Pajara, 12. 24 and 34-EDO and the natural logarithm

I'm working to extend my trapezoid approximation/calculus models
to higher temperaments (that are practical). Just a question,
Is there any value of taking "e-octaves?" (2.71828..) With my
simple double-trapezoid model, ln(2) is 24/35, so "e" would
be 2^(35/24). This works nicely in a quartertone system. Obviously
this is not there in 12-EDO, but it is right there in 24-EDO. And
a P4 and P5 remain exactly the same in 24-EDO.

So using e-octaves, you get 2/5 and 2/7 for a P5 and P4 respectively.

Moving to 34-EDO Pajara, 21/20 and 15/14 are 3 steps and 35/24 is
19 steps. So it differs from a P5 by one step (in 12-EDO, it is
7 steps=P5). I know I am using 35/24 in different contexts here,
but I feel there might be a number-theoretical relationship. Lastly,
although D4 does not come into play in 34-EDO, the 20 steps of 3/2
still map to 21/20 or 15/14 (3 steps) through a "tritone" of 17 steps
and goes to 35/24 by going down 1 step.

Anyway, I know I'm a little off the wall sometimes, but could someone
also verify that this is always the case for Pajara:

Let x= 3/2-half(octave)

5/4=17-2x
6/5=3x
7/6=17-3x
8/7=2x

8/7*6/5=48/35=5x so 24/35=5x-(Octave) and 35/24=(0ctave)-5x, which
approximates "e". You can work with this in different ways, to
approximate the log2 of 24/35 itself, as ln(ln(2)/ln(2) or going the
other way consider the antilog2 of 35/24 to be "e". (I know, I'm
boring, or maybe the perpetual schoolboy:))

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
> ...
> Moving to 34-EDO Pajara, 21/20 and 15/14 are 3 steps and 35/24 is
> 19 steps. So it differs from a P5 by one step (in 12-EDO, it is
> 7 steps=P5). I know I am using 35/24 in different contexts here,
> but I feel there might be a number-theoretical relationship. Lastly,
> although D4 does not come into play in 34-EDO, the 20 steps of 3/2
> still map to 21/20 or 15/14 (3 steps) through a "tritone" of 17
steps
> and goes to 35/24 by going down 1 step.
>
> Anyway, I know I'm a little off the wall sometimes, but could
someone
> also verify that this is always the case for Pajara:
>
> Let x= 3/2-half(octave)
>
> 5/4=17-2x
> 6/5=3x
> 7/6=17-3x
> 8/7=2x

The above is correct, if by "17" you mean ~17/12 or 1/2
octave. "Always the case for Pajara" would be 17 steps only in the
34 division.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
> > ...
> > Moving to 34-EDO Pajara, 21/20 and 15/14 are 3 steps and 35/24 is
> > 19 steps. So it differs from a P5 by one step (in 12-EDO, it is
> > 7 steps=P5). I know I am using 35/24 in different contexts here,
> > but I feel there might be a number-theoretical relationship.
Lastly,
> > although D4 does not come into play in 34-EDO, the 20 steps of 3/2
> > still map to 21/20 or 15/14 (3 steps) through a "tritone" of 17
> steps
> > and goes to 35/24 by going down 1 step.
> >
> > Anyway, I know I'm a little off the wall sometimes, but could
> someone
> > also verify that this is always the case for Pajara:
> >
> > Let x= 3/2-half(octave)
> >
> > 5/4=17-2x
> > 6/5=3x
> > 7/6=17-3x
> > 8/7=2x
>
> The above is correct, if by "17" you mean ~17/12 or 1/2
> octave. "Always the case for Pajara" would be 17 steps only in the
> 34 division.
>
> --George
>
Thanks. I meant to say H instead of 17 to represent half-octave.
Then I think these equations all always true for Pajara.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Thanks. I meant to say H instead of 17 to represent half-octave.
> Then I think these equations all always true for Pajara.

Then usual way to express that on this list would be via the mapping to
primes:

[<2 3 5 6|, <0 1 -2 -2|]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > Thanks. I meant to say H instead of 17 to represent half-octave.
> > Then I think these equations all always true for Pajara.
>
> Then usual way to express that on this list would be via the
mapping to
> primes:
>
> [<2 3 5 6|, <0 1 -2 -2|]

Yes, I guess that's the same thing. I was thrown off by 7 being high
in 34EDO (96 instead of 95) so I was approaching it from the commas
instead. (50/49 and 64/63) Do you just fix 3 as close as possible in
the generator? (And then apply all that RMS, minimax etc etc?)

I know, "Tuning 101"

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Do you just fix 3 as close as possible in
> the generator? (And then apply all that RMS, minimax etc etc?)

