Tuning-Math Digests messages 3525 - 3549

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Message: 3525

Date: Fri, 25 Jan 2002 21:00:09

Subject: Linear temperament consistency?

From: genewardsmith

Does anyone apply the consistency concept to linear temperaments, or even to higher dimensions? It seems like that would make sense also.


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Message: 3526

Date: Fri, 25 Jan 2002 00:54:59

Subject: Re: twintone, paultone

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>>You can't hear consistency, so why is this relevant?
> >> 
> >>You can hear consistency, when neighboring chords involve
> >>different approximations to the same interval.
> > 
> >That wouldn't happen in the case Gene is talking about.
> 
> I didn't say it would.  Gene's was a general dismissal of
> consistency.

You can have neighboring chords involve different approximations to 
the same interval even in a consistent tuning. I see this happening 
in 76-tET, where one could modulate between twintone, meantone, 
double-diatonic, as well as other systems.


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Message: 3527

Date: Fri, 25 Jan 2002 23:23:23

Subject: Re: OUR PAPER

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> The latest list of 5-limit temperaments is fine by me, 

Which list is that exactly?

> though if 
> Graham and Dave are into the idea of a stronger penalty for 
> complexity, sacrificing flatness, I'll side with them against Gene.

I'd describe it as giving a weaker reward for tiny errors (sub-cent),  
but it would have pretty much the same effect.


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Message: 3528

Date: Fri, 25 Jan 2002 01:25:33

Subject: Re: twintone, paultone

From: clumma

>You can have neighboring chords involve different approximations
>to the same interval even in a consistent tuning. I see this
>happening in 76-tET, where one could modulate between twintone,
>meantone, double-diatonic, as well as other systems.

Example?  I don't see how this could happen, unless it involved:

() Switching between subsets of the ET, which is cheating.
() Invoking higher-order approximations (ie, 10:12:15->16:19:24).

-Carl


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Message: 3529

Date: Fri, 25 Jan 2002 02:51:41

Subject: Re: Blackjack standard...

From: paulerlich

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:

> Hi Paul!
> 
> I'm assuming you mean the lattices in the "standard" key C-G-D-A 
that 
> Dave Keenan kindly refined for us, yes?
> 
> *That's* the lattice I'm currently using now...
> 
> Joseph

Hi Joseph . . .

Well, I think I'll tend to use a different, non-diatonic notation 
altogether, as the modified Sims notation is a bit too "hairy" for 
the "pretty" book I'd like to produce. Of course, a separate, more 
practically oriented Blackjack paper would be great too . . . 
someday, someday . . .


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Message: 3530

Date: Fri, 25 Jan 2002 01:31:46

Subject: Re: twintone, paultone

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> I don't see how this could happen, unless it involved:
> 
> () Switching between subsets of the ET, which is cheating.

Cheating? Jeez, can't I modulate from diatonic to diminished to whole-
tone to augmented in 12-tET?


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Message: 3531

Date: Fri, 25 Jan 2002 23:36:11

Subject: Re: twintone, paultone

From: clumma

>>>You can't hear consistency, so why is this relevant?
>> 
>>You can hear consistency, when neighboring chords involve
>>different approximations to the same interval.
> 
> 
>I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in
>55-et, and so I can claim to "hear" inconsistency.

Not really.  For isolated chords, you just always use the best
approximation (say, minimum rms).  As long as you're happy with
that approximation, and you've based it on the chords you
actually want to use, not just the dyads involved (as some
early investigators did), then you're golden.

The "problem" occurs when modulating from the best approx. of
one chord to the best approx. of another, and thereby creating
anomalous (as in, non-existent in JI) commas.  Some people
think commas are a "feature not a bug", others prefer to temper
them out.  Others (apparently both you and I) think both
approaches are valid.  And, as I said...

>>Which is not to say that this is in any way "bad".

...even tempering some commas out while inventing news ones can
probably be interesting.  But for me, as a composer, this is just
too confusing.  Thus, I restrict myself to consistent ets.

Consistency is also useful as a "badness" measure.  It may not
be ideal for looking at ets up to 10 million, as some optimum
"flat" measure may be, but for any kind of goodness per notes
you'd actually care about from a pragmatic standpoint, it is
more than adequate.

