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Message: 3500 - Contents - Hide Contents

Date: Wed, 23 Jan 2002 23:04:34

Subject: Re: Proposed dictionary entry: torsion

From: genewardsmith

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

> torsion
> 
> Torsion describes a condition wherein an independent set of n unison vectors (<uvector.htm>)
fails to define a periodicity block of dimension n...


No agreement has been reached on what peridicity block means, so this
could also read "defines an anomalous periodicity block".

Properties which might or might not be included:

(1) Epimorphic

(2) Convex

(3) Connected

(4) Linf "Max" norm, with paralelipiped blocks.

Clearly (4) ==> (2), but I don't think there are any other
implications, so this leaves a lot up in the air. I was going to try
to nail it down, but people thought I was getting too technical for a
general math dictionary; however it seems to me some precise
definitions should be agreed to.

Message: 3501 - Contents - Hide Contents

Date: Wed, 23 Jan 2002 23:07:20

Subject: Re: Proposed dictionary entry: torsion

From: genewardsmith

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


> Well then your definition doesn't seem to work, because if the basis 
> is the diesis and the schisma, the syntonic comma squared is in the 
> kernel, but not in the basis.


That's why I said "not products of the proposed set of unison vectors" instead of "not members of the proposed set of unision vectors".

Message: 3502 - Contents - Hide Contents

Date: Wed, 23 Jan 2002 04:15:46

Subject: Re: new Dictionary entry: "torsion"

From: genewardsmith

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

> 
> new Dictionary entry: "torsion"


I would suggest adding something to explain why it is called "torsion"--namely,
the existence of torsion elements. If you decide to temper out
6561/6250 and 128/125, then 81/80 is not a unison, but (81/80)^2 is.
It is therefore an element of finite order, which is called a torsion
element.

Here is what Mathworld has to say, which you could link to:

 * [with cont.] 

Message: 3503 - Contents - Hide Contents

Date: Wed, 23 Jan 2002 04:27:29

Subject: Re: the Lattice Theory Homepage

From: genewardsmith

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


>> If you want specific lattices to look at, the root lattice
>> An and its dual An* are the key ones, I think.
> 
> 

> I don't understand that.  Where can these be found?


If you have a university library available, "Sphere Packings, Lattices and Groups" by Conway and Sloane is a good place to read up on it; "A
Course inArithmetic" by Serre might be relevant but I can't find my
copy to check. Don't get sucked into Lie algebras; the Mathworld
definition and things of that ilk will just lead you into all kinds of
irrelevant complexities.

The lattices people often like to draw--with triangles, or in three
dimensions, hexany/octahedrons and tetrahedrons, are A2 and A3.

Message: 3505 - Contents - Hide Contents

Date: Wed, 23 Jan 2002 04:31:17

Subject: Re: new Dictionary entry: "torsion"

From: monz

new Dictionary entry: "torsion"
Definitions of tuning terms: torsion, (c) 2002... * [with cont.]  (Wayb.)

 

> I would suggest adding something to explain why it is called "torsion"



Done.  Thanks, Gene!


-monz


 



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Message: 3506 - Contents - Hide Contents

Date: Wed, 23 Jan 2002 12:46 +0

Subject: Re: twintone, paultone (was something about schoenberg

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <200201231200.OAA67852@xxxxxx.xxx.xxxxx.xxx>
Robert C Valentine wrote:


> More quick definitions needed : twintone, paultone and quick 
> comparison with diaschismic. Any additional commentary regarding
> 34 welcome too.


I'm trying to keep track of these at 
<Catalogue of linear temperaments * [with cont.]  (Wayb.)>.  I do need some updates, 
especially for this Pelog thing.

Diaschismic is any temperament with a period of a half octave, and a 
mapping of [1, -2].  That is, you have a semitone generator.  A perfect 
fifth is a tritone plus a semitone.  A major third is a tritone minus two 
semitones.

Paultone, which now seems to be being called twintone, is a particular 
7-limit diaschismic consistent with 22-equal.  The mapping is [1, -2, -2] 
so a 7:4 is an octave minus two semitones.

34-equal is an accurate, 5-limit diaschismic.  But it isn't consistent in 
the 7-limit.  There are two 7-limit mappings which converge at 34, and 
give good results for different tunings.  Neither of them is paultone.

The typical diaschismic mapping, for me at least, is that consistent with 
46- and 58-equal.  That's [1, -2, -8] and it's accurate to about 6 cents.

A script for finding the linear temperament consistent with (the prime 
mappings of) a pair of equal temperaments can be found at 
<Linear Temperament Finding Home * [with cont.]  (Wayb.)>


                         Graham

Message: 3507 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 13:57 +0

Subject: Re: twintone, paultone

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a2n9fh+hqeh@xxxxxxx.xxx>
Me:

>> 34-equal is an accurate, 5-limit diaschismic.  But it isn't 
>> consistent in the 7-limit.  There are two 7-limit mappings which 
>> converge at 34, and give good results for different tunings.  Neither 
>> of them is paultone.

