Tuning-Math Digests messages 3550 - 3574

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Message: 3550

Date: Sun, 27 Jan 2002 22:44:13

Subject: Re: twintone, paultone

From: genewardsmith

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

> Using either of the other diaschismic mappings of 34-et, the inconsistent 
> chords are those of twintone.  You certainly do get some of them within 
> the 10 note MOS.

What's an inconsistent chord? 

> > It shows 34-et as a part of a range of twintone et possibilities.
> 
> Well, that's a radical idea.  I'm sure I'd never have worked that out 
> myself.

You seemed to be objecting to it strongly, so I don't know what your point is.

> 
> BTW, what do you make the LLL reduction of

> [ 1 1 1]
> [-1 0 2]
> [ 3 5 6]

That might depend on what inner product I use, and I don't know what this is supposed to represent. If I use the standard dot product, I get

[ 0 1 0]
[ 1 0 1]
[-1 0 2]


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Message: 3551

Date: Sun, 27 Jan 2002 01:00:03

Subject: Re: Proposed dictionary entry: torsion (revised)

From: monz

Hi Gene,


> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, January 26, 2002 10:37 PM
> Subject: Re: [tuning-math] Re: Proposed dictionary entry: torsion
(revised)
>
>
> Can you integrate that into the "good" definition in the top
> part of the page?  Then I can delete all the other old junk
> in the bottom part.  Thanks.



Just thought I'd mention ...

Even tho I really still don't understand it, because of what
I see on the lattice I can intuitively sense how torsion works.
And my intuition tells me that torsion is a very important
part of getting a better focus on my model of "finity":

Definitions of tuning terms: finity, (c) 1998 by Joe Monzo *


I'm thinking that the patterns of unison-vectors that one
can see within a torsional block mean something, and this
can be modeled mathematically.

So I'd really like to keep exploring it until I understand it
fully, and to correspondingly expand the Dictionary webpage.
I've had the idea to create a book full of 5-limit
periodicity- and torsional-blocks, and many of these
can go into the webpage.

Please, Gene and the others here who do understand torsion,
feel free to comment profusely, with lots of 5- and higher-limit
examples.  I'll try to diagram all of them and include them
in the definition, or perhaps I'll make a separate analytical
page about torsion.



-monz







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Message: 3552

Date: Sun, 27 Jan 2002 01:27:22

Subject: Re: Proposed dictionary entry: torsion (revised)

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, January 27, 2002 1:00 AM
> Subject: Re: [tuning-math] Re: Proposed dictionary entry: torsion
(revised)
>
>
> Even tho I really still don't understand it, because of what
> I see on the lattice I can intuitively sense how torsion works.
> And my intuition tells me that torsion is a very important
> part of getting a better focus on my model of "finity":
>
> Definitions of tuning terms: finity, (c) 1998 by Joe Monzo *
>
>
> I'm thinking that the patterns of unison-vectors that one
> can see within a torsional block mean something, and this
> can be modeled mathematically.



Well, OK ... actually I can already see that the unison-vectors
inside the torsional-block on my lattice diagram here

Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo *

are exactly the same pair as the bounding vectors of the Duodene

Definitions of tuning terms: duodene, (c) 1998 by Joe Monzo *

... a *real* periodicity-block, and apparently one whose
12 pitches can "stand in" for the 24 of this torsional-block?
Hmmm...



Very curious,

-monz








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Message: 3553

Date: Sun, 27 Jan 2002 12:36 +0

Subject: Re: twintone, paultone

From: graham@xxxxxxxxxx.xx.xx

Me:
> > No.  If you're using a regular temperament, you can't be using 34-et. 

Gene:
> This is just wrong; the whole thing is beginning to seem like another 
> of those "religion" deal.

Close.  It's actually a question of terminology.  Equal and regular 
temperaments are different things.

Me:
> > 34-et is an inconsistent, equal temperament. 

Gene:
> 34-et isn't a regular temperament at all until you define a mapping to 
> primes according to my proposed definition, which I think would help 
> clarify all of this confusion.

34-et can be used *as* a regular temperament, but 34-et *is not* a regular 
temperament.

Me:
>  If you're using one of the 
> > other diaschismic mappings of 34-et, the inconsistent chords will be 
> > simpler than the regular ones.  So what are you going to do?  Pretend 
> > they aren't there?  Pretend they're not really 7-limit?

