List cut-off point

I looked at the step-cents of the 7-limit temperaments on my list,
and they range from 2.09 for ennealimmal to 99.5 for the system

[0 12]
[0 19]
[1 28]
[1 34]

One possibility on the low end is therefore to set 100 sc as the
cut-off; which would leave us needing something else on the high end.

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

> I looked at the step-cents of the 7-limit temperaments on my list,
> and they range from 2.09 for ennealimmal to 99.5 for the system
>
> [0 12]
> [0 19]
> [1 28]
> [1 34]
>
> One possibility on the low end is therefore to set 100 sc as the
> cut-off; which would leave us needing something else on the high
end.

s means scale size and c means error?

Did you have a problem with the (s^2)*c or whatever rule that graham
used? Would you consider (2^s)*c? Or s*(2^s)*c? The latter should
_not_ need a high-end cutoff, as the number of possibilities will be
finite (if my brain is working right) . . .

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

> s means scale size and c means error?

s means steps and c means cents, so sc are units of step-cents.

> Did you have a problem with the (s^2)*c or whatever rule that
graham
> used?

I had no problem with it beyond failing to interpret the units;
however it weighs the smaller systems more heavily, which might be a
good idea from the point of view of practicality.

Would you consider (2^s)*c? Or s*(2^s)*c? The latter should
> _not_ need a high-end cutoff, as the number of possibilities will
be
> finite (if my brain is working right) . . .

I'd consider anything, but 2^s sounds ferocious!

--- In tuning-math@y..., genewardsmith@j... wrote:
>
> I'd consider anything, but 2^s sounds ferocious!

Fokker evaluated equal temperaments using 2^n and found 31-tET best!

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> >
> > I'd consider anything, but 2^s sounds ferocious!
>
> Fokker evaluated equal temperaments using 2^n and found 31-tET best!

If he'd picked a different exponent he might have found something
else to be the best. It's probably pretty easy to make 12 the best.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> > >
> > > I'd consider anything, but 2^s sounds ferocious!
> >
> > Fokker evaluated equal temperaments using 2^n and found 31-tET
best!
>
> If he'd picked a different exponent he might have found something
> else to be the best. It's probably pretty easy to make 12 the best.

My point is that 2^n doesn't go unreasonably far in penalizing large
systems.

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

> Did you have a problem with the (s^2)*c or whatever rule that
graham
> used?

When I use a s^2 c rule, I still get ennealimmal coming out on top.

> Would you consider (2^s)*c?

I tried that, and the best system turned out to be <49/48,25/24>,
which is below your lower cut, and it gives me the impression that
this is too extreme in the other direction. Best and worst were:

<25/24,49/48> (Pretty much 6+4=10)

2^c s = 118.69

<2048/2025,4375/4374> (46,80,92 ets)

2^c s = 1.01 x 10^7.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Did you have a problem with the (s^2)*c or whatever rule that
> graham
> > used?
>
> When I use a s^2 c rule, I still get ennealimmal coming out on top.

What if you were to consider more complex unison vectors? You'd just
keep finding better and better ones, wouldn't you?

> > Would you consider (2^s)*c?
>
> I tried that, and the best system turned out to be <49/48,25/24>,
> which is below your lower cut,

How did that system get in there in the first place? Aren't there
even better systems according to this criterion, say ones using 21/20
as a unison vector?

> and it gives me the impression that
> this is too extreme in the other direction.

What else makes the top 10 with this criterion?

> Best and worst were:
>
> <25/24,49/48> (Pretty much 6+4=10)
>
> 2^c s = 118.69
>
> <2048/2025,4375/4374> (46,80,92 ets)
>
> 2^c s = 1.01 x 10^7.

You mean 2^s c, not 2^c s?

What about something like

2^(s/3)*c?

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

> > When I use a s^2 c rule, I still get ennealimmal coming out on
top.

> What if you were to consider more complex unison vectors? You'd
just
> keep finding better and better ones, wouldn't you?

