Tuning-Math messages 125 - 149

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Message: 125

Date: Mon, 04 Jun 2001 03:46:22

Subject: Re: Constant Structure (was: Temperament program issues)

From: Paul Erlich

--- In tuning-math@y..., carl@l... 
wrote:
> > It's sounds to me as if you're trying to define a property other
> > than  Wilson's CS.  As I understand it, Wilson uses CS to describe
> > a pattern shared  by a group of tunings that can be mapped onto a
> > single scale tree pattern,even if the generator size is only a
> > average value.  For example, he maps the 3(1,3,7,9,11,15 Eikosany
> > (plus two "pigtails"), a scale based on an Indian Sruti model, and
> > 22tet onto a 22-tone keyboard and notation. The subsets described
> > by the scale tree then becomes useful paths for orientation in
> > īthe larger system.
> 
> Amazing, Daniel!  This is just what I guessed CS was, before
> the correspondence from Kraig.  See monz's web site reffed earlier
> in this thread.

Kraig also thought that CS meant 
something different before 
Wilson gave him the definition on 
Monz' web page.
> 
> > (Writing the above, I've got the strong suspicion that
> > constructing a formula that will predict whether and where in
> > the scale tree a given scale will find a CS is probably very hard
> > to construct.)
> 
> On the contrary, unless I mis-understood you, the scale tree
> shows the generator ranges itself.  I know you know this, so
> maybe I _did_ mis-read you.  Can I get you to check monz's
> web page?
> 
> Definitions of tuning terms: constant structure, (c) 1999 by Joe Monzo *
> 
> -Carl

Daniel Wolf didn't post directly to 
this list. So make sure someone 
forwards this to him. Also, maybe 
he wrote Dave privately because 
the idea was that this list should 
cease to exist? Perhaps we 
should be posting all this to the 
tuning list?


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Message: 126

Date: Tue, 05 Jun 2001 03:20:06

Subject: Math models (about Hypothesis)

From: Pierre Lamothe

I was not available at moment the thread Hypothesis was active. I had begun
to write this post but I had urgent tasks forcing me to stop. It is not
complete but since the List will probably terminate and since I miss yet
time I post that unfinished.

I would like to use the Paul's conjecture as starting point to show the
necessity for fine tuned definitions if we want to progress in mathematical
way.

I would have difficulty to express mathematically the given conjecture for
I don't know the mathematical definition used for almost all the terms.


-----------------------
distributional evenness
-----------------------

I begin with this definition appearing in the Monzo site.

<< distributional evenness 

   The scale has no more than two sizes of interval
   in each interval class. >>

What follows is not a criticism of the Monzo's definition but only some
arguments to show that the conjecture has not really a mathematical form.

There exist maybe a light ambiguity with << no more than two >> but the
hard problem would be rather with the link between _scale_ and _class_ terms.

Forgetting for a moment the term scale, let us look at << no more than two
>> as if the property could be applied to a simple set. Supposing that,

<< . . . has no more than two sizes of interval
   in each interval class. >>

might be mathematically written

   A partition of a set S by an equivalence relation R
   is _distributionally even_ if for any class x in S/R

      0 < Card x < 3

where it is only assumed each class has either one or two elements.

Thus the partition {{a},{b},{c}} of a set {a,b,c} would be distributionally
even. For Paul, is that sense allowed?

Besides, << no more than two >> implies that the case

   "for any x, Card x = 2"

is only a possibility among others. So, if the partition {{a},{b},{c}} is
not _distributionally even_, what rule is used to determine allowed
combinations of classes with one or two elements? Is the form << no more
than >> used only to allow the class of unison having only one interval
while all others would have two?

I imagine this light ambiguity is easy to clarified. The next one seems
more serious.


---------------
class and scale
---------------

Is a _scale_ anything else, for Paul, than an ordered finite set of reals.
If not so, what are the essential conditions? Has a _scale_ an essential
link, or only an optional one, with periodicity block and also with
properties often mentioned like consonance possibilities?

Are interval classes in a scale something added to that scale by an
external equivalence relation or are classes "sui generis" in sense that
the classes follows (by a general principle used for any scale) from the
intervals itself given as a whole?  

