Complete rank 2 searches (for real this time)

I've improved my theory for rank 2 searches so I'm sure they're complete. The improvement is in approximating the rank 2 temperament's generator with a ratio, n/d, where for now n and d don't share a common factor. With a generator of g, my rule before was

|g - n/d| < 1/d**2

where "**2" is "squared" today. I guessed the inequality from looking at the scale tree, but it is indeed for all convergents of the continued fraction of g. A more general formula is:

|g - n[k]/d[k]| < 1/d[k]/d[k+1]

where [i] refers to the ith element in the expansion. It comes from equation 24 here:

http://plus.maths.org/issue11/features/cfractions/

What I want to do either get all convergents below a given number, or guarantee that there'll be another convergent to get instead. So the inequality becomes

|g - n/d| < 1/d/d_max

where d_max is the highest denominator we're looking for (that is, the maximum number of notes to the octave in the equal temperaments we're going to seed with). The equation bounding the Tenney-weighted prime RMS error becomes

E**2 < c**2/d**2/d_max**2 + Emax**2

where c is the highest complexity of a rank 2 temperament we're looking for (either the standard deviation of the Tenney-weighted mapping, or half the max-min which is guaranteed to be no smaller) and Emax is the highest error of a rank 2 temperament we want (as Tenney prime RMS near enough, or TOP as this is guaranteed to be no smaller). The complexity takes care of cases where the period divides the octave.

The point of this limit is that every rank 2 temperament within the complexity and error limits must have at one equal temperament of its class that obeys this inequality. At least, provided d_max is above the largest division of the period that might work, which you can check. The generator is always a fraction between 0 and 1/2. As a continued fraction, that means 0/m must be a convergent where m is the number of periods to an octave. The smallest value of m is 1. So we start at 1 and check that either we select an equal temperament or we defer it until later. As we approach d_max the convergents have a likelihood approaching 1 of being selected.

Another way of writing the inequality is

E**2 < b**2/d**2 + Emax**2

where b=c/d_max is a measure of badness. I happen to know that 0.1 gives a thorough search that doesn't produce an overabundance of 1 note equal temperaments. But you can choose any value you like and get d_max as c/b. Then you can to check that d_max is at least larger than the theoretical highest division of the octave within the compelxity limit.

So here's the rule for selecting equal temperaments suitable for putting on T-shirts and the like:

E**2 < b**2/d**2 + Emax**2

where 0 < d <= c/b

That guarantees at least one equal temperament that belongs to each rank 2 class we're interested in. I have code to search all mappings of a single equal temperament but it still needs perfecting.

I use an algorithm similar to how I select equal temperaments. First you try each mapping for a prime and see if it's within the limit. Only if it is, you look at the next prime for each valid mapping. Without the constraint this grows exponentially and for linear temperament mappings consumes huge amounts of memory. To get practical results I have to restrict both the error and complexity for each step. Even then it gets ridiculously slow for general searches in the 13-limit. Probably a badness limit would fix this. But, in its imperfect state, it works provided you're specific enough about the error and complexity you want -- rather like the complete search itself.

So, I think we're in agreement that there's no point in looking at temperaments beyond the 19-limit, and so the 19-limit is the most complicated case we need to worry about. I like the weighted error to be less then about 1 cent per octave (this corresponds to a little over 4 cents for the 19:1 I think, because the interval's a little over 4 octaves). I guessed that there might be something with a max-min weighted complexity under 10 (whatever that means). Using the rule above (and remembering c is 5 not 10) I get 156 equal temperaments to look at. The search takes a few seconds and returns one rank 2 temperament:

5/29

599.628 cents period
103.748 cents generator

mapping by period and generator:
[2, 3, 5, 7, 9, 10, 8]
[0, 1, -2, -8, -12, -15, 1]

mapping by steps:
(12, 19, 28, 34, 42, 45, 49)
(46, 73, 107, 129, 159, 170, 188)

complexity measure: 9.369
RMS weighted error: 0.890 cents/octave
max weighted error: 1.864 cents/octave

That's some kind of diaschismic. I don't know if it's been noticed before -- I don't pay much attention to the higher limits. Raise the complexity cutoff to 14 and there are 264 seed ETs and 32 or *cough* 33 rank 2 temperaments. Here's the one that happens to be at the top:

11/36

600.400 cents period
183.318 cents generator

mapping by period and generator:
[2, 5, 8, 5, 6, 8, 10]
[0, -6, -11, 2, 3, -2, -6]

mapping by steps:
(46, 73, 107, 129, 159, 170, 188)
(26, 41, 60, 73, 90, 96, 106)

complexity measure: 11.209
RMS weighted error: 0.529 cents/octave
max weighted error: 1.071 cents/octave

26+46=72 if that makes it easier to identify the family.