>I can also approxmiate it by 0-6-11-15 or 0-6-11-16 in 19-et;
>can I also claim to hear the 7-inconsistency of the 19-et? Why
>or why not?

Depending on the context, the former chord is more likely to
approximate 4:5:6:7 or 1/1-5/4-3/2-12/7, and the latter chord
1/1-5/4-3/2-9/5 or 12:15:18:22... in other words, I'd guess
these would normally sound like different chords when
juxtaposed.

>What about both 0-10-18-25 and 0-10-18-26 in the 31-et? Can
>you hear inconsistency here?

The former chord is clearly 4:5:6:7.  The latter chord would
attract the same suspects as 0-6-11-16 in 19-tET, and as Paul
points out, may be tuned any number of ways in diatonic music
since it functions as a dissonance there (1/1-5/4-3/2-16/9
often works well).

-Carl


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Message: 3532

Date: Fri, 25 Jan 2002 01:54:35

Subject: OUR PAPER

From: paulerlich

Hello?

Let's push forward, shall we?

Graham, do you agree with the way Gene's doing things?

If so, you guys have a plurality, against Dave and myself, who both 
seem to be resisting in different areas.

It's really time to get this stuff published in some form -- who 
knows how many university course notes it's appearing in already :)

The latest list of 5-limit temperaments is fine by me, though if 
Graham and Dave are into the idea of a stronger penalty for 
complexity, sacrificing flatness, I'll side with them against Gene.

Then we'll need similar lists for {2,3,5,7}, {2,3,7}, {2,5,7}, and 
{3,5,7} -- always keeping the first prime as the interval of 
equivalence, for brevity's sake. Additional useful info would include 
a list of proper and improper MOSs (actually, a horagram might be 
best) and lattices wherever feasible.

And all this is only part IV of our paper . . .


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Message: 3533

Date: Fri, 25 Jan 2002 22:41:29

Subject: Re: twintone, paultone

From: clumma

>>>You can't hear consistency, so why is this relevant?
>> 
>>You can hear consistency, when neighboring chords involve
>>different approximations to the same interval.
> 
> 
>I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in
>55-et, and so I can claim to "hear" inconsistency.

Not really.  For isolated chords, you just always use the best
approximation (say, minimum rms).  As long as you're happy with
that approximation, and you've based it on the chords you
actually want to use, not just the dyads involved (as some
early investigators did), you're golden.

The "problem" occurs when modulating from the best approx. of
one chord to the best approx. of another, which sometimes creates
anomalous (as in, non-existent in JI) commas.  Some people think
commas are a "feature not a bug", others prefer to temper them
out.  Others (apparently both you and I) think both approaches
are valid.  And, as I said...

>>Which is not to say that this is in any way "bad".

...even tempering some commas out while inventing news ones can
probably be interesting.  But for me, as a composer, this is just
too confusing.  Thus, I restrict myself to consistent ets.

Consistency is also useful as a "badness" measure.  It may not be
ideal for looking at ets up to 10 million, as some optimum  "flat"
measure may be, but for any kind of goodness per notes you'd
actually care about from a pragmatic standpoint, it is more than
adequate.

>I can also approxmiate it by 0-6-11-15 or 0-6-11-16 in 19-et;
>can I also claim to hear the 7-inconsistency of the 19-et? Why
>or why not?

Depending on the context, the former chord is more likely to
approximate 4:5:6:7 or 1/1-5/4-3/2-12/7, and the latter chord
1/1-5/4-3/2-9/5 or 12:15:18:22... in other words, I'd guess
these would normally sound like different chords when juxtaposed.

>What about both 0-10-18-25 and 0-10-18-26 in the 31-et? Can
>you hear inconsistency here?

The former chord is clearly 4:5:6:7.  The latter chord would
attract the same suspects as 0-6-11-16 in 19-tET, and as Paul
points out, may be tuned any number of ways in diatonic music
since it functions as a dissonance there (1/1-5/4-3/2-16/9
often works well).

