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Message: 9525 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 21:02:01

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>>> >o particular generator basis is assumed in the TOP complexity 
>>> calculations. Instead, it's a direct measure of how much the 
>>> tempering simplifies the lattice, and reduces (Gene seems to 
>>> imply/agree) to the number of notes per acoustic whatever in the
>>> case of equal (1-dimensional) temperaments.
>> 

>> I thought equal temperaments were "0-dimensional"?
>

>We've always said linear temperaments were actually 2-dimensional,
>so equal temperaments are 1-dimensional, and are generated by their 
>step. Remember we've dropped octave-equivalence to get TOP.


Are you suggesting that TOP "linear" temperaments have a greater
dimensionality than old-style "linear" temperaments?

-Carl

Message: 9526 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 01:39:02

Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
...

>> Number 25 Miracle
>> 
>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>> TOP generators [1200.631014, 116.7206423]
>> bad: 9.8358 comp: 6.793166 err: .631014
> 

> This seems absurdly far down the list. I think mine was better.


I agree with Gene here.

Paul has 

k*comp + err < x

Isn't a log-flat badness cutoff equivalent to

k*log(comp) + log(err) < x

for some k and x?

If so, something in between might have the form

k*(comp**p) + err**p < x

where 0<p<1.

One might try 

k*sqrt(comp) + sqrt(err) < x

for starters.

I'd really like to see these on a chart. Never mind about those Gene's
badness cutoff might have left out (for now). 114 seems like plenty to
choose from.

By the way, it seems that a useful rule of thumb is that the worst
error in the intervals we usually care about is about 5 times the
tenney-weighted minimax error. e.g. you can mentally round it to 2
significant digits, shift the decimal point to the right and divide by
2 to get a rough idea.

Message: 9527 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 02:10:14

Subject: Re: rank complexity explanation updated

From: Carl Lumma

>>>> > also see I had the wrong definition of interval matrix;
>> 

>> That's pretty incredible considering that you've got the
>> Rothenberg papers on the matter.
>

>Paul now tells us I had a correct definition.


If the difference between Paul's way and Manuel's way affects
the definition, somebody's smoking crack.

Wait, let me guess, it affects the def. if you intend to
interpret them as some sort of algebraic thing.  Well
Rothenberg never did this to my knowledge (unless you see
otherwise in his papers) but I'm all ears as to why the
heck anyone would do such a thing.


>Did Rothenberg invent the idea of an interval matrix?


As I am using the term in this thread, yes.  If by "invented"
you mean "defined".

-Carl

Message: 9528 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 21:04:29

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>> >'ve also suggested number of notes in a corresponding Fokker block,
>> but I'll have to defer to Paul on the status of that suggestion.
>

>Again, this is exactly what I've been discussing in this thread and 
>in "Attn: Gene 2", except that you have to replace "Fokker block" 
>with "Fokker strip", "Fokker slice", or whatever is appropriate 
>(block only appropriate for ET). In the latter cases, the number of 
>notes is infinite, but the "size" of the relevant construct in the 
>lattice is still ultra-meaningful, and what I've been itching about 
>here.

Ok.

-Carl

Message: 9529 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 01:54:36

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>> It's the minimax over *all* intervals.
> 

> You mean the 1/log(product_complexity) weighted minimax over all
> intervals in the prime limit.
> 

>>> Obviously you don't do an infinite
>>> number of calculations. 
>> 

>> No, you only need to set the primes, and the rest falls out 
correctly.
>> 
> 

> Yes, It's very clever in that way. But still it flies in the face of
> Partch's "observation one" (or whatever it is called) that as the
> complexity of a frequency ratio increases it must be tuned more and
> more accurately to be perceived as just (or something), until it
> becomes too complex to hear that way even when tuned precisely.


I don't think that's quite what Partch says. Manuel, at least, has 
always insisted that simpler ratios need to be tuned more accurately, 
and harmonic entropy and all the other discordance functions I've 
seen show that the increase in discordance for a given amount of 
mistuning is greatest for the simplest intervals.


> The very fact that TOP cannot distinguish between a good 7-limit
> temperament and a good 9-limit temperament (nor 8 or 10 limit) 
should
> make one suspicious.


Such distinctions may be important for *scales*, but for 
temperaments, I'm perfectly happy not to have to worry about them. 
Any reasons I shouldn't be?