So, this all looks entirely practical unless there are devils lurking in the details of the one-ET search that I haven't worked out yet. I'll keep you informed of that. The problems are that I miss generators of 0 steps and mappings where the best one isn't the simplest one.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> I've improved my theory for rank 2 searches so I'm sure they're
> complete. The improvement is in approximating the rank 2 temperament's
> generator with a ratio, n/d, where for now n and d don't share a common
> factor. With a generator of g, my rule before was
>
> |g - n/d| < 1/d**2

I'm not following this. Since the generator does not itself determine
a temperament mapping, but rather a tuning, how does this help sort
out temperaments?

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> >>I've improved my theory for rank 2 searches so I'm sure they're >>complete. The improvement is in approximating the rank 2 temperament's >>generator with a ratio, n/d, where for now n and d don't share a common >>factor. With a generator of g, my rule before was
>>
>>|g - n/d| < 1/d**2
> > > I'm not following this. Since the generator does not itself determine
> a temperament mapping, but rather a tuning, how does this help sort
> out temperaments?

The tuning is part of the temperament. We want temperaments with a low error. The generator does determine the position on the scale tree, and so the sequence of convergent ETs that will give successively good approximations to the optimal tuning. With this inequality we can ignore the generator (which we don't know) and instead look at the denominator of convergent equal temperaments (which we can search for).

I've got the code finished now, and I found 3 more temperaments in that 19-limit search I did.

http://x31eq.com/temper/regular.zip

There's a lot of hairy code in there do deal with the pre-filtering by complexity and error. So it isn't pretty but it does work. If anybody's interested it should be possible to add badness pre-filtering without much trouble.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > I'm not following this. Since the generator does not itself determine
> > a temperament mapping, but rather a tuning, how does this help sort
> > out temperaments?
>
> The tuning is part of the temperament.

If you say so. But it doesn't tell you what the temperament is; is
19/72 a generator for keemun or for cataclysmic?

We want temperaments with a low
> error. The generator does determine the position on the scale tree,
and
> so the sequence of convergent ETs that will give successively good
> approximations to the optimal tuning.

And that doesn't tell you what the temperament is.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>>I'm not following this. Since the generator does not itself determine
>>>a temperament mapping, but rather a tuning, how does this help sort
>>>out temperaments?
>>
>>The tuning is part of the temperament.
> > > If you say so. But it doesn't tell you what the temperament is; is
> 19/72 a generator for keemun or for cataclysmic?

Please don't break up my paragraphs.

> We want temperaments with a low > >>error. The generator does determine the position on the scale tree,
> > and > >>so the sequence of convergent ETs that will give successively good >>approximations to the optimal tuning. > > And that doesn't tell you what the temperament is.

No, but if you know what temperament it is it tells you the error. Like I said. This is a formula to predict the errors of the ETs.

As it happens, the search is by generator ratios. It's quite possible to have more than one temperament for each ratio. You get everything below the error and complexity cutoffs. It was a tricky problem but it's solved now. I thought you'd been doing this kind of search anyway.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> 5/29

What does 5/29 have to do with it? That's 10/58, of course, but how
does that connect with a generator of 5/58?

> 599.628 cents period
> 103.748 cents generator
>
> mapping by period and generator:
> [2, 3, 5, 7, 9, 10, 8]
> [0, 1, -2, -8, -12, -15, 1]

> That's some kind of diaschismic. I don't know if it's been noticed
> before -- I don't pay much attention to the higher limits.

Indeed it is. It's 46&58 in terms of 17-limit rounded vals, and it has
been noted. What did your program do to find it--step through a Farey
sequence and then do something with that?

Here's the TM basis: {126/125, 136/135, 176/175, 196/195, 256/255}.
Why all these have a denominator divisible by 5 I don't know.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>5/29
> > > What does 5/29 have to do with it? That's 10/58, of course, but how
> does that connect with a generator of 5/58?

It's the generator/period ratio.

