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Message: 7225 Date: Mon, 11 Aug 2003 14:24:23 Subject: Re: tctmo! From: Graham Breed Gene Ward Smith wrote:

> The lattices Carl were talking about were honest mathematical > lattices, and hence groups. Or so I think, anyway. Anyway, this is > the important case which gets the work done in most contexts, but the > work is simply abelian groups at work.

Carl made two comments about your original statement "It's jargon that's confusing." Which may be correct. "Modern Computer Algebra" manages to define these kind of lattices without using group theory. I think they are still groups by that definition, but integers under addition are a group. You wouldn't require group theory for any paper that added numbers, would you? That would be confusing. As our lattices (by analogy to crystallography) can be explained without group theory it would only be confusing to require group theory. "If one can draw a lattice it should be sufficient to understand this, without a rigorous definition." So he's talking about a lattice as something you draw. If distance on a lattice is supposed to be measured along the bases, I don't think they're algebraic lattices anyway, as that isn't a valid inner product. Me:

>>Yes, we should at least target fairly intelligent idiots. But

> there's

>>no reason to assume knowledge of group theory.

Gene:

> Why? Group theory is what works. We don't need anything complicated, > just some very basic stuff. Then, of course, I want to toss some > multilinear algebra into the works...

We shouldn't assume knowledge because most people don't have it! Why do you have so much difficulty with that idea? It should be possible to understand our ideas with high school mathematics, and group theory is way beyond that. If you want to note the connection with group theory as well, that's fine. But we shouldn't pollute our simple explanations with an infrastructure that most people don't know and don't want to know. Graham

Message: 7226 Date: Mon, 11 Aug 2003 23:35:24 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> The error heuristic I use is size/complexity where size is the

magnitude

> of the pitch difference

What pitch difference? A comma has a size, namely itself; we can take the log of that also, of course. and complexity is the smallest number of

> intervals in the relevant odd limit that make up the comma.

A comma is made up of intervals? I thought it *was* an interval. I think

> both are valid metrics, but complexity isn't an inner product.

If you want an inner product, what about the one I use to define geometric complexity?

Message: 7227 Date: Mon, 11 Aug 2003 11:49:18 Subject: Re: tctmo! From: Carl Lumma

>The only special treatement given to JI is that the prime intervals >are supplied for you, there's a formula for expressing any ratio in >terms of them and you can get a list of consonances that correspond >to an odd limit.

Ok, ok. But this is a caveat. We've got JI in our brains and one can only go so far from it.

>> Za?

> >Is that supposed to mean something?

It means "?".

>>>"Straightness" is another thing I never understood.

>> >> It has to do with the angle between the commas. If A and B are >> commas that vanish, and a and b are their lattice points, then the >> interval C between a and b also vanishes. The thing is, A and B >> could both be simple, but if the angle between them is wide, C could >> still be complex. So you have to account for this in a heuristic.

> >So it's an angle on the lattice? It's something that would be nice >to have (and the heuristic certainly won't work without) but I've >never been able to calculate it.

The heuristic works fine without it for linear temperaments. But for more commas, one needs some way of taking into acount the 'difference vector(s)'. Straightness is Paul's idea, but I'm not sure he ever suggested a way to calculate it. Maybe he would like to say something here. -Carl

Message: 7228 Date: Mon, 11 Aug 2003 23:38:32 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> If you want an inner product, what about the one I use to define > geometric complexity?

probably not a bad choice, at least for this community at this point in time. i'm amazed at your inability to understand graham, though!

Message: 7229 Date: Mon, 11 Aug 2003 12:05:30 Subject: Re: Bosanquet keyboards and linear temperaments From: Carl Lumma

>> "12-based"? Bosanquet issued layouts for negative, positive, and >> doubly-positive temperaments, IIRC.

> >"Negative" means the fifth is smaller then that of 12-equal. >"Positive" means the fifth is greater than that of 12-equal. >"Doubly-positive" means the fifth is larger than that of 12-equal, >and the Pythagorean comma divides in 2. This is all based on 2.

Based on 12, you mean. It's simpler to say it's based on the pythagorean comma. What did Bosanquet leave out?

>I had a brief look at Bosanquet's book, and didn't see a >doubly-positive mapping.