> I suspect that while the results of tenney weighting are quite good
> for 5-limit, and may be acceptable for 7-limit, we might find it
> agreeing less and less with our subjective experience when we go to 
11
> and 13 limits. But then again, maybe not. I'll wait and see (or 
hear).


Tenney weighting can be conceived of in other ways than you're 
conceiving of it. For example, if you're looking at 13-limit, it 
suffices to minimize the maximum weighted error of {13:8, 13:9, 
13:10, 13:11, 13:12, 14:13} or any such lattice-spanning set of 
intervals. Here the weights are all very close (13:8 gets 1.12 times 
the weight of 14:13), *all* the ratios are ratios of 13 so simpler 
intervals are not directly weighted *at all*, and yet the TOP result 
will still be the same as if you just used the primes. I think TOP is 
far more robust than you're giving it credit for.


>> There is no need to use base 2 -- the result is the same 
regardless 
>> of which base you use in the logarithms. The error is measured 
using 
>> logarithms, but so is the complexity = log(n*d), so error divided 
by 
>> complexity, which is what you're minimizing the maximum of, is 
>> insensitive to choice of base.
> 

> Paul, that's clearly not what Gene has done in this list, otherwise
> the error figures would be a factor of 1200 smaller than they are.


You mean 1200/log(2), right?


> Gene has in fact already correctly normalised the errors and given
> them in cents.


That's a weird interpretation of the units because it's really only 
in cents for the octave, and often the octave doesn't even achieve 
the reported error.


> It's just that one now has to think like this:

Yes.


>>> Is the weighting the same for the complexity? Minimax where 
gens per
>>> interval is divided by lg2(product_complexity(interval))?
>> 

>> No particular generator basis is assumed in the TOP complexity 
>> calculations. Instead, it's a direct measure of how much the 
>> tempering simplifies the lattice, and reduces (Gene seems to 
>> imply/agree) to the number of notes per acoustic whatever in the 
case 
>> of equal (1-dimensional) temperaments.
> 

> What's an "acoustic whatever"?  Anything that relates to sound??? 
I'm
> being a pedant here in case you didn't guess. :-)


An interval that is fixed in (logarithmic) size, as measured for 
example in cents.


> OK. That sounds alright, but how do I relate it to upper limits (or
> whatever) on the numbers of notes before I get a particular 
>interval?


The word "before" implies some particular generating scheme. In fact 
it seems to imply a *single* generator, which is not even possible 
for planar, etc. temperaments. However, I wouldn't be surprised if it 
is indeed possible to view TOP complexity this way, due to some 
amazing theorem to be proved by Gene.



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Message: 9530 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 10:14:57

Subject: Maximum Error in TOP

From: Kalle Aho

Hi!

How do I get the maximum Tenney-weighted error in a TOP temperament? 
Is it the same as the maximum weighted error of the primes or some 
other set of lattice-spanning intervals? 

Kalle

  



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Message: 9531 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 08:56:35

Subject: Re: What the numbers mean

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
>> wrote:

>>> Here is a no doubt long overdue discussion of what the numbers in 
> the

>>> wedgie for a linear temperament mean.
>> ...
>> 

>> That was a good explanation. Thanks heaps Gene!
>> 

>>> If we take absolute values and normalize each by dividing by the 
> log

>>> base 2 of the two primes involved, we get
>>> 
>>> [6/p3 7/p5 2/p7 25/(p3p5) 20/(p3p7) 15/(p5p7)] =
>>> 
>>> [3.785578521, 3.014735907, .7124143742, 6.793166368, 4.494834256,
>>> 2.301151281]
>>> 
>>> The complexity is defined by the weighted value for 5/3, which is 
> the

>>> worst case (as, in fact, it clearly is.)
>> 

>> You are dividing by the _product_ of the two logs, shouldn't you be
>> dividing by their _sum_ (the log of the product of the primes)?
> 

> Product makes a lot more sense to me than sum, in my vague intuitive 
> understanding of these things. You probably see that Gene's defining 
> p2 as 1. Now the wedgie represents a bivector in 4-dimensional space, 
> which means the relevant basis elements are *directed areas* in 4D, 
> of which there are of course 6 "multilinearly-independent" ones. Now, 
> the unit lengths in the lattice are scaled by (p2,) p3, p5, and p7, 
> so the unit areas would have to be scaled by the 6 possible 
> *products*, not *sums*, of these -- (p2)p3, (p2)p5, (p2)p7, p3p5, 
> p3p7, and p5p7.


