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.