Gene:

> This is getting ridiculous.


Why so?  You haven't addressed anything I said there, so I'll have to do 
it myself.


>> 34-equal is an accurate, 5-limit diaschismic.


The worst-tuned 5-limit interval in 34-equal is 5:4, which is 4 cents 
sharp, or about 1/9 steps of 34.  That's pretty good.


>> But it isn't consistent in the 7-limit.


The worst-tuned 7-limit interval in h34 (34-equal using the best 
approximations to ratios of prime numbers) is 7:4, which is 0.56 steps 
flat.  Hence 34-equal is not 7-limit consistent.


>> There are two 7-limit mappings which 
>> converge at 34, and give good results for different tunings.


h34&h46 (the temperament consistent with 46-equal and the prime mapping of 
34-equal) is accurate to 5.9 cents in the 7-limit with a generator of 
103.9 cents.  h34&h22 is accurate to 6.9 cents in the 7-limit with a 
generator of 107.5 cents.  These are the most accurate 7-limit 
diaschismics I know of.  (You could probably improve on them with much 
more complex mappings, and there's also Shrutar and the like.)  They both 
involve the prime mapping of 34-equal, hence converge at 34-equal.


>> Neither 
>> of them is paultone.


h34&h22 has a period-generator mapping of

[(2, 0), (3, 1), (5, -2), (4, 9)]

h34&h46 has a period-generator mapping of

[(2, 0), (3, 1), (5, -2), (7, -8)]

paultone has a period-generator mapping of

[(2, 0), (3, 1), (5, -2), (6, -2)]

These are clearly three different temperaments.


So what's the problem?


> Here is a comparison of twintone as tuned in h22 and g34, which I think 
> shows g34 is perfectly practical and arguably preferable:


Preferable to what?  All this seems to show is that 34-equal is 
inconsistent in the 7-limit, as I stated above.


> 3: 7.13  3.93
> 5: -4.50 1.92
> 7: 12.99 19.41
> 5/3: -11.63 -2.01
> 7/3: 5.86 15.48
> 7/5: 17.49 17.49
> 
> The 22-et version has better 7/4s and 7/6s, and the 34-et version has
> better 3/2s, 5/4s, and 5/3s; they share the twintone 7/5 of sqrt(2).


Of course g34 has good 5-limit intervals, they're the same as those from 
h34 !  The 7-limit as a whole is much worse, and most errors are in the 
same direction, so a compromise temperament won't improve matters.

As it happens, my optimum for paultone/twintone is a 109.4 cent generator, 
with a worst 7-limit error of 17 cents.  3/34 octaves are 105.9 cents and 
2/22 octaves are 109.1 cents.  So the twintone optimum is not only closer 
to 22 than 34, but falls the other side of 22 than 34.  That makes 
34-equal far less characteristic of paultone/twintone than the other two 
diaschismics I mention above.

So again, what's the problem?


                      Graham

Message: 3508 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 13:57 +0

Subject: Re: Thank you Graham!

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a2ntkf+353u@xxxxxxx.xxx>
paulerlich wrote:


> I don't see any precedent for the kind of stuff we're dealing with in
> our paper, though . . .


It derives a linear temperament by tempering out a unison vector.  I 
thought Fokker only did equal temperaments.


                Graham

Message: 3509 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 19:46:14

Subject: Re: twintone, paultone

From: genewardsmith

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

> In-Reply-To: <a2n9fh+hqeh@e...>


>>> 34-equal is an accurate, 5-limit diaschismic.  But it isn't 
>>> consistent in the 7-limit.  There are two 7-limit mappings which 
>>> converge at 34, and give good results for different tunings.  Neither 
>>> of them is paultone.

> Gene:

>> This is getting ridiculous.
> 

> Why so?  You haven't addressed anything I said there, so I'll have to do 
> it myself.


You say, correctly, that there are two reasonable 34-et 7-limit mappings. You claim, incorrectly, that neither gives twintone. 


>>> But it isn't consistent in the 7-limit.
> 

> The worst-tuned 7-limit interval in h34 (34-equal using the best 
> approximations to ratios of prime numbers) is 7:4, which is 0.56 steps 
> flat.  Hence 34-equal is not 7-limit consistent.


You can't hear consistency, so why is this relevant?


>>> There are two 7-limit mappings which 
>>> converge at 34, and give good results for different tunings.
> 

> h34&h46 (the temperament consistent with 46-equal and the prime mapping of 
> 34-equal) is accurate to 5.9 cents in the 7-limit with a generator of 
> 103.9 cents.  h34&h22 is accurate to 6.9 cents in the 7-limit with a 
> generator of 107.5 cents.  These are the most accurate 7-limit 
> diaschismics I know of.


