Re: [tuning-math] Re: non-1200: Tenney/heuristic meantone temperament

Paul wrote on 5 Jan:
>Now, for all primes r,
>If p contains any factors of r, the r-rungs in the lattice (which
>have length log2(r)) are shrunk from
>cents(r)
>to
>cents(r) - log2(r)*cents(p/q)/log2(p*q).
>If q contains any factors of 2, they are instead stretched to
>cents(r) + log2(r)*cents(p/q)/log2(p*q).

Sorry for being so behind. I noticed that this TOP tempering
can be easily done with Scala. First you need to set the prime
weights to the reciprocal of their log2, like this:

SET HARMCONST 2 1.0
SET HARMCONST 3 0.69092975
SET HARMCONST 5 0.43067656
SET HARMCONST 7 0.35620719
etc., primes not in the comma don't have to be set of course.

Then do PROJECT/TEMPER/WEIGHTED <comma(s)>

>No, it's simply limited to temperaments of codimension 1.

It doesn't have to be, the above command takes any number of
commas and tempers them out simultaneously.

I'll make an extra option for the PROJECT/TEMPER command so
that the SET HARMCONSTs can be omitted.

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Paul wrote on 5 Jan:
> >Now, for all primes r,
> >If p contains any factors of r, the r-rungs in the lattice (which
> >have length log2(r)) are shrunk from
> >cents(r)
> >to
> >cents(r) - log2(r)*cents(p/q)/log2(p*q).
> >If q contains any factors of 2, they are instead stretched to
> >cents(r) + log2(r)*cents(p/q)/log2(p*q).
>
> Sorry for being so behind. I noticed that this TOP tempering
> can be easily done with Scala. First you need to set the prime
> weights to the reciprocal of their log2, like this:
>
> SET HARMCONST 2 1.0
> SET HARMCONST 3 0.69092975
> SET HARMCONST 5 0.43067656
> SET HARMCONST 7 0.35620719
> etc., primes not in the comma don't have to be set of course.
>
> Then do PROJECT/TEMPER/WEIGHTED <comma(s)>

Huh!

> >No, it's simply limited to temperaments of codimension 1.
>
> It doesn't have to be, the above command takes any number of
> commas and tempers them out simultaneously.

Really?? Wow. What do you get for 5-limit 12-equal TOP?

>Really?? Wow. What do you get for 5-limit 12-equal TOP?

(1195.378, 1894.637, 2797.035)

Manuel

Not sure that you expected that answer though.
If I temper out the syntonic comma and the schisma the
result is better:

(1200.051, 1901.874, 2785.782)

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> >Really?? Wow. What do you get for 5-limit 12-equal TOP?
>
> (1195.378, 1894.637, 2797.035)
>
> Manuel

That's not correct. For one thing, it's not equal!

1195.378/12 = 99.6148
1894.637/19 = 99.7177
2797.035/28 = 99.8941

For another thing, the diesis doesn't vanish:

<1195.378 1894.637 2797.035|7 0 -3> = -23.459

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Not sure that you expected that answer though.
> If I temper out the syntonic comma and the schisma the
> result is better:
>
> (1200.051, 1901.874, 2785.782)
>
> Manuel

The syntonic comma isn't being tempered out at all here:

<1200.051 1901.874 2785.782|-4 4 -1> = 21.51

And this certainly isn't an equal temperament!

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> It doesn't have to be, the above command takes any number of
> commas and tempers them out simultaneously.

So at best, there seems to be an error in this procedure.

>That's not correct. For one thing, it's not equal!

Yes, sorry, I forgot again how to use my own program!
The code is correct, but I gave it the wrong parameters.
It should have been (1200.6171 1900.9770 2801.4398).
Step is 100.051421.

Manuel

31-equal TOP is this, identical for 5-limit and 7-limit:
(1201.6366 1899.3611 2790.8979 3372.3350)

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> 31-equal TOP is this, identical for 5-limit and 7-limit:
> (1201.6366 1899.3611 2790.8979 3372.3350)

I get 1201.4675

Gene wrote:
>> 31-equal TOP is this, identical for 5-limit and 7-limit:
>> (1201.6366 1899.3611 2790.8979 3372.3350)

>I get 1201.4675

I don't see how that can be correct. Your twelfth will be
1899.094. Then (1901.955 - 1899.094) / (1201.4675 - 1200.0) =
1.95 which is not log2(3)/log2(2) = 1.585.