Yes. I agree now. My confusion was caused by the omission of the
"p2"s, since this gave different dimensionality (in the physics sense
of "units") for the first three versus the last three.

Message: 9532 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 21:52:42

Subject: Re: What the numbers mean

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


>> Can you show this process in action for the simpler, 3-limit and 
5-
>> limit cases? And why do we take the worst case, instead of some 
> sort 

>> of product (which would appear to get you your spacial measure 
once 
>> you've orthogonalized)?
> 

> "Orthogonalized" is one of those words which is holding up 
> communication, as I can only interpret that in terms of a L2 norm.


Yes, but clearly there are *some* geometrical constructs that hold 
even when you don't define angles, and this is, I believe, 
what "affine geometry" refers to. The determinant gives you the 
correct "area" (where there's one lattice node per unit area) of a 
periodicity block regardless of the angles you assume.

I'll keep playing around on my own in an attempt to understand this. 
If I construct a matrix of unison vectors (say, 3 of them for 12-
equal, or 4 of them for pajara), each of which has a 0 for a 
different prime, so that I have 0s along the diagonal, the triangles 
of numbers above and below the diagonal each look suspiciously like 
the wedgie, and . . .


> The vals, if we assume the Tenney metric, have an L_inf norm, and I 
> am regarding wedgies as multivals, hence the L_inf norm.


But didn't you say you could regard them as multimonzos instead? In 
which case you would apply the L_1 norm? This confusion is why I 
wanted you to provide the mathematical undergirding for my "Attn: 
Gene 2" post, something I'm still strongly hoping you'll do.

Message: 9533 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 10:19:18

Subject: TOP BP = BP

From: Gene Ward Smith

If we look at the appromimations to odd septimal intervals on the BP
site, we discover they are based on two commas, 245/243 and 3125/3087.
Putting them together gives a 7-limit linear temperament. Maybe "bop",
for Bohlen-Pierce? That's a little more dynamic that Number 106,
though it is nice to see that it turned up on the list. These are both
odd ratios, and so 2 will be exactly 2 in the TOP tuning for this
temperament, and as it turns out, the other generator will be
3^(1/13). So, there you have Top Bop. 

Bop can be bopped in 41, 49 or 90 equal if you like, with generators
of 5/41, 6/49 or 11/90. We have DES or MOS or whatever we are calling
it this week of 9, 17 or 25 notes to the octave for any big boppers
out there.

Message: 9534 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 16:46:03

Subject: Re: Crunch algorithm

From: Carl Lumma

>Suppose we have a list of rational numbers greater than one, sorted in
>ascending order, the elements of which should be independent. Define
>the crunch function as follows: take the quotient of the last with the
>next-to-last, and then place it in its proper location in the list,
>and return the new sorted list. If we start from an independent set of
>numbers, and in particular a basis for the p-limit, we crunch down to
>a basis with smaller elements.


Now here's the kind of thing I can understand!


>Here is crunch, starting out on [2, 3, 5, 7]. The next column is the
>top row of the inverse matrix for the monzos, which is a val matrix;
>so these are scale divisions.
>
>I thought of this because Oldlyzko pointed me to an unpublished paper
>of his, which suggested the problem of finding very large scale
>divisions (in the context of the Riemann zeta function, not music!)
>is harder than I believe it to be.


Yes, well Crunch is easy enough for my computer.  How large is very
large?

-Carl

Message: 9535 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 21:57:45

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>>>> No particular generator basis is assumed in the TOP complexity 
>>>> calculations. Instead, it's a direct measure of how much the 
>>>> tempering simplifies the lattice, and reduces (Gene seems to 
>>>> imply/agree) to the number of notes per acoustic whatever in 
the
>>>> case of equal (1-dimensional) temperaments.
>>> 

>>> I thought equal temperaments were "0-dimensional"?
>> 

>> We've always said linear temperaments were actually 2-dimensional,
>> so equal temperaments are 1-dimensional, and are generated by 
their 
>> step. Remember we've dropped octave-equivalence to get TOP.
> 

> Are you suggesting that TOP "linear" temperaments have a greater
> dimensionality than old-style "linear" temperaments?
> 
> -Carl


If we assume total octave-equivalence, and just deal in pitch 
classes, then the dimensionality of everything basically goes down by 
one. But so far, the mathematics of temperament (for example, 
detecting torsion) seems to be a lot more straightforward if we don't 
assume octave-equivalence, and just treat 2 like any other prime 
throughout.

