Yahoo Tuning Groups Ultimate Backup tuning-math Gould Article

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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> ------------------------------------------------------------------------
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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.
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
>
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Gould Article

🔗Mark Gould <mark.gould@argonet.co.uk>

🔗paulerlich <paul@stretch-music.com>

🔗dkeenanuqnetau <d.keenan@uq.net.au>

🔗graham@microtonal.co.uk