Tuning-Math Digests messages 2250 - 2274

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Message: 2250

Date: Thu, 6 Dec 2001 08:59:33

Subject: Re: The wedge invariant commas

From: monz

Hi Gene,

> From: ideaofgod <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, December 05, 2001 5:08 PM
> Subject: [tuning-math] Re: The wedge invariant commas
>
>
> ... so I've messed
> things up around here by introducing multilinear algebra, Baker's
> theorem and what-not, as well as something I (and Pierre) saw as
> relevant already, namely abelian groups (or Z-modules, as Pierre
> prefers to call them), and quadratic forms in connection with
> lattices.


I'm having lots of trouble understanding what's been discussed
on this list since you joined.  But this bit of your post jumped
out at me, and I thought you'd find this profitable:


Mark Lindley & Ronald Turner-Smith.  1993.
  _Mathematical Models of Musical Scales: A New Approach_.
  Orpheus-Schriftenreihe zu Grundfragen der Musik vol. 66,
  Verlag für systematische Musikwissenschaft, Bonn-Bad Godesberg.

Lindley, Mark and Ronald Turner-Smith.
  "An Algebraic Approach to Mathematical Models of Scales",
  Music Theory Online vol. 0 no. 3, June 1993.

http://boethius.music.ucsb.edu/mto/issues/mto.93.0.3/mto.93.0.3.lindley.art *


Lindley/Turner-Smith view tuning systems as abelian groups.
(see especially paragraph [5] of the latter article)



love / peace / harmony ...

-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: 2251

Date: Thu, 06 Dec 2001 00:35:04

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> > An order of growth estimate shows there should be an infinite 
list 
> > for step^2, but not neccesarily for anything higher, and looking 
far 
> > out makes it clear step^3 gives a finite list. What this means, 
of 
> > course, is that in some sense step^2 is the right way to measure 
> > goodness. 
> 
> Yes! Only squared, not cubed.
> 
> > Step^3 weighs the small systems more heavily, and that is 
> > why we see so many of them to start with.
> 
> I believe the way to fix this is not to go to step^3 (I don't think 
there's any human-perception-or-cognition-based justification for 
doing that),

What human-perception-or-cognition-based justification is there for 
using step^2 ???

> Yes. Once the deviation goes past about 20 cents it's irrelevant > 
how big it is,

That's not true -- you're ignoring both adaptive tuning and adaptive 
timbring.

>and a 0.1 cent deviation does not sound 10 times better than a 1.0 
>cent deviation, it sounds about the same.

In my own musical endeavors, this is true, but with all the strict-JI 
obsessed people out there, a 0.1 cent deviation may end up being 10 
times more interesting than a 1.0 cent deviation.

> I suggest this figure-of->demerit.
> 
> step^2 [...]

Again, what on earth does step^2 tell you about how composers and 
performers would rate a temperament? OK, step^2 is the number of 
possible dyads in the typical scale. Step^3 is the number of possible 
triads. Why is the former so much more "human-perception-or-cognition-
based" to you than the latter?

As for the other part, the dissonance measure . . . by doing it 
Gene's way, we're going to end up with all the most interesting 
temperaments for a wide variety of different ranges, from "you'll 
never hear a beat" to "wafso-just" to "quasi-just" to "tempered" 
to "needing adaptive tuning/timbring". Thus our top 30 or whatever 
will have much of interest to all different schools of microtonal 
composers.


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Message: 2252

Date: Thu, 6 Dec 2001 19:03 +00

Subject: Wedge products

From: graham@xxxxxxxxxx.xx.xx

Okay, let's go right back to the beginning.  I already have a tutorial 
online for matrix algebra.  See 
<Matrix tutorial *>.  If you take the basic 
equivalence there, that

5x + 3y + z

is the same as

(5 3 1)(x)
       (y)
       (z)

you can similarly write any row vector (a b c) as

(a b c)(e1)
       (e2)
       (e3)

or

a*e1 + b*e2 + c*e3

where * is a normal multiplication and ei simply means the ith element of 
the basis.  You can then multiply two vectors (a b) and (c d) to get