Message: 9536 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 18:08:18

Subject: Re: Maximum Error in TOP

From: Kalle Aho

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> 
> wrote:

>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
>>> wrote:

>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" 
<kalleaho@m...> 
>>> wrote:
>>>>> Hi!
>>>>> 

>>>>> How do I get the maximum Tenney-weighted error in a TOP 
>>>> temperament? 

>>>>> Is it the same as the maximum weighted error of the primes
>>>> 
>>>> Yes.
>>> 

>>> Thanks, Paul. Can you explain why this is so? 
>>> 
>>> Kalle
> 

> Whoops, I screwed up. Let me try again:
> 
>  It follows from the prime factorization theorem.
>  
>  The maximum Tenney-weighted error being T implies that
>  
>  error(p)/log(p) <= T
> 
> so
> 
>   error(p) <= T*log(p)
> 
>  for all the primes p. If the factors in a chosen ratio are 2^a 
3^b 
>  5^c . . . (each exponent may either be positive or negative), 
then 
>  the error of the chosen ratio cannot be greater than
>  
>  T*(|a|*log(2) + |b|*log(3) + |c|*log(5) . . .)
>  
>  since the errors in the primes that make up the chosen ratio, at 
>  worst, add up without cancellation to the error in the chosen 
ratio. 
>  The complexity of the ratio, meanwhile, is exactly
>  
>  (|a|*log(2) + |b|*log(3) + |c|*log(5) . . .)
>  
>  So the Tenney-weighted error in the ratio cannot be greater than 
the 
>  second-to-last expression divided by the last expression, i.e., T.


Thanks, I understand it now!

Kalle

Message: 9537 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 21:59:35

Subject: Re: Attn: Gene 2

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>> Again, it would really help if he went 
>> through my original post and showed, step by step, how all the 
>> results there generalize to a single formula.
> 

> Which post, what results?


The 3-limit results in the "Attn: Gene 2" post. Then it would make 
sense to wrap our brains around the 5-limit linear and equal 
temperament cases. After that, we might have some more clarity of 
vision as regards 7-limit linear temperaments.

Message: 9538 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 14:02:56

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>> >re you suggesting that TOP "linear" temperaments have a greater
>> dimensionality than old-style "linear" temperaments?
>

>If we assume total octave-equivalence, and just deal in pitch 
>classes, then the dimensionality of everything basically goes down by 
>one. But so far, the mathematics of temperament (for example, 
>detecting torsion) seems to be a lot more straightforward if we don't 
>assume octave-equivalence, and just treat 2 like any other prime 
>throughout.


Agreed but this doesn't seem to address the question.  A TOP
"linear" temperament requires the same number of commas as an
old-style "linear" temperament... oh, but the TOP lattice has
an extra dimension.  Ok.

-Carl

Message: 9539 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 20:57:44

Subject: Re: TOP BP = BP

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:

I went to bed last night thinking that the TOP bop generator can't
possibly be 3^(1/13), and it isn't. Here are some members of the bop
generation:

3^(1/13): 146.304
41-et: 146.341
TOP bop: 146.476
TOP tuned 41-et bop: 146.476
rms bop: 146.535

There's not a hell of a lot of difference between the 41-et bop and
the traditional 3^(1/13).

Message: 9540 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 16:54:07

Subject: Re: 60 for Dave

From: Carl Lumma

>>>> >omplexity, I should think, should definitely be punished more 
>>>> than 1:1.  Using 8 notes instead of 7 notes would seem to demand
>>>> more than eight 7ths the mental energy.
>>> 

>>> We could use a quadratic penalty on the complexity too.
>> 

>> If we square both terms, doesn't this give the same ranking?
>

>No, because following Dave, we're adding the terms, not multiplying 
>them. Dave restated Gene's product as a sum of logs.


Oh.  Why do that?