I'm talking about twintone, which means a generator of a fourth or fifth, or of a half-octave translate. The fifth of 34-et is 705.9
cents, and taking 600 from this gives 105.9 cents.


> So what's the problem?


The
problem is that I'm trying to clear up a misconception, that you can't
do twintone with 34 equal. You can, it works and it makes good sense.


>> Here is a comparison of twintone as tuned in h22 and g34, which I
think 
>> shows g34 is perfectly practical and arguably preferable:
> 

> Preferable to what?  All this seems to show is that 34-equal is 
> inconsistent in the 7-limit, as I stated above.


It's much better for the 5-limit than 22-et, and the 7-limit of
twintone can never be improved beyond the 7/5~10/7 compromise tritone
anyway. If you look at the numbers, the 7-limit is worse, but the
5-limit is much better--so the question is, which do you want?


>> 3: 7.13  3.93
>> 5: -4.50 1.92
>> 7: 12.99 19.41
>> 5/3: -11.63 -2.01
>> 7/3: 5.86 15.48
>> 7/5: 17.49 17.49


> As it happens, my optimum for paultone/twintone is a 109.4 cent
generator, 
> with a worst 7-limit error of 17 cents.  3/34 octaves are 105.9
cents and 
> 2/22 octaves are 109.1 cents.  So the twintone optimum is not only
closer 
> to 22 than 34, but falls the other side of 22 than 34.


This is a weak argument--the rms optimized generator of 708.8 cents
falls between the 22-et fifth of 709.1 cents and the 34-et fifth of
705.9 cents, but of course the whole thing is a one-parameter family
which will give you different optimum values depending on what you
decide to optimize, and what type of optimization you use. Weighting
the 5-limit more heavily will move things in the direction of 34-et.

Message: 3510 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 21:30:06

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.

-Carl

Message: 3511 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 22:08:47

Subject: Re: twintone, paultone

From: clumma

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".

-Carl

Message: 3512 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 19:58:22

Subject: Re: OUR PAPER

From: monz

Just wanted to mention that I'll offer space on my website
for anything anyone wants to publish there regarding this stuff.

I plan to explain as much as I can understand of the tuning-math
proceedings, with *LOTS AND LOTS* of lattice diagrams!


-monz


----- Original Message ----- 
From: jpehrson2 <jpehrson@xxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Thursday, January 24, 2002 6:06 PM
Subject: [tuning-math] Re: OUR PAPER



> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> Yahoo groups: /tuning-math/message/2958 * [with cont.] 
> 
>> Hello?
>> 

>> Let's push forward, shall we?
>> 
>> ...
>

> 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.




-monz




 



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Message: 3513 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 00:04:10

Subject: Re: Proposed dictionary entry: torsion

From: paulerlich

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

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

>> Torsion describes a condition wherein an independent set of n 

unison vectors (<uvector.htm>) fails to define a periodicity block of 
dimension n...

> 
> No agreement has been reached on what peridicity block means, so 

this could also read "defines an anomalous periodicity block".

That would be preferable.

Message: 3514 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 00:05:44

Subject: Re: Proposed dictionary entry: torsion

From: paulerlich

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

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

>> Well then your definition doesn't seem to work, because if the 
basis 
>> is the diesis and the schisma, the syntonic comma squared is in 
the 
>> kernel, but not in the basis.
> 

> That's why I said "not products of the proposed set of unison 
>vectors" instead of "not members of the proposed set of unision 
>vectors".


Oops! I read it wrong, somehow. But "products" might still fail to 
capture a case like a^2/b where a and b are in the basis. Can't we 
think of a better terminology?

Message: 3515 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 02:06:44

Subject: Two papers by Kazuo Kondo

From: genewardsmith

John Chalmers very kindly sent me two papers by Kondo, a professor of something or other in Japan. The basic thesis is that scales "must" be
defined in terms of the non-zero elements of finite fields. His
reasons for claiming this are gibberish; he seems to be saying that
since sine waves are defined in terms of complex exponentials and
complex exponentials live in an infinte field, for a scale to exist
the infinite field (presumably, the complex numbers) must map to a
finite field. This makes no sense, and in any case field homomorphims
of this type don't exist; he would need to shift gears to a ring of
algebraic integers instead, and even so it still would make no sense.

However, once we drop the "undoubtable lemma" he claims to have "theoretically discovered" and simply ask what he is doing, it comes
down to this:

(1) Scales must have a number of degrees equal to p^n - 1 to the
octave, where p is a prime. p=2 is particularly favored, so 7 and 15
note scales are his babies. You'd think 31 would interest him also,
but he seems to have no notion that small integer ratios are in any
way interesting.