It certainly looks that way, but I haven't even thought about a proof.

> > > Would you consider (2^s)*c?
> >
> > I tried that, and the best system turned out to be <49/48,25/24>,
> > which is below your lower cut,
>
> How did that system get in there in the first place? Aren't there
> even better systems according to this criterion, say ones using
21/20
> as a unison vector?

No doubt, but I started with cut-off in my list of generating commas
of 49/48.

> You mean 2^s c, not 2^c s?

Sorry.

Partial results for s^2c on my list of 66 temperaments are:

1. Ennealimmal, 33.46
2. <2401/2400, 3136/3125> system, 89.14
3. Miracle, 94.70

...

65.

[0 12]
[0 19]
[-1 28]
[-1 34]

975.31

66. <2048/2025,4375/4374>, 1187.06

Can you come up with a goodness measure for linear temperaments just
as you did for ETs?

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

> Can you come up with a goodness measure for linear temperaments
just
> as you did for ETs?

Just what I was thinking of--my stuff on goodness of ets connecting
to goodness of generator steps should allow for an estimate, and
hence a more rational measure.

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

> Can you come up with a goodness measure for linear temperaments
just
> as you did for ETs?

Here's an estimate: if we have d odd primes, and two good ets of
about size n, then the error in relative cents is on the order of
n^(-1/d), which from the connection I showed between this and
generator steps means that number of steps is on the order of
n^(1-1/d). Since the error in relative cents is O(n^(-1/d)) for the
ets, the error in absolute cents is O(n^(-1-1/d)). The shift from ets
to linear temperaments leads to some improvement, so the error goes
down, but it doesn't seem to go down a great deal. We end up with an
estimate of (n^(1-1/d))^b * n^(-1-1/d) for step^b cents; this gives
exponent zero for b = (d+1)/(d-1). This is infinity for d=1, where we
have chains of perfect fifths, 3 for d=2 (5-limt), 2 for d=3 (7-
limit), 5/3 for the 11-limit and so forth.

I conclude that we can expect an infinite number of linear
temperaments under some fixed limit in step^2 cents in the 7-limit,
which seems to be the case. I suspect we will get no such thing for
step^3 cents, since the improvement to linear temperaments seems
unlikely to give that much. Here is what I got from my list of 66
temperaments for step^3 cents. Notice that ennealimmal temperament
still does pretty well!

(1) [2,3,1,-6,4,0]
<27/25,49/48> ets: 4,5,9,14 measure: 210.36

(2) [4,2,2,-1,8,6]
<25/24,49/48> ets: 4,6,10 measure: 294.93

(3) [4,4,4,-2,5,-3]
<36/35,50/49> ets: 4,12,16,28 measure: 433.02

(4) [18,27,18,-34,22,1]
<2401/2400,4375/4374> ets: 27,72,99,171,270,441,612 measure: 535.89

The first to pass Paul's lower cut is *still* ennealimmal.

(5) [6,5,3,-7,12,-6]
<49/48,126/125> ets: 4,15,19,34 measure: 642.89

...

(65) [12,12,-6,-19,19]
<50/49,3645/3584> ets: 12, 48 measure: 9556.05
Fails Paul's strong TM condition.

ets: 12,48

Map:

[ 0 12]
[ 0 19]
[-1 28]
[-1 34]

(66) [2,-4,30,81,-42,-11]
<2048/2025,4375/4374> ets: 46, 80 measure: 26079.25

What do you think? This one might work.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> What about something like
>
> 2^(s/3)*c?

Oops -- I was actually thinking

2^(s^(1/3))*c . . .