Gene:
> If you are using a 10-tone subset of 34 et, then they won't be there. 

Bzzt -- yes they will.  Try reading that paragraph a bit more carefully.

This is also the first time you've mentioned a 10-note *subset* which 
would obviously skew towards twintone.

> In any case, this is not a new "problem"; it arises in meantone, where 
> you get augmeted sixth intervals which are much closer to 7/4 than the 
> 64/63 approximation ones intrinsic to diatonic 7-limit harmony, and so 
> one has a connitption fit about it.

Not new at all.  In both cases there's a simplified 7-limit mapping that 
optimises close to one of the extreme ETs -- 12 in meantone, 22 in 
diaschismic.  The difference with diaschismic is that the "normal" range 
is covered by two different more-complex mappings, so we can't talk about 
a "typical" 7-limit diaschismic.  But this isn't new either, it's been on 
my website for a few years.

Me:
> > If you're not going to make use of the inconsistency, I don't see the 
> > point in using 34-equal at all.

Gene:
> The point would be to make use of the superior 5-limit 
> harmonies--compare the major sixth/minor thirds of 34-et to those of 
> 22-et, for instance.

That's not a sufficient reason.  You can get better 5-limit harmonies with 
a 105.2 cent generator (worst error 3.3 cents compared with 3.9 for 
34-equal).  So again, why use 34-equal?

> If we consider 12-et, with a fifth which is two 
> cents *flat* to be capable of producing a sort of twintone, we can 
> certainly accept 34-et. If you look at how the fifth is tempered in 
> various ets, a whole range of possibilities emerge:
> 
> h12: -1.96
> g34: 3.93
> g56: 5.19
> h22: 7.14
> h54: 9.16
> 
> There should be something for everyone in there.

What does this have to do with the price of eggs?


                        Graham


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Message: 3554

Date: Sun, 27 Jan 2002 21:37:22

Subject: Re: twintone, paultone

From: genewardsmith

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

> 34-et can be used *as* a regular temperament, but 34-et *is not* a regular 
> temperament.

What is your definition of "regular temperament"? I gave mine.

> > If you are using a 10-tone subset of 34 et, then they won't be there. 
> 
> Bzzt -- yes they will.  Try reading that paragraph a bit more carefully.

Where?

> This is also the first time you've mentioned a 10-note *subset* which 
> would obviously skew towards twintone.

Twintone is the subject of our discussion--how can we skew towards where we already are?
> > The point would be to make use of the superior 5-limit 
> > harmonies--compare the major sixth/minor thirds of 34-et to those of 
> > 22-et, for instance.
> 
> That's not a sufficient reason.  You can get better 5-limit harmonies with 
> a 105.2 cent generator (worst error 3.3 cents compared with 3.9 for 
> 34-equal).  So again, why use 34-equal?

This is simpl an agument that we should never use equal divisions at all. Should I go into reasons why we might want to?

> > h12: -1.96
> > g34: 3.93
> > g56: 5.19
> > h22: 7.14
> > h54: 9.16
> > 
> > There should be something for everyone in there.
> 
> What does this have to do with the price of eggs?

It shows 34-et as a part of a range of twintone et possibilities.


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Message: 3555

Date: Sun, 27 Jan 2002 21:43:01

Subject: Re: twintone, paultone

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:
> --- In tuning-math@y..., graham@m... wrote:
> 
> > 34-et can be used *as* a regular temperament, but 34-et *is 
not* a regular 
> > temperament.
> 
> What is your definition of "regular temperament"? I gave mine.

Yours seems to fit perfectly with the existing literature. I have no 
idea what definition Graham might be going by.

You may note that many of the decatonic "keys" in the 22-tone 
well-temperament in my paper are quite similar to 34-tET in 
intonation.


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Message: 3556

Date: Sun, 27 Jan 2002 22:02 +0

Subject: Re: twintone, paultone

From: graham@xxxxxxxxxx.xx.xx

genewardsmith wrote:

> What is your definition of "regular temperament"? I gave mine.

In fact, I've noticed that I have no idea what a "regular temperament" is. 
 So, I'd better beat a hasty retreat.  I can't find your definition 
either, though ...

> > > If you are using a 10-tone subset of 34 et, then they won't be 
> > > there. 
> > Bzzt -- yes they will.  Try reading that paragraph a bit more 
> > carefully.
> 
> Where?