(a*e1 + b*e2) * (c*e1 + d*e2)

which comes out as

a*c*e1*e1 + a*d*e1*e2 + b*c*e2*e1 + b*d*e2*e2

The music matrices I usually talk about have a basis H, where each entry 
is a number.  So a normal multiplication really does work

(a b)H * (c d)H

 = (a*log(2) + b*log(3)) * (c*log(2) + d*log(3))

 = a*c*log(2)*log(2) + a*d*log(2)*log(3)
   + b*c*log(3)*log(2) + b*d*log(3)*log(3)

 = a*c*log(2)**2 + (a*d + b*c)*log(2)*log(3) + b*d*log(3)**2

But in general you can't do that, because there isn't a rule for 
multiplying elements of a basis.  The wedge product is a specific rule 
for multipying bases.  NOW PAY ATTENTION!

ei^ej = - ej^ei

According to Gene, that's the full definition of a wedge product.  So, as 
long as you paid attention when I told you you can ignore the rest.  You 
know what a wedge product is.  It follows that

ei^ei = -ei^ei

ei^ei + ei^ei = 0

2*(ei^ei) = 0

ei^ei = 0


So the wedge product of (a b) and (c d) is

(a b)^(c d)
= (a*e1 + b*e2) ^ (c*e1 + d*e2)
= a*c*e1^e1 + a*d*e1^e2 + b*c*e2^e1 + b*d*e2^e2
= a*d*e1^e2 - b*c*e2^e1
= (a*d - b*c)

the same as the determinant

|a b|
|c d|


It gets more complicated in three dimensions

A^B
= (a1*e1 + a2*e2 + a3*e3)^(b1*e1 + b2*e2 + b3*e3)
= a1*b1*e1^e1 + a1*b2*e1^e2 + a1*b3*e1^e3
  + a2*b1*e2^e1 + a2*b2*e2^e2 + a2*b3*e2^e3
  + a3*b1*e3^e1 + a3*b2*e3^e2 + a3*b3*e3^e3

= a1*b2*e1^e2 + a1*b3*e1^e3
  - a2*b1*e1^e2 + a2*b3*e2^e3
  - a3*b1*e1^e3 - a3*b2*e2^e3

= (a1*b2 - a2*b1)*e1^e2 + (a1*b3 - a3*b1)*e1^e3
  + (a2*b3 - a3*b2)*e2^e3

which is the same as the determinant

|e2^e3 e3^e1 e1^e2|
|  a1   a2    a3  |
|  b1   b2    b3  |

That's analogous to the cross product of vectors, where AxB is

|x  y   z|
|a1 a2 a3|
|b1 b2 b3|

and, if A and B are octave-specific 5-limit unison vectors, it happens 
that AxB gives the number of steps to each prime consonance.  Or, if A and 
B are octave-equivalent 7-limit commatic unison vectors, AxB is the 
generator mapping.

I can't be bothered to work out the triple wedge product, but you'll find 
it's all in terms of e1^e2^e3.  So it's something like the determinant

|a1 a2 a3|
|b1 b2 b3|
|c1 c2 c3|

And the right number of wedge products in the right number of dimensions 
will always give a determinant.

As determinants and cross products are already useful in dealing with 
unison vectors, it shouldn't be such a surprise that wedge products in 
general are also useful.  I still don't know how they're useful, but Gene 
assures us that they are.

What is this "wedge invariant" he keeps using, and how do you go from it 
to get a list of unison vectors?


I'm not sure how to implement these things in Python.  I could implement 
vectors like dictionaries, so

A = {e1:a1, e2:a2, e3:a3}
B = {e1:b1, e2:b2, e3:b3}

then (A^B)[ei^ej] = A[ei]*B[ej] - A[ej]*B[ei]

so the next problem is what data type ei and ej should be.  Probably 
something like tuples:

A = {(1,):a1, (2,):a2, (3,):a3}
B = {(1,):b1, (2,):b2, (3,):b3}

that makes them look a bit like lists, so instead of A[1]=a1, we have 
A[1,]=a1.  Then, (A^B)[1, 2] = A[1,]*B[2,] + A[2,]*B[1,].