>> 599.628 cents period
>> 103.748 cents generator
>>
>>mapping by period and generator:
>>[2, 3, 5, 7, 9, 10, 8]
>>[0, 1, -2, -8, -12, -15, 1]
> > >>That's some kind of diaschismic. I don't know if it's been noticed >>before -- I don't pay much attention to the higher limits. > > Indeed it is. It's 46&58 in terms of 17-limit rounded vals, and it has
> been noted. What did your program do to find it--step through a Farey
> sequence and then do something with that?

I think the one I posted came from a pair of ETs, so that you'd get both of them to identify. Presumably 46&12 if the total came to 58. I don't do anything with Farey sequences. The new method is roughly as follows:

1) find some nice looking equal temperaments

2) for each equal temperament, look at all distinct generators

3) for each generator, look at all possible mappings of each prime

4) if it's within the error and complexity limits, keep it

So step (1) now uses the search that's guaranteed to give at least one representative of each temperament class that can beat the error and complexity limits. If the generator and octave size have a common factor, divide the octave accordingly. Hence that temperament really would come from 10/58. The error and complexity tests have to be done early because going through all plausible mappings and then filtering takes far too long. That means steps 3 and 4 aren't really distinct.

Graham

>The new method is roughly as follows:
>
>1) find some nice looking equal temperaments
>
>2) for each equal temperament, look at all distinct generators
>
>3) for each generator, look at all possible mappings of each prime
>
>4) if it's within the error and complexity limits, keep it
>
>So step (1) now uses the search that's guaranteed to give at least one
>representative of each temperament class that can beat the error and
>complexity limits. If the generator and octave size have a common
>factor, divide the octave accordingly. Hence that temperament really
>would come from 10/58. The error and complexity tests have to be done
>early because going through all plausible mappings and then filtering
>takes far too long. That means steps 3 and 4 aren't really distinct.

What do you do to avoid redundancies in the final output (ie, two
different meantone mappings)?

-Carl

Carl Lumma wrote:

> What do you do to avoid redundancies in the final output (ie, two
> different meantone mappings)?

Keep a dictionary indexed by a standard form of the generator mapping (including the period division).

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> Gene Ward Smith wrote:

> > What does 5/29 have to do with it? That's 10/58, of course, but how
> > does that connect with a generator of 5/58?
>
> It's the generator/period ratio.

I figured that out, but what I was asking is what you did with it.

> I think the one I posted came from a pair of ETs, so that you'd get
both
> of them to identify. Presumably 46&12 if the total came to 58.

In terms of rounded vals, it's 46&58, which is the obvious way of
getting it.

> 1) find some nice looking equal temperaments

Nice looking meaning?

> 2) for each equal temperament, look at all distinct generators

OK, I've done that.

> 3) for each generator, look at all possible mappings of each prime

This is of course impossible, since there are an infinite number. You
need error bounds.

> 4) if it's within the error and complexity limits, keep it

So why not just say look at all et vals within certain error and
complexity bounds, and forget the part about "nice" and looking at all
possible mappings?

> So step (1) now uses the search that's guaranteed to give at least one
> representative of each temperament class that can beat the error and
> complexity limits.

But you need two vals to get the temperament. Picking an edo, and then
using that to get a generator and period mapping, will work, but I
don't know that this is what you are describing. What are you describing?

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>Gene Ward Smith wrote:
> > >>>What does 5/29 have to do with it? That's 10/58, of course, but how
>>>does that connect with a generator of 5/58?
>>
>>It's the generator/period ratio.
> > > I figured that out, but what I was asking is what you did with it.

I don't do anything with it. The generator and octave division are calculated in the initialization, so the generator and period can be displayed in the output routine.

>>I think the one I posted came from a pair of ETs, so that you'd get
> > both > >>of them to identify. Presumably 46&12 if the total came to 58.
> > > In terms of rounded vals, it's 46&58, which is the obvious way of
> getting it.

Yes, but the algorithm favors the simplest pair, and 12 will beat 58. If the period's 29 the original ETs were probably 12 and 46.

>>1) find some nice looking equal temperaments
> > Nice looking meaning?

I say that lower down

>>2) for each equal temperament, look at all distinct generators
> > OK, I've done that.

Good! How did you handle zero?

>>3) for each generator, look at all possible mappings of each prime
> > This is of course impossible, since there are an infinite number. You
> need error bounds.

I say that lower down as well.

>>4) if it's within the error and complexity limits, keep it
> > So why not just say look at all et vals within certain error and
> complexity bounds, and forget the part about "nice" and looking at all
> possible mappings?