The one you cut out:

>  If you're using one of the 
> > other diaschismic mappings of 34-et, the inconsistent chords will be 
> > simpler than the regular ones.  So what are you going to do?  Pretend 
> > they aren't there?  Pretend they're not really 7-limit?

Using either of the other diaschismic mappings of 34-et, the inconsistent 
chords are those of twintone.  You certainly do get some of them within 
the 10 note MOS.

> > This is also the first time you've mentioned a 10-note *subset* which 
> > would obviously skew towards twintone.
> 
> Twintone is the subject of our discussion--how can we skew towards 
> where we already are?

What?  At this time of night I can't even understand why that objection's 
bogus.  You're saying because we're discussing something it must be right?

> > > The point would be to make use of the superior 5-limit 
> > > harmonies--compare the major sixth/minor thirds of 34-et to those 
> > > of 22-et, for instance.
> > 
> > That's not a sufficient reason.  You can get better 5-limit harmonies 
> > with a 105.2 cent generator (worst error 3.3 cents compared with 3.9 
> > for 34-equal).  So again, why use 34-equal?
> 
> This is simpl an agument that we should never use equal divisions at 
> all. Should I go into reasons why we might want to?

You could do.

> It shows 34-et as a part of a range of twintone et possibilities.

Well, that's a radical idea.  I'm sure I'd never have worked that out 
myself.


BTW, what do you make the LLL reduction of

[ 1 1 1]
[-1 0 2]
[ 3 5 6]

?

                  Graham


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Message: 3557

Date: Sun, 27 Jan 2002 22:06:35

Subject: Re: twintone, paultone

From: paulerlich

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

> BTW, what do you make the LLL reduction of
> 
> [ 1 1 1]
> [-1 0 2]
> [ 3 5 6]
> 
> ?

Is the first column 2 or 3?

How about the Minkowski reduction?


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Message: 3558

Date: Mon, 28 Jan 2002 06:01:40

Subject: Re: Proposed dictionary entry: torsion (revised)

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> 
> OK, fair enough.  I decided to go ahead and make the lattice
> diagram of your example after all.  Here's the latest definition:
> 
> Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo *

What happened to the really nice definition Gene gave??????????????

This should be _at the top_, rather than omitted entirely!!!!!!!!!!!


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Message: 3560

Date: Mon, 28 Jan 2002 11:14 +0

Subject: consistent mappings, LLL, Re: twintone, paultone

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a31vrt+4sau@xxxxxxx.xxx>
genewardsmith wrote:

> What's an inconsistent chord? 

I meant chords inconsistent with the "official" mapping but which 
approximate some interval better.  So in 34-as-twintone, anything from h34 
is inconsistent.  Using the other diaschismic mappings (h34&h46 or h34&h22 
 which could be written as h46&h58 and h56&h22 respectively using only 
consistent ETs) it's chords from g34 that are inconsistent.

> > > It shows 34-et as a part of a range of twintone et possibilities.
> > 
> > Well, that's a radical idea.  I'm sure I'd never have worked that out 
> > myself.
> 
> You seemed to be objecting to it strongly, so I don't know what your 
> point is.

What am I objecting to now?  I thought it was what I said before but you 
described as "ridiculous".  Still, it was something I hadn't said that was 
ridiculous then as well.

I can't find any more 7-limit diaschismics covering 34 with a complexity 
of less than 34.  The closest is g34&h46 (prime mapping of 46 with 
alternative mapping of 34) which has a complexity of exactly 34 (so 36 
notes for two complete otonalities) and a minimax error of 4 cents. 
Period/generator mapping

[(2, 0), (3, 1), (5, -2), (3, 15)]

and my wedge invariant

(2, -4, 30, -11, 42, 81)

(this is different to Gene's wedge invariants which I still don't know how 
to get).


Oh, time for a new conjecture:

The linear temperament formed by combining two consistent equal 
temperaments will never have a higher complexity than the number of notes 
in the more complex ET.

This is using the same definitions of complexity and consistency as my 
program.  Anybody care to prove/refute it?


> That might depend on what inner product I use, and I don't know what 
> this is supposed to represent. If I use the standard dot product, I get

...

Yes, that's what I meant.  It's one of the examples from the book, and my 
function doesn't get it right.  So thanks for confirming that I'm wrong 
and not the book.