So this is making some sense, but I still haven't worked out all the 
details needed to write the code.


                    Graham


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Message: 2254

Date: Thu, 06 Dec 2001 00:57:16

Subject: Re: The slippery six

From: paulerlich

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > > (6) [-2,4,-30,-81,42,11] ets: 46,80
> > > 
> > > [ 0   2]
> > > [-1   4]
> > > [ 2   3]
> > > [-15 18]
> > > 
> > > a = 33.01588032 / 80 (~4/3); b = 1/2
> > > measure 26079
> > 
> > So this _isn't_ 46+34??
> 
> It is, but it won't show up as the *sum* of ets, only as the 
> difference. This is because the 34-et map in question is
> h80 - h46, and that isn't h34 in the 7-limit, since h34(7)=95 and
> (h80-h46)(7)=96. Maybe adding in a list of differences would be a 
> good idea.

Hmm . . . you keep avoiding my whining about consistency (most 
recently with regard to 21), and this would seem to be a good place 
to bring it up again. You told Graham that something like 46+34 to 
you would be _defined_ so that the 80 would come out right, not 
necessarily the individual ETs. Now you seem to be contradicting 
yourself. What gives?

> The consequence of being 46+34 of course is that this system is a 
> hell of a lot better in the 5-limit than it is in the 7-limit; the 
> 5-limit comma I get from the wedgie is 2048/2025--the diaschisma. 
> Graham devotes a web page to the diaschismic temperament as a 5-
limit 
> temperament, where it makes a lot of sense.

And you brought up 80 when we were discussing ways of extending 
diaschismic to 11-limit, if you recall . . . probably this same 
mapping through the 7-limit.


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Message: 2257

Date: Thu, 06 Dec 2001 01:22:04

Subject: Re: The grooviest linear temperaments for 7-limit music

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> > Yes. Once the deviation goes past about 20 cents it's irrelevant > 
> how big it is,
> 
> That's not true -- you're ignoring both adaptive tuning and adaptive 
> timbring.

You can adaptively tune or timbre just about anything, so it seems 
like we _should_ ignore it.

> >and a 0.1 cent deviation does not sound 10 times better than a 1.0 
> >cent deviation, it sounds about the same.
> 
> In my own musical endeavors, this is true, but with all the 
strict-JI 
> obsessed people out there, a 0.1 cent deviation may end up being 10 
> times more interesting than a 1.0 cent deviation.

A strict JI obsessed person will not be the slightest bit interested 
in linear temperaments, or at least that has been my experience. If 
they are at all interested then think they will be quite happy to have 
a 1c error rather than a 0.1c one if it lets them halve (actually 
divide by 10^(1/3)) the number of notes in the scale. Given that 1c is 
way below the typical accuracy of non-electronic instruments.

> > I suggest this figure-of->demerit.
> > 
> > step^2 [...]
> 
> Again, what on earth does step^2 tell you about how composers and 
> performers would rate a temperament? OK, step^2 is the number of 
> possible dyads in the typical scale. Step^3 is the number of 
possible 
> triads. Why is the former so much more 
"human-perception-or-cognition-
> based" to you than the latter?

Ok. Maybe I don't have good argument for that. Try 

step^3 * exp((cents/k)^2)

> As for the other part, the dissonance measure . . . by doing it 
> Gene's way, we're going to end up with all the most interesting 
> temperaments for a wide variety of different ranges, from "you'll 
> never hear a beat" to "wafso-just" to "quasi-just" to "tempered" 
> to "needing adaptive tuning/timbring". Thus our top 30 or whatever 
> will have much of interest to all different schools of microtonal 
> composers.

I think it has some extreme cases that are of interest to no one. This 
can be fixed.