>> In March '02 I wrote, "I think I'd rather have a smooth pain
>> function, like ms, and a stronger exponent on complexity."
>

>In response to what?


Dunno, but by "stronger" I meant "stronger than whatever we use
on error".  And "exponent" maybe shouldn't be taken literally...
I just meant to say that I'm willing to accept lots of error for
a small savings in notes.


>> By the way, when doing ms error, if an error is less than a cent
>> it will get *smaller* when squared.
>

>No, you can't compare cents to cents-squared. These quantities do not 
>have the same dimension.
>

>> Do you see this as a good
>> thing, should we be ceilinging these to 1 before squarring, or...?
>

>To 1 cent? Definitely not -- there's no justification for treating 1 
>cent as a special error size.
>
>Besides, the errors Gene gave are only in units of cents if you're 
>looking at the error of the octave -- other intervals have different 
>units, since it's minimax Tenney-weighed error we're looking at.


This was a more general question.  When calculating the rms error of
an equal temperament, as we used to do, we just allow things less
than 1 to get smaller and influence the mean?  It seems one is special
whether or not we do anything.

-Carl

Message: 9541 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 21:44:17

Subject: Re: Graef article on rationalization of scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>> Pretty much. We've been through this before, whereupon Gene 
>> defined "block" for convex PB and "semiblock" for something "not 
> too 

>> concave" or something, if my vague recollection is reliable.
> 

> But we never settled anything, did we?


I thought you were solid with your definitions of these. What remains 
unsettled?



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Message: 9542 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 22:22:43

Subject: Re: TOP BP = BP

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> If we look at the appromimations to odd septimal intervals on the BP
> site, we discover they are based on two commas, 245/243 and 
3125/3087.
> Putting them together gives a 7-limit linear temperament. 
Maybe "bop",
> for Bohlen-Pierce?


Sure -- this is certainly better than "beep" for 27/25 vanishing, 
since the latter isn't even how BP works. Looking at this:

Yahoo groups: /tuning_files/files/Erlich/dualx... * [with cont.] 

it might not be a terrible idea to rename "beep" to "mother", though 
that's an awfully prestigious name for this undistinguished 
temperament.

Message: 9543 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 22:35:13

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:


>>> I think Gene may be using the wrong norm to get his complexity 
>>> values. I'll wait until I'm sure they're right or corrected.


Are you sure yet?


>> Well something's wrong. Whether its the badness functions or only 
> the

>> complexity I don't know.
> 

> Would you take a closer look, then? 


This would be a lot easier with an error versus complexity plot.


>I think it's important to rethink 
> the problem each time, to allow old prejudices a chance of dissolving.
> 


Yes. For example, the tempering of the octave may have reduced the
worst error significantly in some cases. And I may be getting used to
the idea of using the maximum function (inf-norm) for the complexity. 

But it's possible that the combination of using maximum _and_ the new
weighting for complexity is too much. i.e. taking it too far from my
(most people's?) subjective judgement of complexity. But then again,
simply reducing k (the penalty for complexity relative to error) in
some badness function might restore sanity for me.


>> But Diaschismic shouldn't be so far down.
> 

> For 7-limit? This is only one of at least 3 possible 7-limit 
> mappings, so it's weird to even use that name.


Oops. Yeah. I was thinking of the most obvious one

pajara               <0 1, -2 -2]

which _is_ appropriately high on the list, but I was looking at

15-limit diaschismic <0 1, -2 -8 ...]

which I agree should be well down the 7-limit list.

Then there's

"56-ET" diaschismic    <0 1, -2 9]

and

shrutar              <0 1, -2 3.5]


>> And where's Shrutar?
> 

> It fell off the end, with a badness of 6.1221 in this formulation. 
> Doesn't shine as a 7-limit linear temperament . . .


But wasn't that it's whole raison d'etre?

Message: 9544 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 17:19:50

Subject: Re: 60 for Dave

From: Carl Lumma

>>>>> >e could use a quadratic penalty on the complexity too.
>>>> 

>>>> If we square both terms, doesn't this give the same ranking?
>>> 

>>> No, because following Dave, we're adding the terms, not
>>> multiplying them. Dave restated Gene's product as a sum of logs.
>> 

>> Oh.  Why do that?
>

>So that he could understand Gene's badness and my linear badness in 
>the same form, and propose a compromise.