The part about "nice" is a kind of error and compelexity bound, if you take the number of notes to the octave as a measure for the complexity of an equal temperament. That inequality again

E**2 < b**2/d**2 + Emax**2

where 0 < d <= c/b

In addition to that, I still reject mappings with a common factor other than 1. That's less important in higher limits.

As to all possible mappings, it's impossible. There are infinitely many ways of mapping each prime for any given period and generator. So there have to be bounds. And I've found the bounds have to be strict and applied early because the number of possibilities grows exponentially with the number of primes.

>>So step (1) now uses the search that's guaranteed to give at least one >>representative of each temperament class that can beat the error and >>complexity limits. > > But you need two vals to get the temperament. Picking an edo, and then
> using that to get a generator and period mapping, will work, but I
> don't know that this is what you are describing. What are you describing?

Yes. One EDO with mapping in, zero or more rank 2 temperaments out.

Graham

P.S. I originally promised 19-limit, but only delivered 17-limit. That was a pure oversight, so here are the temperaments with error below 1 cent/octave and complexity below 12 of whatever that's measured in. I didn't duplicate this search with two ETs so it's a bit harder to work out the melody. Two of them are variants of 46&72. Notice how the generator gets chosen wrongly in the other one.

4/13

600.391 cents period
183.454 cents generator

mapping by period and generator:
(2, 5, 8, 5, 6, 8, 10, 14)
(0, -6, -11, 2, 3, -2, -6, -18)

mapping by steps:
(26, 41, 60, 73, 90, 96, 106, 110)

complexity measure: 11.209
RMS weighted error: 0.762 cents/octave
max weighted error: 1.362 cents/octave

4/13

600.553 cents period
183.372 cents generator

mapping by period and generator:
(2, 5, 8, 5, 6, 8, 10, 10)
(0, -6, -11, 2, 3, -2, -6, -5)

mapping by steps:
(26, 41, 60, 73, 90, 96, 106, 110)

complexity measure: 11.209
RMS weighted error: 0.930 cents/octave
max weighted error: 2.082 cents/octave

0/1

29.289 cents period
-11.125 cents generator

mapping by period and generator:
(41, 65, 95, 115, 142, 152, 168, 174)
(0, 0, 0, 0, 1, 1, 1, 0)

mapping by steps:
(41, 65, 95, 115, 142, 152, 168, 174)

complexity measure: 11.852
RMS weighted error: 0.943 cents/octave
max weighted error: 1.666 cents/octave

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>Two of them are variants of 46&72. Notice how the
> generator gets chosen wrongly in the other one.

What does that mean?

>
> 4/13
>
> 600.391 cents period
> 183.454 cents generator
>
> mapping by period and generator:
> (2, 5, 8, 5, 6, 8, 10, 14)
> (0, -6, -11, 2, 3, -2, -6, -18)

This could be called 26&72 (rounded vals), and like the rest of this
list it isn't something I am ever likely to use. The tuning is such
that you may as well use 72-et for it, and like the rest of this list
it suffers from the problem that the tuning isn't very good. Of
course, you can regard it as a way of organizing 72-et for use in the
19 limit, maybe via a 26 note MOS.

Temperaments where the errors are quite a bit lower would be more
interesting to me, certainly; at least up to the likes of 94&311,
which has 1/3 of a fourth as generator, or 43/311.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>Two of them are variants of 46&72. Notice how the >>generator gets chosen wrongly in the other one.
> > What does that mean?

If you can't see anything wrong with the generator, don't worry about it.

> Temperaments where the errors are quite a bit lower would be more
> interesting to me, certainly; at least up to the likes of 94&311,
> which has 1/3 of a fourth as generator, or 43/311.

That's a harder search. Maybe the initial badness needs to be tweaked. This is the only thing comparable:

4/69

1200.038 cents period
69.457 cents generator

mapping by period and generator:
(1, 5, 4, 9, 10, 7, 13, 5)
(0, -59, -29, -107, -113, -57, -154, -13)

mapping by steps:
(69, 109, 160, 193, 238, 255, 281, 293)

complexity measure: 38.114
RMS weighted error: 0.148 cents/octave
max weighted error: 0.192 cents/octave

Here's one with a smaller error:

5/111

1199.878 cents period
54.020 cents generator

mapping by period and generator:
(1, 1, -2, -3, 4, 1, -1, -3)
(0, 13, 96, 129, -12, 60, 113, 161)

mapping by steps:
(111, 176, 258, 312, 384, 411, 454, 472)

complexity measure: 49.420
RMS weighted error: 0.098 cents/octave
max weighted error: 0.150 cents/octave

one with a bigger error:
29/270

1199.897 cents period
128.879 cents generator

mapping by period and generator:
[1, -4, 2, -6, -9, -5, -3, 5]
[0, 52, 3, 82, 116, 81, 66, -7]

mapping by steps:
(149, 236, 346, 418, 515, 551, 609, 633)
(121, 192, 281, 340, 419, 448, 495, 514)

complexity measure: 35.179
RMS weighted error: 0.159 cents/octave
max weighted error: 0.340 cents/octave

and one based on a friendly equal temperament:

1/2

199.971 cents period
102.169 cents generator

mapping by period and generator:
(6, 9, 17, 24, 32, 36, 23, 26)
(0, 1, -6, -14, -22, -27, 3, -1)

mapping by steps:
(12, 19, 28, 34, 42, 45, 49, 51)

complexity measure: 48.182
RMS weighted error: 0.110 cents/octave
max weighted error: 0.218 cents/octave

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> That's a harder search. Maybe the initial badness needs to be tweaked.
> This is the only thing comparable:
>
> 4/69
>
> 1200.038 cents period
> 69.457 cents generator
>
> mapping by period and generator:
> (1, 5, 4, 9, 10, 7, 13, 5)
> (0, -59, -29, -107, -113, -57, -154, -13)

Right; that one is 121&311 rather than 94&311. Either one could be
used as an entre to the mysterious world of 311, but 94&311 seems like
a better place to start.

> Here's one with a smaller error:
>
> 5/111
>
> 1199.878 cents period
> 54.020 cents generator
>
> mapping by period and generator:
> (1, 1, -2, -3, 4, 1, -1, -3)
> (0, 13, 96, 129, -12, 60, 113, 161)

And 311 strikes yet again; this one is 111&311.

> one with a bigger error:
> 29/270
>
> 1199.897 cents period
> 128.879 cents generator
>
> mapping by period and generator:
> [1, -4, 2, -6, -9, -5, -3, 5]
> [0, 52, 3, 82, 116, 81, 66, -7]

121&149; a departure from 311 at last.

> and one based on a friendly equal temperament:

Seems to me this is true of all of them.

> 1/2
>
> 199.971 cents period
> 102.169 cents generator
>
> mapping by period and generator:
> (6, 9, 17, 24, 32, 36, 23, 26)
> (0, 1, -6, -14, -22, -27, 3, -1)

This time we have 270&282. Is either of those friendly, or at you
looking at the difference between the rounded vals, which is
<12 19 28 34 42 45 49 51|?

Gene Ward Smith wrote:

> Right; that one is 121&311 rather than 94&311. Either one could be
> used as an entre to the mysterious world of 311, but 94&311 seems like
> a better place to start.

311 keeps coming up because it's a good equal temperament, You generally find good R2 temperaments relate to good equal temperaments. That's why a search by equal temperaments is so efficient. 311 and 270 both come out with a lower badness than anything smaller and are both implicated in one of other of the temperaments I posted.

94 doesn't make this shortlist, but 217 does:

72, 111, 217, 243, 270, 282, 311, 354, 364, 373, 400

so unless my errors are unusual in some way, 217&311 might be a better naming than 94&311. Why is it a better temperament class than 121&311?

> This time we have 270&282. Is either of those friendly, or at you
> looking at the difference between the rounded vals, which is
> <12 19 28 34 42 45 49 51|?

Yes, it has 12-equal as a special case. Most people will find 12-equal more friendly than 270 and 282, although 270 and 282 are very efficient. Why do you keep going on about rounded vals?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> 94 doesn't make this shortlist, but 217 does:
>
> 72, 111, 217, 243, 270, 282, 311, 354, 364, 373, 400
>
> so unless my errors are unusual in some way, 217&311 might be a better
> naming than 94&311.

My 19-limit short list is different; if I cherry pick those with my
logflat badness score below 1, I get these:

31, 50, 80, 94, 111, 121, 311, 320, 364, 400

217 gets a logflat badness of 1.08 and 94 of 0.89 by this measure,
which is unweighted max error on the diamond adjusted to make it logflat.

> Why is it a better temperament class than 121&311?

Lower complexity. It has a Graham complexity of 127, and 121&311 comes
in at 173. 121&311 gives 154. There are other systems which beat out
173. Of course say that assumes the tuning is stuck with being 311.

> Why do you keep going on about rounded vals?