In the following set and partitions

   S = {1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9}

   P1 = {{1}, {9/8, 32/27}, {4/3}, {3/2}, {27/16, 16/9}}

   P2 = {{1}, {9/8}, {32/27}, {4/3}, {3/2}, {27/16}, {16/9}}

   P3 = {{1}, {4/3, 27/16}, {9/8, 16/9}, {32/27, 3/2}}

how the definitions of scale, class and distributionally evenness should be
applied?

As you may see, I'm far from knowing clearly what this simple definition
mathematically means, but it could be worst with other definitions. My
discussions with Paul on the Tuning List shows we could differ on
periodicity block and srutis and probably on unison vectors and steps.


-----------------
periodicity block
-----------------


   *** TONE ***
 
   I will use here the term _tone_ in the strict sense 
   of an interval of the first octave as representing
   its class modulo 2.


For me, a normal set of unison vectors in a discrete tones Z-module
determine a periodicity block refering to a canonical set of tones
determined by an oriented object with origin : the hyperparallelopiped
defined by the given unison vectors sharing a common origin.

This canonical set of tones corresponds both to 

   - tones inside the hyperparallelopiped having unison
     as origin and

   - tones on the unison vectors

(*** so neither ambiguity nor double counting ***).

Since the origin of the unison vectors is never inside the
hyperparallelopiped, so keeping the same shape but changing the origin for
another vertices don't correspond (most cases) to a simple translation of
the block in the lattice. One tone, minimally, would be changed compared to
the translated canonical set.

Besides, since Paul and Dave seems to refer rather at periodical shapes, I
would add that a same shape may correspond to distinct sets of unison
vectors. If you know only the tile you don't know forcely the tiling.

Another thing seems mysterious comparing our uses of periodicity block. For
me, a system build from the periodicity block concept implies all modes in
the system have the same amount of tones or degrees. If we use only one
block we may have only one mode. Using N blocks surrounding unison we
obtain for each D classes possibly N intervals. It remains possible after
that to use appropriate temperament (removing commatic vectors or keeping
integrally the structure).

But it seems Paul and Monz would start with periodicity blocks having too
tones, to be interpreted as degrees of a musical mode, in their approach of
the Hindu system where the amount of degrees is 7 (SA RI GA ...) rather
than 22, and it seems it is a temper process that serves to . . . (I don't
understand, so I have some questions).

If you use only one block, which have by definition only one element in
each of the D classes determined by the periodicity, what is the sense of
the presumed new classes which would appear after a temper treatment?  Is
it like that you obtain seven classes in the Hindu system from a block
having a 22 or more periodicity?  Do you have then an explicit epimorphism
transforming an interval in its degree value?


----- As I said, I cut here for I have not much time to write.
      However, the questions already written may have interest.


Pierre


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Message: 132

Date: Tue, 05 Jun 2001 18:32:54

Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning

From: Paul Erlich

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> >>At first I thought a new set of tuning files would be necessary to
> >>accomplish this, but then realized that a weak, but not 
negligible,
> >>tritone spring would also address the goal.  A tritone _must_ be
> >>targeted to 600 cents, because we know nothing of inversion(s) in 
the
> >>tuning-file free approach.
> 
> [Paul E:]
> >I don't understand this . . . I thought your program recognized 
> >octave-equivalence, and hence inversions, independently of the 
tuning 
> >files. 
> 
> Your statement is correct.  But, for example, in 7-limit, a tritone 
is
> NOT 600 cents, yet the program knows which side of the inversion 
it's
> on because of pattern-matching between the notes on the the 
desirable
> intervals specified by the tuning file.

This I understand . . .

>Am I being clear?

Still confused as to the original statement.
> 
> >And by 'weak', do you mean 'strength approaches zero'?
> 
> That would be 'negligible'.  Here, 'weak' means 1/8 of nominal.  Do 
the
> tables make sense?
> 
Very much so. I guess I was just wondering if your concept here was 
similar to those cases where you needed to put a negligible strength 
on some spring for computational reasons (i.e., the matrix 
wouldn't 'relax' correctly otherwise) but conceptually the strength 
was really zero.


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Message: 133

Date: Tue, 05 Jun 2001 19:14:07

Subject: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress

From: Paul Erlich

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
Dave Keenan wrote (privately),

>What's the
>set of 7-limit UV's for Blackjack again?

