Tuning-Math Digests messages 2300 - 2324

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 3

Previous Next

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 2600 2650 2700 2750 2800 2850 2900 2950

2300 - 2325 -



top of page bottom of page down


Message: 2300

Date: Fri, 07 Dec 2001 03:12:24

Subject: Re: More lists

From: dkeenanuqnetau

--- In tuning-math@y..., graham@m... wrote:
> I've updated the script at <Automatically generated temperaments *> 
to 
> produce files using Dave Keenan's new figure of demerit.  That is
> 
> width**2 * math.exp((error/self.stdError*3)**2)

Thanks for doing that Graham.

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.

> The stdError is from some complexity calculations we did before.  I 
forget 
> what, but it's 17 cents.

Actually that looks like the 1% std dev in frequency that came from 
some dude's experiments with actual live humans experiencing actual 
air vibrations. Paul can you remind us who it was and what s/he 
measured?

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.

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.

I realised there's no need to have non-linear functions of _both_ 
num_gens and error (steps and cents) in this badness metric. e.g. 
This will give the same ranking as the above:

  badness = num_gens / gaussian(error/(17c * sqrt(2)))

So all we really want to know is the relationship between error and 
num_gens. What shape is a line of constant badness (an isobad) on a 
plot of number of generators needed for a complete otonality (or 
diamond) against error in cents.

The simplest badness, num_gens * error, would mean the isobads are 
hyperbolas, (and num_gens^2 * error or equivalently num_gens * 
sqrt(error) is of course very similar) but I think it is clear that, 
for constant badness, as error goes to zero, num-gens does _not_ go to 
infinity, but levels off. Even for zero error there is a limit to how 
many generators you can tolerate. I find it difficult to imagine 
anyone being seriously interested in using a temperament that needs 30 
generators to get a single complete otonality, no matter how small the 
error is. And I think this limiting number of generators decreases as 
the odd-limit decreases.

We can introduce this as a sudden limit as Gene suggested, or we can 
use some continuous function to make it come on gradually

An isobad will also have a maximum number of cents error that can be 
tolerated even when everything is approximated by a single generator. 
Notice that the number of generators can't go below 1 (even for rms), 
so we don't care what an isobad does for num_gens < 1.

What's a nice simple badness metric that will give us these effects?


> Oh yes.  Seeing as a 7-limit microtemperament is now causing 
something of 
> a storm, notice that the top 11-limit one is 26+46 (complexity of 
30, 
> errors within 2.5 cents).  And the simplest with all errors below a 
cent 
> is 118+152 (complexity of 74).

Yes, even though we don't consider it a microtemperament at the 
11-limit, Miracle temperament is really a serious 11-limit optimum, by 
any (reasonable) goodness measure. You have to pay an enormous cost in 
extra complexity to get the max error even _slightly_ lower than 
11-limit-Miracle's 3.3 cents, or an enormous cost in cents to get the 
complexity down even slightly below 11-limit-Miracle's 22 generators. 
Is that what you are indicating?


top of page bottom of page up down


Message: 2301

Date: Fri, 07 Dec 2001 06:54:04

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

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > I understand what the slope is in the HE case, but what slope are 
> you 
> > talking about re badness of linear temperament? Badness wrt what?
> 
> What is the problem with a "flat" system and a cutoff?

Dave is trying to understand why this _is_ a flat system.

> It doesn't 
> commit to any particular theory about what humans are like and what 
> they should want, and I think that's a good plan.

Thank you.


top of page bottom of page up down


Message: 2302

Date: Fri, 07 Dec 2001 03:23:14

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

From: dkeenanuqnetau

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

It's nice to have pretty looking (i.e. simple) fomulae but we can 
hardly ignore the fact that we're trying to come up with a list of 
linear temperaments that will be of interest to the largest possible 
number of human beings. Unfortunately human perception and cognition 
is messy to model mathematically, not well established experimentally 
and highly variable between individuals. But I'm sure we can come up 
with something that is both reasonably elegant mathematically and that 
we (in this forum) can all agree isn't too bad. We certainly do it 
without trying some out and looking at the results!