-Carl


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Message: 3534

Date: Fri, 25 Jan 2002 02:06:07

Subject: Re: OUR PAPER

From: jpehrson2

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

Yahoo groups: /tuning-math/message/2958 *

> Hello?
> 
> Let's push forward, shall we?
> 
> Graham, do you agree with the way Gene's doing things?
> 
> If so, you guys have a plurality, against Dave and myself, who both 
> seem to be resisting in different areas.
> 
> It's really time to get this stuff published in some form -- who 
> knows how many university course notes it's appearing in already :)
> 
> The latest list of 5-limit temperaments is fine by me, though if 
> Graham and Dave are into the idea of a stronger penalty for 
> complexity, sacrificing flatness, I'll side with them against Gene.
> 
> Then we'll need similar lists for {2,3,5,7}, {2,3,7}, {2,5,7}, and 
> {3,5,7} -- always keeping the first prime as the interval of 
> equivalence, for brevity's sake. Additional useful info would 
include 
> a list of proper and improper MOSs (actually, a horagram might be 
> best) and lattices wherever feasible.
> 
> And all this is only part IV of our paper . . .


Well, this is all very exciting, and I saw it posted on the Tuning 
List.

However, it magically disappeared, so I figured Paul meant to post it 
to Tuning Math instead.

I would propose (if I may humbly do that for a micromini second, or a 
mathmicromini second) that there are actually *two* papers...

One the "intense" "real" one, and the other a kind of "synopsis" 
along the lines of Paul Erlich's *very* fine... in fact *very, very* 
fine "The Forms of Tonality" which was a very readable and *broadly-
based* effort, directed to the larger microtonal community.  And it 
had nice *pictures* too.

Whaddya say??

Rather than "diluting" the effort, I think it will just *focus* 
things on the new developments.  Or, similarly, such a "preamble" 
or "synopsis" could be on the Web similar to Paul's "Gentle 
Introduction" efforts...

Anyway, that's what I'm hoping for...  Not to "spoil" the progress 
over here... but just to share in the excitement!

Joseph Pehrson


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Message: 3535

Date: Fri, 25 Jan 2002 02:09:42

Subject: Re: OUR PAPER

From: paulerlich

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:

> One the "intense" "real" one, and the other a kind of "synopsis" 
> along the lines of Paul Erlich's *very* fine... in fact *very, 
very* 
> fine "The Forms of Tonality" which was a very readable and *broadly-
> based* effort, directed to the larger microtonal community.  And it 
> had nice *pictures* too.
> 
> Whaddya say??

I'll see to that -- but of course that'll be a paper (or book, 
encompassing "The Forms of Tonality" too) I produce *alone*. As you 
can imagine, the lattices I created for Blackjack and that you are 
already using will appear in it . . .
> 
> Rather than "diluting" the effort, I think it will just *focus* 
> things on the new developments.  Or, similarly, such a "preamble" 
> or "synopsis" could be on the Web similar to Paul's "Gentle 
> Introduction" efforts...

Yup!


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Message: 3536

Date: Fri, 25 Jan 2002 02:17:11

Subject: Re: OUR PAPER

From: jpehrson2

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

Yahoo groups: /tuning-math/message/2960 *

> --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
> 
> > One the "intense" "real" one, and the other a kind of "synopsis" 
> > along the lines of Paul Erlich's *very* fine... in fact *very, 
> very* 
> > fine "The Forms of Tonality" which was a very readable and 
*broadly-
> > based* effort, directed to the larger microtonal community.  And 
it 
> > had nice *pictures* too.
> > 
> > Whaddya say??
> 
> I'll see to that -- but of course that'll be a paper (or book, 
> encompassing "The Forms of Tonality" too) I produce *alone*. As you 
> can imagine, the lattices I created for Blackjack and that you are 
> already using will appear in it . . .
> > 
> > Rather than "diluting" the effort, I think it will just *focus* 
> > things on the new developments.  Or, similarly, such a "preamble" 
> > or "synopsis" could be on the Web similar to Paul's "Gentle 
> > Introduction" efforts...
> 
> Yup!

Great, Paul!

I'll be anxious to see all this!  Go team!

JP


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Message: 3537

Date: Fri, 25 Jan 2002 02:43:51

Subject: Blackjack standard...