The blackjack scale is the result of forming a periodicity block with 
the unison vectors 2401:2400, 225:224, and 36:35; treating the 
2401:2400 and 225:224 as commatic unison vectors and tempering them 
out; and treating 36:35 as a chromatic unison vector and not 
tempering it out.

Confused? Maybe it would help to add that the good old diatonic scale 
in 5-limit is the result of forming a periodicity block with the 
unison vectors 81:80 and 25:24; treating 81:80 as a commatic unison 
vector and tempering it out; and treating 25:24 as a chromatic unison 
vector and not tempering it out.

In other words, the diatonic scale is an infinite 'band' of the 
infinite 2D 5-limit lattice, and the thickness of the band is given 
by the 25:24 interval. This is clearly explained and depicted in my 
paper, _The Forms Of Tonality_.

The blackjack lattice that I posted (and need to correct) shows that 
the blackjack scale is an infinite 'slice' of the infinite 3D 7-limit 
lattice, and the thickness of the slice is given by the 36:35 
interval. You can see that there is no 36:35 interval, which would be 
formed by moving two red connectors to the right, one green connector 
to the lower-left, and one blue connector to the lower left, within 
the blackjack scale. If you were to transpose the blackjack scale by 
this interval, it would fit, with no gaps, on top of its transposed 
self . . . and an infinite number of such 'layers' would fill the 
infinite 3D 7-limit lattice.

So the blackjack scale is a "Form Of Tonality" (perhaps "Form of 
Modality" would be better since no 'tonal center' is necessarily 
implicated) with commatic and chromatic unison vectors very much in 
accordance with the sizes of commatic and chromatic unison vectors in 
the scales I've already described in that paper. Funny how things 
that appear unrelated at first seem to 'fit together'!

Canasta seems less interesting from this point of view because its 
chromatic unison vector is 81:80 . . . which is more like a commatic 
unison vector in size, begging to be tempered out . . . and if you do 
that, you get the wonderful 31-tET . . .

Always using epimoric ratios for the unison vectors has one 
advantage . . . the size of the numbers in the ratio immediately 
tells you both the melodic smallness of the interval, and its taxicab 
distance in the triangular lattice (suitably constructed, as in the 
second-to-last lattice on Kees van Prooijen's page 
lattice orientation *). So tempering out an 
epimoric unison vector that uses numbers N times smaller than another 
one means that the constituent consonances will have to be tempered 
N^2 times as much . . . am I on to something?
--- End forwarded message ---


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Message: 134

Date: Tue, 05 Jun 2001 20:13:26

Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/133 *

Thanks for posting and cross-posting this message, Paul.  I never 
even understood that blackjack is an infinite 7-limit lattice as 
compared with our traditional diatonic being an infinite 5-limit 
lattice...  Somehow I never "got" that before from the previous 
dialogues...  No wonder there are so many "just tetrads" in it!

__________ ___________ _______
Joseph Pehrson


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Message: 136

Date: Tue, 05 Jun 2001 21:09:32

Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning

From: Paul Erlich

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> In the er3/es3, all spring strengths are greater than negligible.  
And
> in truth I could have done the er2/es2 with zero-strength springs 
on 1, 
> 2, and 6; that won't hurt the matrix (it's only infinite strength 
> springs that are trouble-makers!).  I just left the 1% springs in 
place 
> to see those other intervals represented in the table.
> 
So what you're saying is, in practice, you could use zero-strength 
springs on 1, 2, and 6 for most MIDI files, since the size of the 
tritone would be determined by various horizontal & grounding 
considerations . . . but for this single-chord example, you needed to 
use a finite-strength spring because otherwise, there would be no 
solution . . . ? In that case, I would favor using zero-strength 
springs in practice . . .

P.S. Why is this discussion on this list? I thought it was voted that 
this list should not remain separate . . . but it seems to be 
growing! A monster! Perhaps I'd feel like I was respecting the vote 
if I at least transferred ownership of this list to someone 
else . . .