How does that look, Gene?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Can you come up with a goodness measure for linear temperaments
> just
> > as you did for ETs?
>
> Here's an estimate: if we have d odd primes, and two good ets of
> about size n, then the error in relative cents is on the order of
> n^(-1/d), which from the connection I showed between this and
> generator steps means that number of steps is on the order of
> n^(1-1/d). Since the error in relative cents is O(n^(-1/d)) for
the
> ets, the error in absolute cents is O(n^(-1-1/d)). The shift from
ets
> to linear temperaments leads to some improvement, so the error
goes
> down, but it doesn't seem to go down a great deal.

Except in cases like ennealimmal, huh?

> We end up with an
> estimate of (n^(1-1/d))^b * n^(-1-1/d) for step^b cents; this
gives
> exponent zero for b = (d+1)/(d-1). This is infinity for d=1, where
we
> have chains of perfect fifths, 3 for d=2 (5-limt), 2 for d=3 (7-
> limit), 5/3 for the 11-limit and so forth.
>
> I conclude that we can expect an infinite number of linear
> temperaments under some fixed limit in step^2 cents in the
7-limit,
> which seems to be the case. I suspect we will get no such thing
for
> step^3 cents, since the improvement to linear temperaments seems
> unlikely to give that much.

Does step^3 cents come out of one of the considerations above, or is
it merely an arbitrary function that increases more rapidly than
step^2 cents?

> Here is what I got from my list of 66
> temperaments for step^3 cents. Notice that ennealimmal temperament
> still does pretty well!
>
> (1) [2,3,1,-6,4,0]
> <27/25,49/48> ets: 4,5,9,14 measure: 210.36
>
> (2) [4,2,2,-1,8,6]
> <25/24,49/48> ets: 4,6,10 measure: 294.93
>
> (3) [4,4,4,-2,5,-3]
> <36/35,50/49> ets: 4,12,16,28 measure: 433.02
>
> (4) [18,27,18,-34,22,1]
> <2401/2400,4375/4374> ets: 27,72,99,171,270,441,612 measure: 535.89

And you don't think an infinite number of even better ones would be
found if we moved out further in the lattice? I really would like to
cut the cord to the original list of unison vectors, if possible.

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

The shift from
> ets
> > to linear temperaments leads to some improvement, so the error
> goes
> > down, but it doesn't seem to go down a great deal.
>
> Except in cases like ennealimmal, huh?

Well, maybe it doesn't except when it does. :) If you are suggesting
the actual critical exponent is going to be higher than 2, because of
the improvement of linear temperament over ets, then I would not be
surprised, but getting it all the way up to 3 is another matter, I
think. The critical exponent is *at least* 2, however.
> Does step^3 cents come out of one of the considerations above, or
is
> it merely an arbitrary function that increases more rapidly than
> step^2 cents?

It's an arbitary function of polynomial growth--using exponential
growth I suspect to be overkill.

> And you don't think an infinite number of even better ones would be
> found if we moved out further in the lattice?

All I've really argued for is that this will, indeed, happen if we
use steps^2; I *think* steps^3 will kill it off eventually.

I really would like to
> cut the cord to the original list of unison vectors, if possible.

On that note, I made up a list of 990 pairs of ets which I can use to
seed things, but I'm waiting to see what to use for a cut-off.

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

> Except in cases like ennealimmal, huh?

I've been looking at some extreme examples, concocted from high-
powered ets in the 1000-10000 range, and it seems likely that 2
actually is the critical exponent, and pretty well certain that even
if it isn't using steps^3 will lead to a finite list. Ennealimmal may
just be exceptionally good.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Except in cases like ennealimmal, huh?
>
> I've been looking at some extreme examples, concocted from high-
> powered ets in the 1000-10000 range, and it seems likely that 2
> actually is the critical exponent, and pretty well certain that
even
> if it isn't using steps^3 will lead to a finite list. Ennealimmal
may
> just be exceptionally good.

So it's the best 7-limit linear temperament ever discovered according
to steps^2 cents?

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

> So it's the best 7-limit linear temperament ever discovered
according
> to steps^2 cents?