We should probably hone the badness metric using 5-limit, where the 
most experience exists.


top of page bottom of page up down


Message: 2303

Date: Fri, 07 Dec 2001 06:59:08

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:
> 
> > Possibly, though since gens and cents are two dimensions, we 
really 
> > need a shuf-off _curve_, don't we?
> 
> If we bound one of them and gens^2 cents, we've bound the other; 
> that's what I'd do.

Hmm . . . so if we simply put an upper bound on the RMS cents error, 
we'll have a closed search? That doesn't seem right . . .


top of page bottom of page up down


Message: 2304

Date: Fri, 07 Dec 2001 03:38:36

Subject: Re: More lists

From: paulerlich

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

> Actually that looks like the 1% std dev in frequency that came from 
> some dude's experiments with actual live humans experiencing actual 
> air vibrations. Paul can you remind us who it was and what s/he 
> measured?

It measured the typical uncertainties with which sine-wave partials 
in an optimal frequency range were heard, based on the uncertainties 
with which the virtual fundamentals were heard.


top of page bottom of page up down


Message: 2305

Date: Fri, 07 Dec 2001 07:00:21

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

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > I understand what the slope is in the HE case, but what slope are 
> you 
> > talking about re badness of linear temperament? Badness wrt what?
> 
> What is the problem with a "flat" system and a cutoff?

I may be able to answer that when someone explains what is flat with 
respect to what.

 It doesn't 
> commit to any particular theory about what humans are like and what 
> they should want, and I think that's a good plan.

Don't the cutoffs have to be based on a theory about what humans are 
like?

If a "flat" system was miles from anything related what humans are 
like, would you still be interested in it?

I don't think you can avoid this choice. You must publish a finite 
list. If you include more of certain extremes, you must omit more 
of the middle-of-the-road.


top of page bottom of page up down


Message: 2306

Date: Fri, 07 Dec 2001 03:47:20

Subject: Re: More lists

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> What's a nice simple badness metric that will give us these effects?

Hey! What's wrong with simply

  badness = num_gens + error_in_cents

(i.e. steps + cents)

or if that seems too arbitrary, how about agreeing on some value of k 
in

  badness = k * num_gens + error_in_cents, where k ~= 1

or maybe even

  badness = k/odd_limit * num_gens + error_in_cents, where k ~= 5

Wanna give this one a spin Graham?


top of page bottom of page up down


Message: 2307

Date: Fri, 07 Dec 2001 07:03:27

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

From: genewardsmith

--- 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.


top of page bottom of page up down


Message: 2308

Date: Fri, 07 Dec 2001 03:50:46

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

From: paulerlich

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

> But I'm sure we can come up 
> with something that is both reasonably elegant mathematically and 
that 
> we (in this forum) can all agree isn't too bad.

I felt that way about steps^3 cents, except where was 12+14?

> We certainly do it 
> without trying some out and looking at the results!

You mean a priori? The more arbitrary parameters we put into it, the 
more we'll have to rely on particular assumption on how someone is 
going to be making music, and this assumtion will be violated for the 
next person. The top 25 or 40 according to a very generalized 
criterion will best serve to present the _pattern_ of this whole 
endeavor, upon which any musician can base their _own_ evaluation, 
and if they don't want to, at least pick off one or two temperaments 
that interest them.

But I have a nagging suspicion that there are even more "slippery" 
ones out there, especially on the ultra-simple end of things . . .

I suspect we can use step^2 cents and cut it off at some point where 
there's a long gap in the step-cent plane. For example, the next 
point out after Ennealimmal is probably a long way out, so we can 
probably put a cutoff there. As for simple temperaments with large 
errors, I suspect there are more than Gene and Graham have found so 
far that would end up looking good on this criterion, so it may end 
up making sense to place another cutoff there . . . but I want to be 
sure we've caught all the slippery fish before we decide that.