Ah.  Is yours the one from the Attn: Gene post?  It involves taking
determinants, which I haven't fully learned how to do yet.


>>>> In March '02 I wrote, "I think I'd rather have a smooth pain
>>>> function, like ms, and a stronger exponent on complexity."
>>> 

>>> In response to what?
>> 

>> Dunno, but by "stronger" I meant "stronger than whatever we use
>> on error".  And "exponent" maybe shouldn't be taken literally...
>> I just meant to say that I'm willing to accept lots of error for
>> a small savings in notes.
>

>Well, according to log-flat badness, this will only happen at very 
>low complexity values. At high complexity values, log-flat badness is 
>essentially "flat" in error.


That's good to know, but the above is just my value judgement, and
as you point out log-flat badness frees us from those, in a sense.


>> It seems one is special
>> whether or not we do anything.
>

>Incorrect. If you change to a different system of units (say, 
>millicents) so that nothing's smaller than 1, perform the rms 
>calculation, and then change back to the original units, you get the 
>same answer. Try it!


Of course you're right.  Thanks.

-C.

Message: 9545 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 22:49:40

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>> wrote:
> 

>>>> I think Gene may be using the wrong norm to get his 
complexity 
>>>> values. I'll wait until I'm sure they're right or corrected.
> 

> Are you sure yet?


No; but it did seem Gene was suggesting that at least one other 
complexity formulation would result if we treated the wedgies as 
multimonzos instead of as multivals. I'm hoping to get a clear 
intuitive, geometrical, and musical understanding of this stuff, 
which I was itching for with my "Attn: Gene 2" post, which I'm still 
hoping will get some (attention). It may merely require a clear view 
of what these mathematical quantities (wedgies?) are saying in the 3-
limit and 5-limit.


>>> Well something's wrong. Whether its the badness functions or 
only 
>> the

>>> complexity I don't know.
>> 

>> Would you take a closer look, then? 
> 

> This would be a lot easier with an error versus complexity plot.


I'm afraid my arm is up for the task of cutting and pasting all of 
Gene's numbers and names that this would require. If someone puts the 
relevant information in a table, then anyone could make the required 
plot, even you, with Excel.


>> I think it's important to rethink 
>> the problem each time, to allow old prejudices a chance of 
dissolving.
>> 
> 

> Yes. For example, the tempering of the octave may have reduced the
> worst error significantly in some cases. And I may be getting used 
to
> the idea of using the maximum function (inf-norm) for the 
complexity. 


Well, for 3-limit and 5-limit complexity, I've been using the 
weighted L_1 norm of the monzo, i.e., the Tenney harmonic distance of 
the comma. So don't get *too* used to it. :)


> But it's possible that the combination of using maximum _and_ the 
new
> weighting for complexity is too much. i.e. taking it too far from my
> (most people's?) subjective judgement of complexity.


Not unreasonable. I look forward to the day when this is all crystal 
clear and unsubjective.


> But then again,
> simply reducing k (the penalty for complexity relative to error) in
> some badness function might restore sanity for me.


Sure. Of course, I still hope we can do away with badness and simply 
choose by looking at the graph. We should choose some cutoff that 
corresponds to a wide swath of empty space in the graph, so that 
modest changes in the cutoff do not affect which temperaments make it 
in and which don't.


>>> But Diaschismic shouldn't be so far down.
>> 

>> For 7-limit? This is only one of at least 3 possible 7-limit 
>> mappings, so it's weird to even use that name.
> 

> Oops. Yeah. I was thinking of the most obvious one
> 
> pajara               <0 1, -2 -2]
> 
> which _is_ appropriately high on the list, but I was looking at
> 
> 15-limit diaschismic <0 1, -2 -8 ...]
> 
> which I agree should be well down the 7-limit list.
> 
> Then there's
> 
> "56-ET" diaschismic    <0 1, -2 9]
> 
> and
> 
> shrutar              <0 1, -2 3.5]
> 

>>> And where's Shrutar?
>> 

>> It fell off the end, with a badness of 6.1221 in this 
formulation. 
>> Doesn't shine as a 7-limit linear temperament . . .
> 

> But wasn't that it's whole raison d'etre?


11-limit, I thought . . .