That's because it isn't in the book. But you almost certainly were looking at Rasch's edition, in which he includes the doubly- positive mapping (from a later paper by B.) in the introduction. Yep, Wilson definitely showed a lot of mappings that Bosanquet didn't. -Carl

Message: 7230 Date: Mon, 11 Aug 2003 21:37:46 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Here's the doc... > > * [with cont.] (Wayb.)

rather than being "what's been going on on the tuning-math list", it's really quite a small part, and virtually all of it dates back to pre-tuning-math-list days. already i see falsitudes:

> 1.1.1-- A theme played in a different mode keeps >generic > intervals (3rds, etc.) the same while pitches change. > 1.1.1.1-- This is, in fact, only true for > Rothenberg-proper scales, such as the familiar > diatonic scale in 12-tone equal temperament.

the pythagorean diatonic is improper but would seem to have the property you're trying to describe. so would blackjack . . .

> ...Herman, you're welcome to link to the Mizarian Porcupine > Overture at * [with cont.] (Wayb.). Or > if you have a url for it elsewhere, I can use that instead. > > Paul, Glassic is at * [with cont.] (Wayb.).

thanks! the part that sounds vaguely like a repeating mixolydian I- VII-VI-V progression is the part that uses the 7-tone porcupine scale.

Message: 7231 Date: Mon, 11 Aug 2003 21:38:57 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Here's the doc... > > * [with cont.] (Wayb.)

what happened to the corrections/reactions i already posted??

Message: 7232 Date: Mon, 11 Aug 2003 22:09:35 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> You can hardly get more basic. Why are we concerned with error? > >> Error from what? What separates what we do here from all the > >> other kooked-out music theory out there? Acoustics.

> > > >I prefer to make my assumptions clear and leave acoustics out of

it.

> >It's then up to the reader whether or not they agree with it.

> > Everything we do starts from the fundamental assumption of JI. > Or so it seems to me.

is this the same "we" that didn't understand gene's initial keyboard proposal? stop speaking for everyone.

>

> >"Straightness" is another thing I never understood.

> > It has to do with the angle between the commas. If A and B are > commas that vanish, and a and b are their lattice points, then the > interval C between a and b also vanishes. The thing is, A and B > could both be simple, but if the angle between them is wide, C could > still be complex. So you have to account for this in a heuristic. > > -Carl

carl, this is far less important than the opposite case, where A and B are very complex but C is very simple. C can't be any longer than length (A) + length (B), but it can be arbitrarily shorter than either.

Message: 7233 Date: Mon, 11 Aug 2003 00:07:20 Subject: Re: Bosanquet keyboards and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...>

wrote:

> > Gene Ward Smith wrote:

>

> > Bosanquet didn't use hexagons. Erv Wilson did, and you can see

his

> > original papers at > > > > WILSON ARCHIVES * [with cont.] (Wayb.)

> > Do you think Wilson lattices would be a better name than Bosanquet > lattices for these? >

> > > We can now generalize this for any linear temperament...

> > > > Erv did that a long time ago.

> > I'll see if I can figure out what Erv is saying (not always easy

for

> me, since he says it in pictures) and if I agree. Do you have a

more

> exact reference?

I found Xenharmonikon 3. This does generalize Bosanquet somewhat, but it is hardly the same as what I was saying. Is this what you meant?

Message: 7234 Date: Mon, 11 Aug 2003 20:09:08 Subject: Re: tctmo! From: Carl Lumma

>> Here's the doc... >> >> * [with cont.] (Wayb.)

> >what happened to the corrections/reactions i already posted??

They were added. -C.

Message: 7235 Date: Mon, 11 Aug 2003 22:14:32 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> If one can draw a lattice it should be sufficient to understand > this, without a rigorous definition.

graham's definition was far better than gene's, anyway.

Message: 7236 Date: Mon, 11 Aug 2003 20:12:48 Subject: Re: tctmo! From: Carl Lumma

>> Everything we do starts from the fundamental assumption of JI. >> Or so it seems to me.

> >is this the same "we" that didn't understand gene's initial keyboard >proposal?

I dunno. It didn't seem like Graham understood it, until I created a diagram of it. I understood it at first, but then changed my mind after Gene pointed us to a diagram that bore absolutely no resemblance to what he described.

>stop speaking for everyone.

>> Or so it seems to me. >> It has to do with the angle between the commas. If A and B are >> commas that vanish, and a and b are their lattice points, then the >> interval C between a and b also vanishes. The thing is, A and B >> could both be simple, but if the angle between them is wide, C could >> still be complex. So you have to account for this in a heuristic. >> >> -Carl

> >carl, this is far less important than the opposite case, where A and >B are very complex but C is very simple. C can't be any longer than >length (A) + length (B), but it can be arbitrarily shorter than >either.