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Message: 2258

Date: Thu, 06 Dec 2001 01:33:09

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > > Yes. Once the deviation goes past about 20 cents it's 
irrelevant > 
> > how big it is,
> > 
> > That's not true -- you're ignoring both adaptive tuning and 
adaptive 
> > timbring.
> 
> You can adaptively tune or timbre just about anything,

Not true -- in adaptive tuning, you don't want the horizontal shifts 
to be too big, or you lose the melodic coherence of the scale; and in 
adaptive timbring, you don't want the partials to deviate too far 
from a harmonic series, or you'll lose the sense that each note has a 
definite pitch.

> A strict JI obsessed person will not be the slightest bit 
interested 
> in linear temperaments, or at least that has been my experience. If 
> they are at all interested then think they will be quite happy to 
have 
> a 1c error rather than a 0.1c one if it lets them halve (actually 
> divide by 10^(1/3)) the number of notes in the scale.

You don't know that for sure. But look, I myself was trying to get 
Gene to adopt some exponential, rather than polynomial, function of 
the number of notes in the scale. He resisted . . .

> Given that 1c is 
> way below the typical accuracy of non-electronic instruments.

Hey, it won't be the first time a feature of tuning that is highly 
removed from most musicians' possible realm of experience has gotten 
published!

> 
> > > I suggest this figure-of->demerit.
> > > 
> > > step^2 [...]
> > 
> > Again, what on earth does step^2 tell you about how composers and 
> > performers would rate a temperament? OK, step^2 is the number of 
> > possible dyads in the typical scale. Step^3 is the number of 
> possible 
> > triads. Why is the former so much more 
> "human-perception-or-cognition-
> > based" to you than the latter?
> 
> Ok. Maybe I don't have good argument for that. Try 
> 
> step^3 * exp((cents/k)^2)

That's the _last_ conclusion I wanted you to reach!

> I think it has some extreme cases that are of interest to no one. 
This 
> can be fixed.

I tried to argue this point to Gene, but he seems to really like 
Ennealimmal. Hey, if we're getting mathematical elegance with this 
criterion, and all our favorite systems are showing up (I'm still 
waiting for double-diatonic ~26), shouldn't we be willing to pay the 
price of letting the guy who's doing all the work get his favorite 
system in too?


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Message: 2260

Date: Thu, 06 Dec 2001 01:56:20

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Personally I'd feel much better if everyone could somehow agree what
> was the overall most sensible measure regardless of the results!

Fat chance :)

> In Gene's case, I would hope that it would be some elegant internal
> consistency that ties the whole deal together. I'd personally settle
> for that even if the results were a tad exotic.

I feel the same way.

> Of course it might help if I understood it all a bit better too! I
> feel like I'm getting there though, I just wish Gene were a little 
bit
> more generous with the narrative--either that or someone else 
besides
> him were saying the same things slightly differently... that helps 
me
> sometimes too.

I think he's the only one who understands abstract algebra around 
here, so in a lot of cases, that isn't really possible, 
unfortunately . . . of course, I should study up on it, but I should 
also make more music, and get more sleep, and . . .


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Message: 2262

Date: Thu, 06 Dec 2001 02:45:38

Subject: Re: The grooviest linear temperaments for 7-limit music

From: genewardsmith

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

> You don't know that for sure. But look, I myself was trying to get 
> Gene to adopt some exponential, rather than polynomial, function of 
> the number of notes in the scale. He resisted . . .

You wanted to have exponential growth for the "step" factor, and Dave 
for the "cents" factor, which have opposite tendencies; Dave seems to 
want to filter the very things out on the low end that you wanted 
included.

If we added an exponential growth to "cents", I would suggest
trying k sinh (cents/k) for various k.

> > Given that 1c is 
> > way below the typical accuracy of non-electronic instruments.
> 
> Hey, it won't be the first time a feature of tuning that is highly 
> removed from most musicians' possible realm of experience has 
gotten 
> published!

It seems to me it is quite relevant to the strict JI school of 
thought. I got roasted for mentioning Partch in such a connection, 
but it's hard to see what theoretical objection he could raise to 45 
notes of ennealimmal in the 7-limit.