From: jpehrson2

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

Yahoo groups: /tuning-math/message/2960 *

> --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
> 
> > One the "intense" "real" one, and the other a kind of "synopsis" 
> > along the lines of Paul Erlich's *very* fine... in fact *very, 
> very* 
> > fine "The Forms of Tonality" which was a very readable and 
*broadly-
> > based* effort, directed to the larger microtonal community.  And 
it 
> > had nice *pictures* too.
> > 
> > Whaddya say??
> 
> I'll see to that -- but of course that'll be a paper (or book, 
> encompassing "The Forms of Tonality" too) I produce *alone*. As you 
> can imagine, the lattices I created for Blackjack and that you are 
> already using will appear in it . . .
> > 

Hi Paul!

I'm assuming you mean the lattices in the "standard" key C-G-D-A that 
Dave Keenan kindly refined for us, yes?

*That's* the lattice I'm currently using now...

Joseph


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Message: 3538

Date: Fri, 25 Jan 2002 04:17:20

Subject: Re: twintone, paultone

From: genewardsmith

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> I wrote...
> 
> >>You can't hear consistency, so why is this relevant?
> > 
> >You can hear consistency, when neighboring chords involve
> >different approximations to the same interval.
> 
> Which is not to say that this is in any way "bad".

I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in 55-et, and so I can claim to "hear" inconsistency. I can also approxmiate
it by 0-6-11-15 or 0-6-11-16 in 19-et; can I also claim to hear the 
7-inconsistency of the 19-et? Why or why not? What about both
0-10-18-25 and 0-10-18-26 in the 31-et? Can you hear inconsistency here?


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Message: 3539

Date: Fri, 25 Jan 2002 04:21:48

Subject: Re: OUR PAPER

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Hello?
> 
> Let's push forward, shall we?
> 
> Graham, do you agree with the way Gene's doing things?
> 
> If so, you guys have a plurality, against Dave and myself, who both 
> seem to be resisting in different areas.

One possibility would be for me to write up my own approach to the theory part, and have that as a separate paper. Would they publish it?


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Message: 3540

Date: Fri, 25 Jan 2002 09:15:43

Subject: Two more Japanese papers

From: genewardsmith

Here is a brief account of the other two papers John sent me.

The first is another by Kazuo Kondo. Here he decides to construct the
periodicity block for the kleisma and the schisma, and seems to want
to detemper it into 53 equal or something of the sort. However, he
somehow manages to convince himself the PB has 54 tones, even though
it should have 53, so he completely wigs out and removes a corner from
the block, and then treats it as if it was a problem in chemistry
concerning imperfect crystals. He produces a lot of nonsense about
non-Riemannian geometry and screw dislocations, which may be
describing his mental processes, and he ends by an unattributed
quotation saying that tempering out commas is a good idea.

The second paper is non-Kondo, and makes more sense than anything
Kondo seems capable of. The author, Nobuyuki Otsu, takes sixteen
temperaments, such as meantone, Kirnberger 1-3, Werckmeister 1, C and
3, and so forth, and subjects them to various kinds of statistical
analysis designed to show relationships. A cluster analysis gives a
dendrogram, (sometimes called cladogram) of the relationships in the
form of a tree. Factor analysis sorts of methods, involving eigenvalue
decomposition, are also applied, and he shows diagrams of the
projections onto the first few eigenspaces. He also applies the same
analysis to the tones themselves, so that you caan find out if C would
rather hang with G or with F#. The results, unsurprisingly, tell us
that meantones are like other meantones, and well-temperaments like
other well-temperaments, but the details might interest someone.


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Message: 3541

Date: Fri, 25 Jan 2002 11:11 +0

Subject: Re: Our Paper

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a2qdsr+8o3t@xxxxxxx.xxx>
paulerlich wrote:

> Graham, do you agree with the way Gene's doing things?
> 
> If so, you guys have a plurality, against Dave and myself, who both 
> seem to be resisting in different areas.

What differences are you seeing?  I thought we were in broad agreement.