I'm not clear what to do with the function when it is working.  It takes 
square matrices, but the matrices formed by commatic unison vectors aren't 
square.

Do you have an inner product that works for octave-equivalent harmony?  At 
least then we could get a reduced basis for a periodicity block.

Also, the book covers simultaneous Diophantine approximations (of which 
more sometime) but not simultaneous linear Diophantine equations.  So I 
still don't know how to get an original basis without torsion.  Any clues?


                      Graham


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Message: 3561

Date: Mon, 28 Jan 2002 11:40 +0

Subject: Re: twintone, paultone, 34

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <200201280951.LAA65774@xxxxxx.xxx.xxxxx.xxx>
Robert C Valentine wrote:

> Or, (and this is probably the RIGHT answer)
>               
>    2 * best( 5/4 ) = octave_reduced( 4 * best( 4/3 ) )

That looks right, but it's usually done as

  best(5:4) = half_octave_reduced( 2 * best(4:3) )

because all diaschismics are divisible by two (in terms of notes to the 
octave).

> So, I have tried in the past to express it as a system in 3, 5, 
> and 13 (since 13 is notably absent from other EDOs I intend using)
> 
>   3   5 13
> [ 1   0 2  ]   2 * best( 13/8 ) = normalized( best( 4/3 ) ) 
> [ -4  2 0  ]   from above...   
> [          ]   got another ? does it come out to 34 ?

By combining 58 (which is fully 13-limit consistent) and 34, I get

best (16:13) = tritone_reduced(2*best(3:2))

and

best (16:13) = tritone_reduced(4*best(8:5))

but that gets quite hairy.


                  Graham


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Message: 3562

Date: Mon, 28 Jan 2002 18:30:38

Subject: Re: twintone, paultone, 34

From: paulerlich

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:
> 
> Due to my math skills this would probably be more appropriate
> on the main list, but due to the topic here and what I think
> my questions may elicit, I'll put it here.
> 
> First, some "Bob Valentine oriented definitions for EDOs".
> 
> Meantone := best(5/4) = octave_reduced( 4 * best(3/2) )
> Schismic := best(5/4) = octave_reduced( 8 * best(4/3) )
> 
> diaschismic := [SNIP]
> Or, (and this is probably the RIGHT answer)
>               
>    2 * best( 5/4 ) = octave_reduced( 4 * best( 4/3 ) )

That's right! If you ever forget these things, just look up the 
ratios:

Schisma = 32805:32768
Diaschisma = 2048:2025

Then you can always work out the equivalencies.

These apply not only to EDOs, but also to equal temperaments.

Now, can you figure out how "kleismic" is defined? Hint: the kleisma 
= 15625:15552

You can always look up commas here:

Stichting Huygens-Fokker: List of intervals *


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Message: 3563

Date: Mon, 28 Jan 2002 19:46:16

Subject: consistent mappings, LLL, Re: twintone, paultone

From: genewardsmith

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

> [(2, 0), (3, 1), (5, -2), (3, 15)]
> 
> and my wedge invariant
> 
> (2, -4, 30, -11, 42, 81)
> 
> (this is different to Gene's wedge invariants which I still don't know how 
> to
get).

What about a compromise? You change the order of the basis elements,
so that this reads (2, -4, 30, 81, 42, -11). The point is to make the
wedgie calclulated from the period matrix or from two ets the same as
the wedgie calculated from two commas. In return, I could normalize by
your method. Mine was chosen to correspond with the 5-limit, where a
wedgie is just the comma of the temperament, and where the obvious way
to normalize is to make the comma greater than one. However this isn't
going to work in the 11-limit, and we should decide on a single system
which may as well be the one you are using now.

> 
> Oh, time for a new conjecture:
> 
> The linear temperament formed by combining two consistent equal 
> temperaments will never have a higher complexity than the number of
notes 
> in the more complex ET.
> 
> This is using the same definitions of complexity and consistency as
my 
> program.  Anybody care to prove/refute it?

Have you done enough calculations to check its plausibility? It sounds
plausible, though.

> I'm not clear what to do with the function when it is working.  It
takes 
> square matrices, but the matrices formed by commatic unison vectors
aren't 
> square.

You should be able to change the code to nonsquare easily enough. If
that doesn't work, try filling out the matrix with rows of zeros.