Ah, ok. -Carl

Message: 7237 Date: Mon, 11 Aug 2003 22:16:29 Subject: Re: Bosanquet keyboards and linear temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >I don't know. I usually say "Bosanquet/Wilson" when there's any > >doubt. The layout is Bosanquet's but the shape of the keys is > >Wilson's. Bosanquet only considered 12-based temperaments as well, > >so some of the generalisation is Wilsonian.

> > "12-based"? Bosanquet issued layouts for negative, positive, and > doubly-positive temperaments, IIRC. > > -Carl

all of which are defined with respect to 12.

Message: 7238 Date: Mon, 11 Aug 2003 20:22:27 Subject: Re: tctmo! From: Carl Lumma

>It depends on whether you are talking to a mathematician or not.

Gene, It strikes me as quite possible that group theory is a better basis from which to explain this stuff, but can a group theory crash course be fit into a short document? -Carl

Message: 7239 Date: Mon, 11 Aug 2003 22:20:22 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> The heuristic works fine without it for linear temperaments. But > for more commas, one needs some way of taking into acount the > 'difference vector(s)'. Straightness is Paul's idea, but I'm not > sure he ever suggested a way to calculate it. Maybe he would like > to say something here. > > -Carl

i tried my darndest here to get gene, kees, and others to help me figure out which metric the heuristic implies, so that i could generalize it to higher dimensions, but with no luck. is the inner product defined over a triangular-taxicab metric?

Message: 7240 Date: Mon, 11 Aug 2003 20:26:07 Subject: Re: tctmo! From: Carl Lumma

>Now Carl, I don't think Part III of Rothenberg's series should >be in the references. All you consider is propriety and efficiency.

Yeah, I was going to take it out.

>The result of tempering out all commas can be thought of geometrically >as collapsing the (hyper)plane into a (hyper)torus. Periodicity blocks >are different because no commas are tempered out, so one block is >different to another.

The way the document is currently set up, tempered things can still be called blocks. Is that unacceptable to anyone?

>When you talk about lattice distance, you should say how the distance >is measured.

An earlier version mentioned "taxicab", but I pulled it, because Gene's metric isn't taxicab. Paul, I just noticed you said the complexity heuristic doesn't work but with taxicab on a kees lattice. What are the failing cases? -Carl

Message: 7241 Date: Mon, 11 Aug 2003 22:21:25 Subject: Re: Bosanquet keyboards and linear temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> "12-based"? Bosanquet issued layouts for negative, positive, and > >> doubly-positive temperaments, IIRC.

> > > >"Negative" means the fifth is smaller then that of 12-equal. > >"Positive" means the fifth is greater than that of 12-equal. > >"Doubly-positive" means the fifth is larger than that of 12-equal, > >and the Pythagorean comma divides in 2. This is all based on 2.

> > Based on 12, you mean. It's simpler to say it's based on the > pythagorean comma. What did Bosanquet leave out?

tunings based on more than one chain of fifths, like 72, for example.

Message: 7242 Date: Mon, 11 Aug 2003 06:32:16 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Gene Ward Smith wrote:

> > They'd better, since you need groups to define lattices. A

lattice is

> > a free abelian group of finite rank with additional structure,

the

> > additional structure being an embedding into a normed real vector > > space. It is better not to add additional structure unless you

have a

> > use for it; that merely confuses things.

> > No. A lattice, in this context is a "network of points which show

the

> simple translation vectors on which a structure is based." (H. M. > Rosenberg, "The Solid State", third edition pp2-3)

The lattices Carl were talking about were honest mathematical lattices, and hence groups. Or so I think, anyway. Anyway, this is the important case which gets the work done in most contexts, but the work is simply abelian groups at work.

>

> > I don't think it is a for complete idiots subject.

> > Yes, we should at least target fairly intelligent idiots. But

there's

> no reason to assume knowledge of group theory.

Why? Group theory is what works. We don't need anything complicated, just some very basic stuff. Then, of course, I want to toss some multilinear algebra into the works...

> > The p-limit intervals form a group. Equal temperaments are

groups.

> > Linear temperaments are groups. Groups and group homomorphisms

are

> > everywhere on this list. They are absolutely basic.

> > "A scale is a group"?