Message: 9546 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 22:52:44

Subject: Re: 114 7-limit temperaments

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:

>> We've always said linear temperaments were actually 2-dimensional, 
> so 

>> equal temperaments are 1-dimensional, and are generated by their 
>> step. Remember we've dropped octave-equivalence to get TOP.
> 

> You boiled me in oil and rendered me down for lard when I first got 
> here for wanting things this way, and now you claim we've always said 
> it? :)


Hee hee. :-)

What I remember we gave you a hard time about, was not that linear
temperaments are 2-dimensional without octave-equivalence, but that
you wanted to call them "planar" (which would have been too confusing
a departure from the historical usage). We wanted "Linear temperament"
to be the constant name of the musical object which remains
essentially the same while its mathematical models vary in dimensionality.

Message: 9547 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 23:10:49

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:

>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>>> wrote:
>> 

>>>>> I think Gene may be using the wrong norm to get his 
> complexity 

>>>>> values. I'll wait until I'm sure they're right or corrected.
>> 

>> Are you sure yet?
> 

> No; but it did seem Gene was suggesting that at least one other 
> complexity formulation would result if we treated the wedgies as 
> multimonzos instead of as multivals. I'm hoping to get a clear 
> intuitive, geometrical, and musical understanding of this stuff, 
> which I was itching for with my "Attn: Gene 2" post, which I'm still 
> hoping will get some (attention). It may merely require a clear view 
> of what these mathematical quantities (wedgies?) are saying in the 3-
> limit and 5-limit.


If Gene could show how to obtain this complexity figure from a
period-generator prime-mapping in such a way that you get the same
result no matter how you factor the generators (octave, tritave etc.),
this would be a big help to my understanding.



> 

>>>> Well something's wrong. Whether its the badness functions or 
> only 
>>> the

>>>> complexity I don't know.
>>> 

>>> Would you take a closer look, then? 
>> 

>> This would be a lot easier with an error versus complexity plot.
> 

> I'm afraid my arm is up for the task of cutting and pasting all of 
> Gene's numbers and names that this would require. If someone puts the 
> relevant information in a table, then anyone could make the required 
> plot, even you, with Excel.

Gene,


Can you please supply your latest list of 114 as a tab-delimited
table, one line per temperament, with the first row being the column
headings?


>> But it's possible that the combination of using maximum _and_ the 
> new

>> weighting for complexity is too much. i.e. taking it too far from my
>> (most people's?) subjective judgement of complexity.
> 

> Not unreasonable. I look forward to the day when this is all crystal 
> clear and unsubjective.


How could it ever be so, except in a statistical sense, by average the
subjectivity of a lot of humans. It is after all a human perceptual or
cognitive property we're trying to model.

What we're really looking for is something that's mathematically
simple and yet close enough to represent the typical human experience.
Trouble is, not many humans have much experience with many linear
temperaments, including us.


>> But then again,
>> simply reducing k (the penalty for complexity relative to error) in
>> some badness function might restore sanity for me.
> 

> Sure. Of course, I still hope we can do away with badness and simply 
> choose by looking at the graph. We should choose some cutoff that 
> corresponds to a wide swath of empty space in the graph, so that 
> modest changes in the cutoff do not affect which temperaments make it 
> in and which don't.


Yes. I like that idea too. But by "a wide swath" don't you mean one
that it's easy to put a simple smooth curve thru? And you must have
some general idea of which way this intergalactic moat must curve.


>>>> And where's Shrutar?
>>> 

>>> It fell off the end, with a badness of 6.1221 in this 
> formulation. 

>>> Doesn't shine as a 7-limit linear temperament . . .
>> 

>> But wasn't that it's whole raison d'etre?
> 

> 11-limit, I thought . . .


You're probably right.

Message: 9548 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 23:13:03

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:

>>> We've always said linear temperaments were actually 2-
dimensional, 
>> so 

>>> equal temperaments are 1-dimensional, and are generated by 
their 
>>> step. Remember we've dropped octave-equivalence to get TOP.
>> 

>> You boiled me in oil and rendered me down for lard when I first 
got 
>> here for wanting things this way, and now you claim we've always 
said 
>> it? :)
> 

> Hee hee. :-)
> 
> What I remember we gave you a hard time about, was not that linear
> temperaments are 2-dimensional without octave-equivalence, but that
> you wanted to call them "planar" (which would have been too 
confusing
> a departure from the historical usage). We wanted "Linear 
temperament"
> to be the constant name of the musical object which remains
> essentially the same while its mathematical models vary in 
>dimensionality.