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Message: 3564

Date: Mon, 28 Jan 2002 20:01:25

Subject: consistent mappings, LLL, Re: twintone, paultone

From: genewardsmith

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

> > This is using the same definitions of complexity and consistency as my 
> > program.  Anybody care to prove/refute it?
> 
> Have you done enough calculations to check its plausibility? It sounds plausible, though.

It occurs to me that the theorems I've already posted here should suffice for this. I'll get back to it.


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Message: 3565

Date: Tue, 29 Jan 2002 12:15 +0

Subject: Re: consistent mappings, LLL, Re: twintone, paultone

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a349q8+dd3a@xxxxxxx.xxx>
Me:
> > (2, -4, 30, -11, 42, 81)
> > 
> > (this is different to Gene's wedge invariants which I still don't 
> > know how to get).

Gene:

> What about a compromise? You change the order of the basis elements, so 
> that this reads (2, -4, 30, 81, 42, -11). The point is to make the 
> wedgie calclulated from the period matrix or from two ets the same as 
> the wedgie calculated from two commas.

Ah!  Well, in my case this isn't true.  This is the first time I've seen 
you acknowledge the problem.  The wedgie above is

{(2, 3): 81, (0, 1): 2, (1, 3): 42, (0, 3): 30, (0, 2): -4, (1, 2): -11}

which has the invariant

(2, -4, 30, -11, 42, 81)

its complement is

{(2, 3): 2, (1, 2): 30, (1, 3): 4, (0, 3): -11, (0, 2): -42, (0, 1): 81}

with invariant

(81, -42, -11, 30, 4, 2)

How am I supposed to know which of these to take?  They're dimensionally 
identical.  It looks like your invariant function magically removes the 
distinction, but like I said before, I don't know how to do that.


I did make a mistake above.  It's

{(2, 3): 2, (1, 2): 30, (1, 3): 4, (0, 3): -11, (0, 2): -42, (0, 1): 81}

that gives the period/generator mapping

[(2, 0), (0, 1), (11, -2), (-42, 15)]

Whereas

{(2, 3): 81, (0, 1): 2, (1, 3): 42, (0, 3): 30, (0, 2): -4, (1, 2): -11}

gives

[(1, 0), (-74, 81), (38, -42), (10, -11)]

which has a stonking 3256 cent 7-limit minimax, but is still legally 
defined.  How do you tell which is "correct"?  I really don't know.

Gene:
> In return, I could normalize by 
> your method. Mine was chosen to correspond with the 5-limit, where a 
> wedgie is just the comma of the temperament, and where the obvious way 
> to normalize is to make the comma greater than one. However this isn't 
> going to work in the 11-limit, and we should decide on a single system 
> which may as well be the one you are using now.

All I do is take the simpler of a "pair" of wedgies, which is ambiguous 
only in the 7 prime limit.  Then, I sort each basis into ascending order 
and sort the whole thing in ascending order of bases.

Me:
> > The linear temperament formed by combining two consistent equal 
> > temperaments will never have a higher complexity than the number of 
> > notes in the more complex ET.
> > 
> > This is using the same definitions of complexity and consistency as 
> > my program.  Anybody care to prove/refute it?

Gene:
> Have you done enough calculations to check its plausibility? It sounds 
> plausible, though.

I haven't found any exceptions yet, but haven't done a systematic search.  
It's easy to refute if you don't enforce consistency.  Pretty much 
anything can be written as g0&g1 with the right g0, g1 and period.

Me:
> > I'm not clear what to do with the function when it is working.  It 
> > takes square matrices, but the matrices formed by commatic unison 
> > vectors aren't square.

Gene:
> You should be able to change the code to nonsquare easily enough. If 
> that doesn't work, try filling out the matrix with rows of zeros.

The former might work.  I'll have to see if I can get my brain around it.  
The latter certainly won't.  The Gram-Schmidt orthogonalization involves 
dividing by inner products, which will be zero if a row is zero.


                       Graham


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Message: 3566

Date: Tue, 29 Jan 2002 10:19:43

Subject: Re: Proposed dictionary entry: torsion (revised)

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, January 27, 2002 10:01 PM
> Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised)
>
>
> > ... Here's the latest definition:
> > 
> > Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo *
> 
> What happened to the really nice definition Gene gave??????????????
> 
> This should be _at the top_, rather than omitted entirely!!!!!!!!!!!