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Message: 137

Date: Tue, 05 Jun 2001 21:14:17

Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress

From: Paul Erlich

--- In tuning-math@y..., jpehrson@r... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> Yahoo groups: /tuning-math/message/133 *
> 
> Thanks for posting and cross-posting this message, Paul.  I never 
> even understood that blackjack is an infinite 7-limit lattice as 
> compared with our traditional diatonic being an infinite 5-limit 
> lattice...  Somehow I never "got" that before from the previous 
> dialogues...  No wonder there are so many "just tetrads" in it!
> 
Hey Joseph . . . look at blackjack3.bmp in the new bjlatt.ZIP . . . 
then look at Figure 8 in my paper _The Forms of Tonality_ . . . same 
idea, different scale (and different lattice orientation) . . .

You can think of them as infinite, as we are here, or you can think 
of them simply as 'wrapping around' to meet themselves. Remember 
the 'donut'?


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Message: 138

Date: Tue, 05 Jun 2001 21:43:20

Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/137 *

> Hey Joseph . . . look at blackjack3.bmp in the new bjlatt.ZIP . . . 
> then look at Figure 8 in my paper _The Forms of Tonality_ . . . 
same 
> idea, different scale (and different lattice orientation) . . .
> 
> You can think of them as infinite, as we are here, or you can think 
> of them simply as 'wrapping around' to meet themselves. Remember 
> the 'donut'?

Yes, indeed!  It became patently obvious once I was SHOWN it!  :)

Regarding the list, I don't know.  I'm reluctant to post over to 
the "big" list right now, just because it's so "big!" and there have 
been so many complaints.

Maybe it is right to keep a list of more numerical or math items over 
here.  I was even reluctant to post the recent question Monz had 
about fractional remainders after division...  And probably that 
would have been of interest to more people than just Monz and 
myself....

Dunno.  Maybe it's best to keep things running like they are until 
there are "complaints" even though the vote indicated otherwise.  
After all, there were lots of abstentions...

For myself, I'm pretty much getting used to having so many lists 
now... and, since I'm still trying to read most all of them, 
MATHEMATICALLY, it really doesn't make any difference.

See... I *was* going to get to the "math" part...

_________ _________ _______
Joseph Pehrson


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Message: 139

Date: Tue, 05 Jun 2001 21:45:05

Subject: combine with Harmonic Entropy

From: jpehrson@r...

HOWEVER, (and I forgot to mention this), I personally *do* believe 
that the "Harmonic Entropy" list should be "nuked"  (messages saved, 
of course-- we'll have to ask Robert Walker) and any further posts on 
that subject be presented on the "Tuning Math" list...

___________ _________ ________
Joseph Pehrson


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Message: 141

Date: Tue, 05 Jun 2001 23:54:04

Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning

From: jpehrson@r...

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

Yahoo groups: /tuning-math/message/140 *

> 
> If you are determined not to host this list any more, then please do
> hand it off.  I'd rather keep my long, detailed posts off the fat 
list if possible.  Maybe I'll start a boring-details@y...
> 
> JdL

Hi John...

What I was thinking is that this list could include *BOTH* the 
salient details and *ALSO* expressions of exuberance!

Let's say, after a long detailed post, somebody *else* could just 
write, "Dig it!"

Right now, one can't do that on the "big" list, since somebody is 
always counting the words so that people with slow connections don't 
have to download so much!

________ _______ ___
Joseph Pehrson


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Message: 142

Date: Tue, 5 Jun 2001 22:59:19

Subject: keep the tuning-math list

From: monz

> [Paul:]
> >P.S. Why is this discussion on this list? I thought it was voted that 
> >this list should not remain separate . . . but it seems to be 
> >growing! A monster! Perhaps I'd feel like I was respecting the vote 
> >if I at least transferred ownership of this list to someone 
> >else . . . 
> 
> If you are determined not to host this list any more, then please do
> hand it off.  I'd rather keep my long, detailed posts off the fat list
> if possible.  Maybe I'll start a boring-details@y...


I know I'm repeating myself... but I totally agree with John.
I think the microtonal community really needs *precisely this* list.
Please keep it alive.


-monz
Yahoo! GeoCities *
"All roads lead to n^0"


 
-


_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 143

Date: Wed, 6 Jun 2001 10:01 +01

Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . ..

From: graham@m...

In-Reply-To: <9fjjpo+t5e1@e...>
Joseph Pehrson wrote:

> Dunno.  Maybe it's best to keep things running like they are until 
> there are "complaints" even though the vote indicated otherwise.  
> After all, there were lots of abstentions...