I would still like to see the "step" thing weighted -- there should 
be something very mathematically and acoustically elegant about doing 
it that way (if defined correctly) since we are using the Tenney 
lattice after all!
> 
> We should probably hone the badness metric using 5-limit, where the 
> most experience exists.

Yes, I was just going to say we should write the whole paper first in 
the 5-limit.


top of page bottom of page up down


Message: 2309

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

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..., "genewardsmith" <genewardsmith@j...> wrote:

> > It doesn't 
> > commit to any particular theory about what humans are like and 
what 
> > they should want, and I think that's a good plan.
> 
> Don't the cutoffs have to be based on a theory about what humans 
are 
> like?

I'm suggesting we place the cutoffs where we find big gaps, and 
comfortably outside any system that has been used to date.
> 
> If a "flat" system was miles from anything related what humans are 
> like, would you still be interested in it?

Again, any system that is "best" according to a "human" criterion 
will show up as "best in its neighborhood" under a flat criterion.


top of page bottom of page up down


Message: 2310

Date: Fri, 07 Dec 2001 04:18:34

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:
> 
> > But I'm sure we can come up 
> > with something that is both reasonably elegant mathematically and 
> that 
> > we (in this forum) can all agree isn't too bad.
> 
> I felt that way about steps^3 cents, except where was 12+14?
> 
> > We certainly do it 
> > without trying some out and looking at the results!

Oops! That should have been

We certainly _can't_ do it without trying some out and looking at the 
results!

> You mean a priori? The more arbitrary parameters we put into it, the 
> more we'll have to rely on particular assumption on how someone is 
> going to be making music, and this assumtion will be violated for 
the 
> next person.

"Not putting in" an arbitrary parameter is usually equivalent to 
putting it in but giving it an even more arbitrary value like 0 or 1.


top of page bottom of page up down


Message: 2311

Date: Fri, 07 Dec 2001 07:08:44

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:
> 
> > > 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?


top of page bottom of page up down


Message: 2312

Date: Fri, 07 Dec 2001 04:30:35

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:

> > You mean a priori? The more arbitrary parameters we put into it, 
the 
> > more we'll have to rely on particular assumption on how someone 
is 
> > going to be making music, and this assumtion will be violated for 
> the 
> > next person.
> 
> "Not putting in" an arbitrary parameter is usually equivalent to 
> putting it in but giving it an even more arbitrary value like 0 or 
1.

Well, I think Gene is saying that step^2 cents is clearly the right 
measure of "remarkability".


top of page bottom of page up down


Message: 2313

Date: Fri, 07 Dec 2001 07:11:53

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:
> 
> > > Because those measures give an overall "slope" to the results, 
in 
> > > analogy to what the Farey series seeding does to harmonic 
entropy.
> > 
> > What's objective about that? A certain slope may be _real_. i.e. 
> > humans on average may experience it that way, in which case 
> the "flat" 
> > case will really be favouring one extreme.
> 
> But I don't feel comfortable deciding that for anyone.

But you _are_ deciding it. You can't help but decide it, unless you 
intend to publish an infinite list. No matter what you do there will 
be someone who thinks there's a lot of fluff in there and you missed 
out some others. They aren't going to be impressed by any argument 
that "our metric is 'objective' or 'flat'".

> > I understand what the slope is in the HE case, but what slope are 
> you 
> > talking about re badness of linear temperament? Badness wrt what?
> 
> Both step and cent.

Huh? Obviously any badness metric _must_ slope down towards (0,0) on 
the (cents,gens) plain. If you make the gens and cents axes 
logarithmic then badness = gens^k * cents is simply a tilted plane. 
The only way you can decide on whether it should tilt more towards 
gens or cents (the exponent k) is through human considerations.


top of page bottom of page up down


Message: 2314

Date: Fri, 07 Dec 2001 04:35:25

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

From: genewardsmith

--- 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.