Nope.

Message: 7243 Date: Mon, 11 Aug 2003 22:29:37 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> So it's an angle on the lattice? It's something that would be nice

to

> have (and the heuristic certainly won't work without) but I've

never

> been able to calculate it.

If we have a Euclidean lattice, we also have a corresponding quadratic and bilinear form. The bilinear form gives us the dot product, and hence angles. The question now becomes, what lattice are we talking about, and is it Euclidean?

Message: 7244 Date: Mon, 11 Aug 2003 08:42:30 Subject: Re: Bosanquet keyboards and linear temperaments From: Graham Breed Gene Ward Smith wrote:

> Do you think Wilson lattices would be a better name than Bosanquet > lattices for these?

I don't know. I usually say "Bosanquet/Wilson" when there's any doubt. The layout is Bosanquet's but the shape of the keys is Wilson's. Bosanquet only considered 12-based temperaments as well, so some of the generalisation is Wilsonian.

> I'll see if I can figure out what Erv is saying (not always easy for > me, since he says it in pictures) and if I agree. Do you have a more > exact reference?

I think that's what the multi-keyboard grid iron is about. It's only octave-based scales, but we can always define temperaments by the period. He doesn't line them up so that the octaves are all in a horizontal line, so you may have some new geometry. I also wrote this some time ago: Notating and keyboarding musical scales * [with cont.] (Wayb.) Graham

Message: 7245 Date: Mon, 11 Aug 2003 20:33:04 Subject: Re: tctmo! From: Carl Lumma

>> Ok, ok. But this is a caveat. We've got JI in our brains and one >> can only go so far from it.

> >The theory shouldn't have to rely on empirical details about the >human brain.

This is about music, right?

>> It means "?".

> >The heuristics involves numerators and denominators, and interval >vectors in general don't have numerators and denominators. But, as I >saw one of them derived, it's only an approximation to something that >you can calculate directly for all interval vectors as I use them. So >there's an avoidable loss of generality.

What's an interval vector?

>> The heuristic works fine without it for linear temperaments. But >> for more commas, one needs some way of taking into acount the >> 'difference vector(s)'. Straightness is Paul's idea, but I'm not >> sure he ever suggested a way to calculate it. Maybe he would like >> to say something here.

> >For 5-limit linear temperaments.

Right; sorry. -Carl

Message: 7246 Date: Mon, 11 Aug 2003 22:33:18 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...>

wrote:

>

> > So it's an angle on the lattice? It's something that would be

nice

> to

> > have (and the heuristic certainly won't work without) but I've

> never

> > been able to calculate it.

> > If we have a Euclidean lattice, we also have a corresponding > quadratic and bilinear form. The bilinear form gives us the dot > product, and hence angles. The question now becomes, what lattice

are

> we talking about, and is it Euclidean?

we haven't yet been able to pin down the tempering rules on which the error heuristic works, though it seems we must use the triangular- taxicab metric on kees van prooijen's lattice to get the complexity heuristic to work.

Message: 7247 Date: Mon, 11 Aug 2003 00:45:48 Subject: Re: Bosanquet keyboards and linear temperaments From: Carl Lumma

>I don't know. I usually say "Bosanquet/Wilson" when there's any >doubt. The layout is Bosanquet's but the shape of the keys is >Wilson's. Bosanquet only considered 12-based temperaments as well, >so some of the generalisation is Wilsonian.

"12-based"? Bosanquet issued layouts for negative, positive, and doubly-positive temperaments, IIRC. -Carl

Message: 7248 Date: Mon, 11 Aug 2003 20:37:43 Subject: Re: tctmo! From: Carl Lumma

>The error heuristic I use is size/complexity where size is the >magnitude of the pitch difference

That sounds the same as Paul's heuristic.

>and complexity is the smallest >number of intervals in the relevant odd limit that make up the >comma.

This sounds like taxicab distance. Paul uses d, but d is supposed to be like taxicab distance (at least, on the right lattice)...

>The error measured is exactly the worst case of the worst tuning >of any interval in the relevant limit for an optimized temperament >with the given comma.

You seem to like worst error more than average error. Any reason? -Carl

Message: 7249 Date: Mon, 11 Aug 2003 22:34:31 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >

> > If one can draw a lattice it should be sufficient to understand > > this, without a rigorous definition.

> > graham's definition was far better than gene's, anyway.

It depends on whether you are talking to a mathematician or not.

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