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> >That's not correct. For one thing, it's not equal!
>
> Yes, sorry, I forgot again how to use my own program!
> The code is correct, but I gave it the wrong parameters.
> It should have been (1200.6171 1900.9770 2801.4398).
> Step is 100.051421.
>
> Manuel

Incorrect, I'm afraid.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
> <manuel.op.de.coul@e...> wrote:
> >
> > 31-equal TOP is this, identical for 5-limit and 7-limit:
> > (1201.6366 1899.3611 2790.8979 3372.3350)
>
> I get 1201.4675

Yes, to two places I get [1201.47 1899.09 2790.51 3371.86].

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Gene wrote:
> >> 31-equal TOP is this, identical for 5-limit and 7-limit:
> >> (1201.6366 1899.3611 2790.8979 3372.3350)
>
> >I get 1201.4675
>
> I don't see how that can be correct. Your twelfth will be
> 1899.094. Then (1901.955 - 1899.094) / (1201.4675 - 1200.0) =
> 1.95 which is not log2(3)/log2(2) = 1.585.
>
> Manuel

But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.6366 -
1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure
what your point is.

Paul wrote:
>But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 -
>1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure
>what your point is.

Yes, never mind, I shouldn't be posting when in a hurry.
Today I found your post where you explain the equal tempered case,
so I understand it now. So the method in Scala is only compatible
with TOP when tempering out a single comma.
I could implement it for more than one comma sometime, but it's
more complicated than the existing procedure.

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 -
> >1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure
> >what your point is.
>
> Yes, never mind, I shouldn't be posting when in a hurry.
> Today I found your post where you explain the equal tempered case,
> so I understand it now.

Not so fast -- the single-comma case is the only one where the TOP
tempering is motivated geometrically, so I'd be interested if there
are other methods of tempering you'd come up with for multiple commas.

> So the method in Scala is only compatible
> with TOP when tempering out a single comma.
> I could implement it for more than one comma sometime, but it's
> more complicated than the existing procedure.

If you have a method that doesn't require searching 2^n corners or
whatever, I'd be most interested in learning it.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Not so fast -- the single-comma case is the only one where the TOP
> tempering is motivated geometrically...

Did you read my TOP tuning page? Clearly, this is false.

Paul wrote:
>Not so fast -- the single-comma case is the only one where the TOP
>tempering is motivated geometrically, so I'd be interested if there
>are other methods of tempering you'd come up with for multiple commas.

What I have is not so good as TOP, it needs experimenting with the
prime weights to get a good result for more than one comma.

>If you have a method that doesn't require searching 2^n corners or
>whatever, I'd be most interested in learning it.

I had indeed the brute force approach in mind. I thought it was a
linear programming problem, and Gene's TOP and Tenney space page
confirmed it.

Manuel

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Not so fast -- the single-comma case is the only one where the
TOP
> > tempering is motivated geometrically...
>
> Did you read my TOP tuning page?

Yes, several times.

>Clearly, this is false.

?

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >Not so fast -- the single-comma case is the only one where the TOP
> >tempering is motivated geometrically, so I'd be interested if there
> >are other methods of tempering you'd come up with for multiple
commas.
>
> What I have is not so good as TOP, it needs experimenting with the
> prime weights to get a good result for more than one comma.
>
> >If you have a method that doesn't require searching 2^n corners or
> >whatever, I'd be most interested in learning it.
>
> I had indeed the brute force approach in mind. I thought it was a
> linear programming problem, and Gene's TOP and Tenney space page
> confirmed it.

Amazingly, Gene's page would have you believe you need to search even
in the codimension 1 case!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Amazingly, Gene's page would have you believe you need to search
even
> in the codimension 1 case!

You want I should derive the codimension 1 formula instead?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Amazingly, Gene's page would have you believe you need to search
> even
> > in the codimension 1 case!
>
> You want I should derive the codimension 1 formula instead?

Why not?