The definition I now have at the top (apparently the same one
you commented on here) is the latest one Gene sent to the
tuning-math list:


Message 2973
From:  "genewardsmith" <genewardsmith@j...> 
Date:  Fri Jan 25, 2002  2:46 pm
Subject:  Proposed dictionary entry: torsion (revised)
Yahoo groups: /tuning-math/message/2973 *
 

(... did you click "refresh/reload"?)



-monz



 




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Message: 3567

Date: Tue, 29 Jan 2002 20:27:15

Subject: Re: Proposed dictionary entry: torsion (revised)

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Sunday, January 27, 2002 10:01 PM
> > Subject: [tuning-math] Re: Proposed dictionary entry: torsion 
(revised)
> >
> >
> > > ... Here's the latest definition:
> > > 
> > > Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo *
> > 
> > What happened to the really nice definition Gene 
gave??????????????
> > 
> > This should be _at the top_, rather than omitted 
entirely!!!!!!!!!!!
> 
> 
> 
> The definition I now have at the top (apparently the same one
> you commented on here) is the latest one Gene sent to the
> tuning-math list:
> 
> 
> Message 2973
> From:  "genewardsmith" <genewardsmith@j...> 
> Date:  Fri Jan 25, 2002  2:46 pm
> Subject:  Proposed dictionary entry: torsion (revised)
> Yahoo groups: /tuning-math/message/2973 *
>  
> 
> (... did you click "refresh/reload"?)
> 
> 
> 
> -monz

Who is going to understand that definition? You, Monz? I meant the 
definition in

Yahoo groups: /tuning-math/message/2937 *

but possibly changing "products of the proposed set of unison 
vectors" to "members of the group generated by the unison vectors".


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Message: 3568

Date: Tue, 29 Jan 2002 13:15:27

Subject: new cylindrical meantone lattice

From: monz

I've added an important new lattice to my "meantone"
Dictionary entry, at the bottom:

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This attempts to show visually how a meantone "wraps"
the lattice into a cylinder, thus closing one of the
theoretically infinite dimensions of the JI lattice.


Does anyone know the math that will apply sines and
cosines, to warp the lattice-lines and points so that
they actually *look* like they're sitting on the
curved face of a cylinder?



-monz


 



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Message: 3569

Date: Tue, 29 Jan 2002 21:35:37

Subject: Re: new cylindrical meantone lattice

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>  
> I've added an important new lattice to my "meantone"
> Dictionary entry, at the bottom:
> 
> Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *
> 
> 
> This attempts to show visually how a meantone "wraps"
> the lattice into a cylinder, thus closing one of the
> theoretically infinite dimensions of the JI lattice.
> 
> 
> Does anyone know the math that will apply sines and
> cosines, to warp the lattice-lines and points so that
> they actually *look* like they're sitting on the
> curved face of a cylinder?

Well, assuming you're looking at the cylinder in a direction 
perpendicular to its axis, it's pretty easy -- first of all, only use 
a slice of the lattice running perpendicular to, and with width of 
one, syntonic comma . . . then divide that width into 360 
degrees . . . then, for each point, you can use either the sine or 
the cosine of that angle to determine the final horizontal position, 
while the vertical position remains the same! Of course, that will 
have no "perspective" effect, nor will it impart any sense 
of "translucency" or "opacity" to the surface of the cylinder . . .


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Message: 3570

Date: Tue, 29 Jan 2002 21:39:18

Subject: Re: new cylindrical meantone lattice

From: paulerlich

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >  
> > I've added an important new lattice to my "meantone"
> > Dictionary entry, at the bottom:
> > 
> > Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *

I still have a problem with the way you're plotting the "meantone 
chain", i.e.,

"each meantone chain itself . . ."

What is the meaning of it? And what is it for meantones like 31-tET 
or 69-tET or LucyTuning? Does it still exist?


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Message: 3571

Date: Tue, 29 Jan 2002 13:48:34

Subject: Re: Proposed dictionary entry: torsion (revised)

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, January 29, 2002 12:27 PM
> Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised)
>
>
> > > > ... Here's the latest definition:
> > > > 
> > > > Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo *
> > > 
> > 
> > The definition I now have at the top (apparently the same one
> > you commented on here) is the latest one Gene sent to the
> > tuning-math list:
> > 
> > 
> > Message 2973
> > From:  "genewardsmith" <genewardsmith@j...> 
> > Date:  Fri Jan 25, 2002  2:46 pm
> > Subject:  Proposed dictionary entry: torsion (revised)
> > Yahoo groups: /tuning-math/message/2973 *
> >  
> > 
> > (... did you click "refresh/reload"?)
> > 
> > 
> > 
> > -monz
> 
> Who is going to understand that definition? You, Monz? I meant the 
> definition in
> 
> Yahoo groups: /tuning-math/message/2937 *
> 
> but possibly changing "products of the proposed set of unison 
> vectors" to "members of the group generated by the unison vectors".


