Tuning-Math Digests messages 8350 - 8374

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Message: 8350

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?


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Message: 8351

Date: Tue, 18 Nov 2003 22:38:58

Subject: contravariant vs. covariant vectors

From: Paul Erlich

403 Forbidden *


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Message: 8352

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 *

"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."


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Message: 8353

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?


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Message: 8354

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?


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Message: 8355

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?


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Message: 8356

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.


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Message: 8357

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?


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Message: 8358

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.


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Message: 8360

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 its 
significance . . .
> 
> 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.


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Message: 8361

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 channel

You might be wrong about that, because the tracks in .seq files don't
correspond with midi channels.
So if you want to get rid of bothering with pitch-bend events, it
should be quite easy. You can transform the Cakewalk midi file to a
.seq file, where each midi channel will be mapped to a different
track. Then add the tuning to the .seq file, and transform it back to
a midi file. If the note numbers don't correspond exactly with the
scale degrees, you can use a keyboard mapping as well.

Manuel


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Message: 8363

Date: Wed, 19 Nov 2003 14:33:50

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>> Are you entering notes from a keyboard?
>> 
>> -C.
>
>No, I have to mouse back and forth around the screen, clicking on a 
>note duration in one place (if it needs to be changed from what was 
>set for the previous note) and then clicking on the staff in the 
>appropriate place to draw the note.

That's the only composition method I've ever used with a computer.

>This part would go much faster 
>if Cakewalk allowed me to use the keyboard to change the note 
>durations (with the left hand; only about a half-dozen different keys 
>would be needed) while I inserted the notes with the mouse

When you say "keyboard"... many notation packages support computer
keyboard in such a fashion.  And even Cakewalk supports a MIDI key-
board for choosing the notes.

>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


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Message: 8364

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!


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Message: 8366

Date: Wed, 19 Nov 2003 14:49:07

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>> There'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


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Message: 8367

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/bookpdf *
TheComplement.pdf

The page cannot be displayed


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Message: 8368

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.


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Message: 8369

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


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Message: 8371

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)?".


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Message: 8372

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?


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