... * [with cont.] (Wayb.) TheComplement.pdfThe page cannot be displayed
Message: 8368 - Contents - Hide Contents Date: Wed, 19 Nov 2003 05:29:25 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Going strictly by alphabetical ordering, this would be > > [-4 4 -1> ^ [-11 4 2> = [[28 -19 12>> > > after which > > [[28 -19 12>>* = [12 19 28> > > and we have the cross product.I'm still confused here. So the complement operation keeps the braket pointing in the same direction? So <12 19 28] is not the complement of [[28 -19 12>> but is simply _equal_ to it (because it has a complementary basis)? Likewise in 3-limit, <12 19] is equal to [19 -12>? I like the prefix tilde for complement since it supports De-Morgan-like intuitions from Boolean algebra.
Message: 8369 - Contents - Hide Contents Date: Wed, 19 Nov 2003 17:29:12 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> GABLE gives 28*e2^e3 + 12*e3^e5 + 19*e5^e2, where "e" is the unit >> vector. >> Sounds great. What's GABLE?it's that matlab program you turned me on to -- Geometric AlgeBra Learning Environment.> Now you know why I tried to sweep all of this under the rug. he he
Message: 8370 - Contents - Hide Contents Date: Wed, 19 Nov 2003 23:16:08 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> Gene, did you intend the term "epimorphic" to apply to temperaments > as well as rational scales? If so, then is it possible that "qn" > *could* be an irrational number?I thought about making the definition more general, but I decided it would be confusing enough as it was. However, it *can* easily be generalized to a scale whose degrees are expressed in a certain fixed tuning of a regular temperament, whether linear, planar, or what. The regular temperament of dimension n-1, which means rank n, has homomorphic mappings to the integers; if there exists a mapping which orders the scale degrees correctly (here is where the tuning comes into it) we could call such a scale epimorphic. The diatonic scale, for example, would be epimorphic if we adopted this definition.
Message: 8371 - Contents - Hide Contents Date: Wed, 19 Nov 2003 06:03:41 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I'm still confused here. > > So the complement operation keeps the braket pointing in the same > direction? > > So <12 19 28] is not the complement of [[28 -19 12>> but is simply > _equal_ to it (because it has a complementary basis)?Sorry. That should have been "(because it has a _reciprocal_ basis)?".
Message: 8372 - Contents - Hide Contents Date: Wed, 19 Nov 2003 17:30:28 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:>> Lexicographic ("alphabetical") ordering (in both of the above > cases),>> is something that's easy to remember. It's what Browne uses in his >> Mathematica package. And it seems like it might give rise to a > uniform>> complementation rule of "negate every second one and reverse the > order". >> It doesn't. It is, however, what we've been using ever since Graham > got me to switch by basis choice to conform to his.what was your original basis choice, and what do the patterns of signs for duals look like under it?
Message: 8373 - Contents - Hide Contents Date: Wed, 19 Nov 2003 23:20:38 Subject: Re: Vals? From: Gene Ward Smith --- 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).I'm willing to adopt a prefix tilde and not a postfix asterisk. Paul?
Message: 8374 - Contents - Hide Contents Date: Wed, 19 Nov 2003 08:13:37 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:>> [[28 -19 12>>* = [12 19 28> >> >> and we have the cross product. >> I'm still confused here. > > So the complement operation keeps the braket pointing in the same > direction?Sorry, that should have been [[28 -19 12>>* = <12 19 28]> I like the prefix tilde for complement since it supports > De-Morgan-like intuitions from Boolean algebra.Postfix seems more natural to me; that's normally how these things are done.
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