It appears that I need to join Tuning Math, as for some unknown reason,

pleople have mentioned a certain article.

Please feel free to lambast/berate/vilify etc, for any inaccuracies in the

theory within.

Mark

> From: tuning-math@yahoogroups.com

> Reply-To: tuning-math@yahoogroups.com

> Date: 20 Mar 2002 14:23:20 -0000

> To: tuning-math@yahoogroups.com

> Subject: [tuning-math] Digest Number 322

>

>

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>

> ------------------------------------------------------------------------

>

> There are 6 messages in this issue.

>

> Topics in this digest:

>

> 1. Re: Wedge product understood at last (was: Grassmann Algebra, by Joh

> From: graham@microtonal.co.uk

> 2. Re: Wedge product understood at last (was: Grassmann Algebra, by Joh

> From: "genewardsmith" <genewardsmith@juno.com>

> 3. Re: A common notation for JI and ETs

> From: David C Keenan <d.keenan@uq.net.au>

> 4. Re: A common notation for JI and ETs

> From: "genewardsmith" <genewardsmith@juno.com>

> 5. Re: A common notation for JI and ETs

> From: "dkeenanuqnetau" <d.keenan@uq.net.au>

> 6. Re: A common notation for JI and ETs

> From: "genewardsmith" <genewardsmith@juno.com>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 1

> Date: Tue, 19 Mar 2002 16:12 +0000 (GMT)

> From: graham@microtonal.co.uk

> Subject: Re: Wedge product understood at last (was: Grassmann Algebra, by Joh

>

> In-Reply-To: <a762uk+mcqd@eGroups.com>

> dkeenanuqnetau wrote:

>

>> If anyone is still struggling, as I was until a few minutes ago, to

>> understand what the heck the exterior product is (I prefer this name

>> to "wedge product"), and what the point of it is, then just read the

>> first 6 pages of the Introduction. The result was a great "aha!" for

>> me.

>

> Yes, the introduction already covers a lot of ground. I'm working through

> the whole thing now.

>

>> It is really quite a beautiful conception. I still say we don't "need"

>> it, but I can certainly see that some things will be described much

>> more elegantly _with_ it.

>

> Once you know the formalism, it's easier to calculate the temperaments

> using exterior algebra than matrices. More people will already be

> familiar with matrices, but I suppose most musicians won't be with either.

> One problem is that you can think of the matrix approach as solving an

> equation.

>

> The alternative model is that the wedgie allows you to tell which

> intervals are equivalent to a unison, and can be generated from a sample

> set of such intervals. I'm not sure why it should also give mappings to

> convert prime coordinates to tempered ones, other than by analogy with

> matrices.

>

>> Thanks guys for all your attempts. What I was missing was

>> (a) Its geometrical interpretation

>> "thus enabling us to bring to bear the power of geometric

>> visualization and intuition into our algebraic manipulations."

>

> Yes, it has big geometric and physical applications I didn't know about.

>

>

> The thing I was calling the complement is also called the complement in

> this book, so that saves me making any changes to my program. Although I

> could change "Wedgable" to "ExteriorElement". From the regressive product

> being like an intersection, we should have

>

> (h31^h41)v(h19^h41) = h41

>

> So if you set

>

> miracle = (h31^h41).complement()

> magic = (h19^h41).complement()

>

> then miracle.complement() v magic.complement() = +/- h41

>

> or equivalently

>

> (miracle ^ magic).complement() = +/- h41

>

> Hence we have an answer to the question asked some time ago about how you

> find the equal temperament common to a pair of linear temperaments.

> Unfortunately, it doesn't work beyond the 5-limit. Still, I can get at

> the 7-limit result like this:

>

>>>> (~(~magic^{(3,):1}) ^ ~(~miracle^{(3,):1})).invariant()

> (41, 65, 95)

>>>> (~(~magic^{(2,):1}) ^ ~(~miracle^{(2,):1}) ^ {(2,):1}).invariant()

> (41, 65, 0, 115)

>

> where I've experimentally redefined the ~ operator to find the complement.

> The first calculation is (miracle^magic).invariant() for the 5-limit

> subset. The invariant() method already gets the complement and sign

> right. The second calculation is the same idea but removing the prime 5

> from the reckoning. The two results between them give the correct mapping

> [41 65 95 115].

>

> Starting from the 11-limit, you'd have to knock out two primes at a time.

>

>

> Graham

>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 2

> Date: Tue, 19 Mar 2002 20:21:51 -0000

> From: "genewardsmith" <genewardsmith@juno.com>

> Subject: Re: Wedge product understood at last (was: Grassmann Algebra, by Joh

>

> --- In tuning-math@y..., graham@m... wrote:

>

>> One problem is that you can think of the matrix approach as solving an

>> equation.

>

> Why is that a problem? You can generalize Cramer's rule with wedges, for that

> matter.