Becuase it is an extremely efficient way to characterize the vast
majority of linear temperaments, and both you and Paul are being
obstructionist about defining n&m to simply *mean* rounded vals are
being used, so I have to blather on. It would really help if you two
would drop your objections, or suggest something better.

Gene Ward Smith wrote:

> My 19-limit short list is different; if I cherry pick those with my
> logflat badness score below 1, I get these:
> > 31, 50, 80, 94, 111, 121, 311, 320, 364, 400
> > 217 gets a logflat badness of 1.08 and 94 of 0.89 by this measure,
> which is unweighted max error on the diamond adjusted to make it logflat.

That's an unweighted, odd-limit, max-error list in a weighted, prime-limit, rms-error context. Surprisingly enough it's different to my weighted, prime-limit, rms-error limited list.

>>Why is it a better temperament class than 121&311?
> > Lower complexity. It has a Graham complexity of 127, and 121&311 comes
> in at 173. 121&311 gives 154. There are other systems which beat out
> 173. Of course say that assumes the tuning is stuck with being 311.

In an unweighted odd-limit, yes, but not in a weighted prime-limit.

>> Why do you keep going on about rounded vals?
> > Becuase it is an extremely efficient way to characterize the vast
> majority of linear temperaments, and both you and Paul are being
> obstructionist about defining n&m to simply *mean* rounded vals are
> being used, so I have to blather on. It would really help if you two
> would drop your objections, or suggest something better.

It's more efficient to use the best mapping. But I'm obstructing a standard precisely because you'll use things like n&m without defining them, and expect people to know what you mean.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> It's more efficient to use the best mapping. But I'm obstructing a
> standard precisely because you'll use things like n&m without defining
> them, and expect people to know what you mean.

I've explained precisely what I mean by "m&n" and "standard val" a
number of times over the years and have always stuck to the exact
meaning specified. I could wish more people followed that example.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@> wrote:
> > Gene Ward Smith wrote:
>
> > > What does 5/29 have to do with it? That's 10/58, of course, but
how
> > > does that connect with a generator of 5/58?
> >
> > It's the generator/period ratio.
>
> I figured that out, but what I was asking is what you did with it.
>
> > I think the one I posted came from a pair of ETs, so that you'd
get
> both
> > of them to identify. Presumably 46&12 if the total came to 58.
>
> In terms of rounded vals, it's 46&58, which is the obvious way of
> getting it.
>
> > 1) find some nice looking equal temperaments
>
> Nice looking meaning?
>
> > 2) for each equal temperament, look at all distinct generators
>
> OK, I've done that.
>
> > 3) for each generator, look at all possible mappings of each prime
>
> This is of course impossible, since there are an infinite number.
You
> need error bounds.

I think you mean complexity bounds?

In any case, Graham implied in his post that steps (2) and (3) are in
fact done together with a filtering for error and complexity, so
there would not in factbe an infinite number.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"

> both you and Paul are being
> obstructionist about defining n&m to simply *mean* rounded vals are
> being used,

Are you saying something about me here? Because if you are, I can't
figure out what it is, let alone whether it's true.

> so I have to blather on. It would really help if you two
> would drop your objections, or suggest something better.

How about "naive" instead of "standard"?

>> so I have to blather on. It would really help if you two
>> would drop your objections, or suggest something better.
>
>How about "naive" instead of "standard"?

Does anybody like my suggestion of "immediate"?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> so I have to blather on. It would really help if you two
> >> would drop your objections, or suggest something better.
> >
> >How about "naive" instead of "standard"?
>
> Does anybody like my suggestion of "immediate"?

I think "patent" would be better than either:

Main Entry: pat·ent
Pronunciation: 'pat-&nt 3 also 'pAt-
Function: adjective
Etymology: Anglo-French, from Latin patent- patens, from present
participle of patEre to be open

3 : readily seen, discovered, or understood <a patent defect> <if no
bad faith or abuse is patent> —compare LATENT —pat·ent·ly adverb

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> so I have to blather on. It would really help if you two
> >> would drop your objections, or suggest something better.
> >
> >How about "naive" instead of "standard"?
>
> Does anybody like my suggestion of "immediate"?
>
> -Carl

I like it but it's not my favorite.

>> >> so I have to blather on. It would really help if you two
>> >> would drop your objections, or suggest something better.
>> >
>> >How about "naive" instead of "standard"?
>>
>> Does anybody like my suggestion of "immediate"?
>
>I think "patent" would be better than either:

Foo.

-Carl