OK, well, the definition I previously put at the top is
still the latest revision from Gene.  I haven't heard from 
him on this for a while now, so the Dictionary entry for
"torsion" continues to grow and get messier.

I've now included Paul's favorite definition at the top,
and labeled it as such, with everything else still intact
below it.

Any comments on the lattice diagram and description of it
which I added to illustrate Gene's example?  I found it
interesting that the torsion element appears as a line
which divides the PB in half.  Is this typical?  Omnipresent?


-monz


 



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Message: 3572

Date: Tue, 29 Jan 2002 14:02:05

Subject: Re: new cylindrical meantone lattice

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, January 29, 2002 1:39 PM
> Subject: [tuning-math] Re: new cylindrical meantone lattice
>
>
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >  
> > > I've added an important new lattice to my "meantone"
> > > Dictionary entry, at the bottom:
> > > 
> > > Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *
> 
> I still have a problem with the way you're plotting the "meantone 
> chain", i.e.,
> 
> "each meantone chain itself . . ."
> 
> What is the meaning of it?


I'm trying to show that, even acknowledging that the meantone
cylinder is *always the same* (which I mention on the webpage),
the way each fraction-of-a-comma meantone "cuts" across the
cylinder is different.

I don't know if there's any special meaning to that, and I
know that you've argued that there isn't.  But the actual
amount of tempering in a meantone is an acoustical reality
which can be modeled as a mathematical property that's easy
to see on this lattice as the angle of spiral which the
meantone chain produces across the face of the cylinder.

Would you like me to create a 1/6-comma example for contrast?

Using the view that's on my webpage, the 1/4-comma chain
has an angle something like this:

-._
   '-._
       '-._

Whereas 1/6-comma is nearly vertical.



> And what is it for meantones like 31-tET 
> or 69-tET or LucyTuning? Does it still exist?


Well ... the lattice I'm using here is "8ve-equivalent",
so I can't put any EDOs on them.  But my formula can easily
include 2 as a prime-factor, in which case the whole lattice
is stretched vertically, and EDOs simply form vertical chains.

But I'm not sure how well it works, because in my lattice
formula, 2 is the smallest step in ratio-space, and so
the entire 69-EDO, for example, would be crammed into a
space smaller than that which separates 1:1 from 3:2,
making it hard to give any real visual relevance.

But of course I could easily change the "step sizes" of
my lattice metrics too.  Could even reverse it, and make
2 the *largest* step.  Hmmm.... I'm thinking that that
might be a really useful idea... then it'd be easy to show
EDOs.


As for LucyTuning, I don't know ... other than the fact
that it's audibly indistinguishable from 3/10-comma MT,
which I could easily plot.  But as far as actually putting
pi on my lattice, I don't have a clue.


-monz



 



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Message: 3573

Date: Tue, 29 Jan 2002 06:46:57

Subject: question for Gene

From: paulerlich

Is there any way to directly compare the badnesses of equal 
temperaments and linear temperaments and meaningfully ask the 
question: Which of the linear temperaments that you found (in the 5-
limit, and whatever other cases you've completed) could be expressed 
by an equal temperament, without pushing the badness over the limit 
you've computed? 'Cents/error' will always increase, 
and 'gens/complexity' will often increase as well, but may 
conceivably decrease . . . or maybe this isn't meaningful at 
all. . . .?


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Message: 3574

Date: Tue, 29 Jan 2002 22:06:01

Subject: Re: Proposed dictionary entry: torsion (revised)

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Is this typical? 

Yes -- try the 24-tone {diesis, schisma} case.

> Omnipresent?

No -- it depends which UVs you use to construct the parellelepiped. 
For example, you could restate your 24-tone {diesis, schisma} one 
with a {6561:6400, 128:125} basis, and then the syntonic-comma-
squared will _not_ go from one diagonal of the block to the other. 
Try it!


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