On suggestion I would like to make, though.  If the intention of this 
paper is to concentrate on unison vectors, I'd like it to avoid mentioning 
the method of constructing linear temperaments from equal temperaments.  
So long as acknowledgements are in place, you can leave me off that one.  
Then I can write the ET method up, along with whoever's interested, for a 
future issue of Xenharmonikon.  This would concentrate more on the 
practicality than the maths (most of which you'll already have covered) 
and so has to be left until I have more practical experience of the 
scales.

> The latest list of 5-limit temperaments is fine by me, though if 
> Graham and Dave are into the idea of a stronger penalty for 
> complexity, sacrificing flatness, I'll side with them against Gene.

That's all froth as far as I'm concerned.

> Then we'll need similar lists for {2,3,5,7}, {2,3,7}, {2,5,7}, and 
> {3,5,7} -- always keeping the first prime as the interval of 
> equivalence, for brevity's sake. Additional useful info would include 
> a list of proper and improper MOSs (actually, a horagram might be 
> best) and lattices wherever feasible.

I can calculate these, with whatever metrics you like, if you can't work 
out the Python code.  I'm thinking of adding CGIs to do this kind of 
thing, but it would mean restructuring the code.

I still have to do the automatic search on unison vectors as well.  Is 
that a priority?

> And all this is only part IV of our paper . . .

It is getting bloated.  I suggest you decide what really needs to be 
published now, and get cracking.  Or perhaps an introduction to some of 
the new temperaments for the imminent Xenharmonikon, and leave the 
mathematical details for a formal journal (but get your foot in the door 
as soon as possible).


                    Graham


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Message: 3542

Date: Sat, 26 Jan 2002 13:32 +0

Subject: Re: twintone, paultone

From: graham@xxxxxxxxxx.xx.xx

Carl:
> > The "problem" occurs when modulating from the best approx. of
> > one chord to the best approx. of another, and thereby creating
> > anomalous (as in, non-existent in JI) commas. 

Gene:
> That won't happen if you confine yourself to a regular temperamemt, 
> such as the twintone version of 34-et, so I don't think it is relevant.

No.  If you're using a regular temperament, you can't be using 34-et.  
34-et is an inconsistent, equal temperament.  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?

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


                   Graham


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Message: 3543

Date: Sat, 26 Jan 2002 13:32 +0

Subject: Re: Linear temperament consistency?

From: graham@xxxxxxxxxx.xx.xx

genewardsmith wrote:

> Does anyone apply the consistency concept to linear temperaments, or 
> even to higher dimensions? It seems like that would make sense also.

I've thought about it.  You'd have to confine it to a particular scale, 
because any non-just interval can be approximated better with enough steps 
of an irrational generator.

One detail is that the 31 note MOS of 1/4 comma meantone would be 
inconsistent.  The unofficial fifth is tuned better than the official one.

Linear mappings of equal temperaments will tend to be ambiguous, because 
you can get to the same intervals by going round the circle in the other 
direction.


                         Graham


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Message: 3544

Date: Sat, 26 Jan 2002 22:37:38

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

From: monz

Hey Gene,


> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, January 26, 2002 8:04 PM
> Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised)
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > Is there some special reason to use the ...
> > 
> > UVs =
> > <648/625, 2048/2025> = [3 4 -4], [11 -4 -2]
> > 
> > adj =
> > [-24  0  0]
> > [-38 -2  4]
> > [-56  4  4]
> > 
> > ... PB as an example, instead of the one I already put into
> > the definition? 
> 
> I wanted an example, and I cooked this one up, that's all.
> The only advantage of it I can see is that it uses simpler commas.


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 *


I'd like to leave in the bit which explains how to calculate
the torsion factor from the gcd of the determinants of the minors.
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.



-monz



 



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Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 3545

Date: Sat, 26 Jan 2002 13:09:45

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

From: monz

Hi Gene,


> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, January 25, 2002 11:46 AM
> Subject: [tuning-math] Proposed dictionary entry: torsion (revised)
>
>
> torsion
> 
> Torsion describes a condition wherein an 
> independent set of n unison vectors {u1, u2, ..., un}
> (<uvector.htm>) defines a non-epimophic (epimorphic.htm>)
> periodicity block, because of the existence a torsion
> element, meaning an interval which is not the product
> 
> u1^e1 u2^e2 ... un^en 
> 
> of the set of unison vectors raised to (positive,
> negative or zero) integral powers, but some integer
> power of which is. An example would be a block
> defined by 648/625 and 2048/2025; here 81/80 is
> not a product of these commas, but 
> (81/80)^2 = (648/625) (2048/2025)^(-1).