Unfortunately for us, 'linear temperament' has probably never 
referred to a multiple-chains-per-octave system (like pajara, 
diminished, augmented, ennealimmal . . .) before we started using it 
that way, and some of the original users of the term (say, Erv 
Wilson) might be rather upset with this slight generalization.

(Now hiding head under sand while Gene throws a fit :)

Message: 9549 - Contents - Hide Contents

Date: Fri, 30 Jan 2004 23:24:28

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>> wrote:

>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
>> wrote:

>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" 
<d.keenan@b...> 
>>>> wrote:
>>> 

>>>>>> I think Gene may be using the wrong norm to get his 
>> complexity 

>>>>>> values. I'll wait until I'm sure they're right or 
corrected.
>>> 

>>> Are you sure yet?
>> 

>> No; but it did seem Gene was suggesting that at least one other 
>> complexity formulation would result if we treated the wedgies as 
>> multimonzos instead of as multivals. I'm hoping to get a clear 
>> intuitive, geometrical, and musical understanding of this stuff, 
>> which I was itching for with my "Attn: Gene 2" post, which I'm 
still 
>> hoping will get some (attention). It may merely require a clear 
view 
>> of what these mathematical quantities (wedgies?) are saying in 
the 3-
>> limit and 5-limit.
> 

> If Gene could show how to obtain this complexity figure from a
> period-generator prime-mapping in such a way that you get the same
> result no matter how you factor the generators (octave, tritave 
etc.),
> this would be a big help to my understanding.


Where did he fall short?


> 

>>> But it's possible that the combination of using maximum _and_ 
the 
>> new

>>> weighting for complexity is too much. i.e. taking it too far 
from my
>>> (most people's?) subjective judgement of complexity.
>> 

>> Not unreasonable. I look forward to the day when this is all 
crystal 
>> clear and unsubjective.
> 

> How could it ever be so, except in a statistical sense, by average 
the
> subjectivity of a lot of humans. It is after all a human perceptual 
or
> cognitive property we're trying to model.


With the complexity measure, I'm hoping it will be an affine-
geometrical property (in the Tenney lattice), purely mathematical but 
as unobjectionable as the 3-limit cases I already illustrated.


> What we're really looking for is something that's mathematically
> simple and yet close enough to represent the typical human 
experience.
> Trouble is, not many humans have much experience with many linear
> temperaments, including us.


OK, so I meant unsubjective to me. :)


>>> But then again,
>>> simply reducing k (the penalty for complexity relative to 
error) in
>>> some badness function might restore sanity for me.
>> 

>> Sure. Of course, I still hope we can do away with badness and 
simply 
>> choose by looking at the graph. We should choose some cutoff that 
>> corresponds to a wide swath of empty space in the graph, so that 
>> modest changes in the cutoff do not affect which temperaments 
make it 
>> in and which don't.
> 

> Yes. I like that idea too. But by "a wide swath" don't you mean one
> that it's easy to put a simple smooth curve thru?

Yes.

> And you must have
> some general idea of which way this intergalactic moat must curve.


True. Even though it's out-Keenaning Keenan with respect to Smith, I 
still think a straight line -- if not a *convex* curve, perish the 
thought -- makes some sense. Both error and complexity are things 
that we typically judge and compare in a *linear* fashion, so 
performing various operations on them seems arbitrary at best. At 
least, it seems that if there's zero error, doubling the complexity 
should double the badness; and if there's zero complexity, doubling 
the error should double the badness. 


>>>>> And where's Shrutar?
>>>> 

>>>> It fell off the end, with a badness of 6.1221 in this 
>> formulation. 

>>>> Doesn't shine as a 7-limit linear temperament . . .
>>> 

>>> But wasn't that it's whole raison d'etre?
>> 

>> 11-limit, I thought . . .
> 

> You're probably right.


Well, you came up with the final tuning, so you should know :) I 
recall you tempering out 896:891 (though I think this was replaced 
with 176:175 in Gene's TM reduction), so it would seem to be 11-
limit . . .

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