> I tried to argue this point to Gene, but he seems to really like 
> Ennealimmal. Hey, if we're getting mathematical elegance with this 
> criterion, and all our favorite systems are showing up (I'm still 
> waiting for double-diatonic ~26), shouldn't we be willing to pay 
the 
> price of letting the guy who's doing all the work get his favorite 
> system in too?

I think the only way you will get rid of Ennealimmal is to have an 
upper-end cut-off, and you said you wanted none. Sorry, you are stuck 
with it, and it has nothing to do with my liking it really. I've 
never even tried it!


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Message: 2263

Date: Thu, 06 Dec 2001 02:53:30

Subject: Re: The grooviest linear temperaments for 7-limit music

From: genewardsmith

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> In Gene's case, I would hope that it would be some elegant internal
> consistency that ties the whole deal together. I'd personally settle
> for that even if the results were a tad exotic.

Elegant internal consistency suggests to me steps^2 cents as a 
measure, but that would need an upper cut-off. We do it for ets, 
however, so I don't see that as a bif deal myself.

> Of course it might help if I understood it all a bit better too! I
> feel like I'm getting there though, I just wish Gene were a little 
bit
> more generous with the narrative--either that or someone else 
besides
> him were saying the same things slightly differently... that helps 
me
> sometimes too.

I'm hoping Paul will absorb it all and start coming out with his own 
interpretations, but I can't get him to compute a wedge product. :)


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Message: 2264

Date: Thu, 06 Dec 2001 02:59:36

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

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

> You wanted to have exponential growth for the "step" factor, and 
Dave 
> for the "cents" factor,

I think you misunderstood Dave -- he wanted the *goodness* for the 
cents factor to be a Gaussian.


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Message: 2265

Date: Thu, 06 Dec 2001 03:00:40

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> 
> > In Gene's case, I would hope that it would be some elegant 
internal
> > consistency that ties the whole deal together. I'd personally 
settle
> > for that even if the results were a tad exotic.
> 
> Elegant internal consistency suggests to me steps^2 cents as a 
> measure, but that would need an upper cut-off. We do it for ets, 
> however, so I don't see that as a bif deal myself.

Who's we?
> 
> > Of course it might help if I understood it all a bit better too! I
> > feel like I'm getting there though, I just wish Gene were a 
little 
> bit
> > more generous with the narrative--either that or someone else 
> besides
> > him were saying the same things slightly differently... that 
helps 
> me
> > sometimes too.
> 
> I'm hoping Paul will absorb it all and start coming out with his 
own 
> interpretations, but I can't get him to compute a wedge product. :)

I'll take a look at it again when I get a chance.


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Message: 2266

Date: Fri, 07 Dec 2001 06:00:40

Subject: Re: The grooviest linear temperaments for 7-limit music

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > Well, I think Gene is saying that step^2 cents is clearly the 
> right 
> > > measure of "remarkability".
> > 
> > Huh? "Remarkability" sounds like a kind of goodness. Step^2 * 
cents 
> is 
> > obviously a form of badness.
> 
> Right, but it's the _objective_ kind. Not the kind that has anything 
> to do with any particular musician's desiderata.

Paul! You seem to have ignored the most of the rest of my message.

What the heck is _objective_ about deciding that a doubling of the 
number of generators is twice as bad as a doubling of the error. It's 
completely arbitrary.

> It's the only 
> measure that doesn't favor a certain range of acceptable values for 
> error or for complexity. It only favors the best examples within 
each 
> range.

What _objective_ reason is there, to choose it over gens^3 * cents or 
gens^2.3785 * cents?


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Message: 2267

Date: Fri, 07 Dec 2001 08:03:05

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > Huh? Obviously any badness metric _must_ slope down towards 
(0,0) 
> > on 
> > > the (cents,gens) plain.
> > 
> > The badness metric does, but the results don't. The results have 
a 
> > similar distribution everywhere on the plane, but only when 
gens^2 
> > cents is the badness metric.
> 
> You're not making any sense. The results are all just discrete 
points 
> in the badness surface with respect to gens and cents, so they have 
> exactly the same slope. The results have a similar distribution of 
> what? Everywhere on what plane?