I've got a new list of 505 temperaments, we'll see if anything beats
it. However, it is what you might call a "capstone temperament" in
the 7-limit. That is, it uses the two smallest superparticular commas
to define itself. Similarly, meantone would be a linear temperament
capstone in the 5-limit, and <3025/3024,4375/4374,9801/9800> a
capstone for the 11-limit. Since 7 is the higher of a prime pair
(5 and 7) its smallest superparticular commas are relatively small,
so I suppose capstone temperaments in the 13-limit would be more to
the point, and maybe planar or 3D more than linear.

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

> Since 7 is the higher of a prime pair
> (5 and 7) its smallest superparticular commas are relatively small,
> so I suppose capstone temperaments in the 13-limit would be more to
> the point, and maybe planar or 3D more than linear.

Can you explain this sentence? I don't understand it at all.

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

> Can you explain this sentence? I don't understand it at all.

It's simply conjecture on my part that the higher of a pair of twin
primes should have a comparitively larger largest superparticular
ratio associated to it than the lower, but I think I'll investigate
it as a question in number theory.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Can you explain this sentence? I don't understand it at all.
>
> It's simply conjecture on my part that the higher of a pair of twin
> primes should have a comparitively larger largest superparticular
> ratio associated to it than the lower,

Assuming this is true, can you explain the sentence?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > > Can you explain this sentence? I don't understand it at all.
> >
> > It's simply conjecture on my part that the higher of a pair of
twin
> > primes should have a comparitively larger largest superparticular
> > ratio associated to it than the lower,
>
> Assuming this is true, can you explain the sentence?

The superparticular ratio commas are rather special ones, coming in
more profusion than with other differences "a" in (b+a)/b, and so if
there are expecially large ones, I would expect the associated
temperaments to be especially good. I'd expect something more cooking
in the 13-limit than the 11-limit, therefore.

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

> The superparticular ratio commas are rather special ones, coming in
> more profusion than with other differences "a" in (b+a)/b, and so
if
> there are expecially large ones, I would expect the associated
> temperaments to be especially good. I'd expect something more
cooking
> in the 13-limit than the 11-limit, therefore.

The jump from the longest 7-limit superparticular to the longest 11-
limit superparticular, you're saying, is not nearly as great as the
jump from the longest 11-limit superparticular to the largest 13-
limit superparticular? I bet John Chalmers on the tuning list could
immediately verify whether that's true. He might be interested to
learn of a mathematical explanation of this fact.

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

> The jump from the longest 7-limit superparticular to the longest 11-
> limit superparticular, you're saying, is not nearly as great as the
> jump from the longest 11-limit superparticular to the largest 13-
> limit superparticular? I bet John Chalmers on the tuning list could
> immediately verify whether that's true. He might be interested to
> learn of a mathematical explanation of this fact.

Yes, take the ratio log(T(superparticular))/log(T(prime)) and I'm
guessing 7,13,19 stick out. 23 even more so--it is an isolate, with a
distance of 4 to 19 and 6 to 29.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > The jump from the longest 7-limit superparticular to the longest
11-
> > limit superparticular, you're saying, is not nearly as great as
the
> > jump from the longest 11-limit superparticular to the largest 13-
> > limit superparticular? I bet John Chalmers on the tuning list
could
> > immediately verify whether that's true. He might be interested to
> > learn of a mathematical explanation of this fact.
>
> Yes, take the ratio log(T(superparticular))/log(T(prime)) and I'm
> guessing 7,13,19 stick out. 23 even more so--it is an isolate, with
a
> distance of 4 to 19 and 6 to 29.

John Chalmers calculated all the superparticulars with numerator and
denominator less than 10,000,000,000 (IIRC), for numerator and
denominator up to 23. Can he verify this?

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

> John Chalmers calculated all the superparticulars with numerator
and
> denominator less than 10,000,000,000 (IIRC), for numerator and
> denominator up to 23. Can he verify this?

That would be a very useful thing to upload to the files area or
stick on a web page.