(2) The chords of the scale are defined by its structure as a
projective space.

In particular, the diatonic scales "must" have the structure of a
projective plane of order 7, so that the seven triads of it in one
form are FCD, CGA, GDE, DAB, AEF, EBC, and BFG if I am understanding
him correcly. I don't guarantee I am, since he's none too clear. The
idea of trying to jam musical meaning into finite combinatorial
structures with nice automorphism groups such as finite projective
spaces or designs is one I've had also, but it is rather artificial.

Message: 3516 - Contents - Hide Contents

Date: Thu, 24 Jan 2002 03:04:47

Subject: Thank you Graham!

From: paulerlich

Thanks for the papers you sent me, Graham!

I don't see any precedent for the kind of stuff we're dealing with in
our paper, though . . .

Message: 3517 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 18:11:23

Subject: Re: twintone, paultone

From: paulerlich

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

> What about both
> 0-10-18-25 and 0-10-18-26 in the 31-et?


The first is a 7-limit tetrad or German augmented sixth, the second 
is a classical dominant seventh (I know, we argued about this a while 
ago, you do not believe this, but I sure do).

Message: 3518 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 10:29:43

Subject: Re: Proposed dictionary entry: torsion

From: monz

Hi Gene,

Two things.

1) I love your new definition of "torsion".  What exactly
should I replace in my old definition?  Everything?  Please
be as specific as possible.


2) The last post on this subject was this one from Paul,
with a question to which you have not yet responded:



> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, January 23, 2002 4:05 PM
> Subject: [tuning-math] Re: Proposed dictionary entry: torsion
>
>
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

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

>>> Well then your definition doesn't seem to work, because if the 
> basis 

>>> is the diesis and the schisma, the syntonic comma squared is in 
> the 

>>> kernel, but not in the basis.
>> 

>> That's why I said "not products of the proposed set of unison 
>> vectors" instead of "not members of the proposed set of unision 
>> vectors".
> 

> Oops! I read it wrong, somehow. But "products" might still fail to 
> capture a case like a^2/b where a and b are in the basis. Can't we 
> think of a better terminology?



I'd appreciate your help in getting the definiton webpage
of "torsion" finished.  Thanks.



-monz



 



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Message: 3519 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 18:16:02

Subject: Re: Our Paper

From: paulerlich

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


> Or perhaps an introduction to some of 
> the new temperaments for the imminent Xenharmonikon,


It's not as imminent as some have made it out to be. Despite reports 
to the contrary, no papers have yet been accepted for the next issue 
of XH.

Maybe we should all write separate papers that fit together and 
reference one another.

Message: 3520 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 08:10:56

Subject: Approximate consonances of Parch's 43 tone scale

From: genewardsmith

I searched for all instances where one tone of this scale differed from
another by an amount within 5 cents of an 11-limit consonance; this
led to four commas: 385/384, 8019/8000, 441/440, and 540/539. These
commas are linearly independent, and define an equal temperament (as
well as a PB, incidentally.) The et they define is
(drum roll please) 72 et.

The linear temperaments obtained by leaving out one of the commas were
<385/384,441/440,540/539>, miracle; <540/539,8019/8000,385/384>,
catakleismic; and two unnamed temperaments with half-octave period:

#5 on my list, <540/539,441/440,8019/8000> with wedgie 
[12,34,20,30,26,-2,6,-49,-48,15] and generators a=4.9919/72, b=1/2;

<441/440,8019/8000,385/384> with wedgie
[12,22,-4,-6,7,-40,-51,-71-90,-3] and generators a=10.9910/72, b=1/2.

Message: 3521 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 18:12:38

Subject: Re: OUR PAPER

From: paulerlich

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

> --- 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?


I have a feeling John Chalmers would probably publish it in 
Xenharmonikon -- he's a chemist, so he might be a little more 
favorably disposed to something so heavily mathematical.

Message: 3522 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 00:30:18

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.

Message: 3523 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 19:46:24

Subject: Proposed dictionary entry: torsion (revised)

From: genewardsmith

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).


Torsion may be tested by forming the n by n+1 matrix whose rows
correspond to the unison vectors, and calculating the
gcd(<<http://mathworld.wolfram.com/GreatestCommonDiv... * [with cont.] >>) 
of the minors
(<<http://mathworld.wolfram.com/Minor.html * [with cont.] >>)
of the matrix. If the rows are linearly independent but the gcd is not
one, we have torsion. 

The term comes from mathematical usage, see 
<< * [with cont.] >>.

Message: 3524 - Contents - Hide Contents

Date: Fri, 25 Jan 2002 00:50:40

Subject: Re: twintone, paultone

From: clumma

>>> >ou 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.

-Carl

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