How about insisting that, as the list already exists, an absolute majority 
of the Tuning List is required to vote it away?  Then, the overwhelming 
number of people who don't bother to vote are sure to carry the day!

                 Graham


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Message: 146

Date: Wed, 06 Jun 2001 13:30:20

Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . ..

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/143 *

> In-Reply-To: <9fjjpo+t5e1@e...>
> Joseph Pehrson wrote:
> 
> > Dunno.  Maybe it's best to keep things running like they are 
until 
> > there are "complaints" even though the vote indicated otherwise.  
> > After all, there were lots of abstentions...
> 
> How about insisting that, as the list already exists, an absolute 
majority 
> of the Tuning List is required to vote it away?  Then, the 
overwhelming 
> number of people who don't bother to vote are sure to carry the day!
> 
>                  Graham

That's pretty funny, Graham.  I can see you're good at "politics" 
too... (!!)

Looks like the "Math List" won!

___________ ________ ________
Joseph Pehrson


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Message: 147

Date: Wed, 06 Jun 2001 13:33:11

Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning

From: jpehrson@r...

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

Yahoo groups: /tuning-math/message/145 *

> [I wrote:]
> >>If you are determined not to host this list any more, then please 
do
> >>hand it off.  I'd rather keep my long, detailed posts off the fat 
> >>list if possible.  Maybe I'll start a boring-details@y...
> 
> [Joseph Pehrson:]
> >What I was thinking is that this list could include *BOTH* the 
> >salient details and *ALSO* expressions of exuberance!
> 
> >Let's say, after a long detailed post, somebody *else* could just 
> >write, "Dig it!"
> 
> Oh, I agree!  When I say "boring details", I just mean that I know 
> _some_ will be bored, but I'm excited, and if others are as well, 
it's 
> great to hear it!
> 
> >Right now, one can't do that on the "big" list, since somebody is 
> >always counting the words so that people with slow connections 
don't 
> >have to download so much!
> 
> I suppose if this list ever gets real fat, we'll have to be careful
> here as well.
> 
> JdL

I suppose that's true, John... but at the moment it still isn't the 
case... so I'm for lots of big numbers and stuff that I can't 
understand (well I'll get PART of it!) and EXUBERANCE!

_________ ________ ________
Joseph Pehrson


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Message: 148

Date: Wed, 06 Jun 2001 18:51:15

Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning

From: Paul Erlich

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> [Paul:]
> >So what you're saying is, in practice, you could use zero-strength 
> >springs on 1, 2, and 6 for most MIDI files, since the size of the 
> >tritone would be determined by various horizontal & grounding 
> >considerations . . . but for this single-chord example, you needed 
to 
> >use a finite-strength spring because otherwise, there would be no 
> >solution . . . ? In that case, I would favor using zero-strength 
> >springs in practice . . .
> 
> Are we clear that the 0% and the 1% springs (the latter represented
> by er2/es2 files) would be almost identical in result, and that 
both 
> will be different from the er3/es3 I've been describing?

Sure.

> Are you making
> a statement about your preference as to one group vs. the other, or
> commenting on how you would do the former group?

I'm asking a question, and you didn't answer (see above).
> 
> [Paul:]
> >P.S. Why is this discussion on this list? I thought it was voted 
that 
> >this list should not remain separate . . . but it seems to be 
> >growing! A monster! Perhaps I'd feel like I was respecting the 
vote 
> >if I at least transferred ownership of this list to someone 
> >else . . . 
> 
> If you are determined not to host this list any more, then please do
> hand it off.  

Are you volunteering?


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Message: 149

Date: Wed, 06 Jun 2001 18:55:38

Subject: Re: linear approximation

From: Paul Erlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> It's exactly the same as the best LS generator for
> a Pythagorean scale, with the generator being one step here.
> 
The best LS generator . . . Manuel, there are several _strange_ 
features in this calculation. First of all, you're including _both_ 
the 1/1 _and_ the 2/1, while every other pitch class appears only 
once. Second, you're forcing the fit line to pass through the 1/1. 
These are very odd features and I'm not sure how you'd justify them. 
As far as I can tell, if you do it correctly, the best LS step has to 
equal the mean step.


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