I see Gene is, at this very moment, doing a good job explaining these 
issues to you; meanwhile, my brain is toast.


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Message: 2268

Date: Fri, 07 Dec 2001 20:47:31

Subject: Re: The grooviest linear temperaments for 7-limit music

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> So ... What is n? What is a 7-limit et? How does one use n^(4/3) to 
> get a list of them? How would one check to see whether the list 
> favours high or low n.

"n" is how many steps to the octave, or in other words what 2 is 
mapped to. By a "7-limit et" I mean something which maps 7-limit 
intervals to numbers of steps in a consistent way. Since we are 
looking for the best, we can safely restrict these to what we get by 
rounding n*log2(3), n*log2(5) and n*log2(7) to the nearest integer, 
and defining the n-et as the map one gets from this.

Let's call this map "h"; for the 12-et, h(2)=12, h(3)=19, h(5)=28 and
h(7)=34; this entails that h(5/3) = h(5)-h(3) = 9, h(7/3)=15 and
h(7/5)=6. I can now measure the relative badness of "h" by taking the 
sum, or maximum, or rms, of the differences of |h(3)-n*log2(3)|, 
|h(5)-n*log2(5)|, |h(7)-n*log2(7)|, |h(5/3)-n*log2(5/3)|, 
|h(7/3)-n*log2(7/3)| and |h(7/5)-n*log2(7/5)|. 

This measure of badness is flat in the sense that the density is the 
same everywhere, so that we would be selecting about the same number 
of ets in a range around 12 as we would in a range around 1200. I 
don't really want this sort of "flatness", so I use the theory of 
Diophantine approximation to tell we that if I multiply this badness
by the cube root of n, so that the density falls off at a rate of
n^(-1/3), I will still get an infinite list of ets, but if I make it 
fall off faster I probably won't. I can use either the maximum of the 
above numbers, or the sum, or the rms, and the same conclusion holds; 
in fact, I can look at the 9-limit instead of the 7-limit and the 
same conclusion holds. If I look at the maximum, and multiply by 1200
so we are looking at units of n^(4/3) cents, I get the following list 
of ets which come out as less than 1000, for n going from 1 to 2000:

1     884.3587134
2     839.4327178
4     647.3739047
5     876.4669184
9     920.6653451
10    955.6795096
12    910.1603254
15    994.0402775
31    580.7780905
41    892.0787789
72    892.7193923
99    716.7738001
171   384.2612749
270   615.9368489
342   968.2768986
441   685.5766666
1578  989.4999106

This list just keeps on going, so I cut it off at 2000. I might look 
at it, and decide that it doesn't have some important ets on it, such 
as 19,22 and 27; I decide to put those on, not really caring about 
any other range, by raising the ante to 1200; I then get the 
following additions:

3     1154.683345
6     1068.957518
19    1087.886603
22    1078.033523
27    1108.589256
68    1090.046322
130   1182.191130
140   1091.565279
202   1143.628876
612   1061.222492
1547  1190.434242

My decision to add 19,22, and 27 leads me to add 3 and 6 at the low 
end, and 68 and so forth at the high end. It tells me that if I'm 
interested in 27 in the range around 31, I should also be interested 
in 68 in the range around 72, in 140 and 202 around 171, 612 around 
441, and 1547 near 1578. That's the sort of "flatness" Paul was 
talking about; it doesn't favor one range over another.





> But no matter what you come up with I can't see how you can get 
past 
> the fact that gens and cents are fundamentally incomensurable 
> quantities, so somewhere there has to be a parameter that says how 
bad 
> they are relative to each other.

"n" and cents are incommeasurable also, and n^(4/3) is only right for 
the 7 and 9 limits, and wrong for everything else, so I don't think 
this is the issue if we adopt this point of view.

Why not 
> use k*gens + cents. e.g. if badness was simply gens + cents and you 
> listed everything with badness not more than 30 then you don't need 
> any additional cutoffs. You automatically eliminate anything with 
gens 
> > 30 or cents > 30 (actually cents > 29 because gens can't go below 
> 1).

