Here are comma lists for the 7 and 11 limits. Each comma is less than fifty cents, and each has the property that if the comma is p/q>1 in reduced form, then ln(p-q)/ln(q) < .5 in the 7-limit, and < .3 in the 11-limit. I've found this weaking of the superparticularity condition useful in the past, and it occurred to me it would be one way of getting a finite list of temperaments a la Dave--we could simply require it to have a basis of commas passing such a condition. The lists below may be complete; at least, I haven't been able to add to them.

Seven limit list, ln(p-q)/ln(q)<1/2, cents < 50

[1029/1000, 250/243, 36/35, 525/512, 128/125, 49/48, 50/49,

3125/3072, 686/675, 64/63, 875/864, 81/80, 3125/3087, 2430/2401, 2048/2025, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024, 15625/15552, 225/224, 19683/19600, 16875/16807, 10976/10935, 3136/3125, 5120/5103, 6144/6125, 65625/65536, 32805/32768, 703125/702464, 420175/419904, 2401/2400, 4375/4374,

250047/250000, 78125000/78121827]

Eleven limit list ln(p-q)/ln(q) < .3, cents < 50

[36/35, 77/75, 128/125, 45/44, 49/48, 50/49, 55/54, 56/55, 64/63,

81/80, 245/242, 99/98, 100/99, 121/120, 245/243, 126/125, 1331/1323, 176/175, 896/891, 1029/1024, 225/224, 243/242, 3136/3125, 385/384, 441/440, 1375/1372, 6250/6237, 540/539, 4000/3993, 5632/5625, 43923/43904, 2401/2400, 3025/3024, 4375/4374, 9801/9800, 151263/151250, 3294225/3294172]

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

> Here are comma lists for the 7 and 11 limits. Each comma is less

>than fifty cents, and each has the property that if the comma is

>p/q>1 in reduced form, then ln(p-q)/ln(q)

that's my complexity heuristic -- did you arrive at it independently?

> < .5 in the 7-limit, and < .3 in the 11-limit. I've found this

>weaking of the superparticularity condition useful in the past, and

>it occurred to me it would be one way of getting a finite list of

>temperaments a la Dave--we could simply require it to have a basis

>of commas passing such a condition.

unfortunately, that doesn't properly take "straightness" into

account -- the commas may be at different "angles" to one another.

but this is very much along the lines of what i was suggesting with

the whole "heuristic" thread, in which i seemed to lose everyone . . .

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

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

> > Here are comma lists for the 7 and 11 limits. Each comma is less

> >than fifty cents, and each has the property that if the comma is

> >p/q>1 in reduced form, then ln(p-q)/ln(q)

>

> that's my complexity heuristic -- did you arrive at it independently?

I thought your complexity heuristic was (p-q)/(q ln q); in any case I came up with this and some other things similar (but not, I think identical) to your heuristic independently.

If you consider this to be a complexity heuristic, it gives a complexity of 0 to any superparticular ratio, which I don't think is what you wanted to do. It works more like a weakened superparticularity condition than complexity, I think.

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

> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

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

wrote:

>

> > > Here are comma lists for the 7 and 11 limits. Each comma is

less

> > >than fifty cents, and each has the property that if the comma is

> > >p/q>1 in reduced form, then ln(p-q)/ln(q)

> >

> > that's my complexity heuristic -- did you arrive at it

independently?

>

> I thought your complexity heuristic was (p-q)/(q ln q);

actually, we were both wrong -- my complexity heuristic is merely ln

(q). you're probably thinking of my error heuristic?

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

> I've found this weaking of the superparticularity condition useful

>in the past, and it occurred to me it would be one way of getting a

>finite list of temperaments a la Dave--we could simply require it to

>have a basis of commas passing such a condition.

a more pointed reply to this is that we've already settled on a

complexity measure and an error measure, so we shouldn't regress

backwards to less accurate evaluations of these quantities. what we

may want to consider is a different formula for combining the two

into a badness function. at this point, i'm not too concerned about

what that function is, as long as it's quite simple -- and, as i've

suggested before, we should provide a plot showing all the

temperaments in our list as points on a complexity vs. error graph,

and our badness criteria will be shown as a bounding curve on that

graph. as long as we do this, i'm not too concerned about the

particulars.

On Mon, 17 Jun 2002 18:31:08 -0000 "emotionaljourney22"

<paul@stretch-music.com> writes:

> a more pointed reply to this is that we've already settled on a

> complexity measure and an error measure, so we shouldn't regress

> backwards to less accurate evaluations of these quantities.

We've settled on this only for linear temperaments, actually.

what we

> may want to consider is a different formula for combining the two

> into a badness function.

I suppose the most obvious possibility is to adjust the exponent of

complexity.

What did you think of my comma lists just as comma lists? The condition I

put on them has a certain logic to it, and I think the results were

pretty good, in a finite-list kind of way.

--- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:

> What did you think of my comma lists just as comma lists?

they seemed fine, and would be excellent ways of deriving a list of

temperaments having only a single commatic unison vector, for example

linear temperaments in 5-limit, planar temperaments in 7-limit, etc.