> > The pair of unisons 
> > returned in this way can be LLL reduced by the "com7" function, 
> which 
> > takes a pair of intervals and LLL reduces them.
> 
> Why not TM-reduce them?

I'd always LLL reduce them first.


top of page bottom of page up down


Message: 2315

Date: Fri, 07 Dec 2001 07:16:27

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:
> > If a "flat" system was miles from anything related what humans are 
> > like, would you still be interested in it?
> 
> Again, any system that is "best" according to a "human" criterion 
> will show up as "best in its neighborhood" under a flat criterion.

But some neighbourhoods may be so disadvantaged that their best 
doesn't even make it into the list.


top of page bottom of page up down


Message: 2316

Date: Fri, 07 Dec 2001 04:56:08

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:
> 
> > 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.
> 
> 
> > > The pair of unisons 
> > > returned in this way can be LLL reduced by the "com7" function, 
> > which 
> > > takes a pair of intervals and LLL reduces them.
> > 
> > Why not TM-reduce them?
> 
> I'd always LLL reduce them first.

How come?


top of page bottom of page up down


Message: 2317

Date: Fri, 07 Dec 2001 07:25:50

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

From: genewardsmith

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

> Hmm . . . so if we simply put an upper bound on the RMS cents 
error, 
> we'll have a closed search? That doesn't seem right . . .

I was suggesting a *lower* bound on RMS cents as one possibility.

If with all quantities positive we have g^2 c < A and c > B, then
1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it probably 
makes more sense to use g>=1, so that if g^2 c <= A then c <= A.


top of page bottom of page up down


Message: 2318

Date: Fri, 07 Dec 2001 05:17:35

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

From: genewardsmith

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

> > I'd always LLL reduce them first.
> 
> How come?

Because it makes the TM reduction dead easy.


top of page bottom of page up down


Message: 2319

Date: Fri, 07 Dec 2001 07:26:24

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:
> > 
> > > > Because those measures give an overall "slope" to the 
results, 
> in 
> > > > analogy to what the Farey series seeding does to harmonic 
> entropy.
> > > 
> > > What's objective about that? A certain slope may be _real_. 
i.e. 
> > > humans on average may experience it that way, in which case 
> > the "flat" 
> > > case will really be favouring one extreme.
> > 
> > But I don't feel comfortable deciding that for anyone.
> 
> But you _are_ deciding it. You can't help but decide it, unless you 
> intend to publish an infinite list. No matter what you do there 
will 
> be someone who thinks there's a lot of fluff in there and you 
missed 
> out some others. They aren't going to be impressed by any argument 
> that "our metric is 'objective' or 'flat'".

We won't be missing out on anyone's "best" (unless they are really 
far out on the plane, beyond the big gap where we will establish the 
cutoff). Then they can come up with their own criterion and get their 
own ranking. But at least we'll have something for everyone.

> > > I understand what the slope is in the HE case, but what slope 
are 
> > you 
> > > talking about re badness of linear temperament? Badness wrt 
what?
> > 
> > Both step and cent.
> 
> 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.


top of page bottom of page up down


Message: 2320

Date: Fri, 07 Dec 2001 05:18:59

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

From: dkeenanuqnetau

--- 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. I think I've already explained why no 
product of poynomials of these two things will ever be acceptable to 
me, at least not without cutoffs applied to them first. And I 
understand Gene to be saying that he wants at least an upper cutoff on 
"steps" (which seems like a bad name to me since it suggests scale 
steps, I prefer "num_gens" or just "gens").

  gens^2 * cents 
gives exactly the same ranking as 
  log(gens^2 * cents)     [where the log base is arbitrary]
because log(x) is monotonically increasing. Right?
Now
  log(gens^2 * cents)
= log(gens^2) + log(cents)
= 2*log(gens) + log(cents)

So this says that a doubling of the number of generators is twice as 
bad as a doubling of the error. And previously someone suggested it 
was 3 times as bad. You've arbitrarity decided that only the 
logarithms are comparable (when cents is already a logarithmic 
quantity) and you arbitrarily decided that the constant of 
proportionality between them must be an integer!