Gens^3 cents also automatically cuts things off, but I rather like 
the idea of keeping it "flat" in the above sense and doing the 
cutting off deliberately, it seems more objective.


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Message: 2269

Date: Fri, 07 Dec 2001 06:23:45

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Paul! You seem to have ignored the most of the rest of my message.

Not at all.

> > It's the only 
> > measure that doesn't favor a certain range of acceptable values 
for 
> > error or for complexity. It only favors the best examples within 
> each 
> > range.
> 
> What _objective_ reason is there, to choose it over gens^3 * cents 
or 
> gens^2.3785 * cents?

Because those measures give an overall "slope" to the results, in 
analogy to what the Farey series seeding does to harmonic entropy.


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Message: 2270

Date: Fri, 07 Dec 2001 08:06:28

Subject: Re: The grooviest linear temperaments for 7-limit music

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > 
> > > > We could search (16/15)^a (25/24)^b (81/80)^c to start out 
> with, 
> > > and 
> > > > go to something more extreme if wanted.
> > > 
> > > More extreme? I'm not getting this.
> > 
> > (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives 
the 
> > 5-limit, but is better for finding much smaller commas, to take a 
> > more or less random example.
> 
> Once a, b, and c are big enough, the original choice of commas will 
> do little to induce any tendency of smallness or largeness in the 
> result, correct?

(78732/78125)^53 (32805/32768)^(-84) (2109375/2097152)^65 = 2

I wouldn't search that far myself.


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Message: 2271

Date: Fri, 7 Dec 2001 21:14 +00

Subject: Re: Wedge products

From: graham@xxxxxxxxxx.xx.xx

genewardsmith@xxxx.xxx (genewardsmith) wrote:

> First you order the basis so that a wedge product taken from two ets 
> or two unison vectors will correspond:
> 
> Yahoo groups: /tuning-math/message/1553 *

I think I've done that.  All comes down to bubblesort -- one in the eye 
for those who say it's of no practical use!

> Then you put the wedge product into a standard form, by
> 
> (1) Dividing through by the gcd of the coefficients, and

Okay, that's easy enough.

> (2) Changing sign if need be, so that the 5-limit comma (or unison)
> 2^w[6] * 3^(-w[2])*5^w[1] where w is the wedgie, is greater than 1. 
> If it equals 1, go on to the next invariant comma, which leaves out 
> 5, and if that is 1 also to the one which leaves out 3. See
> 
> Yahoo groups: /tuning-math/message/1555 *
> 
> for the invariant commas. The result of this standardization is the 
> wedge invariant, or wedgie, which uniquely determins the temperament.

I'll need to study that a bit more.

> > I'm not sure how to impleme
> nt these things in Python. 
> 
> The above should do for the 7-limit; in general is another matter.

Oh, it needs to be done in general.  And it's almost working now.  I still 
need to get my wedge invariants to look like yours.  Also to divide 
through by common factors, but I've done that before.

To check:

>>> wedge.wedgeProduct(h12,h22)
{(2, 3): 2, (0, 1): 2, (1, 3): -12, (0, 3): -4, (0, 2): -4, (1, 2): -11}

That's numbering from 0 as the 2-direction, 1 as the 5-direction, etc.  
Then for the commas

>>> for i in range(4):
	print wedge.interval(
	    reduce(
	        wedge.wedgeProduct,
	        ((zeros[:i]+(1,)+zeros[i+1:]), h12, h22)))
	        
	        
[0, -2, -12, 11]
[2, 0, 4, -4]
[12, -4, 0, -2]
[-11, 4, 2, 0]

which looks right.  I'll look at it some more tomorrow.  I'm still don't 
know how to do the generator mapping.