>

>> The thing I was calling the complement is also called the complement in

>> this book, so that saves me making any changes to my program.

>

> I didn't know you knew about compliments. This is all connected to the jabber

> I giving about Poincare duality, where vals map to compliments of 1-intervals

> and wedges of vals map to vee products of compliments.

>

> By the way, it seems people are adopting the terminology of this book.

> The wedge product is probably most often called "exterior product" (four

> syllables rather than one, and I didn't learn it that way, so I like wedge

> better) but I think "regressive product" goes all the way back to

> Grassmann--at least, its usually called the "intersection product", but I like

> "vee product" myself. There's also something called the "Grassmann product" in

> case this wasn't confusing enough as it is.

>

>

>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 3

> Date: Tue, 19 Mar 2002 16:32:52 -0800

> From: David C Keenan <d.keenan@uq.net.au>

> Subject: Re: A common notation for JI and ETs

>

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>>

>>> I don't know of a name, but I expect it is listed in John

> Chalmer's

>>> lists of superparticulars at a given prime-limit. Anyone know a

>>> URL? I was astonished to learn that such lists seem to be finite.

>>

>> I proved that a while back using Baker's theorem on this list.

>

> Good work. Did you (or can you) prove that the smallest on each of John's

> lists is the smallest there is?

>

> Here's John Chalmers list of superparticulars

> /tuning-math/message/1687

> -- Dave Keenan

> Brisbane, Australia

> http://dkeenan.com

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 4

> Date: Wed, 20 Mar 2002 03:34:36 -0000

> From: "genewardsmith" <genewardsmith@juno.com>

> Subject: Re: A common notation for JI and ETs

>

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>

>> Good work. Did you (or can you) prove that the smallest on each of John's

>> lists is the smallest there is?

>

> Baker's theorem would give an effective bound, but not a good one.

> Proving that the seemingly smallest one is in fact the smallest one

> looks to me, as a number theorist, to be a difficult problem in number theory,

> though certainly not hard in the 3-limit. Maybe I should try showing 81/80 is

> the smallest in the 5-limit.

>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 5

> Date: Wed, 20 Mar 2002 04:50:04 -0000

> From: "dkeenanuqnetau" <d.keenan@uq.net.au>

> Subject: Re: A common notation for JI and ETs

>

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>>

>>> Good work. Did you (or can you) prove that the smallest on each of

> John's

>>> lists is the smallest there is?

>>

>> Baker's theorem would give an effective bound, but not a good one.

>> Proving that the seemingly smallest one is in fact the smallest one

>> looks to me, as a number theorist, to be a difficult problem in

> number theory, though certainly not hard in the 3-limit. Maybe I

> should try showing 81/80 is the smallest in the 5-limit.

>>

>

> If it looks hard, forget it. I'm sure you've got better things to do

> with your time.

>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 6

> Date: Wed, 20 Mar 2002 07:01:15 -0000

> From: "genewardsmith" <genewardsmith@juno.com>

> Subject: Re: A common notation for JI and ETs

>

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

>> If it looks hard, forget it. I'm sure you've got better things to do

>> with your time.

>

> If I *could* prove it, it might be enough for a paper.

>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

>

>

> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

>

>

>

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> It appears that I need to join Tuning Math, as for some unknown

reason,

> pleople have mentioned a certain article.

unknown reason . . . well, you asked on the tuning list for reactions

to your article, and since most of the generalizing-diatonicity work

goes on over here, it's not too surprising that it first got

mentioned here.

if you're interested in generalizing diatonicity to higher harmonic

limits, you're in the right place. hope you will participate and be

patient during the weeks/months/years it often takes people to learn

one another's languages.

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> It appears that I need to join Tuning Math, as for some unknown

reason,

> pleople have mentioned a certain article.

Paul, I think Mark is being facetious here. :-)

> Please feel free to lambast/berate/vilify etc, for any inaccuracies

in the

> theory within.

Hi Mark,

It seems we must first berate you for failing to remove the quote of

an entire tuning-math digest from the end of your post. :-)

Regards,

-- Dave Keenan

In-Reply-To: <B8BE8615.3860%mark.gould@argonet.co.uk>

Mark Gould wrote:

> It appears that I need to join Tuning Math, as for some unknown reason,

> pleople have mentioned a certain article.

It's an article on tuning and mathematics, isn't it? That suggests it's

fairly relevant.

> Please feel free to lambast/berate/vilify etc, for any inaccuracies in

> the

> theory within.

As you're being negative, I'll suggest your use of "generated scale" is

wrong, or at least inconsistent with Carey & Clampitt. You say something

like generated scales will only have two step sizes, which is the extra

criterion that makes a generated scale MOS or WF.

I don't understand either of the second criteria you use for choosing

diatonics. (That is the second numbered point on both the lists.) Can

you clarify them?

Graham