Thanks for the revised definition!
Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo *



-24  0  0
-38 -2  4
-56  4  4

Is there some special reason to use the ...

UVs =
<648/625, 2048/2025> = [3 4 -4], [11 -4 -2]

adj =
[-24  0  0]
[-38 -2  4]
[-56  4  4]

... PB as an example, instead of the one I already put into
the definition? -- that one is also the same one which Paul
used as an illustration when the torsion discussion first
began on this list:

UVs =
<128/125, 32805/32768> = [7 0 -3], [-15 8 1]

adj =
[24  0  0]
[38  1  3]
[56 -8  0]


My website already has several webpages and lattice diagrams
of this PB, and I'd like to link to them.  For example, see
the first graphic at:
more on the duodene *


If I use your PB in the definition, I'll have to create new
diagrams for it.  ... Not that I don't want to do that anyway ...
but since I already have diagrams of a torsional PB, I'd
like to employ them right away as illustration.



Also, please tell me if I should keep anything that appears
below the row of asterisks in the definition.  Otherwise it's
trash, but I don't fully understand torsion yet, so I'm being
careful and only deleting what you tell me to delete.



-monz




 



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Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 3546

Date: Sat, 26 Jan 2002 13:30:58

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

From: monz

----- Original Message ----- 
From: monz <joemonz@xxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Saturday, January 26, 2002 1:09 PM
Subject: Re: [tuning-math] Proposed dictionary entry: torsion (revised)


> Thanks for the revised definition!
> Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo *
> 
> 
> 
> -24  0  0
> -38 -2  4
> -56  4  4
> 
> Is there some special reason to use the ...


Oops ... my bad.  That matrix didn't belong there,
it means nothing where it is, and should have been deleted.
Sorry.


-monz


 



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Message: 3547

Date: Sat, 26 Jan 2002 22:19:51

Subject: Re: twintone, paultone

From: genewardsmith

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

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

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

> 34-et
is an inconsistent, equal temperament. 

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.

 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?

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

In the 34-et twintone, 96 is mapped to 96/54 mod 17 = -2, the familiar
64/63 approximation, whereas 95 maps to 95/54 mod 17 = -8, which isn't
even a part of the 10 or 12 note twintone scales. This is quite
analogous to the situation with the diatonic and standard septimal
versions of 7-limit meantone; if g31 is the map [31,49,72,88] instead
of the usual h31 of [31,49,72,87], then h12^g31 gives the temperament
[-1,-4,2,16,-6,-4] rather than h12^h31 = [-1,-4,-10,-12,13,-4]. The
first is the diatonic version of 7-limit meantone, and may be regarded
as the standard Western temperament of the last few centuries; the
second uses the much better version of 7/4 which the 31-et allows, but
it makes no appearance on the diatonic scale, as a glance at the
wedgie shows. 31 equal can deal with either.

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

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. 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.


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Message: 3548

Date: Sat, 26 Jan 2002 02:18:05

Subject: Re: twintone, paultone

From: genewardsmith

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> The "problem" occurs when modulating from the best approx. of
> one chord to the best approx. of another, and thereby creating
> anomalous (as in, non-existent in JI) commas. 

That won't happen if you confine yourself to a regular temperamemt, such as the twintone version of 34-et, so I don't think it is relevant.


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Message: 3549

Date: Sun, 27 Jan 2002 04:04:43

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

From: genewardsmith

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

> Is there some special reason to use the ...
> 
> UVs =
> <648/625, 2048/2025> = [3 4 -4], [11 -4 -2]
> 
> adj =
> [-24  0  0]
> [-38 -2  4]
> [-56  4  4]
> 
> ... PB as an example, instead of the one I already put into
> the definition? 

I wanted an example, and I cooked this one up, that's all. The only advantage of it I can see is that it uses simpler commas.


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