Here's the library code:


def wedgeProduct(a, b):
    result = {}
    for base1, value1 in makewedgable(a).items():
        for base2, value2 in makewedgable(b).items():
            value = value1*value2
            for element in base1:
                if element in base2:
                    break
            else:
                base, value = wedgeEquivalent(base1+base2, value)
                result[base] = result.get(base, 0)+value
    return result

def interval(wedgie):
    result = []
    bases = wedgie.keys()
    for i in range(len(wedgie)):
        for base in bases:
            if i not in base:
                if i%2:
                    result.append(-wedgie[base])
                else:
                    result.append(wedgie[base])
    return result

def wedgeEquivalent(base, value):
    workingBase = list(base)
    for i in range(len(base)):
        for j in range(i,0,-1):
            if workingBase[j]<workingBase[j-1]:
                workingBase[j-1:j+1] = [
                    workingBase[j],
                    workingBase[j-1]]
                value = -value
    return tuple(workingBase), value

def addWedges(a, b):
    x = makewedgable(a)
    y = makewedgable(b)
    result = {}
    for element in x.keys()+y.keys():
        result[element]=0
    for key in x.keys():
        result[key] = result[key] + x[key]
    for key in y.keys():
        result[key] = result[key] + y[key]
    return result

def makewedgable(thing):
    if isinstance(thing, type({})):
        return thing
    else:
        result = {}
        for i in range(len(thing)):
            result[i,]=thing[i]
        return result


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Message: 2272

Date: Fri, 07 Dec 2001 06:34:59

Subject: Re: The grooviest linear temperaments for 7-limit music

From: genewardsmith

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > The solutions represent?
> 
> I take the 5-limit comma defined by the temperament, and then find 
> another comma 2^p 3^q 5^r 7 such that the wedgie of this and the 5-
> limit comma is the correct wedgie, that means these two commas 
define 
> the temperament.

This should be 2^p 3^q 5^r 7^s where s is gcd(a,b,c), and the 5-limit
comma is 2^a 3^b 5^c.


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Message: 2273

Date: Fri, 07 Dec 2001 08:20:04

Subject: Re: The grooviest linear temperaments for 7-limit music

From: paulerlich

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

> > > (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives 
> the 
> > > 5-limit, but is better for finding much smaller commas, to take 
a 
> > > more or less random example.
> > 
> > Once a, b, and c are big enough, the original choice of commas 
will 
> > do little to induce any tendency of smallness or largeness in the 
> > result, correct?
> 
> (78732/78125)^53 (32805/32768)^(-84) (2109375/2097152)^65 = 2
> 
> I wouldn't search that far myself.

How do you know you wouldn't be missing any good ones?


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Message: 2274

Date: Fri, 7 Dec 2001 21:14 +00

Subject: Re: More lists

From: graham@xxxxxxxxxx.xx.xx

Dave Keenan wrote:


> I note that Graham is using maximum width and (optimised) maximum 
> error where Gene is using rms width and (optimised) rms error. It will 
> be interesting to see if this alone makes much difference to the 
> rankings. I doubt it.

I've implemented RMS error now.  It's actually faster than the minimax, so 
I've made it the default.  I've uploaded new copies of the .txt and .gauss 
files.  There are also other changes to the code to make it more 
efficient.  As it stands, the ET matching is broken.  I've fixed that, but 
not uploaded.

You could implement the RMS width easily enough, but I expect it'll slow 
down execution, so you can do it on your own time.

> So I see that while the gaussian with std error of 17 cents seems to 
> do the right thing in eliminating temperaments with tiny errors but 
> huge numbers of generators, it is too hard on those with larger 
> errors. Notice that Ennealimmal is still in the 7-limit list (about 
> number 22). The problem is that Paultone isn't there at all! It has 
> 17.5 c error with 6 gens per tetrad.

I'm dividing the 17 cents by 3 in this case, to give a figure more like 
what you asked for.

> Those lists don't contain any temperament with errors greater than 10 
> cents. The 5-limit 163 cent neutral second temperament has the largest  
> at 9.8 cents, with 5 generators per triad.
> 
> So I have to agree with Paul that 
>   badness = num_gens^2 / gaussian(error/17c) 
> doesn't work.

It works fine.  You asked for errors of around 6 cents, so why should you 
expect errors greater than 10 cents?


                         Graham


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