I'm not sure what you mean.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > Do you just fix 3 as close as possible in
> > the generator? (And then apply all that RMS, minimax etc etc?)
>
> I'm not sure what you mean.
>
Well, I've noticed the mapping for 3 is always mapped as close as
possible in any EDO. 7 is high in Pajara, in 34-EDO at least, but
it's not the generator. (I shouldn't have mentioned RMS, minimax,
etc.) I was just trying to see how values are determined in any EDO
for a temperament. Finally, have been working on Graham's online
script page, and both mapping by period and generator / mapping by
steps makes complete sense, based on comma(s), (I would love to see
the code for mapping by steps) Paul E sent me his code showing how
to find all the 7-limit temperaments up to 35-EDO. I suppose one
could just apply the vectors, and filter out, for example,
everything but Pajara? I suppose also mapping by steps could also
be key to finding all the right combinations that produce it.

I suppose it might just be luck that the generator representing
3/2 or 4/3 or 3/2 /(half-octave period)maps correctly is because rms
or minimax come so close anyway, of course, it would be hard to
imagine 3 being a step off in any EDO less than 100.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Well, I've noticed the mapping for 3 is always mapped as close as
> possible in any EDO.

No it isn't; for exmaple I was just talking on the main list about
using the sharp fifth of 81-edo for porcupine.

7 is high in Pajara, in 34-EDO at least, but
> it's not the generator. (I shouldn't have mentioned RMS, minimax,
> etc.) I was just trying to see how values are determined in any EDO
> for a temperament.

The whole point of calling it an "edo" is that they aren't determined.
Given a temperament, there is usually one stand-out mapping for it for
a given edo for the smaller sized divisions, but even that isn't always
true.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > Well, I've noticed the mapping for 3 is always mapped as close
as
> > possible in any EDO.
>
> No it isn't; for exmaple I was just talking on the main list about
> using the sharp fifth of 81-edo for porcupine.

I realized this later. Even 3 cannot be pinned down, so to speak...

> 7 is high in Pajara, in 34-EDO at least, but
> > it's not the generator. (I shouldn't have mentioned RMS, minimax,
> > etc.) I was just trying to see how values are determined in any
EDO
> > for a temperament.
>
> The whole point of calling it an "edo" is that they aren't
determined.
> Given a temperament, there is usually one stand-out mapping for it
for
> a given edo for the smaller sized divisions, but even that isn't
always
> true.

Right, working with the list I generated from Paul Erlich's program,
one can filter out the non-Pajara tunings, though. I will run that
tomorrow (it's at work.) This is the program that finds valid 7-
limit temperaments up to 35-edo. I want to also see if taking
Graham's "mapping by steps" for 50/49&64/63 will also generate the
same list, taking positive integer multiples of the two step sizes,
and seeing if they correspond. About half way done with "Prime Based
Error and Complexity Measures" BTW. Question: If you only took
temperament maps that come closest to the primes, would that kill
off most temperaments? Obviously, 34-EDO would be out for Pajara...

On 05/02/07, Paul G Hjelmstad <paul_hjelmstad@allianzlife.com> wrote:

> Right, working with the list I generated from Paul Erlich's program,
> one can filter out the non-Pajara tunings, though. I will run that
> tomorrow (it's at work.) This is the program that finds valid 7-
> limit temperaments up to 35-edo. I want to also see if taking
> Graham's "mapping by steps" for 50/49&64/63 will also generate the
> same list, taking positive integer multiples of the two step sizes,
> and seeing if they correspond. About half way done with "Prime Based
> Error and Complexity Measures" BTW. Question: If you only took
> temperament maps that come closest to the primes, would that kill
> off most temperaments? Obviously, 34-EDO would be out for Pajara...

As there are infinitely many temperaments, only a finite proportion of
them using closest-primes mappings, you can kill effectively all of
them! But mostly you kill the lousy ones. Generally the higher the
prime limit and the lower the error the more it's a problem. The best
mappings also tend to work better than nearest-primes mappings.

Graham

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Question: If you only took
> temperament maps that come closest to the primes, would that kill
> off most temperaments? Obviously, 34-EDO would be out for Pajara...

Nope. You need to go to high error, high prime limit, or high badness
to have much hope of finding something supported by no patent val.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > Question: If you only took
> > temperament maps that come closest to the primes, would that kill
> > off most temperaments? Obviously, 34-EDO would be out for Pajara...
>
> Nope. You need to go to high error, high prime limit, or high badness
> to have much hope of finding something supported by no patent val.

Thanks, but isn't 34-EDO Pajara (34, 54, 79, 96) non-patent val because
7 is mapped to 96 instead of 95?
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> > Nope. You need to go to high error, high prime limit, or high
badness
> > to have much hope of finding something supported by no patent val.
>
> Thanks, but isn't 34-EDO Pajara (34, 54, 79, 96) non-patent val
because
> 7 is mapped to 96 instead of 95?

True but not relevant, as pajara is supported by the patent vals for 2,
10, 12, 22, 32,

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Thanks, but isn't 34-EDO Pajara (34, 54, 79, 96) non-patent val
because
> 7 is mapped to 96 instead of 95?

Sorry, that got posted by accident while I was still writing it. The
list of patent vals supporting 7-limit pajara goes 2, 10, 12, 22, 32,
54, with contorted support from 20 and 44.