So what's wrong with k*steps + cents? The basic idea here is that the 
unit of badness is the cent and we decide for a given odd-limit how 
many cents the error would need to be reduced for you to tolerate an 
extra generator in the width of your tetrads (or whatever), or how 
many generators you'd need to reduce the tetrad (or whatever) width by 
in order to tolerate another cent of error.

Or maybe you think that a _doubling_ of the number of generators is 
worth a fixed number of cents. i.e. badness = k*log(gens) + cents

But always you must decide a value for one parameter k that gives the 
proportionality between gens and cents because there is no 
relationship between their two units of measurement apart from the one 
that comes through human experience. Or at least I can't see any.


top of page bottom of page up down


Message: 2321

Date: Fri, 07 Dec 2001 07:28:22

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:
> > > If a "flat" system was miles from anything related what humans 
are 
> > > like, would you still be interested in it?
> > 
> > Again, any system that is "best" according to a "human" criterion 
> > will show up as "best in its neighborhood" under a flat criterion.
> 
> But some neighbourhoods may be so disadvantaged that their best 
> doesn't even make it into the list.

That won't happen -- that's the point of the "flat" criterion. Only 
the neighborhoods outside our cutoff will be disadvantaged, but at 
least this will be explicit.


top of page bottom of page up down


Message: 2322

Date: Fri, 07 Dec 2001 05:22:38

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

From: genewardsmith

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

> Yes, I was just going to say we should write the whole paper first 
in 
> the 5-limit.

There's not much to the 5-limit--it basically is a mere comma search, 
and that can be done expeditiously using a decent 5-limit notation.


top of page bottom of page up down


Message: 2323

Date: Fri, 07 Dec 2001 07:34:36

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:
> 
> > Hmm . . . so if we simply put an upper bound on the RMS cents 
> error, 
> > we'll have a closed search? That doesn't seem right . . .
> 
> I was suggesting a *lower* bound on RMS cents as one possibility.

Oh . . . well I don't think we should frame it _that_ way!

> If with all quantities positive we have g^2 c < A and c > B, then
> 1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it probably 
> makes more sense to use g>=1, so that if g^2 c <= A then c <= A.

Are you saying that using g>=1 is enough to make this a closed search?


top of page bottom of page up down


Message: 2324

Date: Fri, 07 Dec 2001 05:34:10

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:
> > 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. 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. The particular users of our findings can then decide what 
range suits them best. Within any narrow range, all reasonable 
measures will give the same ranking.

This is kind of like using Tenney complexity to determine the seed 
set for harmonic entropy -- with different complexity measures the 
overall slope of the curve changes, changing the consonance ranking 
of intervals of different sizes, but the consonance ranking of nearby 
intervals remains the same regardless of how complexity is defined 
(as long as the 2-by-2 matrix formed by the numbers in adjacent seed 
fractions always has a determinant of 1).

> I think I've already explained why no 
> product of poynomials of these two things will ever be acceptable 
to 
> me, at least not without cutoffs applied to them first.
> And I 
> understand Gene to be saying that he wants at least an upper cutoff

Yes -- I discussed the situation a few messages back. We use an 
objective measure, and cut things off in a nice wide gap.

> on 
> "steps" (which seems like a bad name to me since it suggests scale 
> steps, I prefer "num_gens" or just "gens").

Yes.


top of page bottom of page up

Previous Next

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 2600 2650 2700 2750 2800 2850 2900 2950

2300 - 2325 -

top of page